Lieb-Robinson bounds show that the speed of propagation of information under the Heisenberg dynamics in a wide class of nonrelativistic quantum lattice systems is essentially bounded. We review works of the past dozen years that has turned this fundamental result into a powerful tool for analyzing quantum lattice systems. We introduce a unified framework for a wide range of applications by studying quasilocality properties of general classes of maps defined on the algebra of local observables of quantum lattice systems. We also consider a number of generalizations that include systems with an infinite-dimensional Hilbert space at each lattice site and Hamiltonians that may involve unbounded on-site contributions. These generalizations require replacing the operator norm topology with the strong operator topology in a number of basic results for the dynamics of quantum lattice systems. The main results in this paper form the basis for a detailed proof of the stability of gapped ground state phases of frustrationfree models satisfying a local topological quantum order condition, which we present in a sequel to this paper.

Topics
Lattice models, Dyson-series

Quantum many-body theory comes in two flavors. The first is the relativistic version generically referred to as Quantum Field Theory (QFT), used for particle physics, and the second is non-relativistic many-body theory, which serves as the basic framework for most of condensed matter physics. The close physical and mathematical similarities between the two have long been recognized and exploited with great success. Bogoliubov’s theory of superfluidity and the BCS theory of superconductivity serve as definitive proof that quantum fields are a useful and even fundamental concept for understanding nonrelativistic many-body systems. Condensed matter theorists have developed field theory techniques that are now omnipresent in the subject.1,3,42,62,79,86,125 See Refs. 51 and 117 for reviews on recent progress in the mathematics of Bogoliubov’s theory and the BSC theory of superfluidity.

The absence of Lorentz-invariance (and the associated constant speed of light c leading to the all-important property of locality in the sense of Haag48) in nonrelativistic many-body theories is the most obvious difference between the two perspectives. Given the importance of this invariance in QFT, which plays an essential role in deriving many of the fundamental properties, and the strong constraints it imposes on its mathematical structure, one would expect that its absence would prevent any close analogy between the relativistic and the nonrelativistic setting to hold true. Contrary to this expectation, successful applications of QFT to problems of condensed matter physics have been numerous. Quantum field theories have provided accurate descriptions as effective theories describing important aspects such as excitation spectra and derived quantities. This typically involves a scaling limit of some type. Conformal field theories have been spectacularly effective in describing and classifying second order phase transitions. Also here a scaling limit is often implied.

The quasilocality properties that are the subject of study in this paper partly explain the closer-than-expected similarities between QFT and the nonrelativistic many-body theory of condensed matter systems. More importantly, they make it possible to prove that much of the mathematical structure of QFT can be found in nonrelativistic many-body systems in an approximate sense. Instead of asymptotic statements and qualitative comparisons, we can prove quantitative estimates: the quasilocality of the dynamics is characterized by an approximate light-cone with errors that can be bounded explicitly. These are the Lieb-Robinson bounds, which have been an essential ingredient in a large number of breakthrough results in the past dozen years.

Although the result of Lieb and Robinson dates back to the early 1970s,75,112 the impetus for the recent flurry of activity and major applications came from the work of Hastings on the Lieb-Schultz-Mattis theorem in arbitrary dimension.54 The possibility of adapting some of the major results of QFT to (nonrelativistic) quantum lattice systems was anticipated by others. For example, Wreszinski studied the connection between the Goldstone theorem,73 charges, and spontaneous symmetry breaking.127 A rigorous proof of a nonrelativistic exponential clustering theorem, long known in QFT,43,114 did not appear until the works.57,92 The time-evolution of quantum spin systems turns local observables into quasilocal ones. Lieb-Robinson bounds were applied to approximate such quasilocal observables by strictly local ones with an error bound in Refs. 23, 88, and 91. These (sequences of) strictly local approximations are those that are used in many concrete applications and also have a conceptual appeal. Further extensions of Lieb-Robinson bounds and a sampling of interesting applications are discussed in Sec. III.

Apart from offering a review of the state of the art of quasilocality estimates, in this paper, we also extend existing results in the literature in several directions. First, for most of the results, we allow the quantum system at each lattice site to be described by an arbitrary infinite-dimensional Hilbert space. For many results, the single-site Hamiltonians may be arbitrary densely defined self-adjoint operators. Another generalization in comparison to the existing literature, made necessary by the consideration of unbounded Hamiltonians, is that time-dependent perturbations are assumed to be continuous with respect to the strong operator topology instead of the operator norm topology. In order to handle this more general situation, a number of technical issues need to be addressed related to the continuity properties of operator-valued functions and of the dynamics generated by strongly continuous time-dependent interactions. These technical issues cascade through the better part of this paper. We will understand if the reader is surprised by the length of the paper, since we were taken aback ourselves as we were completing the manuscript. Many proofs can be shortened if one is only interested in particular cases. Indeed, in many cases, results for more restricted cases exist in the literature. There are also places, however, where the published results in the literature provide only weaker estimates or have incomplete proofs.

In Sec. II, we review the construction and basic properties of quantum dynamics in the context of this work. This includes a careful presentation of analysis with operator-valued functions using the strong operator topology. Section III is devoted to Lieb-Robinson bounds and their application to proving the existence of the thermodynamic limit of the dynamics. We also derive an estimate on the dependence of the dynamics on the interactions and introduce a notion of convergence of interactions that implies the convergence of the infinite-volume dynamics. Section IV is devoted to the approximation of quasilocal observables by strictly local ones by means of suitable maps called conditional expectations. Because they are needed for our applications, the continuity properties of a class of such maps are studied in detail. A general notion of quasilocal maps is introduced in Sec. V, and we study the properties of several operations involving such maps that are used extensively in applications. In Sec. VI, we construct an auxiliary dynamics called the spectral flow (also called the quasiadiabatic evolution), which is the main tool in recent proofs of the stability of spectral gaps and gapped ground state phases. A first application of the spectral flow is the notion of automorphic equivalence, discussed in Sec. VII, which allows us to give a precise definition of a gapped ground state phase as an equivalence class for a certain equivalence relation on families of quantum lattice models. In the  Appendix, we collect a number of arguments that are used throughout this paper.

Our original motivation for this work was to supply all the tools needed for the results of Paper II.96 However, this work can now be read as a stand-alone review article about quasilocality estimates for quantum lattice systems. Since the sequel of Paper II96 will be devoted to applying the quasilocality bounds and the spectral flow results from this work to prove the stability of gapped ground state phases, the examples and applications here will be chosen in support of the presentation of the general results.

Throughout this paper, we focus on so-called bosonic lattice systems, for which observables with disjoint support commute. Virtually, all results carry over to lattice fermion systems with only minor changes. This is discussed in some detail in Ref. 97. Another extension of quasilocality techniques not covered in this paper is the case of so-called extended operators. An important example is the half-infinite string operators that create the elementary excitations in models with topological order such as the Toric Code model.71 Lieb-Robinson bounds for such non-quasi-local operators are used in Ref. 29.

In this paper, the primary object of study is the Heisenberg dynamics acting on a suitable algebra of observables for a finite or infinite lattice system. For finite systems, this dynamics is expressed with a unitary propagator U(t, s), s≤t∈R⁠, on a separable Hilbert space H⁠. However, in some cases, for example, when one is interested in the excitation spectrum and dynamics of perturbations with respect to a thermal equilibrium state of the system, the generator of the dynamics is not semibounded and the Hilbert space may be nonseparable. Therefore, in general, we will not assume that H is separable or that the Hamiltonian is bounded below.

As described in the Introduction, we consider finite and infinite lattice systems with interactions that are sufficiently local. We allow for an infinite-dimensional Hilbert space at each site of the lattice. However, we impose conditions on the interactions that permit us to prove quasilocality bounds of Lieb-Robinson type (in terms of the operator norm) for bounded local observables. This means that we will allow for the possibility of unbounded “spins” and unbounded single-site Hamiltonians, but require that the interaction be given by bounded self-adjoint operators that satisfy a suitable decay condition at large distances (see Sec. III for more details). We do not consider lattice oscillator systems with harmonic interactions in this paper, since one should not expect bounds in terms of the operator norm for this class of systems (see Refs. 4 and 89). An interesting model that does fit in the framework presented here is the so-called quantum rotor model, which has an unbounded Hamiltonian for the quantum rotor at each site, but the interactions between rotors are described by a bounded potential.72,77,115

We will use the so-called interaction picture to describe the dynamics of Hamiltonians with unbounded on-site terms. This requires that we also consider time-dependent interactions. Time-dependent Hamiltonians are, of course, of interest in their own right, for instance, in applications of quantum information theory. Therefore, we begin with a discussion of the Schrödinger equation for the class of time-dependent Hamiltonians considered in this work.

Let H be a complex Hilbert space and B(H) denote the bounded linear operators on H⁠. Let I⊆R be a finite or infinite interval. In this section, we review some basic properties of the dynamics of a quantum system with a time-dependent Hamiltonian of the form

(2.1)

where H0 is a time-independent self-adjoint operator with dense domain D⊂H⁠, and for t ∈ I, Φ(t)*=Φ(t)∈B(H) and t ↦ Φ(t) is continuous in the strong operator topology. This means that for all ψ∈H⁠, the function t ↦ Φ(t)ψ is continuous in the Hilbert space norm. From these assumptions, it follows that for all t ∈ I, H(t) is self-adjoint with time-independent dense domain D [see Ref. 126 (Theorem 5.28)].

We will often consider operator-valued and vector-valued functions of one or more real (or complex) variables and impose various continuity assumptions, which we now briefly review. An operator-valued function is said to be norm continuous (norm differentiable) if it is continuous (differentiable) in the operator norm and strongly continuous (strongly differentiable) if it is continuous (differentiable) in the strong operator topology. With a slight abuse of terminology, we will refer to Hilbert space-valued functions as strongly continuous (strongly differentiable) if they are continuous (differentiable) in the Hilbert space norm. For transparency, when we consider maps defined on a linear space of operators, we will indicate the relevant topology and continuity assumptions explicitly.

The dynamics of a system described in (2.1) is determined by the following Schrödinger equation:

(2.2)

For bounded H(t), through a standard construction, we will see that there exists a family of unitaries U(t,s)∈B(H)⁠, s, t ∈ I, that is jointly strongly continuous with ψ(t) = U(t, t0)ψ0 being the unique solution of (2.2) for all ψ0∈H⁠. It follows that the family U(t, s) has the cocycle property: for r ≤ s ≤ t ∈ I, U(t, r) = U(t, s)U(s, r) and U(t,t)=1⁠. In the case that H(t) = H0 + Φ(t), where H0 is an arbitrary unbounded self-adjoint operator and Φ(t) is bounded, we will make use of the well-known interaction picture dynamics to construct an analogous unitary cocycle. This cocycle will, in particular, generate the unique weak solution of the Schrödinger equation. To this end, we first discuss some other aspects of strongly continuous operator-valued functions that we will need.

In this section, we review some terminology and discuss a number of properties of operator-valued functions that will be used extensively in the rest of this paper.

Let I⊂R be a finite or infinite interval and A:I→B(H) be strongly continuous, i.e., for all ψ∈H⁠, t ↦ A(t)ψ is continuous with respect to the Hilbert space norm. By the uniform boundedness principle, if A is strongly continuous, then A is locally bounded, meaning if J ⊂ I is compact, then

(2.3)

The strong continuity of t ↦ A(t) implies that t ↦ ∥A(t)ψ∥ is continuous for all ψ∈H⁠, and from the above, the map t ↦ ∥A(t)∥ is locally bounded. However, strong continuity does not imply that t ↦ ∥A(t)∥ is continuous [see Ref. 95 (Sec. 2) for a counterexample].

We note that in this paper we use the notations ∥A(t)∥ and ∥A∥(t) interchangeably. For ease of later reference, we now state a simple proposition.

Proposition 2.1.

Let I⊂R be a finite or infinite interval and H and K be Hilbert spaces.

  • If A,B:I→B(H) are strongly continuous (strongly differentiable), then (t, s) ↦ A(t)B(s) is jointly strongly continuous (separately strongly differentiable).

  • If A:I→B(H) and B:I→B(K) are strongly continuous (strongly differentiable), then (t, s) ↦ A(t) ⊗ B(s) is jointly strongly continuous (separately strongly differentiable).

  • If A:I→B(H) is strongly continuous, then the function t ↦ ∥A(t)∥ is lower semicontinuous, measurable, locally bounded and, hence, locally integrable.

It is clear that an analog of Proposition 2.1 holds when strongly is replaced with norm in the statements above. Moreover, an argument similar to the one found in Proposition 2.1(i) shows that if A:I→B(H) and ψ:I→H are both strongly continuous (strongly differentiable), then (t, s) ↦ A(t)ψ(s) is jointly strongly continuous (separately strongly differentiable). As will be clear from the proof, we note that the conclusions of part (iii) of this proposition continue to hold even for weakly continuous A(t).

Proof.

We prove the statements above in the case of strong continuity; the strong differentiability claims follow similarly.

For (i), let ψ∈H and t0, s0 ∈ I be fixed. Given that A and B are strongly continuous, and hence locally bounded, it follows that A(t)B(s)ψ → A(t0)B(s0)ψ as (t, s) → (t0, s0) since
‖A(t)B(s)ψ−A(t0)B(s0)ψ‖≤‖A(t)‖‖(B(s)−B(s0))ψ‖+‖(A(t)−A(t0))B(s0)ψ‖.
To prove (ii), we first show that if t↦A(t)∈B(H) is strongly continuous and K is another Hilbert space, then the map t↦A(t)⊗1∈B(H⊗K) is also strongly continuous. To see this, note that for all ψ∈H⊗K⁠, there exist two countable sets of vectors ψn∈H and ϕn∈K with {ϕn}n orthonormal such that ψ = ∑nψn ⊗ ϕn, and ∑n‖ψn‖2=‖ψ‖2⁠. Fix s ∈ I, and let J ⊆ I be a compact interval that contains a neighborhood of s. Using the orthonormality of ϕn, we find that for all t ∈ J,
‖((A(t)−A(s))⊗1)ψ‖2=∑n‖(A(t)−A(s))ψn‖2.
For any ϵ > 0, choose N large enough so that ∑n>N‖ψn‖2≤ϵ/8MJ2⁠, where MJ > 0 satisfies (2.3). By the strong continuity of A(t), there exists a δ > 0 such that for all t ∈ (s − δ, s + δ) ⊆ J and 1 ≤ n ≤ N, one has ‖(A(t)−A(s))ψn‖2<ϵ/2N⁠. Putting these together, if |t − s| < δ, then
(2.4)
Since A(t) ⊗ B(s) = (A(t) ⊗ 1⁠)(⁠1 ⊗ B(s)), by (i), the tensor product of two strongly continuous maps t↦A(t)∈B(H) and s↦B(s)∈B(K) is jointly strongly continuous.
To prove (iii), we start by noting that, by virtue of the strong continuity of A(t), the function ∥A(t)∥ can be expressed as a supremum of continuous functions,
‖A(t)‖=sup{⟨ϕ,A(t)ψ⟩∣ϕ,ψ∈H,‖ϕ‖=‖ψ‖=1}.
Recall that a function f:I→R is lower semicontinuous if and only if for all s∈R⁠, f−1((s, ∞)) is an open subset of I. Now, if f is the supremum of a family of functions { fα}, we have that f−1((s,∞))=⋃αfα−1((s,∞))⁠. In our case, the fα are indexed by a pair of unit vectors in H and each fα is continuous. Therefore, fα−1((s,∞)) is open for all α and so is ⋃αfα−1((s,∞))⁠. This shows the lower semicontinuity.

Since we have that f−1((s, ∞)) is open, this set is also Borel measurable, for all s∈R⁠. By a standard lemma in measure theory,41 this implies that f is measurable.

We already noted above that ∥A(t)∥ is bounded on compact intervals by the uniform boundedness principle. This concludes the proof.

We will make frequent use of integrals of vector-valued and operator-valued functions. It is straightforward to define such integrals in the weak sense. In fact, if A:I→B(H) is locally bounded and weakly measurable, i.e., for all ϕ,ψ∈H⁠, t ↦ ⟨ϕ, A(t)ψ⟩ is measurable, then for any compact J ⊂ I the integral of A over J is defined as the operator BJ∈B(H) corresponding to the bounded sesquilinear form
(2.5)
We routinely use the notation BJ = ∫JA(t) dt to denote this operator. For strongly continuous functions A(t), the same integral can be interpreted in the strong sense
(2.6)
where the RHS is understood to be the Riemann integral of a strongly continuous, Hilbert space-valued function. Since the range of any strongly continuous, Hilbert space-valued function belongs to a separable subspace [even if H is not separable, see, e.g., Ref. 129 (Sec. V.4)], this integral also exists in the sense of Bochner.
The following well-known inequalities hold for all A:I→B(H)⁠, strongly continuous and J ⊂ I compact:
(2.7)
and thus,
(2.8)
In particular, we obtain
(2.9)
The first inequality extends to infinite J if, e.g., A(t) = B(t)w(t), with B(t) strongly continuous and bounded and w ∈ L1(J). Finally, it is easy to see that if A is strongly continuous, then B(t)=∫t0tA(s)ds is norm continuous and strongly differentiable with ddtB(t)=A(t)⁠. As such, the fundamental theorem of calculus also holds in the strong sense.

In this section, we review some well-known facts about Dyson series and from them obtain the Schrödinger dynamics generated by a bounded, time-dependent Hamiltonian. A standard result in this direction can be summarized as follows: Let H be a Hilbert space, I⊂R be a finite or infinite interval, and H:I→B(H) be strongly continuous and pointwise self-adjoint, i.e., H(t)* = H(t), for all t ∈ I. Under these assumptions (see, e.g., Theorem X.69 of Ref. 109) for each t0 ∈ I and every initial condition ψ0∈H⁠, the time-dependent Schrödinger equation

(2.10)

has a unique solution in the sense that there is a unique, strongly differentiable function ψ:I→H which satisfies (2.10). This solution can be characterized in terms of a two-parameter family of unitaries {U(t,s)}s,t∈I⊂B(H) such that

(2.11)

These unitaries are often referred to as propagators, and an explicit construction of them is given by the Dyson series. Specifically, for any s, t ∈ I and each ψ∈H⁠, the Hilbert space-valued series

(2.12)

is easily seen to be absolutely convergent in norm. One checks that U(t, s), as defined in (2.12), satisfies the differential equation

(2.13)

which is to be understood in the sense of strong derivatives. Of course, under the stronger assumption that H:I→B(H) is norm continuous, then (2.13) also holds in norm.

The additional observation we want to make here is that U(t, s) is not only the unique strong solution of (2.13); it is also the case that any bounded weak solution of (2.3) necessarily coincides with U(t, s). By weak solution, we mean that for all ϕ,ψ∈H and any s, t ∈ I, U(t, s) satisfies

(2.14)

A proof of this fact is contained in the following proposition:

Proposition 2.2.
Let A:I→B(H) be strongly continuous, and consider the differential equation
(2.15)
The following statements hold:

  • There is a unique strong solution V:I→B(H) of (2.15), and V is norm continuous.

  • Any locally norm-bounded, weak solution of (2.15) coincides with the strong solution.

  • Let D⊂H be dense. Suppose V:I→B(H) is strongly continuous and satisfies

(2.16)
for all ψ∈D and t ∈ I. Then, V is the unique strong solution.
  • (iv)

    If V0 is invertible, the strong solution V of (2.15) is invertible for all t ∈ I. Moreover, in this case, the inverse of V is the unique strong solution of

(2.17)
  • (v)

    If H:I→B(H) is strongly continuous and pointwise self-adjoint, then the strong solution V of (2.15) with the choice A = −iH and V0 = 1 is unitary for all t ∈ I. Moreover, the map U:I×I→B(H) given by U(t, s) = V(t)V(s)* is the unique strong solution to (2.13).

Before giving the proof of this proposition, we first comment on the content of part (v) and its relationship to the Dyson series from (2.12). A simple consequence of (i) is that the mapping U(t, s), defined in (v), is jointly norm continuous. As stated in (v), this U(t, s) is the unique strong solution of (2.13); it is also strongly differentiable in s, and by (iv), this strong derivative is
(2.18)
In addition, one readily checks that U(t, s)−1 = V(s)V(t)*, for all s, t ∈ I, and thus U(t, s)−1 = U(t, s)* = U(s, t), for all s, t ∈ I, so that U(t, s) is a two-parameter family of unitaries. This family of unitaries satisfies the cocycle property: if r ≤ s ≤ t, then
(2.19)
Finally, arguing as in the proof of (i), one sees that the Dyson series (2.12) is a strong solution of (2.13). Combining this with the uniqueness proven in (v), we conclude that U(t, s) = V(t)V(s)* must coincide with the Dyson series constructed in (2.12).

Proof.
(i) Define a map V:I→B(H) via a Dyson series, i.e., for any ψ∈H and each t ∈ I, set
(2.20)
We now argue that V is the unique strong solution of (2.15).
First, we show that V is well-defined. The integrals appearing as terms in this series are well-defined due to the strong continuity of A. More precisely, for any n ≥ 1, the product A(t1)⋯A(tn) is jointly strongly continuous in the variables t1, …, tn, and thus, the integrands are locally integrable. Next, for any t ≥ t0, the bound
(2.21)
holds. Here, we note that we are using the alternate notation ∥A∥(t) for ∥A(t)∥. As it is clear that a similar argument holds for t < t0, we see that V is well-defined as an absolutely convergent (in norm) series.
Next, we prove that V is a strong solution of (2.15). To see this, define recursively a sequence of operators {Vn}n≥1⁠, Vn:I→B(H) by setting
(2.22)
With respect to this notation, it is clear that
(2.23)
and moreover, for any h ≠ 0,
(2.24)
Using the recursive definition, i.e., (2.22), it is clear that the first term on the right-hand-side above is A(t)V(t)ψ. A dominated convergence argument, using an estimate like (2.21), guarantees that the remainder term goes to zero in norm, and hence, V is a strong solution.
Finally, we prove uniqueness. Let V1 and V2 be two strong solutions of (2.15). For any t ∈ I, set t+ = max{t, t0} and t− = min{t, t0}. Given ψ∈H⁠, we have that
‖(V1(t)−V2(t))ψ‖=∫t0tdds(V1(s)−V2(s))ψ ds=∫t0tA(s)(V1(s)−V2(s))ψ ds≤∫t−t+‖A‖(s)‖(V1(s)−V2(s))ψ‖ ds
and uniqueness follows from Gronwall’s lemma.
One also sees that this V is norm continuous. In fact, let t, t0 ∈ I and ψ∈H⁠. Clearly,
(2.25)
and norm continuity of V follows.
  • (ii)

    The uniqueness statement in (ii) is proven similarly. In fact, let V1 and V2 be two locally norm-bounded weak solutions of (2.15). In this case, for any ϕ,ψ∈H and each t ∈ I,

(2.26)
where we have used the notation t± as above. Taking the supremum of (2.26) over all normalized ϕ,ψ∈H gives
(2.27)
Again, uniqueness follows from Gronwall’s lemma. Since any strong solution is also a locally bounded weak solution, the unique weak and unique strong solutions must coincide.
  • (iii)

    We will show that any V satisfying the assumptions of (iii) is actually the locally bounded weak solution. To this end, note that for any ϕ∈H⁠, (2.16) implies

(2.28)
holds for all ψ∈D and any t ∈ I. Let ψ∈H and take any sequence {ψn}n≥1 in D with ψn converging to ψ. Consider the sequence of functions fn:I→C defined by
(2.29)
and set f(t) = limn→∞fn(t). Note that these pointwise limits exist as ψn converges to ψ and V is locally bounded. One also sees that f(t) = ⟨ϕ, V(t)ψ⟩ for all t ∈ I. From (2.28), it is clear that fn′(t) = ⟨ϕ, A(t)V(t)ψn⟩. Since AV is strongly continuous, and hence locally bounded, the same argument shows that g:I→C with g(t) = limn→∞fn′(t) = ⟨ϕ, A(t)V(t)ψ⟩ is well-defined. Observing further that fn′ converges to g uniformly on compact subsets of I, it is clear that the conditions of Ref. 113 (Theorem 7.17) are satisfied. We conclude that f′(t) = g(t) for all t ∈ I, and hence, V is the unique locally bounded weak solution. By the result proven in (ii), V also coincides with the unique strong solution.
  • (iv)

    Arguing as in the proof of (i), the function W:I→B(H) defined by setting

(2.30)
for any ψ∈H is a strong solution of the initial value problem
(2.31)
Now, with V being the strong solution of (2.15), consider the function Y:I→B(H) given by Y(t) = V(t)W(t). One checks that
(2.32)
holds in the strong sense. It is clear that Y(t)=1 solves the above initial value problem. A Gronwall argument, similar to those we have proven before, shows that this constant solution is the unique strong solution of (2.32), and thus, W is a right inverse of V for all t ∈ I. Noting that the function Z:I→B(H) defined by Z(t) = W(t)V(t) satisfies the trivial initial value problem,
(2.33)
we conclude W(t) = V(t)−1 as claimed. In fact, the uniqueness of the strong solution of (2.31) follows.
  • (v)

    One sees that V is unitary by noting that the adjoint of the operator defined by (2.20) agrees with the Dyson series for V−1 given in (2.30). For each s ∈ I fixed, the map t ↦ V(t)V(s)* defines a strong solution of (2.13) and by (i) it is unique.

We conclude this section with an estimate on the solution of certain dynamical equations that will be useful in the proof of the Lieb-Robinson bound in Sec. III.

Lemma 2.3.
Let H be a Hilbert space, I⊂R be a finite or infinite interval, and A,B:I→B(H) be strongly continuous with A pointwise self-adjoint, i.e., A(t)* = A(t) for all t ∈ I. For each t0 ∈ I and V0∈B(H), the initial value problem
(2.34)
has a unique strong solution. In particular,
(2.35)
where t+ = max{t, t0}, t− = min{t, t0}. Moreover, any locally bounded weak solution of (2.34) coincides with the strong solution and, therefore, satisfies the estimate in (2.35).

Proof.
By Proposition 2.2(v), the unique strong solution of
(2.36)
is unitary for all t ∈ I. As a product of strongly differentiable maps, V:I→B(H) given by
(2.37)
is strongly differentiable. In fact, a short calculation shows that this V is a strong solution of (2.34), and moreover, the bound claimed in (2.35) is clear. Arguments involving Gronwall’s lemma, similar to those found in the proof of Proposition 2.2(i) and (ii), verify the claimed uniqueness results.

1. On the interaction picture dynamics

Proposition 2.4 is an important application of Proposition 2.2. As explained in the remarks of Sec. X.12 of Ref. 109, applying the interaction picture representation to Hamiltonians with the form H = H0 + Φ, even if Φ is time-independent, leads one to study a dynamics with time-dependent Hamiltonians. In this situation, one often produces Hamiltonians that are strongly continuous, but not norm continuous. This leads us to consider Hamiltonians of the form H(t) = H0 + Φ(t), where H0 is a self-adjoint operator with dense domain D and Φ(t) is a bounded, pointwise self-adjoint operator that is strongly continuous in t.

Proposition 2.4.

Let H be a Hilbert space and H0 be a self-adjoint operator with dense domain D⊂H. Let I⊂R be a finite or infinite interval and Φ:I→B(H) be strongly continuous and pointwise self-adjoint. Then, there is a two parameter family of unitaries {U(t, s)}s,t∈I associated with the self-adjoint operator H(t) = H0 + Φ(t) for which

  • (t, s) ↦ U(t, s) is jointly strongly continuous,

  • U(t, s) satisfies the cocycle property (2.19),

  • U(t, s) generates the unique, locally bounded weak solutions of the Schrödinger equation associated with H(t), i.e., for any t0 ∈ I and ψ0∈H, ψ:I→H given by ψ(t) = U(t, t0)ψ0 satisfies
    (2.38)
    for all ϕ∈D and t ∈ I.

Proof.
Since H0 is self-adjoint, Stone’s theorem implies that {eitH0}t∈R is a strongly continuous, one-parameter unitary group. In this case, the map H̃:I→B(H) given by
(2.39)
is clearly pointwise self-adjoint and strongly continuous. Using Proposition 2.2(v), we conclude that the unique strong solutions of
(2.40)
form a two-parameter family of unitaries {Ũ(t, s)}s,t∈I which satisfy the cocycle property (2.19). In terms of this family, we define U:I×I→B(H) by setting
(2.41)
One checks that {U(t, s)}s,t∈I is a two-parameter family of unitaries satisfying (i) and (ii) above.
To prove (iii), let t0 ∈ I and ψ0∈H⁠. Define ψ:I→H by setting ψ(t) = U(t, t0)ψ0. Observe that for any ϕ∈D and each t ∈ I,
(2.42)
with the right-hand-side being a differentiable function of t. One calculates that
(2.43)
as claimed.

We need to justify only the uniqueness of the locally bounded weak solutions. Let t0 ∈ I, let ψ0∈H⁠, and suppose ψ1 and ψ2 are two locally bounded solutions of the initial value problem (2.38). Consider the functions ψ̃1(t)=eitH0ψ1(t) and ψ̃2(t)=eitH0ψ2(t)⁠. It is easy to check that these functions are locally bounded weak solutions of the Schrödinger equation associated with the bounded Hamiltonian H̃(t) in (2.39). As such, they are unique, which may be argued as in the proof of Proposition 2.2, and therefore, so too are ψ1 and ψ2.

In this work, we define the Heisenberg dynamics on a suitable algebra of observables in terms of the strongly continuous propagator U(t, s) whose existence is guaranteed by Proposition 2.4. We work under assumptions that guarantee the uniqueness of bounded weak solutions. Strictly speaking, the uniqueness of the weak solution and the possible absence of a strong solution to the Schrödinger equation in the Hilbert space will play no role in our analysis. More information about the solutions and their uniqueness could, however, be important for the unambiguous interpretation of our results. Additional results exist in the literature if one is willing to make additional assumptions on H0 and Φ(t). For example, the following theorem establishes the existence of an invariant domain for the generator and, consequently, the existence of a unique strong solution for the situation where H0 is semibounded and Φ(t) is Lipschitz continuous, which is a common physical situation. As explained in the Introduction, there are important applications of the methods in this paper to situations where these additional assumptions are not satisfied.

Theorem 2.5.
Let H0 be a self-adjoint operator with dense domain D⊂H and suppose H0 ≥ 0. Suppose Φ:R→B(H) is pointwise self-adjoint and “Lipschitz” continuous in the sense that for any bounded interval I⊂R, there exists a constant C such that for all s, t ∈ I, we have
(2.44)
Then, there exists a strongly continuous propagator U(t, s), such that U(t,s)D⊂D, for all s ≤ t ∈ I, and such that t ↦ U(t, t0)ψ0 is the unique strong solution of
(2.45)
for all ψ0∈D⁠.

In Ref. 119 (Theorem II.21), Simon credits a version of this theorem to Yosida, who proved it in a more general Banach space context (Ref. 129, Sec. XIV.4), but with the Lipschitz condition replaced by a boundedness condition on the derivative of Φ(t). Yosida gives credit to Kato65,66 and Kisyński.70 

2. A Duhamel formula for bounded perturbations depending on a parameter

In this section, we consider families of Hamiltonians Hλ(t) which depend on a time-parameter t ∈ I and an auxillary parameter λ ∈ J. For such families, we will prove a version of the well-known Duhamel formula (Proposition 2.6) and use it to derive various continuity properties of the corresponding dynamics (Proposition 2.7).

Let H0 be a densely defined, self-adjoint operator on a Hilbert space H and denote by D⊂H the corresponding dense domain. Let I,J⊂R be intervals and consider the family of Hamiltonians Hλ(t), t ∈ I and λ ∈ J, acting on D⊂H given by

(2.46)

where for each t ∈ I and λ ∈ J, Φλ(t)*=Φλ(t)∈B(H)⁠. The self-adjointness of Hλ(t) on the common domain, D⁠, is clear. We will assume that (t, λ) ↦ Φλ(t) is jointly strongly continuous. We will also assume that for each fixed t ∈ I, the mapping λ ↦ Φλ(t) is strongly differentiable and that the corresponding derivative, which we denote by Φ′λ(t), satisfies that the map (t, λ) ↦ Φ′λ(t) is jointly strongly continuous.

Under these assumptions, Proposition 2.4 guarantees that for each λ ∈ J, there exists a two parameter family of unitaries {Uλ(t,s)}s,t∈I which generates the weak solutions of the Schrödinger equation associated with Hλ(t) [see (2.38)]. Our goal here is to show that for fixed s, t ∈ I, the map λ ↦ Uλ(t, s) is strongly differentiable, and moreover,

(2.47)

We will obtain this bound as a corollary of Proposition 2.6, which gives a Duhamel formula for the derivative in this setting. Although the Duhamel formula is well-known, we give an explicit proof here that allows us to clarify the continuity properties implied by our assumptions. In the proof we avoid taking derivatives with respect to t or s which, in general, are unbounded operators.

Proposition 2.6
(Duhamel formula). Let Hλ(t) be a family of self-adjoint operators as in (2.46), and let Uλ(t, s) denote the corresponding unitary propagator. Then, for all s, t ∈ I with s ≤ t, we have that
(2.48)
where the derivative and the integral are to be understood in the strong sense.

With stronger assumptions, one can prove (2.48) holds in norm. In fact, arguing as below, if

  • the map (t, λ) ↦ Φλ(t) is jointly norm continuous,

  • for each t ∈ I, the map λ ↦ Φλ(t) is norm differentiable, with the derivative denoted by Φ′λ(t), and

  • the map (t,λ)↦Φλ′(t) is jointly norm continuous,

then λ ↦ Uλ(t, s) is norm differentiable and its derivative satisfies (2.48).

Proof.
Recall that the unitary propagator Uλ(t, s), as defined in the proof of Proposition 2.4, is
(2.49)
where Ũλ(t, s) is the unique strong solution of
(2.50)
We first prove the analog of (2.48) for Ũλ(t, s), i.e.,
(2.51)
Given (2.51), the λ-derivative of (2.49) is easily seen to satisfy (2.48).
We now show (2.51). The unique strong solution of (2.50) is given by the Dyson series
(2.52)
To each n ≥ 1, define a map Ψλ:In→B(H) by setting
(2.53)
With (t1, …, tn) ∈ In fixed, our assumptions imply that λ ↦ Ψλ(t1, …, tn) is strongly differentiable, and moreover,
(2.54)
The joint strong continuity of (t,λ)↦Φ̃λ(t) and (t,λ)↦Φ̃λ′(t) can be used to justify term-by-term differentiation of the Dyson series (2.52), and we obtain
(2.55)

The proof of (2.51) is now completed by demonstrating that upon inserting the Dyson series for Ũλ(t, r) and Ũλ(r, s) into the integral on the right-hand-side of (2.51), the result simplifies to the expression on the right-hand-side of (2.55).

Note that upon substitution of (2.52) into the right-hand-side of (2.51), we find
(2.56)
Here, p (respectively, q) is the index of the terms in the series for the first (respectively, second) propagator, and we have taken as integration variables t1, …, tp and tp+2, …, tp+q+1. Each integrand above is the product of n = p + q + 1 ≥ 1 operators. Since the goal is to rewrite the above as in (2.55), we now reindex by writing p = k − 1 and q = n − k for n ≥ 1 and 1 ≤ k ≤ n. One sees that
(2.57)
Identity (2.51) now follows by comparing, term by term, the integration domains on the right-hand-sides of (2.55) and (2.57). Equality follows, e.g., by reordering the iterated integrals in (2.57).
For each λ ∈ J, the Heisenberg dynamics τt,sλ⁠, s, t ∈ I, associated with the family of Hamiltonians in (2.46) is the cocycle of automorphisms of B(H) given by
(2.58)
As it will be convenient for later applications, we summarize various continuity properties of this dynamics in Proposition 2.7.

Proposition 2.7.

Let Hλ(t) be a family of Hamiltonians as described in (2.46). The corresponding dynamics, as in (2.58), has the following properties:

  • For each λ ∈ J and A∈B(H), the map (s,t)↦τt,sλ(A) is jointly strongly continuous.

  • For each s, t ∈ I and A∈B(H), the map λ↦τt,sλ(A) is strongly differentiable (and hence strongly continuous). Moreover, one has the estimate

(2.59)
  • (iii)

    For fixed s, t ∈ I and λ ∈ J, the map τt,sλ(⋅):B(H)→B(H) is continuous on bounded sets when both its domain and codomain are equipped with the strong operator topology. This continuity is uniform for λ in compact subsets of J.

Proof.

The statement in (i) follows from Proposition 2.4 as τt,sλ(A) [see (2.58)] is the product of jointly strongly continuous mappings.

To prove (ii), we use Duhamel’s formula from Proposition 2.6 to calculate the derivative. Specifically, note that if s ≤ t, then
(2.60)
An estimate of the form in (2.59) is now clear.
To prove (iii), fix s, t ∈ I and let [a, b] ⊂ J. Without loss of generality, assume that s ≤ t. Let ϵ > 0. Since (r,λ)↦Φλ′(r) is jointly strongly continuous,
(2.61)
Take δ > 0 so that
(2.62)
By compactness, there is some N ≥ 1 and numbers λ1, …, λN ∈ [a, b] for which the balls of radius δ centered at λi, 1 ≤ i ≤ N, cover [a, b]. Using the result in (ii), we see that for every λ ∈ [a, b], there is some 1 ≤ i ≤ N for which
(2.63)
Now, to prove the continuity statement claimed, let {An}n≥1⊆B(H) be a bounded sequence that converges to A∈B(H) in the strong operator topology. Let B < ∞ be such that supn≥1∥An∥ ≤ B. Using (2.58) and the strong convergence of An to A, it is easy to verify that for any ψ∈H and any 1 ≤ i ≤ N, the sequence {τt,sλi(An)ψ}n≥1 converges to τt,sλi(A)ψ in H⁠. Pick n0 ≥ 1 so that for all n ≥ n0 and each 1 ≤ i ≤ N,
(2.64)
In this case, for any λ ∈ [a, b], there is an i for which
‖τt,sλ(An)ψ−τt,sλ(A)ψ‖≤‖τt,sλ(An)ψ−τt,sλi(An)ψ‖+‖τt,sλi(An)ψ −τt,sλi(A)ψ‖+‖τt,sλi(A)ψ −τt,sλ(A)ψ‖≤3ϵB‖ψ‖
whenever n ≥ n0. This proves that the strong convergence is uniform for λ ∈ [a, b], or in other words, that the family of maps {τt,sλ(⋅)∣λ∈[a,b]}⁠, for s, t ∈ I fixed, is equicontinuous on bounded sets in B(H) with respect to the strong operator topology.

The scope of this paper is lattice models with possibly unbounded single-site Hamiltonians and bounded interactions that, in general, may be time-dependent. This is the setting in which one expects to obtain Lieb-Robinson bounds with estimates in terms of the operator norm of the observables. A well-known example of this situation is the quantum rotor model. We will not consider lattice models with unbounded interactions in this work. The only systems with unbounded interactions that have been studied so far are oscillator lattice systems for which the interactions are quadratic34 or bounded perturbations of quadratic interactions.4,89

In this paper, the “lattice” in lattice systems is understood to be a countable metric space (Γ, d) (not necessarily a lattice in the sense of the linear combinations with integer coefficients of a set of basis vectors in Euclidean space). Typically, Γ is infinite (or more specifically, has infinite diameter), and models are given in terms of Hamiltonians for a family of finite subsets of Γ. After an initial analysis of the finite systems, we study the thermodynamic limit through sequences of increasing and absorbing finite volumes {Λn}, i.e., Λn↑ Γ. Often, the goal is to obtain estimates for the finite systems defined on Λn that are uniform in n. The definitions below prepare for this goal. It is possible to consider a finite set Γ and apply the results derived in this paper to finite systems. We note that some of the conditions imposed are trivially satisfied for finite systems.

The points of Γ, also called sites of the lattice, label a family of “small” systems, which are often, but not necessarily, identical copies of a given system such as a spin, a particle in a confining potential such as a harmonic oscillator, or a quantum rotor. The quantum many-body lattice systems of condensed matter physics are of this type. A wide range of interesting behaviors arises due to interactions between the component systems. It is a central feature of extended physical systems that interactions have a local structure, meaning that the strength of the interactions decreases with the distance between the systems. Often, each system only interacts directly with its nearest neighbors in the lattice. The mean-field approximation ignores the geometry of the ambient space and it is often a good first approximation. In more realistic models, however, the interactions between different components depend on the distance between them. In this section, we derive a fundamental property of the dynamics of quantum lattice systems that is intimately related to the local structure of the interactions. This property is referred to as quasilocality and its basic feature is a bound on the speed of propagation of disturbances in the system, which is known as a Lieb-Robinson bound.

Lieb and Robinson were the first to derive bounds of this type.75 In the years following the original article, a number of further important results appeared, e.g., by Radin107 and, in particular, by Robinson112 who gave a new proof of the theorem of Lieb and Robinson (which is included in Ref. 20). Robinson also showed that Lieb-Robinson bounds can be used to prove the existence of the thermodynamic limit of the dynamics and used the bounds to derive fundamental locality properties of quantum lattice systems. It was only much later, however, that Hastings who pointed out how the Lieb-Robinson bounds could be used to prove exponential clustering in gapped ground states in a paper where he provided the first generalization of the Lieb-Schultz-Mattis theorem to higher dimensions.54 Mathematical proofs then followed by Nachtergaele and Sims,92 Hastings and Koma,57 and Nachtergaele, Ogata, and Sims.88 The new approach to proving Lieb-Robinson bounds developed in these works leading to Ref. 94 yields a better prefactor with a more accurate dependence on the support of the observables. This was important for certain applications such as the proof of the split property for gapped ground states in one dimension by Matsui.81,82

Further extensions of the Lieb-Robinson bounds in several directions quickly followed: Lieb-Robinson bounds for lattice fermions,24,57,97 Lieb-Robinson bounds for irreversible quantum dynamics,53,98,105 a bound for certain long-range interactions,45,111,124 anomalous or zero-velocity bounds for disordered and quasiperiodic systems,27,35,36,52 propagation estimate for lattice oscillator systems,4,26,34,89 and other systems with unbounded interactions,106 including classical lattice systems.28,61,108

The list of applications of Lieb-Robinson bounds includes a broad range of topics: Lieb-Schultz-Mattis theorems,54,93 the entanglement area law in one dimension,55 the quantum Hall effect,6,44,58 quasiadiabatic evolution (spectral flow and automorphic equivalence) including stability and classification of gapped ground state phases,12,21,22,59,83,96 the stability of dissipative systems,19,76 the quasiparticle structure of the excitation spectrum of gapped systems,10,50 a stability property of the area law of entanglement,78 the efficiency of quantum thermodynamic engines,118 the adiabatic theorem and linear response theory for extended systems,7,8 the design and analysis of quantum algorithms,49 and the list continues to grow.5–7,25,30,38,47,64,84,123

In order to express the locality properties of the interactions and the resulting dynamics, we introduce some additional structure on the discrete metric space (Γ, d) in Sec. III A.

1. General setup

As described above, we will study quantum lattice models with possibly unbounded single-site Hamiltonians but bounded, in general, time-dependent interactions. In this section, we give the framework for quantum lattice systems and describe the bounded interactions of interest. We will consider the addition of unbounded on-site Hamiltonians in Sec. III B.

The lattice models we consider are defined over a countable metric space (Γ, d). To each site x ∈ Γ, we associate a complex Hilbert space Hx and denote the algebra of all bounded linear operators on Hx by B(Hx)⁠. Let P0(Γ) be the collection of all finite subsets of Γ. For any Λ∈P0(Γ)⁠, the Hilbert space of states and algebra of local observables over Λ are denoted by

(3.1)

where we have chosen to define the tensor product of the algebras B(Hx) so that the last equality holds (i.e., the spatial tensor product, corresponding to the minimal C*-norm116). For any two finite sets Λ0 ⊂ Λ ⊂ Γ, each A∈AΛ0 can be naturally identified with A⊗1Λ\Λ0∈AΛ⁠. With respect to this identification, the algebra of local observables is then defined as the inductive limit

(3.2)

and the C*-algebra of quasilocal observables, which we denote by AΓ⁠, is the completion of AΓloc with respect to the operator norm. We will use the phrase quantum lattice system to mean the countable metric space (Γ, d) and quasilocal algebra AΓ⁠.

A model on a quantum lattice system is given in terms of an interaction Φ. In the time-independent case, this is a map Φ:P0(Γ)→AΓloc such that Φ(Z)*=Φ(Z)∈AZ for all Z∈P0(Γ). The quantum lattice model associated with Φ is the collection of all local Hamiltonians of the form

(3.3)

We will also consider time-dependent interactions. Let I⊂R be an interval. A map Φ:P0(Γ)×I→AΓloc is said to be a strongly continuous interaction if

  • To each t ∈ I, the map Φ(⋅,t):P0(Γ)→AΓloc is an interaction.

  • For each Z∈P0(Γ)⁠, Φ(Z,⋅):I→AZ is strongly continuous.

Given such a strongly continuous interaction Φ, we will often denote by Φ(t) the interaction Φ(·, t) as in (i) above and define the corresponding local Hamiltonians

(3.4)

Similarly, a corresponding time-dependent quantum lattice model may be defined. By our assumptions on the interaction, it is clear that for each t ∈ I, HΛ(t) is a bounded, self-adjoint operator on HΛ⁠. Moreover, by Proposition 2.1, HΛ:I→AΛ is strongly continuous. In this case, Proposition 2.2 demonstrates that there exists a two-parameter family of unitaries {UΛ(t,s)}s,t∈I⊂AΛ⁠, defined as the unique strong solution of the initial value problem

(3.5)

In terms of these unitary propagators, we define a Heisenberg dynamics τt,sΛ:AΛ→AΛ by setting

(3.6)

In some applications, including Theorem 3.1, we will also consider the inverse dynamics,

(3.7)

where the final equality follows from Proposition 2.2(iv).

As discussed above, Lieb-Robinson bounds approximate the speed of propagation of dynamically evolved observables through a quantum lattice system, and this estimate is closely tied to the locality of the interaction in question. To quantify the locality of an interaction, we introduce the notion of an F-function. An F-function on (Γ, d) is a nonincreasing function F:[0, ∞) → (0, ∞), satisfying the following two properties:

  • F is uniformly integrable over Γ, i.e.,

(3.8)
  • (ii)

    F satisfies the convolution condition

(3.9)

An equivalent formulation of (ii) is that there exists a constant C < ∞ such that

(3.10)

Let F be an F-function on (Γ, d) and g : [0, ∞) → [0, ∞) be any nondecreasing, subadditive function, i.e., g(r + s) ≤ g(r) + g(s) for all r, s ∈ [0, ∞). Then, the function

(3.11)

also satisfies (i) and (ii) with ∥Fg∥ ≤ ∥F∥ and CFg≤CF⁠. We call any F-function of this form a weighted F-function.

It is easy to produce examples of these F-functions when Γ=Zν for some ν ≥ 1 and d(x, y) = |x − y| is the ℓ1-distance. In fact, for any ϵ > 0, the function

(3.12)

is an F-function on Zν⁠. It is clear that this function is uniformly integrable, i.e., (3.8) holds. Moreover, one may verify that

(3.13)

In the special case of g(r) = ar, for some a ≥ 0, we obtain a very useful family of weighted F-functions, which we denote by Fa, given by Fa(r) = e−ar/(1 + r)ν+ϵ. See Subsections  1–3 of the  Appendix for other examples and properties of F-functions.

We use these F-functions to describe the decay of a given interaction. Let F be an F-function on (Γ, d) and Φ:P0(Γ)→AΓloc be an interaction. The F-norm of Φ is defined by

(3.14)

It is clear from the above equation that for all x, y ∈ Γ,

(3.15)

Note that for any Z∈P0(Γ)⁠, there exist x, y ∈ Z for which d(x, y) = diam(Z), the latter being the diameter of Z. In this case, a simple consequence of (3.15) is

(3.16)

We will be mainly interested in situations where the quantity in (3.14) is finite. In this case, the bound (3.16) demonstrates that the F-function governs the decay of an individual interaction term, and moreover, the estimate (3.15) generalizes this notion of decay by including all interaction terms containing a fixed pair of points x and y.

When Γ is finite, then ∥Φ∥F is finite for any interaction Φ and any function F. For infinite Γ, the set of interactions Φ for which ∥Φ∥F < ∞ depends on F. It is easy to check that ∥·∥F is a norm on the set of interactions for which it is finite. In terms of this norm, we define the Banach space

(3.17)

Of course, BF depends on Γ and on the single-site Hilbert spaces Hx⁠, but that information will always be clear from the context.

We introduce an analog of (3.14) for time-dependent interactions as follows: Consider a quantum lattice system composed of (Γ, d) and AΓ⁠. Let I⊂R be an interval and Φ:P0(Γ)×I→AΓloc be a strongly continuous interaction. Given an F-function on (Γ, d), we will denote by BF(I) the collection of all strongly continuous interactions Φ for which the mapping

(3.18)

is locally bounded. As with the operator norm, we will sometimes use the alternate notation ∥Φ∥F(t) for the quantity defined in (3.18). The function t ↦ ∥Φ∥F(t) is measurable since it is the supremum of a countable family of measurable functions. As such, ∥Φ∥F is locally integrable. As in the time-independent case, (3.18) implies that for all t ∈ I and x, y ∈ Γ,

(3.19)

a bound which will appear in many of our estimates. See Subsection 4 of the  Appendix for more useful estimates involving interactions Φ∈BF(I)⁠.

2. Lieb-Robinson estimates for bounded interactions

In Theorem 3.1, we demonstrate that the finite volume Heisenberg dynamics τt,sΛ⁠, as defined in (3.6), associated with any Φ∈BF(I) satisfies a Lieb-Robinson bound. Such bounds provide an estimate for the speed of propagation of dynamically evolved observables in a quantum lattice system. One can use these bounds to show that for small times the dynamically evolved observable is well approximated by a local operator. For this reason, Lieb-Robinson bounds and other similar results are often referred to as quasilocality estimates.

Before we state the result, two more pieces of notation will be useful. First, to each X∈P0(Γ)⁠, we denote by ∂ΦIX⊂X the Φ-boundary of X,

(3.20)

In some estimates, it may be useful to restrict the time interval used to define the Φ-boundary. For instance, given Φ∈BF(R) one could find that ∂ΦRX=X for some X, while ∂ΦIX is strictly smaller for a subinterval I⊂R⁠. From now on, we will drop the time-interval I from the notation and simply write ∂ΦX. We note also that in many situations, not much is lost by using X instead of ∂ΦX in the following estimates.

Second, for Φ∈BF(I)⁠, and s, t ∈ I, the quantity It,s(Φ) defined by

(3.21)

will appear in many results we provide, including Theorem 3.1. Clearly, if CF∥Φ(r)∥F ≤ M, for all r ∈ [min(t, s), max(t, s)], we have It,s(Φ) ≤ |t − s|M. For example, we see that

It,s(Φ)≤CF|t−s|ΦF,

with

(3.22)

Theorem 3.1
(Lieb-Robinson bound). Let Φ∈BF(I) and X,Y∈P0(Γ) with X ∩ Y = ∅. For any Λ∈P0(Γ) with X ∪ Y ⊂ Λ and any A∈AX and B∈AY, we have
(3.23)
for all t, s ∈ I. Here, CF is the constant in (3.9), and the quantity D(X, Y) is given by
(3.24)

It is easy to see that with the definition F1(r)=CF−1F(r)⁠, F1 is a new F-function in terms of which the bound (3.23) slightly simplifies in the sense that CF1=1⁠. This is a general feature of our estimates involving F-functions and the associated norms on the interactions. In what follows, a variety of different F-functions will be used. Often, new F-functions are obtained by elementary transformations of old ones, see, e.g., Subsection 3 of the  Appendix. Instead of figuring out the normalization constants that make CF = 1 for each of the F-functions, we note that the final result can be expressed with a renormalized F-function such that CF = 1.

Before moving on to the proof of the theorem, we make two simple remarks that are implicit in many applications of the Lieb-Robinson bounds. First, one trivially has τt,sΛ(A),B≤2‖A‖‖B‖⁠. Second, in the case that Φ∈BFg(I)⁠, for a weighted F-function Fg(r) = e−g(r)F(r), we can further estimate
(3.25)
where d(X, Y) is the distance between X and Y. When g(r) = ar for some a > 0 [i.e., Φ∈BFa(I) with Fa(r) = e−arF(r)], it makes sense to define the quantity va=2a−1CFaΦFa which is often referred to as the Lieb-Robinson velocity, or more correctly a bound for the speed of propagation of any type of disturbance or signal in the system. In terms of va⁠, (3.23) implies the more transparent estimate
(3.26)
Note that the RHS of the bounds in (3.23) and (3.26) are expressed in terms of quantities defined over the system on Γ, and in particular, these estimates are uniform in the choice of the finite set Λ ⊂ Γ. This fact will be vital in many applications.
Before we prove Theorem 3.1, we first prove a lemma. For this lemma and later use, we define the “surface” of X in the volume Λ, denoted SΛ(X), as follows:
(3.27)
It is simply the set of supports of the interaction terms that connect X and Λ\X. We will also use the following notation for X,Y∈P0(Γ)⁠:
(3.28)

Lemma 3.2.
Let Φ∈BF(I). Fix Y∈P0(Γ), B∈AY, and Λ∈P0(Γ) with Y ⊂ Λ. For any X ⊂ Λ, the family mappings gt,sX,B:AX→AΛ for t, s ∈ I, defined by
(3.29)
are norm-continuous; more precisely, (s,t)↦gt,sX,B is jointly continuous in the norm on B(AX,AΛ). Moreover, for fixed t and s, the mapping gt,sX,B satisfies
(3.30)
The continuity of gt,sX,B follows directly from the joint norm continuity of (s, t) ↦ UΛ(t, s) as proven in Proposition 2.2(v), see also statements following. In fact, for any A∈AX⁠, one has the simple estimate
(3.31)
In general, this continuity does not carry over to the thermodynamic limit. Of course, we always have
(3.32)

We also note that the map gt,sX,B equals the restriction of gt,sΛ,B to AX⁠. It is useful, however, to consider them as separate maps for each X ⊂ Λ because the estimates for their norms depend crucially on X through SΛ(X). Also note that each gt,sX,B only depends on interaction terms Φ(Z, r) such that Z ⊂ Λ and r ∈ [min(t, s), max(t, s)].

Proof of Lemma 3.2.
Fix X ⊂ Λ, A∈AX⁠, and s ∈ I. Recall that the inverse dynamics is given by
(3.33)
where the unitary mappings UX(t, s) are defined as in (3.5) [see also (3.4)] with Λ = X. Consider the function fs:I→AΛ given by fs(t)=gt,sX,B(τ^t,sXs(A))⁠. It follows that fs(t)=[τt,sΛ◦τ^t,sX(A),B] is strongly differentiable in t and a short calculation shows that
(3.34)
where for the first equality, we have used that the adjoint of the unitary propagator has a strong derivative which can be calculated using (2.18); for the second equality, we have used that supp(τ^t,sX(A))⊂X⁠; and for the last equality, we used the Jacobi identity. Hence,
(3.35)
where
(3.36)
Since C and D are finite sums and products of strongly continuous functions with C(t) = C(t)*, they satisfy the assumptions on A and B, respectively, in Lemma 2.3 with t0 = s. Thus, we have
(3.37)
As fs(t)=gt,sX,B(τ^t,sX(A))⁠, the bound claimed in (3.30) follows by applying (3.37) to Ã=τt,sX(A)⁠.

Proof of Theorem 3.1.
Below, we will prove that ‖[τt,sΛ(A),B]‖ satisfies the estimate (3.23) with
(3.38)
Since we also have that
(3.39)
the bound in (3.23) with D(X, Y) defined to be the minimum in (3.24) is also clear.
Let X, Y, Λ, A, and B be as in Theorem 3.1. An application of Lemma 3.2 demonstrates that
(3.40)
for all s, t ∈ I. As such, it suffices to consider the case s ≤ t. Applying the bound (3.32) to the integrand in (3.40), it is clear that we may iteratively apply Lemma 3.2. As a result, for any N ≥ 1,
(3.41)
where
(3.42)
and
(3.43)
The remainder term RN+1(t) is estimated as follows: First, we observe that
(3.44)
Next, we note that the sums above are in fact sums over chains of sets (Z1, Z2, …, ZN+1) which satisfy Z1 ∩ ∂ΦX ≠ ∅ and Zj ∩ Zj−1 ≠ ∅ for 2 ≤ j ≤ N + 1. As such, there are points w1⁠, w2⁠, …, wN+1 ∈ Λ such that w1 ∈ Z1 ∩ ∂ΦX and wj ∈ Zj ∩ Zj−1 for all 2 ≤ j ≤ N + 1. A simple upper bound on these sums is then obtained by overcounting,
(3.45)
where * denotes arbitrary non-negative quantities. We have also used that the set ZN+1 must contain more than one point since ZN+1 ∈ SΛ(ZN). As Φ∈BF(I)⁠, (3.19) implies that
(3.46)
for each 1 ≤ k ≤ N + 1. Using this bound as well as (3.8) and (3.9), we conclude that
(3.47)
We note that, in the last inequality, we performed the integration over the simplex. Since ∥Φ∥F is locally integrable on I, this remainder clearly goes to 0 as N → ∞.
A similar estimate can be applied to the terms an(t). In fact, these terms are also sums over chains; however, there is a restriction: only those chains whose final link Zn satisfies Zn ∩ Y ≠ ∅ contribute to the sum. Recalling that It,s(Φ)=CF∫st‖Φ‖F(r)dr⁠, the bound
(3.48)
then follows as above. Since δY(X) = 0 and n ≥ 1, the bound in (3.23) is now clear.

As we now discuss, the methods in Subsection III A extend to models with unbounded on-site terms. Consider a quantum lattice system composed of (Γ, d) and AΓ⁠. Let I⊂R be an interval, F be an F-function on (Γ, d), and Φ∈BF(I) be a time-dependent interaction. To each z ∈ Γ, fix a self-adjoint operator Hz with dense domain Dz⊂Hz⁠. For any Λ∈P0(Γ) and t ∈ I, consider the finite-volume Hamiltonian

(3.49)

The noninteracting Hamiltonian

(3.50)

is essentially self-adjoint with domain

(3.51)

[see Ref. 110 (Theorem VIII.33 and Corollary)]. Since the time-dependent terms are bounded, it follows from Ref. 126 (Theorem 5.28) that for each t ∈ I, HΛ(t) is essentially self-adjoint on HΛ with domain DΛ⁠. We proceed by using the notations HΛ(0) and HΛ(t) for the corresponding self-adjoint closures.

As Φ∈BF(I)⁠, it is a strongly continuous interaction, and so for any Λ∈P0(Γ)⁠, Proposition 2.4 guarantees the existence of a finite volume unitary propagator corresponding to HΛ(t). Let us briefly review this in order to motivate our definition of the finite volume dynamics. By Stone’s theorem, the noninteracting self-adjoint Hamiltonian HΛ(0) generates a free-dynamics

(3.52)

in terms of a group of strongly continuous unitaries UΛ(0)(t,0)=e−itHΛ(0)⁠. In this case,

(3.53)

is pointwise self-adjoint with H̃Λ:I→AΛ strongly continuous. By Proposition 2.2(v), there is a unique strong solution of the initial value problem

(3.54)

for each s ∈ I. In terms of these solutions, we introduce

(3.55)

for any s, t ∈ I. As is demonstrated in the proof of Proposition 2.4, the operators {UΛ(t,s)}s,t∈I form a two-parameter family of unitaries. They satisfy the cocycle property (2.19), and generate the unique locally norm bounded weak solutions of the time-dependent Schrödinger equation corresponding to HΛ(t). We use these unitaries to define a dynamics associated with HΛ(t), namely, for any s, t ∈ I, we take τt,sΛ:AΛ→AΛ as

(3.56)

One readily checks that the family {τt,sΛ∣t,s∈I} of automorphisms on AΛ satisfies the cocycle property and that the following analog of Theorem 3.1 holds for this dynamics.

Theorem 3.3.
Let {Hz}z∈Γ be a collection of densely defined self-adjoint operators, Φ∈BF(I), and τt,sΛ be the dynamics given in (3.56). Let X,Y∈P0(Γ) be disjoint sets. For any Λ∈P0(Γ) with X ∪ Y ⊂ Λ and any A∈AX and B∈AY, the bound
(3.57)
holds for all t, s ∈ I. Here, CF is the constant in (3.9), and the quantities It,s(Φ) and D(X, Y) are as discussed earlier [see (3.21) and (3.24), respectively].

Proof.
By construction, it is clear that
(3.58)
Here, τ^s(0) and τ̃t,sΛ are the inverse free-dynamics and the interaction-picture dynamics, i.e.,
(3.59)
In this case,
(3.60)
Note that for all t ∈ I, τt(0)(A)∈AX and τt(0)(B)∈AY⁠. Moreover, the interaction-picture dynamics is generated by the strongly continuous interaction Φ̃ with terms
(3.61)
for any Z∈P0(Γ) and t ∈ I. Since Φ̃(Z,t) and Φ(Z, t) have the same support and the same norm, it is clear that ‖Φ̃‖F(t)=‖Φ‖F(t) for all t ∈ I. In this case, the bound in (3.57) follows from Theorem 3.1 applied to the interaction-picture dynamics Φ̃⁠.
In Sec. II C 2, we considered a family of Hamiltonians [see (2.46)] with bounded interactions which depend not only on time but also on an auxillary parameter. Within the context of quantum lattice models, the corresponding finite-volume Hamiltonian may have the form
(3.62)
If (λ, t) ↦ Φλ(X, t) is jointly strongly continuous and strongly differentiable with respect to λ, Proposition 2.7 applies to the corresponding finite-volume dynamics
(3.63)
To be clear, we note that the unitaries UΛλ(t,s) [in (3.63)] are constructed as in (3.55) by first solving the analog of (3.54) with
(3.64)
With no further assumptions on the interaction terms, the bound provided by Proposition 2.7 on the λ-derivative of τt,sΛ,λ(A) [see (2.59)] will generally depend on the volume Λ. However, under the additional assumption that Φλ,Φλ′∈BF(I)⁠, one obtains a better, volume independent estimate on the derivative. In fact, arguing as in the proof of Proposition 2.7, one finds that if s ≤ t, then
(3.65)
compare with (2.60). Since Φλ∈BF(I)⁠, the Lieb-Robinson bound (3.57) holds for the dynamics τt,sΛ,λ⁠. In this case, an application of Corollary A.5 shows that for all A∈AX⁠,
(3.66)
the right-hand-side of which is independent of Λ.

In this section, we will prove several convergence and continuity results for the Heisenberg dynamics associated with interactions Φ∈BF(I) that make use of Lieb-Robinson bounds. As is well-known (see, e.g., Ref. 20), Lieb-Robinson bounds can be used to prove the existence of a dynamics in the thermodynamic limit for sufficiently short-range interactions. In Theorem 3.5, we show that given an interaction Φ∈BF(I) the dynamics corresponding to finite-volume restrictions of Φ converge in the thermodynamic limit. To prove the existence of the thermodynamic limit, we will apply Theorem 3.4, which establishes that the Heisenberg dynamics is continuous in the interaction space. For example, in the case of time-independent interactions, Theorem 3.4 implies that the difference between the dynamical evolution of a local observable A with respect to two different interactions Φ,Ψ∈BF is small if ∥Φ − Ψ∥F is small. The statement of this result for finite-volume Heisenberg dynamics is the content of Theorem 3.4, with the analogous thermodynamic limit statement given in Corollary 3.6. Finally, given a sequence of interactions which converge locally in F-norm [see Definition 3.7], we show that the corresponding dynamics (which necessarily exist by Theorem 3.5) converge as well; this is the content of Theorem 3.8. In particular, this can be used to prove that the thermodynamic limit of the Heisenberg dynamics is unchanged by the addition of (sufficiently local) boundary conditions. If the interactions are norm continuous, the cocycle of automorphisms describing the infinite-volume dynamics is differentiable with a strongly continuous generator. This is shown in Theorem 3.9.

We now begin with the continuity statement. For this result, we will once again make use of the quantity It,s(Φ), which is defined in (3.21).

Theorem 3.4.
Consider a quantum lattice system composed of (Γ, d) and AΓ. Let I⊂R be an interval, F be an F-function on (Γ, d), and Φ,Ψ∈BF(I) be time-dependent interactions. Fix a collection of densely defined, self-adjoint on-site Hamiltonians {Hz}z∈Γ, and for any Λ∈P0(Γ), define Hamiltonians
(3.67)
as well as their corresponding dynamics, τt,sΛ and αt,sΛ, for each s, t ∈ I, respectively.
  • For any X,Λ∈P0(Γ) with X ⊂ Λ, the bound
    (3.68)
    holds for all A∈AX and s, t ∈ I.
  • For any X,Λ0,Λ∈P0(Γ) with X ⊂ Λ0 ⊂ Λ, the bound
    (3.69)
    holds for all A∈AX and s, t ∈ I.

Using (3.8), the estimates in (3.68) and (3.69) can be interpreted as bounds on the norm of the difference of two dynamics, thought of as maps from AX to AΛ⁠, that are uniform in Λ but grow linearly in |X|. Since the dynamics are, of course, automorphisms, the bounds of Theorem 3.4 are only nontrivial if the RHS of (3.68) and (3.69) are smaller than 2∥A∥. As is well known, this will be true for both cases if |t − s| is sufficiently small. Additionally, the bound in (3.68) will be nontrivial if ∥Φ − Ψ∥F is small, and the bound in (3.69) will be nontrivial if d(X, Λ\Λ0) is small. Note that even if a map is bounded on all of AΓloc⁠, as is the case in Theorem 3.4, norm bounds for their local restriction can be very useful.

Proof.
To prove (i), we first consider the case that Hz = 0 for all z ∈ Γ. For any Λ∈P0(Γ)⁠, X ⊂ Λ, s, t ∈ I, and A∈AX⁠, we write the corresponding difference as
(3.70)
where we have introduced the local Hamiltonian
(3.71)
Note that the equality in (3.70) is to be understood in the strong sense. When s ≤ t, a simple norm bound then shows that
(3.72)
An application of Corollary A.5 with R = 0 then gives
(3.73)
and moreover,
(3.74)
The bound (3.68) follows from observing that one could instead have estimated the difference αt,sΛ(A)−τt,sΛ(A)⁠, which corresponds to exchanging τt,sΛ and αt,sΛ in (3.70)–(3.74).
To extend the result to the situation with nontrivial Hz, we argue with the interaction picture dynamics as was done in the proof of Theorem 3.3. Using (3.58), it is clear that
(3.75)
and since τt(0)(A)∈AX⁠, the proof in the general situation reduces to the previous case.
The proof of (ii) is nearly identical. In fact, again in the case that Hz = 0 for all z ∈ Γ, the analog of (3.72) is
(3.76)
where we have taken advantage of cancellations and used (3.27). The bound in (3.69) now follows from similar estimates to those used in Proposition A.4 and Corollary A.5.

A first application of Theorem 3.4 is a proof that given any collection of self-adjoint on-sites {Hz}z∈Γ and an interaction Φ∈BF(I)⁠, there is a corresponding infinite-volume dynamics on AΓ⁠. We obtain this infinite-volume dynamics as a limit of finite-volume dynamics. With this in mind, we will say that a sequence {Λn}n≥1⊂P0(Γ) is increasing and exhaustive if Λn ⊂ Λn+1 for all n ≥ 1 and given any X∈P0(Γ)⁠, there is an N ≥ 1 for which X ⊂ ΛN.

Theorem 3.5.
Under the assumptions of Theorem 3.4, for each A∈AΓloc and any s, t ∈ I,
(3.77)
exists in norm and the convergence is uniform for s and t in compact subsets of I. The limit may be taken along any increasing, exhaustive sequence of finite subsets of Γ and is independent of the sequence. Moreover, τt,s is a cocycle of automorphisms of AΓ. If there exists M ≥ 0 such that ∥Hz∥ ≤ M for all z ∈ Γ, then (t, s) ↦ τt,s(A) is norm continuous for all A∈AΓ⁠.

For unbounded Hz, the continuity of τt,s is limited by the continuity of the on-site dynamics τt(0)⁠. In a suitable representation of AΓ on a Hilbert space, one can retrieve weak continuity of the dynamics. See Ref. 90 for an example. Continuity properties of τt,s and other families of maps will be further discussed in Sec. IV B 1.

Proof.
Let X∈P0(Γ)⁠, let A∈AX⁠, and consider {Λn}n≥0 as any increasing, exhaustive sequence of finite subsets of Γ. Since the sequence is exhaustive, there exists N ≥ 1 for which X ⊂ Λn for all n ≥ N. For any integers n, m with N ≤ m ≤ n, Theorem 3.4(ii) implies
(3.78)
where, again, It,s(Φ) is as defined in (3.21). By (3.8), the RHS converges to 0 as n, m → ∞. As such, for any [a, b] ⊂ I, the sequence of observables {τt,sΛn(A)}n≥0 is Cauchy in norm, uniformly for s, t ∈ [a, b].

The proof of the remaining facts in the statement of this theorem is standard and proceeds in the same way as is done, e.g., in Ref. 120 for quantum spin models with time-independent interactions.

Combining Theorems 3.4 and 3.5, we obtain the following useful estimates for the infinite volume dynamics:

Corollary 3.6.

Under the assumptions of Theorem 3.4,

  • For any X, Y∈P0(Γ) such that X ∩ Y = ∅, the bound
    ‖[τt,s(A),B]‖≤2‖A‖‖B‖CF(e2It,s(Φ)−1)D(X,Y)
    holds for all A∈AX⁠, B∈AY, and t, s ∈ I.
  • For any X∈P0(Γ), the bound
    (3.79)
    holds for all A∈AX and s, t ∈ I.
  • For any X,Λ∈P0(Γ) with X ⊂ Λ, the bound
    (3.80)
    holds for all A∈AX and s, t ∈ I.

We now prove a convergence result for the dynamics associated with interactions in BF(I)⁠. First, we introduce some notation and terminology associated with extensions and restrictions of interactions, and then state the result.

In certain applications, e.g., when considering boundary conditions, rather than considering a single interaction Φ:P0(Γ)→AΓloc⁠, we may want to consider a family of strongly continuous interactions {ΦΛ:P0(Λ)×I→AΛloc | Λ∈P0(Γ)}⁠. In this situation, a single mapping ΦΛ can be extended to an interaction on all of Γ by declaring that ΦΛ(Z, t) = 0 for any Z∈P0(Γ) with Z ∩ (Γ\Λ) ≠ ∅. We call this new mapping the extension of ΦΛ to Γ and continue to denote it by ΦΛ. Similarly, given ΦΛ and Λ0 ⊂ Λ, we define ΦΛ↾Λ0:P0(Λ0)×I→AΛ0loc by
(3.81)
We call the mapping ΦΛ↾Λ0 the restriction of ΦΛ to Λ0. If the dynamics associated with ΦΛ↾Λ0 exists, we will denote it by τt,sΦΛ,Λ0⁠. Of course, if Λ0 is finite, such a dynamics always exists and it is generated by the time-dependent Hamiltonian
(3.82)

We now introduce the notion of local convergence in F-norm.

Definition 3.7.

Let (Γ, d) and AΓ be a quantum lattice system, and let I⊂R be an interval. We say that a sequence of interactions {Φn}n≥1 converges locally in F-norm to Φ if there exists an F-function, F, such that

  • Φn∈BF(I) for all n ≥ 1,

  • Φ∈BF(I)⁠,

  • for any Λ∈P0(Γ) and each [a, b] ⊂ I,

(3.83)
Moreover, if F is an F-function for which (i)–(iii) are satisfied, we say that Φn converges locally in F-norm to Φ with respect to F.
Recall that a strongly continuous interaction Φ∈BF(I) with ∥Φ∥F : I → [0, ∞) given by
(3.84)
is locally bounded. In this situation, Theorem 3.5 demonstrates that there exists a cocycle of automorphisms of AΓ⁠, which we denote by τt,sΦ and refer to as the dynamics associated with Φ. Note that we have taken the self-adjoint on-site terms to be identically zero, i.e., Hz = 0 for all z ∈ Γ (see comments following the proof of Theorem 3.8).

Theorem 3.8.

Let {Φn}n≥1 be a sequence of time-dependent interactions on Γ with Φn converging locally in the F-norm to Φ with respect to F.

  • If for every [a, b] ⊂ I,
    (3.85)
    then, for any X∈P0(Γ)⁠, A∈AX⁠, and s, t ∈ I, s ≤ t, we have convergence of the dynamics
    (3.86)
    Moreover, the convergence is uniform for s, t in compact intervals, and the dynamics is continuous,
    (3.87)
  • If, in addition, for all Λ∈P0(Γ), and r ∈ I, we also have pointwise local convergence,
    (3.88)
    and uniform boundedness of the interactions on compact intervals I0 ⊂ I,
    (3.89)
    then the generators converge uniformly for t in compacts: for all compact I0 ⊂ I and A∈AΓloc, we have
    (3.90)
    where
    δr(n)(A)=∑Z∈P0(Γ)[Φn(Z,r),A], δr(A)=∑Z∈P0(Γ)[Φ(Z,r),A].

Proof.
(i) Let X∈P0(Γ) and [a, b] ⊂ I be fixed. For any Λ∈P0(Γ) with X ⊂ Λ, define the restricted interactions Φn↾Λ and Φ↾Λ. By Theorem 3.5, a dynamics may be associated with each of these interactions and the estimate
(3.91)
holds for all A∈AX and s, t ∈ [a, b].
An application of Corollary 3.6(iii) shows that
(3.92)
and similarly
(3.93)
For the middle term above, we apply Theorem 3.4(i) to find that
(3.94)

By assumption (3.85), it is clear that supnIt,s(Φn) is finite for all s, t ∈ [a, b]. In this case, for any ϵ > 0, the estimates in (3.92) and (3.93) can be made arbitrarily small for all n sufficiently large (for example, less than ϵ/3) with a sufficiently large, but finite choice of Λ ⊂ Γ. For any such choice of Λ, the bound (3.94) can be made equally small with large n by using local convergence in F-norm.

To prove the bound (3.87), we use the convergence established above, the existence of the thermodynamic limit (Theorem 3.5), and the differentiability of the finite-volume dynamics, as follows:
‖τt,sΦ(A)−A‖=limΛlimn‖τt,sΦn,Λ(A)−A‖≤lim supΛlimn∑Z⊂Λ‖∫stτt,sΦn,Λ([Φn(Z,r),A])dr‖≤lim supΛlim supn2|X|‖A‖‖F‖∫st‖Φn↾Λ‖F(r)dr≤2|X|‖A‖‖F‖lim supΛ∫st‖Φ↾Λ‖F(r)dr+lim supΛlim supn∫st‖(Φ−Φn)↾Λ‖F(r)dr.
The second term between the square brackets vanishes because the interactions converge locally in F-norm and the first term gives the desired estimate.
  • (ii)

    The interactions Φn(t) and Φ(t) have finite F-norm. Hence, the corresponding derivations, δt(n) and δt, are well-defined on AΓloc⁠. For any Λ∈P0(Γ)⁠, we then have

(3.95)
Therefore, applying (A41) with R = 0 to the first term and using a similar argument for the second term, we get
(3.96)
The first term on the RHS vanishes in the limit n → ∞. Therefore,
(3.97)
By taking Λ↑Γ, the convergence of the generators now follows. The estimate (3.96) shows that the convergence is uniform for t in a compact interval.

The dynamics considered in the proof of Theorem 3.8 corresponds to the one whose existence is established in the proof of Theorem 3.5 in the special case that the on-sites Hz = 0 for all z ∈ Γ. By going to the interaction picture, as is done in the proof of Theorem 3.4, it is clear that the convergence results (3.87) and (3.90) hold in the case of arbitrary self-adjoint on-site terms.

Theorem 3.8 establishes sufficient conditions for the convergence of the sequence of cocycles τt,sΦn to the cocycle τt,sΦ⁠, as well as the convergence of the generators δt(n) to densely defined derivations δt. These conditions are by no means necessary, but will serve our purposes well.

We may now ask whether the dynamics satisfies additional properties and, in particular, whether it is differentiable with the derivative given by the derivation δt. Theorem 5.9 addresses this question.

Theorem 3.9.
For all t in an interval I, let Φ(t)∈BF be interactions such that t ↦ Φ(Z, t) is norm-continuous for all Z∈P0(Γ) and such that ∥Φ(t)∥F is bounded on all compact intervals I0 ⊂ I. Let τt,s denote the strongly continuous dynamics generated by Φ(t). Define, for all t ∈ I,
(3.98)
Then, t → δt(A) is norm-continuous for all A∈AΓloc, and for all s, t ∈ I, s < t,
(3.99)

Proof.
First, note that the conditions of Theorem 3.8, parts (i) and (ii), are satisfied for the sequence Φn=Φ↾Λn associated with any sequence of increasing and absorbing finite volumes Λn. Therefore, we have
(3.100)
and
(3.101)
To prove the continuity of δt(A) as a function of t, note that for s,t∈I,X∈P0(Γ),A∈AX⁠, and Λ ⊂ Γ, we have
‖δt(A)−δs(A)‖=∑Z∈P0(Γ)Z∩X≠∅[Φ(Z,t)−Φ(Z,s),A]≤2‖A‖∑Z⊂ΛZ∩X≠∅‖Φ(Z,t)−Φ(Z,s)‖+        ∑Z∈P0(Γ)Z∩(Γ\Λ)≠∅,Z∩X≠∅        ‖Φ(Z,t)‖+‖Φ(Z,s)‖≤2‖A‖∑Z⊂ΛZ∩X≠∅‖Φ(Z,t)−Φ(Z,s)‖+2supr∈I0‖Φ(r)‖F∑x∈X∑y∈Γ\ΛF(d(x,y)).
Continuity follows from this estimate by first choosing Λ large enough to make the second term small and then, for that Λ, using the fact that the first term is a finite sum of continuous functions that vanish for t = s.
To prove differentiability, we first show that the finite-volume derivatives converge to the desired limit, uniformly on compact intervals. Consider
(3.102)
For the first term on the RHS, we use the uniformity of the convergence of the derivation obtained in Theorem 3.8(ii) and the boundedness of τt,s, and we estimate the second term using Corollary 3.6(iii). This produces
(3.103)
Since ‖δt(n)(A)‖ is uniformly bounded in t and n, the RHS vanishes as n → ∞, uniformly for s, t in any compact interval I0. This shows the uniform convergence of the derivative, i.e., for all X∈P0(Γ) and A∈AX⁠,
(3.104)
By the continuity of τt,s(δt(A)) in t and the usual argument using the fundamental theorem of calculus, it now follows that τt,s(A) is differentiable in t with the derivative given by iτt,s(δt(A)).

We conclude this section with a few comments.

It is clear how to modify Definition 3.7 in such a way to describe sequences that are locally Cauchy in F-norm. Given any such Cauchy sequence which also satisfies (3.85), an ϵ/3-argument almost identical to the one in the proof of Theorem 3.8 shows that the corresponding dynamics converge to a cocycle of automorphisms of AΓ⁠. With this understanding, one sees that Theorem 3.8 implies Theorem 3.5. In fact, let Φ∈BF(I) and take {Λn}n≥1 to be an increasing, exhaustive sequence of finite subsets of Γ. Define the sequence of interactions Φn=Φ↾Λn and extend Φn to all of Γ as indicated above. In this case, it is clear that {Φn}n≥1 is locally Cauchy in the F-norm defined by F, and moreover, (3.85) holds. Thus, the corresponding dynamics converge. Since the sequence {Φn}n≥1 also converges locally to Φ in the F-norm defined by F, we know what the generator of the limiting dynamics is by Theorem 3.8. In this manner, we recover the fact that the limiting dynamics is independent of the increasing, exhaustive sequence of finite-volumes. Moreover, we also see that this limiting dynamics is invariant under a class of finite-volume boundary conditions.

As a final comment we note that one can easily find conditions under which the Duhamel formula (3.66) is also inherited by the infinite-volume dynamics. As this will not be needed in this work, we do not discuss this further.

In Sec. III A 2, we proved a Lieb-Robinson bound for the finite volume dynamics generated by an interaction Φ∈BF(I)⁠. Such bounds provide an estimate for the speed of propagation in a quantum lattice system. More specifically, such bounds can be used to show that while the support of a local observable evolved under the Heisenberg dynamics is nonlocal, at any fixed time t, the observable essentially acts as the identity outside of a finite region of space. It is often useful to approximate these dynamically evolved observables by strictly local observables. It is further desirable that the operation of taking these local approximations has good continuity properties. This is the topic of this section.

We first review how the support of a local observable can be identified using commutators. For any Hilbert space H⁠, the algebra B(H) has a trivial center; in the case of a finite-dimensional Hilbert space, this is known as Schur’s Lemma. A first generalization of this fact is that for any two Hilbert spaces H1 and H2⁠, the commutant of B(H1)⊗1 in B(H1⊗H2) is given by 1⊗B(H2) (see, e.g., Ref. 63, Chap. 11). Given this, and the structure of the quantum models introduced in Sec. III A 1, one concludes the following: given Λ∈P0(Γ) and X ⊂ Λ, if A∈AΛ satisfies [A, B] = 0 for all B∈AΛ\X⁠, then A∈AX⁠. In other words, vanishing commutators can be used to identify the support of local observables. If the commutator [A, B] is small but not necessarily vanishing (in norm), Lemma 4.1, which is proved in Ref. 91, shows that A can be well-approximated (up to error ϵ) by an observable A′∈AX⁠.

Lemma 4.1.

Let H1 and H2 be complex Hilbert spaces. There is a completely positive linear map E:B(H1⊗H2)→B(H1) with the following properties:

  • For all A∈B(H1), E(A⊗1)=A.

  • Whenever A∈B(H1⊗H2) satisfies the commutator bound
    [A,1⊗B]≤ϵ‖B‖  for all B∈B(H2),
    E(A) satisfies the estimate
(4.1)
  • (iii)

    For all C,D∈B(H1) and A∈B(H1⊗H2), we have

E((C⊗1)A(D⊗1))=CE(A)D.

A completely positive linear map E with the properties (i) and (iii) is called a conditional expectation [see, e.g., Ref. 103 (Sec. 9.2)]. If H2 is finite-dimensional, one can take E=id⊗tr⁠, where tr denotes the normalized trace over H2⁠. In this case, it is straightforward to verify the properties listed in the lemma (see, e.g., Refs. 23 and 88). For general H2⁠, a normalized trace does not exist, but using Lemma 4.1 it is easy to show that, at the cost of a factor 2 in the RHS of (4.1), we can replace E with id ⊗ ρ for an arbitrary state ρ on B(H2)⁠. This is the content of Lemma 4.2 from Ref. 91.

Lemma 4.2.
Let H1 and H2 be two complex Hilbert spaces and ρ be a state on B(H2). The map Πρ = id ⊗ ρ satisfies properties (i) and (iii) of Lemma 4.1. Moreover, if A∈B(H1⊗H2) is such that there is an ϵ ≥ 0 for which
(4.2)
then
(4.3)

Proof.
It is clear that Πρ satisfies properties (i) and (iii) in Lemma 4.1. Moreover, since ∥Πρ∥ = 1, we also have that for any A∈B(H1⊗H2)⁠,
(4.4)
where we have used (4.1). The estimate (4.3) now follows:
(4.5)
Note that the state ρ in Lemma 4.2 is not required to be normal, i.e., defined by a density matrix. However, in applications, it will often be useful to take ρ normal (or locally normal in the case of infinite systems, see Sec. IV B). For normal ρ, we can give an explicit expression for Πρ(A) in terms of its matrix elements. Since ρ is given by a density matrix, there is a countable set of orthonormal vectors ξn∈H2 and positive numbers ρn with ∑nρn = 1 so that ρ(A′) = ∑n≥1ρn⟨ξn, A′ξn⟩ for all A′∈B(H2)⁠. The matrix elements of Πρ(A) are then given by
(4.6)

A number of further comments are in order. First, although the mapping Πρ depends on the state ρ, the “error” estimate in (4.3) is independent of ρ. Next, if H2 is finite dimensional, then ρ can be taken to be the normalized trace and we already know that the factor of two in (4.3) is not needed. The bound for Πρ therefore appears to be nonoptimal. The map E from Lemma 4.1 is only known to be bounded (specifically, ‖E‖=1⁠) and hence continuous with respect to the operator norm topology. As such, it is not guaranteed that E is continuous when both the domain and codomain are endowed with the strong operator topology. However, by choosing a normal state ρ, we get a map Πρ that is continuous on bounded subsets of the domain when both B(H1⊗H2) and B(H1) are endowed with their strong (or weak) operator topologies. The case of the strong operator topology is the content of Proposition 4.3. The case of the weak topology follows by a similar argument.

Recall that K:B(H1)→B(H2) is continuous on bounded subsets with both its domain and codomain considered with the strong operator topology if given any bounded net Aα∈B(H1) that converges strongly to A∈B(H1)⁠, the net K(Aα)∈B(H2) converges strongly to K(A)∈B(H2)⁠.

Proposition 4.3.

Let H1 and H2 be two complex Hilbert spaces, and ρ be any normal state on B(H2). The following maps, when restricted to arbitrary bounded subsets of their domain, are continuous when both the domain and codomain are equipped with the strong operator topology:

  • Πρ=id⊗ρ:B(H1⊗H2)→B(H1)⁠,

  • Π̃ρ:B(H1⊗H2)→B(H1⊗H2),A↦Πρ(A)⊗1.

Proof.
(i) To prove that Πρ is continuous on bounded sets, without loss of generality (WLOG), let {Aα∣α ∈ I} be a net in the unit ball of B(H1⊗H2) that converges to A∈B(H1⊗H2) in the strong operator topology, i.e., for all ψ∈H1⊗H2⁠, the net Aαψ converges to Aψ with respect to the Hilbert space norm. Since ∥Aα∥ ≤ 1 for all α ∈ I, we necessarily have that ∥A∥ ≤ 1. Let {ξn, n ≥ 1} denote the orthonormal set of eigenvectors of ρ corresponding to its nonzero eigenvalues ρn. Let ϕ∈H1 with ∥ϕ∥ = 1. We use (4.6) to show that Πρ(Aα)ϕ → Πρ(A)ϕ. Note that
‖Πρ(A−Aα)ϕ‖=supη∈H1,‖η‖=1|⟨η,Πρ(A−Aα)ϕ⟩|=supη∈H1,‖η‖=1∑n≥1ρn|⟨η⊗ ξn,(A−Aα)(ϕ⊗ξn)⟩|.
For any ϵ > 0, choose N so that ∑n>Nρn < ϵ/4 and pick α0 ∈ I so that for all α ≥ α0 and any n = 1, …, N,
‖(A−Aα)(ϕ⊗ξn)‖≤ϵ/(2N).
Then, for any α ≥ α0,
(4.7)
which establishes the desired continuity of Πρ.
  • (ii)

    Let {Aα |α ∈ I} be a net in the unit ball of B(H1⊗H2) that converges to A∈B(H1⊗H2)⁠. By (i), we know that Bα = Πρ(Aα) converges to B = Πρ(A) in the strong operator topology on B(H1)⁠. By Proposition 2.1(ii) [see also (2.4) and the preceding discussion], it follows that {Bα ⊗ 1l∣α ∈ I} strongly converges to B ⊗ 1l in B(H1⊗H2)⁠.

We now extend the results of Subsection IV A to infinite quantum lattice systems on Γ. In this setting, states cannot be defined in terms of a single density matrix. Moreover, as explained below, we will want to define a consistent family of conditional expectations with values in AΛ⁠, for all Λ∈P0(Γ)⁠. To this end, we consider a locally normal product state ρ, i.e., for each site x ∈ Γ, we fix a normal state ρx on Ax=B(Hx) and take the unique state ρ on AΓ such that ρ↾AΛ=⊗x∈Λρx for all finite Λ ⊂ Γ. Then, given X⊂Λ∈P0(Γ)⁠, we define conditional expectations ΠρX,Λ:AΛ→AX similar to those in Lemma 4.2 by

(4.8)

Here, as before, we have taken idX as the identity map on AX⁠. In our applications, the dependence of these maps on ρ is of minor consequence. Moreover, it will be convenient to view these maps as elements of B(AΛ)⁠. For these reasons, we suppress the dependence on ρ and define ΠXΛ:AΛ→AΛ by

(4.9)

For fixed X, these projections are compatible in the sense that if X ∪ Λ′ ⊂ Λ and Λ∈P0(Γ)⁠, then

(4.10)

We summarize this relation and other consistency relations in Proposition 4.5. First, however, we describe how, given a fixed finite volume X, one can extend the maps ΠXΛ⁠, Λ∈P0(Γ)⁠, to an operator ΠX on AΓloc (and consequently, AΓ⁠).

For any Λ′ ⊆ Λ, recall that we can identify AΛ′ as a subalgebra of AΛ⁠, and so we can write (4.10) as ΠXΛ↾AΛ′=ΠX∩Λ′Λ′⁠. In particular, if X ⊆ Λ′ ⊆ Λ, one has that ΠXΛ↾AΛ′=ΠXΛ′⁠, from which we see that the following map ΠX:AΓloc→AΓloc is well-defined:

(4.11)

Since this map is bounded, in fact of norm one, ΠX has a unique extension to AΓ which we also denote by ΠX. We note that ΠX(A) = A if A∈AX⁠. We refer to the family of conditional expectations ΠXΛ⁠, respectively, ΠX, X∈P0(Γ)⁠, as a localizing family. By construction, the finite volume local approximations ΠXΛ all satisfy the conditions of Lemma 4.2. Corollary 4.4 shows that the results of Lemma 4.2 also extend to ΠX:AΓ→AX⁠.

Corollary 4.4.
Let X∈P0(Γ) and ΠX:AΓ→AX be the extension of the map defined in (4.11). Suppose ϵ ≥ 0 and A∈AΓ are such that
(4.12)
Then,
‖ΠX(A)−A‖≤2ϵ.

Proof.
Let A∈AΓ⁠. Then, for any δ > 0, there exists Λ∈P0(Γ) and A′∈AΛ such that ∥A − A′∥ < δ. From (4.12), it follows that
‖[A′,B]‖≤(ϵ+2δ)‖B‖, for all B∈AΛ\Xloc.
Since ΠX(A′)=ΠXΛ(A′)⁠, Lemma 4.2 implies
‖ΠX(A′)−A′‖≤2(ϵ+2δ).
Therefore,
‖ΠX(A)−A‖≤‖ΠX(A′)−A′‖+‖(ΠX−id)(A−A′)‖≤2(ϵ+2δ)+2δ,
and since δ > 0 is arbitrary, the result follows.

We now state several consistency properties of the finite and infinite volume conditional expectations. To facilitate the statement of the properties, we use ΠXΓ to denote ΠX:AΓ→AX⁠.

Proposition 4.5.

Fix a locally normal product state ρ on AΓ, and let X, Y, Λ′∈P0(Γ) and Λ∈P0(Γ)∪{Γ} be such that X, Y, and Λ′ are all subsets of Λ. The following properties hold for the localizing maps defined with respect to ρ:

  • If A∈AX, then ΠXΛ(A)=A.

  • ΠXΛ↾AΛ′=ΠX∩Λ′Λ′.

  • ΠXΛ◦ΠYΛ=ΠYΛ◦ΠXΛ=ΠX∩YΛ.

  • If X ⊆ Λ′, then ΠXΛ′◦ΠΛ′Λ=ΠXΛ.

  • ΠXΛ(A)*=ΠXΛ(A*) for all A∈AΛ.

The proofs of these properties for Λ∈P0(Γ) are all elementary and follow from the definition of ΠX,Λρ and the fact that ρ ↾Λ is a product state. The statements for Λ = Γ follow from taking finite volume limits Λ′ ↑Γ of ΠXΛ′(A) for A∈AΓloc and using the norm bound ∥ΠX∥ ≤ 1 to extend to A∈AΓ in the usual manner.

An immediate consequence of Proposition 4.5(i) is that
(4.13)
for any sequence of increasing and absorbing finite volumes Λn, and any A∈AΓloc⁠. Since AΓloc is dense in AΓ⁠, (4.13) extends to any A∈AΓ⁠.
With the aid of the maps ΠXΛ⁠, we construct local decomposition of any observable A∈AΛ⁠. Let X ⊂ Γ be finite. For any n ≥ 0, denote by X(n) ⊂ Γ the set
(4.14)
Note that X(0) = X. For finite Λ ⊂ Γ with X ⊂ Λ and each integer n ≥ 0, we define ΔX(n)Λ:AΛ→AΛ by
(4.15)
for any n ≥ 1. Note that in contrast to the maps ΠXΛ⁠, ΔX(n)Λ does not only depend on the set X(n), but on X and n separately. This slight abuse of notation will not lead to confusion. As discussed above, ΔX(n)Λ has a range contained in AX(n)∩Λ⁠. Moreover, as a difference of two projections, they satisfy ‖ΔX(n)Λ‖≤2⁠.
Of course, as discussed above, one can also extend these bounded linear maps to AΓ⁠. In fact, for each finite X ⊂ Γ and any n ≥ 0, the maps ΔX(n):AΓ→AΓloc are defined by
(4.16)
for any n ≥ 1. We note again that the range of ΔX(n) is contained in AX(n) regarded as a sub-algebra of AΓloc⁠.
A typical use of the local decompositions is as follows. Fix Λ⊂P0(Γ)⁠, A∈AΛ⁠, and X ⊂ Λ, and denote by N the smallest integer n for which X(n) ∩ Λ = Λ. Clearly, N depends on X and Λ, and ΔX(n)Λ=0 for any n > N. Then, one can write
(4.17)
where this telescopic sum has terms with explicit, local support.
For a quasilocal observable A∈AΓ and X∈P0(Γ)⁠, the conditional convergence of the infinite-volume analog of (4.17), namely,
(4.18)
follows from noticing that ΠX(N)(A)=∑n=0NΔX(n)(A) and invoking (4.13). In Sec. V A, we will discuss situations in which (4.18) converges absolutely. The remainder of this section is concerned with continuity properties and basic estimates for the local approximations ΠX.

1. Continuity of local approximations

Given finite sets X ⊂ Λ ⊂ Γ, Proposition 4.3(ii) implies that the projection map ΠXΛ preserves continuity in the strong operator topology. In particular, if t↦A(t)∈AΛ is strongly continuous for all t in an interval I⊆R⁠, then t↦ΠXΛ(A(t))∈AΛ is also strongly continuous. In applications, we will be interested in a sequence of strongly continuous functions t↦AΛn(t)∈AΛn⁠, with Λn↑ Γ, that converges to a bounded map t↦A(t)∈AΓ⁠. It will then be desirable that the localizing projections ΠY(A(t)), Y∈P0(Γ)⁠, also satisfy certain continuity properties.

While we do not have the standard von Neumann algebra setting where the notion of locally normal is more natural, it is convenient to define a similar notion in our setting with C*-algebras without reference to a representation.

Definition 4.6.

A linear map K:AΓloc→AΓ is called locally normal if there exists an increasing, exhaustive sequence {Λn}n≥0 of finite subsets of Γ and corresponding bounded linear transformations KΛn∈B(AΛn) with the following properties:

  • For all n, KΛn:AΛn→AΛn is continuous on bounded subsets when both its domain and codomain are considered with the strong operator topology;

  • Local uniform convergence of KΛn to K⁠: For all X ⊂ Γ finite and any ϵ > 0, there exists N such that for all n ≥ N we have

(4.19)

Note that local uniform convergence implies that KΛn converges strongly to K⁠. However, since N is allowed to depend on X, this convergence is in general not uniform in P0(Γ)⁠. If HΛn is finite-dimensional, property (i) is automatically satisfied. Let us now consider an example satisfying Definition 4.6.

Example 4.7.
Consider a quantum lattice system composed of (Γ, d) and AΓ. Let F be an F-function on (Γ, d) and Φ∈BF be an interaction. The map K:AΓloc→AΓ given by
(4.20)
is locally normal in the sense of Definition 4.6. In fact, let {Λn}n≥0 be any sequence of nonempty, finite subsets of Γ that are increasing and exhaustive. For each n ≥ 0, define KΛn:AΛn→AΛn by setting
(4.21)
Fix n ≥ 0, let X ⊂ Λn and A∈AX. One checks that
(4.22)
and therefore,
(4.23)
holds for any A∈AX, where we have used that Φ∈BF. Taking X = Λn, one sees that KΛn∈B(AΛn). With n ≥ 0 fixed again, let Z ⊂ Λn. It is clear that A ↦ [Φ(Z), A] satisfies Definition 4.6(i) on AΛn. As a finite sum of such terms, it is clear that KΛn satisfies Definition 4.6(i) as well. Finally, let X∈P0(Γ), A∈AX and N ≥ 1 be sufficiently large so that X ⊂ ΛN. Again, one checks that for any n ≥ N,
(4.24)
and therefore,
(4.25)
Since |X| < ∞, Definition 4.6(ii) holds as F is summable.

A simple consequence of Definition 4.6(i) is the following: for each n ≥ 0, t↦KΛn(A(t)) is strongly continuous if t ↦ A(t) is strongly continuous. Lemma 4.8 establishes that the same property holds for the composition ΠY(K(A(t))) for any Y∈P0(Γ)⁠.

Lemma 4.8.

Let X,Y∈P0(Γ), ΠY:AΓ→AY be the extension of the map defined in (4.11), and let K:AΓloc→AΓ be a locally normal map. Then, for every strongly continuous map t↦A(t)∈AX defined on an interval I⊂R, the function t↦ΠY(K(A(t)))∈AY is also strongly continuous.

Proof.
For ψ∈HY⁠, define f(t)=ΠY(K(A(t)))ψ⁠. We will prove that f:I→HY is continuous by showing that on compact intervals it is the uniform limit of a sequence of continuous functions. Let KΛn be a sequence of maps of the type described in Definition 4.6. For n large enough so that X ⊂ Λn, define fn(t)=ΠYΛn(KΛn(A(t)))ψ⁠. Since K is locally normal, each fn is continuous; here, we are using Proposition 4.3(i) and that t↦KΛn(A(t)) is strongly continuous by Definition 4.6(i). Using compatibility [see Proposition 4.5(ii)], it is clear that fn(t)=ΠY(KΛn(A(t)))ψ⁠. Now for any compact set J ⊂ I, the estimate
(4.26)
follows from ∥ΠY∥ = 1 and local uniform convergence. Since A(t) is locally bounded and J is compact, this proves the claim.

Two comments are in order. First, by Proposition 4.3(ii), for any Z∈P0(Γ) such that Y ⊂ Z, t↦ΠY(K(A(t))) considered as a map into AZ is also strongly continuous. Second, if t ↦ A(t) is, in fact, continuous in the norm topology on AX (in particular, if dimHX<∞⁠), and K is bounded, the result of Lemma 4.8 is trivial since the bounded linear map ΠY◦K preserves the norm-continuity.

We will also encounter one-parameter families of locally normal maps, {Ks∣s∈I}⁠, that are strongly continuous in s and uniformly locally normal in the sense of the following definition:

Definition 4.9.

Let I⊂R be an interval. A family of linear maps Ks:AΓloc→AΓ⁠, s ∈ I, is called a strongly continuous family of uniformly locally normal maps if there exists an increasing, exhaustive sequence {Λn}n≥0 of finite subsets of Γ and families of bounded linear maps KsΛn∈B(AΛn) strongly continuous in s, with the following properties:

  • For all n and s, KsΛn:AΛn→AΛn is continuous on bounded subsets when both its domain and codomain are considered with the strong operator topology, and this continuity is uniform for s ∈ I.

  • Uniform local convergence of KsΛn to Ks⁠: For all X ⊂ Γ finite and any ϵ > 0, there exists N such that for all n ≥ N, we have

(4.27)
In (i), uniform for s ∈ I means that given a bounded net {Aα}α∈I converging strongly to A and ϵ > 0, there exists a choice of α0 ∈ I, independent of s, so that
‖KsΛn(Aα)ψ−KsΛn(A)ψ‖<ϵ, for all α≥α0.

For families Ks⁠, s ∈ I with I being an infinite interval, the uniformity asked for in part (ii) of this definition will typically not hold and one is led to consider subfamilies parametrized by s ∈ I0 ⊂ I for compact intervals I0. Also note that the properties of a strongly continuous family of uniformly locally normal maps imply that s→Ks is strongly continuous by the usual ϵ/3 argument. We have not assumed, however, that the maps Ks are bounded. In general, Ks is only locally bounded and cannot be extended to all of AΓ⁠.

We now discuss two examples. The first is for a model with uniformly bounded on-sites, while the second does not require this assumption.

Example 4.10.
Consider a quantum lattice system composed of (Γ, d) and AΓ. Let F be an F-function on (Γ, d), I⊂R be an interval, and Φ∈BF(I) be a strongly continuous interaction. For each s0 ∈ I and any compact I0 ⊂ I, we claim that Kt:AΓloc→AΓ given by
(4.28)
is a strongly continuous family of uniformly locally normal maps in the sense of Definition 4.9. Here, the dynamics in (4.28) we are using is the infinite volume dynamics corresponding to Φ, from Theorem 3.5, with Hz = 0 for all z ∈ Γ.
To see that this is an example of Definition 4.9, let {Λn}n≥0 be any nonempty sequence of finite subsets of Γ that are increasing and exhaustive. For each n ≥ 0 and any t ∈ I0, define KtΛn:AΛn→AΛn by setting
(4.29)
the finite-volume dynamics associated with Φ.
Fix n ≥ 0, t0 ∈ I0, and A∈AΛn. It is clear that
(4.30)
holds, in the strong sense, for any t ∈ I0. Thus,
(4.31)
and therefore, KtΛn∈B(AΛn) is strongly continuous in t for each n ≥ 0. Here, we have argued as in the proof of (A47) in Corollary A.5 using that ∥Φ∥F is locally integrable.

For each n ≥ 0, one can show that property (i) of Definition 4.9 holds by arguing as in the proof of Proposition 2.7(iii) and using that I0 is compact and ∥Φ∥F is locally bounded.

Finally, we observe that (ii) is a simple consequence of Corollary 3.6(iii).

Example 4.11.
Consider a quantum lattice system composed of (Γ, d) and AΓ. Fix a collection of densely defined, self-adjoint on-site Hamiltonians {Hz}z∈Γ. Let F be an F-function on (Γ, d), I⊂R be an interval, and take Φs∈BF(R) for each s ∈ I. In this case, for any w∈L1(R), the family {Ks}s∈I of linear maps with Ks:AΓloc→AΓ given by
(4.32)
is well-defined. Here, for each fixed s ∈ I, τts is the infinite-volume dynamics corresponding to Φs whose existence is proven in Theorem 3.5.
We will show that, under some additional assumptions on Φs, for each compact I0 ⊂ I, {Ks}s∈I0 is a strongly continuous family of uniformly locally normal maps in the sense of Definition 4.9. These assumptions are as follows:

  • For each Z∈P0(Γ), (s, t) ↦ Φs(Z, t) is jointly strongly continuous on I0×R.

  • For each Z∈P0(Γ) and t∈R, s ↦ Φs(Z, t) is strongly differentiable, and its derivative (s,t)↦Φs′(Z,t) is jointly strongly continuous on I0×R.

  • For each s ∈ I0, Φs′∈BF(R) and moreover, for each T > 0,

(4.33)
To prove the above claim, choose any sequence {Λn}n≥0 of nonempty, increasing, exhaustive finite subsets of Γ. For each such n ≥ 0 and any s ∈ I0, define approximating maps KsΛn:AΛn→AΛn by setting
(4.34)
Here, τtΛn,s(A)=UΛns(t,0)*AUΛns(t,0) is the dynamics generated by the finite volume Hamiltonian
(4.35)

We first show that for each n ≥ 0, the map KsΛn∈B(AΛn) is strongly continuous. In fact, the argument below demonstrates that KsΛn is uniformly continuous in the operator norm on B(AΛn)⁠.

Fix n ≥ 0. Let s ∈ I0 and A∈AΛn⁠. Assumptions (i) and (ii) above guarantee that the strong derivative of the finite-volume dynamics satisfies
(4.36)
Using Assumption (iii) and the estimate (3.66), for any X ⊂ Λn, each A∈AX, and any T > 0, there exists M > 0 such that
(4.37)
Now, let s0 ∈ I0 and take ϵ > 0. Since w∈L1(R), it is clear that there exists T > 0 for which
(4.38)
A short calculation shows that
(4.39)
and thus, for s sufficiently close to s0,
(4.40)
This proves the claimed continuity of KsΛn as a function of s.
We now prove (i). Again fix n ≥ 0. Let {Aα}α∈I be a bounded net in AΛn which converges in the strong operator topology to A. Denote by B=supα∈I‖Aα‖<∞, and note that ∥A∥ ≤ B. Let ϵ > 0. Take T > 0 as in (4.38) and define δ > 0 by requiring that
(4.41)
For this choice of δ > 0, compactness of I0 implies that there is some N ≥ 1 and numbers s1, s2, …, sN ∈ I0 for which the balls of radius δ centered at sj (1 ≤ j ≤ N) cover I0.
For each 1 ≤ j ≤ N, it is clear that KsjΛn(Aα)→KsjΛn(A) in the strong operator topology. In this case, for any ψ∈HΛn and ϵ as above, there is an α0∈I for which
(4.42)
for all 1 ≤ j ≤ N whenever α ≥ α0.
In this case, for any s ∈ I0, there is a value of j for which
(4.43)
and we have proven (i).
Finally, we need to verify uniform local convergence of KsΛn to Ks. Fix X∈P0(Γ) and A∈AX. Let ϵ > 0. Choose T > 0 as in (4.38). For any n ≥ 0 such that X ⊂ Λn, Corollary 3.6 (iii) implies
(4.44)
Since F is summable, for each x ∈ X, there exists Λx∈P0(Γ) such that
(4.45)
For n ≥ 0 sufficiently large that X ⊂ Λn and ⋃x∈XΛx ⊂ Λn, one has that
(4.46)
from which the claim is proved.

Our next result shows that given a strongly continuous family of uniformly locally normal maps Ks and any Y∈P0(Γ)⁠, the map (s,t)↦ΠY(Ks(A(t))) is jointly strongly continuous whenever t ↦ A(t) is strongly continuous. In particular, we have continuity on the diagonal t = s. The result, of course, also applies to the finite-volume setting where we can take Y = Λ = Γ.

Lemma 4.12.

Let I and J be intervals, X,Y∈P0(Γ), ΠY:AΓ→AY be the extension of the map defined in (4.11), and Ks:AΓloc→AΓ, s ∈ I be a strongly continuous family of uniformly locally normal maps. Then, for every strongly continuous t↦A(t)∈AX, t ∈ J, the function (s,t)↦ΠY(Ks(A(t)))∈AY is jointly strongly continuous in t and s.

Proof.

By the assumptions, there exists an increasing, exhaustive sequence {Λn}n≥0 of finite subsets of Γ and bounded linear transformations KsΛn∈B(AΛn)⁠, strongly continuous in s, that approximate Ks as in Definition 4.9.

Let X,Y∈P0(Γ)⁠, ψ∈HY⁠, and t→A(t)∈AX be strongly continuous. Define f(s,t)=ΠY(Ks(A(t)))ψ∈HY⁠. We prove that f(s, t) is jointly continuous. Without loss of generality, we may assume that I is compact. Fix (s0, t0). Since
‖ f(s,t)−f(s0,t0)‖≤‖f(s,t)−f(s,t0)‖+‖f(s,t0)−f(s0,t0)‖,
the joint continuity of f(t, s) can be obtained by proving the following two properties:

  • f(s, t0) is continuous in s.

  • f(s, t) is an equicontinuous family of functions of t parameterized by s ∈ I.

For (a), let ϵ > 0. Using Definition 4.9 (ii), pick n so that
‖Ks(A(t0))−KsΛn(A(t0))‖≤ϵ for all s∈I.
Using that ∥ΠY∥ = 1, the continuity of f(·, t0) follows from the strong continuity of KsΛn as
‖f(s,t0)−f(s0, t0)‖≤2ϵ‖ψ‖+‖KsΛn(A(t0))−Ks0Λn(A(t0))‖‖ψ‖.
For (b), let ϵ > 0. We show that there exists δ > 0 such that |t − t0| < δ implies
(4.47)
To see this, let fn(s,t)=ΠY(KsΛn(A(t)))ψ and J be a compact interval containing a neighborhood of t0. Since ∥A(t)∥ is uniformly bounded for t ∈ J, using Definition 4.9(ii), choose n so that
(4.48)
Now consider the family of functions fn(s, t) parameterized by s ∈ I. By Definition 4.9(i), since ∥A(t)∥ is bounded on J, KsΛn(A(t)) is strongly continuous in t. Since sups∈I‖KsΛn‖<∞ by the uniform boundedness principle, it follows that ‖KsΛn(A(t))‖ is bounded on I × J, and so
ΠY(KsΛn(A(t)))=ΠYΛn∪Y(KsΛn(A(t)))
is strongly continuous in t by Proposition 4.3. The argument used in Proposition 4.3 shows that the strong continuity of ΠY(KsΛn(A(t))) is uniform in s ∈ I. In particular, there is a δ > 0 such that |t − t0| < δ implies
(4.49)
The equicontinuity of f(s, t) follows from (4.48) and (4.49).

In all the proofs above, we have used results on the finite volume local approximates to obtain results for the infinite volume local approximates. However, there may be instances where one wants to work in a suitable representation of the infinite-volume algebra. We conclude this section with a result regarding the GNS representation of a locally normal state.

Proposition 4.13.

Let HΛ,Λ∈P0(Γ), be the family Hilbert spaces defined in (3.1), ω be a locally normal state on AΓloc, and πω:AΓloc→B(Hω) its corresponding GNS representation. The map πω|AΛ:B(HΛ)→B(Hω) is continuous on arbitrary bounded subsets of its domain with respect to the strong operator topology on both its domain and codomain.

Proof.
Let {Aα∣α ∈ I} be a net in the unit ball of B(HΛ) that converges to A in the strong operator topology. To prove the claim, it is sufficient to verify that πω(A − Aα)ψ → 0, for all ψ in the dense subspace of Hω given by vectors of the form πω(B)Ωω, where B∈AΓloc and Ωω is the cyclic vector representing ω. To this end, note that
‖πω((A−Aα)B)Ωω‖2=ω(B*(A−Aα)*(A−Aα)B).
Since A and the Aα belong to AΛ and B∈AΓloc⁠, there exists Λ′∈P0(Γ) such that B*(A−Aα)*(A−Aα)B∈AΛ′⁠. Since ω is locally normal, its restriction to AΛ′ is given by a density matrix ρΛ′ on HΛ′⁠. By writing ρΛ′ = ∑n≥1ρn|ξn⟩⟨ξn| in terms of an orthonormal set of eigenvectors ξn∈HΛ′⁠, it follows that
‖πω(A−Aα)πω(B)Ωω‖2=TrρΛ′B*(A−Aα)*(A−Aα)B=∑n≥1ρn‖(A−Aα)Bξn‖2.
The result follows from using the analogous arguments from the proof of Proposition 4.3.

In Sec. III, we proved a Lieb-Robinson bound for the dynamics associated with a sufficiently local interaction. In addition to estimating the speed of propagation of a dynamically evolved observable, these bounds imply that the dynamics for a fixed time t is quasilocal. As a result, they can be well approximated by local observables as shown in Corollary 4.4. In recent years, other quasilocal maps have played a key role in proving both locality estimates of the spectral flow12,54,59,93 and spectral gap stability of frustrationfree quantum lattice models.21,22,56,83,85,96,97,130 While we will consider both of these topics, the former in Sec. VI and the latter in Paper II,96 the focus of this section is the general study of quasilocal maps [see (5.1)], and the investigation of the key properties that will be useful in the above-mentioned applications. There exists a broad range of other applications that we will not discuss here.5–9,46

We begin by showing how to apply the techniques from Sec. IV to obtain estimable local approximations of quasilocal maps. In Sec. V B, we provide a number of examples that will arise in our applications, including the difference of two dynamics. We discuss compositions of two quasilocal maps in Sec. V C and prove sufficient conditions for which the composition is again quasilocal. In Sec. V D, consider the composition of a quasilocal map with an interaction. We show that under suitable conditions such a composition can be rewritten as a local interaction. Moreover, we quantify the decay of the resulting interaction in terms of the decays of the original interaction and the quasilocal map. If the transformed interaction has sufficient decay, then the theory developed in Sec. III applies and an infinite volume dynamics exists. We conclude in Sec. V E by returning to the example of the difference of two dynamics and proving a continuity result.

Let (Γ, d) and AΓ be a quantum lattice system defined as in Sec. III A 1. A linear map K:AΓloc→AΓ is said to satisfy a quasilocality bound of order q ≥ 0 if there is C < ∞ and a nonincreasing function G : [0, ∞) → [0, ∞) with limr→∞G(r) = 0 such that for all X,Y∈P0(Γ)⁠, and A∈AX⁠, B∈AY⁠,

(5.1)

Any linear mapping K satisfying (5.1) will be referred to as quasilocal. When relevant, we will denote by CK(q,G) the smallest constant for which (5.1) holds. Since the function G in (5.1) governs the decay of the quasilocal map, we may refer to G as a decay function associated with K⁠. In this work, we will always assume that quasilocal maps are linear. However, there may be other contexts in which it might be appropriate to generalize this definition.

The dependence of the bound in (5.1) on the support of the observable A through the factor |X|q is a choice we made based on the applications we have in mind. However, under appropriate assumptions, most of the estimates proved in this section also hold for quasilocal maps with more general functions of |X|.

In most of our applications, the metric space (Γ, d) is equipped with an F-function F. In this case, one can often estimate a quasilocal map K as follows: there is C < ∞ such that for all X,Y∈P0(Γ)⁠, any A∈AX⁠, and B∈AY⁠, we have

(5.2)

As we will see below, in certain estimates, the bound (5.2) has advantages over (5.1). A simple over-counting argument shows that

GF(X,Y)≤|X|G(d(X,Y)), whereG(r)≔supx∈Γ∑y∈Γ:d(x,y)≥rF(d(x,y)).

It follows from the uniform integrability of F [see (3.8)] that G(r) → 0 as r → ∞. When the corresponding F-function is weighted, i.e., F = Fg as defined (3.11), one has that

(5.3)

and so, in this case, an estimate of the form (5.2) reduces to that of (5.1). For more detailed information on F-functions, including weighted F-functions, see Subsections  1–3 of the  Appendix.

We now demonstrate an important estimate concerning quasilocal maps. For this result, we use the concepts introduced in Sec. IV B and, in particular, the localizing maps ΠX and ΔX(n), defined with respect to a locally normal product state ρ, as in (4.11) and (4.16), respectively.

Lemma 5.1.
Let (Γ, d) and AΓ be a quantum lattice system, and ρ be a locally normal product state on AΓ. Let K:AΓloc→AΓ be a quasilocal map. For any X∈P0(Γ) and n ≥ 0,
(5.4)
In particular, if the decay function associated with K is summable, i.e., ∑n≥0G(n) < ∞, then for any X∈P0(Γ) and each A∈AX⁠,
(5.5)
The series on the right-hand-side above is absolutely convergent in norm with a bound that is uniform in the choice of locally normal product state ρ.

Note that the result above, of course, also applies to finite systems. In particular, for any finite Λ, the result holds for any quasilocal map K:AΛ→AΛ⁠.

Proof.
To see that (5.4) is true, fix X∈P0(Γ)⁠, A∈AX⁠, and n ≥ 0. Observe that for any B∈AΓ\X(n)loc⁠, the estimate
(5.6)
follows from (5.1). In this case, an application of Corollary 4.4 with ϵ = C|X|q∥A∥G(n) implies (5.4) as claimed.
Recalling (4.16), for any integer n ≥ 1, one can write
(5.7)
and therefore, an immediate consequence of (5.4) is the estimate
(5.8)
which is valid for any X∈P0(Γ)⁠, A∈AX⁠, and n ≥ 1.
Since G(n) → 0 as n → ∞, (5.4) implies
(5.9)
It is clear from (4.16) that the local decompositions have telescopic sums, i.e., for any N ≥ 1,
(5.10)
and thus, the series on the right-hand-side of (5.5) is norm convergent. Note also that
(5.11)
and so the series is also absolutely convergent, with norm bound independent of ρ, as claimed.
If in Lemma 5.1 we had assumed K satisfies (5.2) instead of (5.1), (5.4) would become
(5.12)
where the right-hand-side above is finite and nonincreasing (in n) by the uniform summability of the F-function F.

In this section, we discuss a few of the most common examples of quasilocal maps [defined as in (5.1)]. In applications, quasilocal maps are often constructed as the thermodynamic limit of appropriate finite-volume maps. We first describe this class of quasilocal maps as a general example. Each of the more concrete examples we present later in this section will be of this general form.

Example 5.2

(A general example). Let q ≥ 0, C < ∞, and G be a nonincreasing function G : [0, ∞) → [0, ∞) with limr→∞G(r) = 0. Let {Λn}n=1∞ be a sequence of increasing and exhaustive finite subsets of Γ. In particular, this means that Λn ⊂ Λn+1 for all n ≥ 1, and given any x ∈ Γ, there exists N ≥ 1 for which x ∈ ΛN. Suppose that for each n ≥ 1, there is a linear map Kn:AΛn→AΛn for which

  • Given any sets X, Y ⊂ Λn, the bound
    (5.13)
    holds for all observables A∈AX and B∈AY⁠.

  • For each finite X ⊂ Γ and any ϵ > 0, there is an N ≥ 1 for which

(5.14)
In this case, the local Cauchy assumption in part (ii) above, implies that a linear map K:AΓloc→AΓ is well-defined by setting
(5.15)
with the limit above being in norm. Moreover, for any finite sets X, Y ⊂ Γ, there is N ≥ 1 large enough so that X ∪ Y ⊂ ΛN since the sequence of sets is exhaustive. In this case, for any A∈AX⁠, B∈AY⁠, and n ≥ N,
(5.16)
Here, we have used the uniform quasilocality estimate in (i) [see (5.13)]. As the final term above vanishes in the limit as n → ∞, it is clear that the map K defined in (5.15) is quasilocal [see (5.1)] and satisfies the same uniform quasilocal estimates as the collection {Kn}n≥1⁠.

For later applications, we now describe a variant of the estimate in Lemma 5.1 which is particularly relevant to quasilocal maps of the type from Example 5.2.

Corollary 5.3.
Let ρ be a locally normal product state on AΓ, and suppose that K and {Kn}n=1∞ are maps that satisfy the uniform quasilocal and local Cauchy conditions from (5.13) and (5.14). For any X∈P0(Γ), m ≥ 1, and n ≥ 1 large enough so that X(m) ⊂ Λn, the bound
(5.17)
holds for any A∈AX, where ΔX(m)Λn:AΛn→AX(m) and ΔX(m):AΓ→AX(m) are the local decomposition maps defined in Sec. IV B [see (4.15) and (4.16)].

The bound above is particularly useful as it expresses the decay of the quantity on the left-hand-side above in both large n and m.

Proof.
The proof uses two separate estimates. First, using consistency of the local decompositions [see Proposition 4.5(ii)], it is clear that
(5.18)
and so the first part of the estimate holds since ΠX(m) is a norm one map. Note that this establishes that the LHS of (5.17) decays in n. For the second part of the argument, we use that each term on the LHS of (5.17) can be estimated using Lemma 5.1. In fact,
‖ΔX(m)(K(A))−ΔX(m)Λn(Kn(A))‖≤‖ΔX(m)(K(A))‖+‖ΔX(m)Λn(Kn(A))‖≤8 C|X|q‖A‖G(m−1),
as all maps considered satisfy the same quasilocal bound.

Example 5.4.
Here, we return to Example 4.7 and show that it has the form of the general example discussed in Example 5.2. Consider a quantum lattice system composed of (Γ, d) and AΓ. Let F be an F-function on (Γ, d) and Φ∈BF be an interaction. As in (4.21), let {Λn}n≥0 be an increasing, exhaustive sequence of finite subsets of Γ and define Kn:AΛn→AΛn by setting
(5.19)
To see that (5.13) holds, let X,Y∈P0(Γ)⁠. For any A∈AX and n ≥ 0 large enough so that X ⊂ Λn, one has that (4.22) holds, and therefore,
(5.20)
where we use the notation GF of (5.2). As any F-function is uniformly integrable, the bound above is uniform in the finite volume since GF(X, Λn) ≤ |X|∥F∥. For n ≥ 0 large enough so that X ∪ Y ⊂ Λn, for each A∈AX, and any B∈AY, a simple commutator bound then yields
(5.21)
When X ∩ Y = ∅, a better estimate is achieved by observing that
(5.22)
This gives a quasilocality estimate (uniform in the finite volume) of the type in (5.13).
To see (5.14), fix a finite set X ⊂ Γ, A∈AX and take m ≤ n with m large enough so that X ⊂ Λm. One checks that
(5.23)
and therefore,
(5.24)
Since F-functions are integrable, the locally Cauchy estimate (5.14) follows.
Under appropriate assumptions, one can generalize this example to prove quasilocality estimates for mappings of the form
(5.25)
For example, the Lindblad generator of a quantum dynamical semigroup is of this form. See Refs. 98 and 105.

Example 5.5
(The dynamics associated with Φ∈BF(I)⁠). Let (Γ, d) and AΓ be a quantum lattice system, and let I⊂R be an interval. Given an F-function, F, recall that a strongly continuous interaction Φ:P0(Γ)×I→Aloc belongs to BF(I) if the function ∥Φ∥F : I → [0, ∞) defined by
(5.26)
is locally bounded on I [see (3.18)]. Fix Φ∈BF(I) and let {Hz}z∈Γ be a collection of densely defined, self-adjoint on-site Hamiltonians. For any Λ∈P0(Γ)⁠, considered the finite-volume Hamiltonians
(5.27)
and associate with them the dynamics {τt,sΛ}s,t∈I, defined in (3.56). Theorem 3.3 demonstrates that these dynamics are quasilocal maps. In fact, this result shows that for any X, Y ⊂ Λ with X ∩ Y = ∅,
(5.28)
for all A∈AX⁠, B∈AY, and s, t ∈ I. Here, the number CF is the convolution constant associated with the F-function F on Γ [see (3.9)] and
(5.29)
Moreover, the bound proven in Theorem 3.4 (ii) shows that for any finite sets X ⊂ Λ0 ⊂ Λ,
(5.30)
holds for all A∈AX and s, t ∈ I. Again, this suffices to establish that these dynamics are locally Cauchy in the sense of (5.14), and so again this example is of the general form in Example 5.2.

Example 5.6
(The difference of two dynamics). A quasilocal map that comes up in our applications related to stability in Paper II96 is the difference of two dynamics. More precisely, consider the setting from Example 5.5. Fix a collection of densely defined, self-adjoint on-site Hamiltonians {Hz}z∈Γ and two interactions Φ,Ψ∈BF(I). For any Λ∈P0(Γ), consider the Hamiltonians
(5.31)
and their corresponding dynamics, which we denote by τt,sΛ and αt,sΛ⁠, respectively. For any s, t ∈ I, a linear map Kt,sΛ:AΛ→AΛ is defined by
(5.32)
Since both of the dynamics used in the definition of this map are automorphisms, it is clear that ‖Kt,sΛ‖≤2. A different estimate is provided by the local norm bound in Theorem 3.4(i), namely,
(5.33)
where It,s(Φ − Ψ) is defined as in (3.21). The above bound better reflects the fact that this difference is small if either Φ is close to Ψ in BF(I) or |s − t| is small. The bound in Theorem 3.4(ii) [see also (5.30)] allows one to establish that these maps are locally Cauchy in the sense of (5.14). In Sec. V E, we prove that these maps Kt,sΛ are uniformly quasilocal in the sense of (5.13) with constant prefactors that once again decay if either Φ is close to Ψ in BF(I) or |s − t| is small; as is the case in (5.33).

Example 5.7
(Weighted integrals of quasilocal maps). We briefly mention another interesting class of examples, which includes the spectral flow introduced in Sec. VI C. Let (Γ, d) and AΓ be a quantum lattice system and μ be a measure on R. Suppose that for each t∈R, there is a quasilocal map Kt:AΓloc→AΓ for which the map K:AΓloc→AΓ given by
(5.34)
is well-defined. If the family {Kt}t∈R is sufficiently quasilocal and ∫|t|≥xdμ(t) decays sufficiently fast as x → ∞, then the mapping defined in (5.34) is quasilocal with explicit decay estimates.
For example, let w∈L1(R) and Φ∈BFa(R) where we recall that Fa is a weighted F-function of the form Fa(r) = e−arF(r) with a > 0. Let τtΛ≔τt,0Λ be the finite-volume dynamics given in (3.6) that is associated with the local Hamiltonians of the form
HΛ(t)=∑X⊂ΛΦ(X,t).
Then, using Lieb-Robinson bounds, one can see that the map K:AΛ→AΛ defined by
K(A)=∫RτtΛ(A)w(t)dt
satisfies the following bound: for all A∈AX⁠, B∈AY with X ∪ Y ⊆ Λ and X ∩ Y = ∅, ‖[K(A), B]‖≤2‖A‖‖B‖|X|G(d(X,Y)), where G is the decreasing function,
G(r)=‖w‖L1‖F‖CFae−ar/2+∫|t|>r/2va|w(t)|dt,
and va is the Lieb-Robinson velocity [see the discussion following Theorem 3.1]. To see this, note that for all T∈R⁠,
‖[K(A), B]‖≤∫|t|≤T‖[τtΛ(A),B]‖|w(t)|dt+∫|t|>T‖[τtΛ(A),B]‖|w(t)|dt.
With the choice of T = d(X, Y)/2va, the bound is attained by applying (3.26) to the integral over |t| ≤ T, and using the trivial bound ‖[τtΛ(A),B]‖≤2‖A‖‖B‖ for the integral over |t| > T.

In applications, we find it useful to recognize certain mappings as the composition of quasilocal maps. When Γ is finite, these compositions are well-defined and estimates, as indicated below, follow readily. For sets Γ of infinite cardinality, more care must be taken when defining such compositions. This section discusses two classes of examples where these compositions are well-defined and we describe the estimates that follow.

It will be convenient to make an additional assumption on the metric space (Γ, d). We say that (Γ, d) is ν-regular if the cardinality of balls in Γ grows at most polynomially, i.e., there exist non-negative κ and ν for which

(5.35)

More comments about ν-regular metric spaces (Γ, d) can be found in Subsection 1 of the  Appendix. Under this assumption, given any X∈P0(Γ) and n > 0, the cardinality of X(n), the inflation of X defined in (4.14), satisfies the following rough estimate:

(5.36)

Let (Γ, d) and AΓ be a quantum lattice system on a ν-regular metric space. We will say that a linear map K:AΓloc→AΓ is locally bounded if there are non-negative numbers p and B for which

(5.37)

More general growth in X, i.e., the support of A, could be allowed, but the above moment condition covers all of the applications we have in mind. As discussed in Sec. V A, we say that a linear map K:AΓloc→AΓ is quasilocal if there are non-negative numbers q and C as well as a nonincreasing function G, G:[0, ∞) → [0, ∞), with limr→∞G(r) = 0 for which given any X,Y∈P0(Γ)⁠,

(5.38)

We will refer to C, q, and G as the parameters of the quasilocal map.

We first consider compositions of linear maps for the following situation. Suppose that K1:AΓloc→AΓ is locally bounded and quasilocal in that it satisfies both (5.37) and (5.38). Furthermore, assume that K2:AΓ→AΓ is linear, bounded, and quasilocal. So, in particular, there are non-negative numbers q2 and C2 and a decay function G2 for which the analog of (5.38) holds for K2⁠. In many applications, the mapping K2 arises as the unique linear extension of a bounded, quasilocal map K̃2:AΓloc→AΓ⁠. In this situation, we can define the composition K:AΓloc→AΓ in the usual way, i.e.,

(5.39)

Moreover, any such map satisfies the following estimate.

Lemma 5.8.

Let (Γ, d) be ν-regular, K1:AΓloc→AΓ be a locally bounded, quasilocal map, and K2:AΓ→AΓ be a bounded, quasilocal map. For i = 1 or 2, denote by Bi, Ci, pi, qi, Gi the corresponding parameters from (5.37) and (5.38). Then, the following holds for the composition K=K2◦K1:

  • K is locally bounded: for any A∈AX with X∈P0(Γ),
    (5.40)
    where one may take B̃=B1‖K2‖ and p = p1.
  • For any A∈AX and B∈AY, where X,Y∈P0(Γ),
    (5.41)
    where the numbers B̃ and p may be taken as in (5.40), one may take q = p1 + q2, and
(5.42)

We note that if the function G described in (5.42) above is nonincreasing and satisfies limr→∞G(r) = 0, then the above estimates show that K is quasilocal.

Proof.
To prove (i), note that given any X∈P0(Γ)⁠,
(5.43)
This proves (5.40).
The proof of (ii) follows from two observations. First, the bound in (5.40) implies a rough estimate on the commutator for any X,Y∈P0(Γ)⁠. In fact, whenever A∈AX and B∈AY⁠, one has
(5.44)
Next, we note that when d(X, Y) > 0, we obtain better estimates. Under this additional constraint, set m = d(X, Y)/2. For any locally normal product state ρ on AΓ⁠, the estimate
(5.45)
follows from an application of Lemma 5.1. Denoting by Am=ΠX(m)(K1(A))⁠, we find that
(5.46)
Since K2 is quasilocal, we have that
(5.47)
where we have used the local bound for K1⁠, (5.36), and the fact that d(X, Y) ≤ 2d(X(m), Y). The second term in (5.46) satisfies
(5.48)
This completes the proof of (5.41).
For some of our applications, the estimates proven in Lemma 5.8 do not suffice. Briefly, some information is lost when estimating the outer mapping K2 with a rough norm bound. Due to this, we consider more general compositions in the proposition below. First, we introduce some notation. Let G:[0, ∞) → [0, ∞) and m ≥ 0, we say that G has a finite m-th moment if
(5.49)

Proposition 5.9.
Let (Γ, d) be a ν-regular metric space. For i = 1, 2, let Ki:AΓloc→AΓ be locally bounded, quasilocal maps. Suppose that G1, the decay function in (5.38) associated with K1, has a finite νp2-th moment. Then, for any choice of locally normal product state ρ on AΓ, the composition Kρ:AΓloc→AΓ given by
(5.50)
is well-defined. The series above is absolutely convergent and may be estimated uniformly in the choice of locally normal product state ρ. In fact, the mapping Kρ is independent of the choice of ρ.

Proof.
Fix a locally normal product state ρ on AΓ⁠. Lemma 5.1 shows that for each X∈P0(Γ) and any A∈AX⁠, we have that
(5.51)
and the series above is absolutely convergent. To obtain this series representation, it is only required that the decay function associated with K1 is summable, see, e.g., (5.11) which is independent of the choice of ρ.
We now claim that under the additional finite moment condition, for each X∈P0(Γ) and any A∈AX⁠, the series defining Kρ(A) in (5.50) is also absolutely convergent. In fact, the bound
(5.52)
can be obtained as follows. For the first inequality above, we use that K2 is locally bounded. For the second inequality, we first partition the sum into n = 0 and n > 0. For n = 0, we use that ΔX = ΠX and K1 is locally bounded. For n > 0, we apply (5.8) using the quasilocality of K1 and invoke (5.36).
Now, let ρ1 and ρ2 be any two locally normal product states on AΓ⁠. We show that for each fixed X∈P0(Γ) and any ϵ > 0, one can estimate
(5.53)
and hence prove that the mapping Kρ is independent of the choice of ρ.
By the absolute convergence proven in (5.52) and the finite moment condition, it is clear that for any ϵ > 0, there is some N ≥ 1, independent of ρ, for which
(5.54)
For N as above, we write
(5.55)
Based on N, it follows from (5.52) that
(5.56)
Using linearity of K2 and the telescopic properties of the sums [see (5.10)], we also have that
(5.57)
(5.58)
where we have used the local bound for K2 and inserted (and removed) K1(A) to apply Lemma 5.1. The final bound above is clear from (5.54). The claim in (5.53) now follows.

From Proposition 5.9, we now have conditions under which there is a well-defined composition of two locally bounded, quasilocal maps. Lemma 5.10 provides local bounds and quasilocal estimates for the resulting composition.

Lemma 5.10.

Let (Γ, d) be a ν-regular metric space. For i = 1, 2, let Ki:AΓloc→AΓ be locally bound, quasilocal maps. Suppose that G1, the decay function in (5.38) associated with K1, has a finite νp2-th moment and let K:AΓloc→AΓ denote the composition from (5.50).

  • K is locally bounded: for any A∈AX with X∈P0(Γ)⁠,
    (5.59)
    where one may take p = p2 + max{p1, q1} and
    (5.60)
  • For any A∈AX and B∈AY where X,Y∈P0(Γ), one has that
    (5.61)
    where one may take C=max{κq2B1C2, 8κp2C1B2}⁠, q = max{p1, q1} + max{p2, q2}, and
    (5.62)

Again we stress that the bounds above demonstrate that the composition is quasilocal if the function G in (5.62) is nonincreasing with limr→∞G(r) = 0.

Proof.
The bound (5.59) is a consequence of (5.52) found in the proof of Proposition 5.9. To prove (5.61), we argue as in the proof of Lemma 5.8(ii). We need only to consider the case when d(X, Y) > 0, and therein, we set m = ⌊d(X, Y)/2⌋. For A∈AX⁠, we write
(5.63)
Here, we have used an expansion as in (5.50) and the telescopic property (5.10). Moreover, we have dropped the dependence of ρ from the notation since (5.50) is invariant under the choice of locally normal product state by Proposition 5.9. The estimate
(5.64)
readily follows.
As K2 is quasilocal, it is clear that
(5.65)
To estimate the remaining term, for each n ≥ m + 1, we find
(5.66)
where we have used (5.8). This proves (5.61).

Important applications of quasilocal maps arise in the classification of gapped ground state phases12,14,16,99,100,101 and recent proofs of stability of the spectral gap.21,22,83,85,96,97 In these proofs, key insights come from analyzing the composition of a quasilocal map with an interaction, K◦Φ:P0(Γ)→AΓ⁠. It is important to note that such maps are not necessarily interactions themselves, as the image lies in the quasilocal algebra, AΓ⁠, rather than the algebra of local observables, AΓloc⁠. In our applications, the interaction and quasilocal map often depend on an auxiliary parameter and we allow for this in our construction and results. In what follows, we provide a general framework under which one can construct a bona fide interaction from such a composition and derive estimates that determine conditions under which these transformed interactions have a finite F-norm.

We begin with a general description of transformed interactions in Sec. V D 1. In Sec. V D 2, we prove estimates on these transformed interactions in finite volume. In Sec. V D 3, we give conditions under which the finite-volume results proven in Sec. V D 2 extend to the thermodynamic limit. A concrete application of these results will be given in Sec. VI E, where we show that the spectral flow automorphisms can be realized as the dynamics generated by a time-dependent interaction with good decay properties.

1. Transformed finite-volume Hamiltonians

To investigate the spectral properties of a given Hamiltonian H, it is often convenient to work with a unitarily equivalent Hamiltonian H′ = U*HU. When the original Hamiltonian is a sum of local terms, the strict locality of these terms is typically not preserved under the mapping H ↦ H′. In recent applications, most notably the proof of stability, it is shown that locality based arguments, such as Lieb-Robinson bounds, still apply to H′ if the automorphism implemented by the unitary U is sufficiently quasilocal.

In this section, we discuss this situation more generally. Specifically, we analyze the transformation of a given interaction by a quasilocal map. Briefly, we first argue that the composition of a quasilocal map with an interaction can, using the methods of Sec. IV B, still be realized as an interaction with strictly local terms. Moreover, we show that the spatial decay associated with this new interaction can be estimated in terms of the decays of the original interaction and the quasilocal map. Finally, we discuss quasilocality estimates for the dynamics of this transformed interaction.

To establish some notation, let us first consider a simple, time-independent case in finite volume. As before, fix a quantum lattice system composed of (Γ, d) and AΓ⁠. Let Φ be an interaction on AΓ and recall that for any finite Λ ⊂ Γ, we denote by

(5.67)

the finite-volume Hamiltonian generated by Φ. Our goal here is to analyze the transformation of this local Hamiltonian HΛΦ by a linear map K:AΛ→AΛ⁠. In particular, we consider

(5.68)

Generally, the map K will not preserve locality, and in such cases, each term in (5.68) will be global in the sense that supp(K(Φ(X)))=Λ for each X ⊂ Λ. For this reason, the sum on the right-hand-side of (5.68) does not represent an interaction in the sense defined in Sec. III A.

Using the methods of Sec. IV B, one can rewrite the right-hand-side of (5.68) as a sum of strictly local terms. To see this, fix a locally normal product state ρ on AΓ⁠. In this finite-volume context, we only use the restriction of ρ to AΛ and again refer to it as ρ. In terms of ρ, we have defined local decompositions with respect to any X ⊂ Λ and each n ≥ 0 as the maps ΔX(n)Λ:AΛ→AΛ given by (4.15). Recall further that ΔX(n)Λ has a range contained in AX(n)∩Λ (using again the identification of the former as a subalgebra of AΛ⁠), and moreover, ‖ΔX(n)Λ‖≤2⁠. In this case, each term K(Φ(X)) appearing in (5.68) can be written as a finite telescopic sum as in (4.17) by

(5.69)

For any Z ⊂ Λ, define

(5.70)

with the understanding that empty sums are taken to be zero. By construction, ΨΛ(Z)∈AZ and under the additional assumption that K(A)*=K(A*) for all A∈AΛ⁠, we see that ΨΛ is a well-defined (finite-volume) interaction in the sense of Sec. III A. Moreover,

(5.71)

In words, using the notation from (5.67), the final equalities in (5.71) show that the transformed Hamiltonian in (5.68) may be rewritten as the Hamiltonian generated by the interaction ΨΛ.

2. Finite-volume results

In this section, we give a finite-volume analysis of the transformed interactions briefly introduced at the end of Sec. V D 1. In Sec. V D 3, we will discuss appropriate conditions under which these results will extend to the thermodynamic limit. For many of our applications, both the interaction and the quasilocal map will be time-dependent. As a consequence, we state and prove our estimates for families of interactions and quasilocal maps.

We make two useful observations in this section. First, we indicate a set of continuity assumptions under which a finite-volume transformed interaction corresponds to a well-defined dynamics. These assumptions will also guarantee that the interaction which generates this transformed interaction is strongly continuous in the sense of Sec. III A 1. Next, we will show that certain decay assumptions on the initial interaction Φ and quasilocal map K lead to explicit estimates on the decay of the interaction ΨΛ; here, we are using the notation introduced at the end of Sec. V D 1. Technically, the continuity and decay assumptions are independent; however, in most applications, the models we consider satisfy both sets of assumptions simultaneously.

Let (Γ, d) and AΓ be a quantum lattice system, and I⊂R be an interval. We once again work with strongly continuous interactions Φ:P0(Γ)×I→AΓloc⁠, meaning that, for all X∈P0(Γ)⁠,

  • Φ(X,t)*=Φ(X,t)∈AX for all t ∈ I.

  • The map t ↦ Φ(X, t) is continuous in the strong operator topology on AX=B(HX)⁠.

For any Λ∈P0(Γ)⁠, we define a finite-volume, time-dependent Hamiltonian associated with Φ by

(5.72)

From the assumptions, it is clear that HΛΦ is also pointwise self-adjoint and strongly continuous as it is the finite sum of such terms.

For the remainder of this subsection, we fix a finite volume Λ∈P0(Γ) and are interested in studying time-dependent transformed finite-volume Hamiltonians analogous to those considered in Sec. V D 1. Specifically, given any family of linear maps {Kt:AΛ→AΛ}t∈I⁠, we consider the set of all operators of the form

(5.73)

and will refer to such collections as a finite-volume family of transformed interactions. Under assumptions which guarantee that t↦Kt(HΛΦ(t)) is pointwise self-adjoint and strongly continuous, the methods of Sec. II B [see also Sec. III A 1] demonstrate that these transformed interactions correspond to a dynamics. More precisely, Proposition 2.2 shows that for any s, t ∈ I, the strong solution UΛ(t,s)∈AΛ of

(5.74)

defines a two-parameter family of unitaries, and thus, a cocycle of automorphisms τt,sΛ of AΛ with

(5.75)

We refer to τt,sΛ as the dynamics corresponding to the transformed Hamiltonian in (5.73).

A main goal of this subsection is to establish assumptions under which the dynamics in (5.75) satisfies a quasilocality estimate, also known as a Lieb-Robinson bound (see Theorem 3.1). In order to do so, we first rewrite the family of transformed interactions in (5.73) as a sum of strictly local terms. Fix a locally normal product state ρ on AΓ⁠. For any Z ⊂ Λ and each t ∈ I, set

(5.76)

where, as in Sec. V D 1, we have made local decompositions of the global terms on the right-hand-side of (5.73) [compare with (5.69) and (5.70)]. We stress that for all t ∈ I, we make local decompositions with respect to the same locally normal product state ρ. As in (5.71), it is clear that for each t ∈ I,

(5.77)

We now introduce a set of assumptions on the family of functions {Kt:AΛ→AΛ}t∈I which guarantee that (i) the dynamics in (5.75) is well-defined and (ii) the mapping ΨΛ:P0(Λ)×I→AΛ is a strongly continuous interaction in the sense of Sec. III A 1.

Assumption 5.11.

We assume that the collection of finite-volume linear maps {Kt:AΛ→AΛ}t∈I is a strongly continuous family of strongly continuous transformations that are compatible with the involution in the sense that

  • for each t ∈ I, Kt(A)*=Kt(A*)⁠, for all A∈AΛ⁠;

  • for each A∈AΛ⁠, the function t↦Kt(A) is norm continuous;

  • for each t ∈ I, the map Kt:AΛ→AΛ is continuous on bounded subsets when both its domain and codomain are equipped with the strong operator topology, and moreover, this continuity is uniform on compact subsets of I.

Assumption 5.11(i), together with Proposition 4.5(v), is used to ensure that the terms defined in (5.76) are pointwise self-adjoint. This is important in defining the unitary propagator, but it plays no role in establishing various continuity properties. Next, as is discussed in Sec. IV B 1, Assumption 5.11 (ii) and (iii) guarantee that t↦Kt(HΛΦ(t)) is strongly continuous. As such, the finite-volume dynamics associated with this transformed interaction [see (5.74) and (5.75)] is well-defined. In particular, this dynamics is independent of the choice of ρ. Note further that Assumption 5.11 (ii) and (iii) also ensure that for each X ⊂ Λ, t↦Kt(Φ(X,t)) is strongly continuous. Given this, Proposition 4.3 shows that each of the finitely many terms on the right-hand-side of (5.76) is strongly continuous as well, and as a result, ΨΛ is a strongly continuous interaction. The interaction ΨΛ, which does depend on the choice of ρ, will be useful in proving a quasilocality bound on the finite-volume dynamics in (5.75).

The goal for the remainder of this section is to quantify a quasilocality estimate for the finite-volume dynamics in (5.75) in terms of decay properties of the original interaction Φ and the finite-volume transformations {Kt}t∈I⁠. For these results, we assume that (Γ, d) is ν-regular and equipped with an F-function F.

Let us again fix an interval I⊂R and a finite-volume Λ∈P0(Γ)⁠. We make the following decay assumptions on a family of finite volume transformations:

Assumption 5.12.

We assume that the family of finite-volume linear maps {Kt:AΛ→AΛ}t∈I is a time-dependent family of locally bounded, quasilocal maps in the sense that

  • There is some p ≥ 0 and a measurable, locally bounded function B : I → [0, ∞) so that given any X ⊂ Λ,

(5.78)
  • (ii)

    There is some q ≥ 0, a nonincreasing function G : [0, ∞) → [0, ∞) with G(r) → 0 as r → ∞, and a measurable, locally bounded function C : I → [0, ∞) for which given any sets X, Y ⊂ Λ, one has that

(5.79)
For the initial interaction, we impose decay assumptions which compensate for the factors of |X| found in (5.78) and (5.79). More precisely, for any time-dependent interaction Φ and each m ≥ 0, we define a new interaction Φm, which we call the m-th moment of Φ, with terms
(5.80)
To prove the result in Theorem 5.13, we will assume that the initial interaction Φ satisfies Φm∈BF(I) for m = max{p, q} with p and q as in (5.78) and (5.79), respectively. Recall that an interaction Φ∈BF(I) if and only if Φ:P0(Γ)×I→AΓloc is a strongly continuous interaction and the map ∥Φ∥F:I → [0, ∞) given by
(5.81)
is locally bounded. An immediate consequence of (5.81) is that for any finite volume Λ∈P0(Γ) and any pair x, y ∈ Λ, the bound
(5.82)
holds for all t ∈ I. We refer to Sec. III A 1 for more details on BF(I)⁠.
Finally, before we state our first result, we review some notation. Recall that a non-negative function G : [0, ∞) → (0, ∞) has a finite mth moment if
(5.83)
Note that, in this case, the tail of the series r ↦ ∑n=⌊r⌋(1 + n)mG(n) is a non-negative, nonincreasing function for which
(5.84)

We state our basic estimate on these finite-volume transformed interactions. In the statement, we make use of the quantities p, q, and G from Assumption 5.12.

Theorem 5.13.
Consider a quantum lattice system composed of ν-regular metric space (Γ, d) and quasilocal algebra AΓ. Let F be anF-function on (Γ, d), I⊂R be an interval, and Λ∈P0(Γ). Assume that {Kt:AΛ→AΛ}t∈I is a quasilocal family of transformations satisfying Assumption 5.12, and Φ is a strongly continuous interaction such that Φm∈BF(I) for m = max{p, q}. If the decay function G associated with the family {Kt}t∈I has a finite 2ν + 1 moment, then for any locally normal state ρ and each choice of x, y ∈ Λ, the mapping ΨΛ defined in (5.76) satisfies the estimate
(5.85)
where the time-dependent prefactors C1 and C2 may be taken as
(5.86)
and C2(t)=4κ‖F‖C(t)‖Φq‖F(t)⁠.

It is clear from the statement, as well as the proof, that the estimate proven in Theorem 5.13 does not require that the mappings Kt satisfy Assumption 5.11. As indicated previously, Assumption 5.11 is convenient because it guarantees that the mapping ΨΛ satisfies the continuity requirements needed to be a strongly continuous interaction, as defined in the beginning of this subsection.

Proof.
Fix Z ⊂ Λ and t ∈ I. A simple norm estimate, using (5.76), shows that
(5.87)
Here, we first used that ΔZ(0)Λ=ΠZΛ and that ‖ΠZΛ‖≤1 (see Sec. IV B for more details). Next, we used the local bound on Kt⁠, i.e., (5.78), for the first term on the right-hand-side above. For the remaining terms, we combined the quasilocal bound on Kt⁠, i.e., (5.79), with the estimate (5.8) as proven in Lemma 5.1.
We conclude that
(5.88)
An application of Lemma A.9 completes the proof.
We finish this subsection with a quasilocality estimate for the finite-volume dynamics (5.75). Such a result is an immediate consequence of Theorem 3.1 once we obtain that ΨΛ∈BF̃(I) for some F-function F̃ on (Γ, d). In concrete applications, the existence of such a function F̃ will depend on the original F-function F and quasilocal decay function G. Rather than making further assumptions on F and G from, e.g., Theorem 5.13, let us assume there is an F-function F̃ on (Γ, d) for which
(5.89)
We note that in many applications the initial decay functions are weighted F-functions in the sense of Subsection 2 of the  Appendix, and therefore, explicit choices for F̃ are readily determined by manipulating the weights. In any case, given such a function F̃⁠, the bound in (5.85) implies an explicit pointwise estimate on ‖ΨΛ‖F̃ [see, e.g., (5.91)].

We end this subsection with the following corollary:

Corollary 5.14.
Under the assumptions of Theorem 5.13, suppose further that the family {Kt}t∈I satisfies Assumption 5.11 and that F̃ is an F-function on (Γ, d) satisfying (5.89). Then, ΨΛ∈BF̃(I), and the finite-volume dynamics in (5.75) associated with ΨΛ satisfies the following bound: given any A∈AX, B∈AY where X, Y ⊂ Λ with X ∩ Y = ∅,
(5.90)
holds for all s, t ∈ I. Moreover, for s ≤ t,
(5.91)
where It,s(ΨΛ) is as in (3.21) with F replaced by F̃⁠.

3. Results in infinite volume

In this section, we show how the results of Sec. V D 2 extend to the thermodynamic limit. We begin with an assumption on a collection of quasilocal maps {Kt:AΓloc→AΓ}⁠. In essence, this definition combines the notion of uniformly locally normal from Definition 4.9 with Assumptions 5.11 and 5.12. As always, we consider a quantum lattice system composed of (Γ, d) and AΓ⁠, and let I⊂R be an interval.

Assumption 5.15.

We assume that the family of linear maps {Kt:AΓloc→AΓ}t∈I is strongly continuous, uniformly locally normal, and uniformly quasilocal in the following sense: there is an increasing, exhaustive sequence {Λn}n≥1 of finite subsets of Γ with a family of linear maps {Kt(n):AΛn→AΛn}t∈I for each n ≥ 1 such that

  • For each n ≥ 1, the family {Kt(n):AΛn→AΛn}t∈I satisfies Assumption 5.11.

  • There is some p ≥ 0 and a measurable, locally bounded function B : I → [0, ∞) for which given any X∈P0(Γ) and n ≥ 1 large enough so that X ⊂ Λn,

(5.92)
  • (iii)

    There is some q ≥ 0, a non-negative, nonincreasing function G with G(x) → 0 as x → ∞, and a measurable, locally bounded function C : I → [0, ∞) for which given any sets X,Y∈P0(Γ) and n ≥ 1 large enough so that X ∪ Y ⊂ Λn,

(5.93)
  • (iv)

    There is some r ≥ 0, a non-negative, nonincreasing function H with H(x) → 0 as x → ∞, and a measurable, locally bounded function D : I → [0, ∞) for which given any X∈P0(Γ) there exists N ≥ 1 such that for n ≥ N,

(5.94)

Before proving the theorem, we make the following comments: First, if {Kt}t∈I is a family of linear maps which satisfies Assumption 5.15, then for any compact I0 ⊂ I, the family {Kt}t∈I0 is clearly a strongly continuous family of uniformly locally normal maps in the sense of Definition 4.9. Moreover, conditions (ii) and (iii) of Assumption 5.15 guarantee that the sequence of finite-volume approximates {Kt(n)}n≥1 satisfies Assumption 5.12 with estimates that are uniform in n. In Sec. VI, an explicit family of weighted integral operators of the type discussed in Example 4.11 will be shown to satisfy all conditions of Assumption 5.15.

Let us now return to the discussion of transformed interactions. Let I⊂R be an interval, Φ:P0(Γ)×I→AΓloc be a strongly continuous interaction, and {Kt}t∈I be a family of transformations satisfying Assumption 5.15. Let {Λn}n≥1 be the increasing, exhaustive sequence of finite subsets of Γ whose existence is guaranteed by Assumption 5.15. For each n ≥ 1, we will denote by HΛnΦ(t) the finite-volume, time-dependent Hamiltonian associated with Φ defined as in (5.72). Let us further denote by
(5.95)
the corresponding finite-volume transformed Hamiltonian. Our assumptions, specifically Assumption 5.15 (i), guarantee that the transformed Hamiltonian in (5.95) is still a Hamiltonian in the sense that t↦Kt(n)(HΛnΦ(t)) is strongly continuous and pointwise self-adjoint. As a result, a finite-volume dynamics may be defined by solving
(5.96)
and then using the corresponding unitary propagator to declare that
(5.97)
is the finite-volume time evolution.
A main goal of this section is to show that, under appropriate decay assumptions, the finite volume dynamics in (5.97) converge to a limiting dynamics as n → ∞. To be more precise, let us introduce some further notation. Fix a locally normal product state ρ on AΓ⁠. As in (5.76), with respect to this fixed ρ, for any n ≥ 1, Z ⊂ Λn, and t ∈ I, set
(5.98)
These finite volume maps Ψn are constructed in such a way that
(5.99)
and moreover, as discussed in Sec. V D 2, under the assumptions above, each Ψn is a strongly continuous interaction in the sense of Sec. III A 1. With respect to the same locally normal state ρ, we can also define a map Ψ:P0(Γ)×I→AΓloc by setting
(5.100)
Since the family of transformations {Kt} locally satisfies Definition 4.9, it is clear that Lemma 4.12 applies, and hence, Ψ is a strongly continuous interaction as well.

In the remainder of this section, we will show that if the initial interaction Φ decays sufficiently fast, then the transformed interactions {Ψn}n≥1 converge locally in F-norm to Ψ in the sense of Definition 3.7. Moreover, our assumptions will allow for an application of Theorem 3.8 from which we will conclude that the finite-volume dynamics in (5.97) converge. For ease of later reference, let us declare the relevant decay of Φ as an assumption.

Assumption 5.16.

Given a ν-regular metric space (Γ, d), and a family of maps {Kt:AΓ→AΓ}t∈I satisfying Assumption 5.15, we assume Φ is a strongly continuous interaction such that Φm∈BF(I) for m = max{p, q, r}, where p, q, and r are the numbers in Assumption 5.15.

We can now state the main result of this section, for which it will be useful to review Definition 3.7.

Theorem 5.17.

Consider a quantum lattice system composed of a ν-regular metric space (Γ, d) and quasilocal algebra AΓ. Let I⊂R be an interval, and let F be an F-function on (Γ, d). Assume that {Kt}t∈I is a family of linear maps satisfying Assumption 5.15, Φ is an interaction satisfying Assumption 5.16, and ρ is a locally normal product state on AΓ.

  • Suppose the quasilocal decay function G from (5.93) has a finite 2ν + 1 moment and F̃ is an F-function on (Γ, d) satisfying (5.89), then Ψ∈BF̃(I).

  • Suppose there is some 0 < α < 1 for which Gα has a finite 2ν + 1 moment, where G is as in (5.93). Suppose also that F̃ is an F-function on (Γ, d) satisfying (5.89) with G replaced by Gα. Then, Ψ∈BF̃(I) and Ψn converges locally in F-norm to Ψ with respect to F̃.

Some comments are in order. First, under the assumptions of Theorem 5.17(i), the finite-volume interactions Ψn, as defined in (5.98), satisfy the assumptions of Theorem 5.13 and hence the estimate (5.85). In this case, for any F-function F̃ on (Γ, d) satisfying (5.89), the corresponding finite-volume dynamics, i.e., the automorphisms τt,s(n) defined in (5.97), satisfy the quasilocality bound proven in Corollary 5.14 [see (5.90)]. A main point of Theorem 5.17(i) is that both of these observations extend to the thermodynamic limit. In fact, the assumptions of Theorem 5.17(i) also guarantee that the arguments in Theorem 5.13, and hence an analog of the bound (5.85), also apply to the infinite-volume interaction Ψ as defined in (5.100). Here, we are using that the uniform local convergence in (5.94) guarantees that both the local bound [see (5.92)] and the quasilocal bound [see (5.93)] extend to the limiting map Kt⁠, and in this case, Lemma 5.1 applies. Given this, for any F-function F̃ on (Γ, d) satisfying (5.89), one concludes that Ψ∈BF̃(I)⁠. As a result, we can apply Theorem 3.5, where we take the case of trivial on-sites Hz = 0 for all z ∈ Γ. This then shows that there exists an infinite volume dynamics, which we denote by τt,s, associated with Ψ. By construction, this infinite-volume dynamics τt,s also satisfies Corollary 5.14.

Theorem 5.17(ii) implies that, under the slightly stronger decay assumptions, the finite-volume dynamics τt,s(n) converge to the infinite-volume dynamics τt,s in the sense given by Theorem 3.8. Since the interactions Ψn are constructed using finite-volume local decompositions [see (5.98)], they are not finite-volume restrictions of Ψ, and so an additional argument is required here. We remark that the decay assumptions in Theorem 5.17(ii) imply the decay assumed in Theorem 5.17(i). As a result, the better quasilocality estimates for the dynamics, which follow as a consequence of the assumptions in Theorem 5.17(i), may be used generally.

Next, a careful look at the proof of Theorem 5.17(i) shows that we actually only require Φm∈BF(I) for m = max(p, q). The proof of Theorem 5.17(ii) requires the stronger condition of Assumption 5.16, namely, Φm∈BF(I) for m = max(p, q, r).

Finally, we note that if the decay function G in (5.93) is a weighted F-function, the arguments below can be simplified a bit.

Proof.

The proof of Theorem 5.17(i) is argued in the paragraphs above.

To prove Theorem 5.17(ii), first note that as 0 < α < 1, it is clear that finiteness of the 2ν + 1 moment of Gα implies finiteness of the 2ν + 1 moment of G. In this case, the estimate proven in Theorem 5.13 [see (5.85)] holds for each finite-volume interaction Ψn as well as for Ψ. Since G is non-negative and nonincreasing, G(n) ≤ G1−α(0)Gα(n), and therefore, given any [a, b] ⊂ I,
(5.101)
holds for any F-function F̃ satisfying the conditions of Theorem 5.17(ii). An analogous bound holds for the infinite volume interaction Ψ.
We need only to show that Ψn converges locally in F-norm to Ψ with respect to F̃ [see Definition 3.7]. Let Λ∈P0(Γ) and take n ≥ 1 large enough so that Λ ⊂ Λn. For any Z ⊂ Λ and each t ∈ I, we estimate
(5.102)
where for the terms corresponding to m = 0 in (5.98) and (5.100), we have set
(5.103)
we have collected the bulk of the terms in
(5.104)
and finally, we have denoted any boundary terms by
(5.105)
It is now clear that
(5.106)
To complete this proof, we will show that each of the 3 sums above are bounded by a product of (a) a measurable, locally bounded function of t; (b) F̃(d(x,y))⁠; and (c) a quantity that vanishes as n → ∞. Given this, it is clear that Ψn converges to Ψ locally in F-norm.
Consider the first collection of terms. By consistency of the projections,
(5.107)
Here, we have also applied Assumption 5.15(iv). Since H is nonincreasing, the bound
(5.108)
follows as Φr∈BF(I)⁠. Since F̃ maximizes F, this completes the argument for the first set of terms.
We now consider the bulk terms. An application of Corollary 5.3 [see (5.17)] yields
(5.109)
To obtain an estimate with explicit decay in both n and m, we use the naive bound min{a, b} ≤ a1−αbα which is valid for any 0 < α < 1 and all non-negative a and b. If we denote dn = d(Λ0, Γ\Λn) and p′ = max(q, r), then the right-hand-side of (5.109) may be further estimated by
(5.110)
Using this, we conclude that
(5.111)
and Lemma A.9 applies. Recalling how F̃ is defined, this completes the argument for the second collection of terms.
For the final collection of terms, each nonzero contribution must correspond to values of m ≥ 1 large enough so that X(m) ∩ (Γ\Λn) ≠ ∅. As such, using the notation above, one checks that m ≥ dn = d(Λ0, Γ\Λn). The bound (5.8) applies to each term and thus
(5.112)
Exploiting again that G = G1−αGα and using its nonincreasing behavior, we obtain decay in n. Estimating what remains using Lemma A.9, we have completed the proof of Theorem 5.17(ii).

In this section, we prove a quasilocality estimate for the difference of two dynamics as discussed in Example 5.6 of Sec. V B.

Theorem 5.18.
Let (Γ, d) be a ν-regular metric space. Fix a collection of densely defined, self-adjoint on-site Hamiltonians {Hz}z∈Γ and two time-dependent interactions Φ,Ψ∈BF(I). For any Λ∈P0(Γ) and each t ∈ I, consider the Hamiltonians
(5.113)
For any s, t ∈ I, denote by τt,sΛ and αt,sΛ the dynamics corresponding to HΛ(Φ) and HΛ(Ψ), respectively, and define Kt,sΛ:AΛ→AΛ by
(5.114)
If F has a finite 2ν-moment, i.e., ∑n=0∞(1+n)2νF(n)<∞, then for any X, Y ⊂ Λ,
(5.115)
for any A∈AX⁠, B∈AY⁠, and t, s ∈ I. Here, one may take
(5.116)
and with R = d(X, Y), we find that
(5.117)

Proof.
To begin, we note that for any X,Y∈P0(Γ)⁠, the naive commutator bound
(5.118)
holds for any A∈AX⁠, B∈AY⁠, and s, t ∈ I. In this case, the local bound proven in Theorem 3.4(i) [see also (5.33)] provides a rough estimate, which is linear in It,s(Φ − Ψ). This explains the first part of the minimum in (5.115). Given this, we need only to consider the case of d(X, Y) > 0. Moreover, as is clear from the arguments given in the proof of Theorem 3.3, we need only to consider the case of trivial on-sites, i.e., Hz = 0 for all z ∈ Γ.
Let X,Y∈P0(Γ) satisfy d(X, Y) > 0 and, for convenience, assume that s ≤ t. Writing Kt,sΛ(A) as in (3.70), the bound
(5.119)
follows readily [see also (3.72)]. Here, as in (3.71), we have denoted by Θ the time-dependent interaction with terms Θ(Z, r) = Φ(Z, r) − Ψ(Z, r).

In the estimates below, we use an argument similar to that of Theorem 3.4 [see, in particular, (3.74)] to show that the claim holds with e2It,s(Ψ) replacing Ct,s(1) in the definition of Ct,s(2)⁠. However, by reordering the dynamics in (5.114) [or equivalently, by considering −Kt,sΛ(A)], we see that the analog of (5.119) holds with the roles of the dynamics τt,s and αt,s interchanged. Since the argument given below applies equally well in this case, it will be clear that Ct,s(2) can then be expressed in terms of Ct,s(1)⁠. We now continue with our estimate of the right-hand-side of (5.119).

To prove (5.115), we first consider those terms on the right-hand-side of (5.119) corresponding to Z ⊂ Λ with d(Z, X) > d(X, Y)/2. Since τr,sΛ is an automorphism, the commutator bound
(5.120)
is clear. By Theorem 3.3, the dynamics αt,rΛ corresponding to HΛ(Ψ) satisfies a quasilocality bound, in particular, we may estimate as in (5.28). In this case, an application of Corollary A.5 with R = d(X, Y)/2 shows that
(5.121)
We need only to estimate those terms on the right-hand-side of (5.119) corresponding to Z ⊂ Λ with d(Z, X) ≤ d(X, Y)/2. For these terms, we first make a strictly local approximation of the inner-most dynamics, i.e., αt,rΛ⁠. Given the quasilocality estimate (5.28) for αt,rΛ⁠, an application of Lemma 5.1 shows that
(5.122)
where we have set AR(r)=ΠX(R)Λ(αt,rΛ(A)) and found an upper bound independent of s ≤ r ≤ t.
In this case, for any s ≤ r ≤ t,
(5.123)
For the second term on the right-hand-side of (5.123), it is clear that
(5.124)
and therefore, the bound
(5.125)
follows from an application of Proposition A.4 [see (A41)].
With the remaining terms, i.e., those corresponding to the first term on the right-hand-side of (5.123), we find it useful to further subdivide the sets Z into those of relative “large” and “small” diameter. More precisely, we will estimate using
(5.126)
For the terms with ‘small’ diameter, we apply the quasilocality estimate for the outer dynamics τt,sΛ⁠, again we use the form found in (5.28), to obtain
(5.127)
Clearly, X(R) ∪ Z ⊂ X(3R/2) for any Z with Z ∩ X(R) ≠ ∅ and diam(Z) ≤ R/2. In this case,
(5.128)
follows immediately from the arguments in Proposition A.6 [ee (A55)].
The remaining terms have relatively large diameters, and so we make the naive estimate
(5.129)
As a consequence,
(5.130)
follows from Proposition A.6 [see (A56)].

Collecting the estimates in (5.121), (5.125), (5.128), and (5.130), we find (5.115) as claimed.

In this section, we consider a family of finite volume quantum lattice Hamiltonians HΛ(s) acting on a Hilbert space HΛ that depend smoothly on a parameter s ∈ [0, 1]. We assume that the spectrum of HΛ(s) can be decomposed into two nonempty disjoint sets, i.e., spec(HΛ(s)) = Σ1(s) ∪ Σ2(s), where Σ1(s) is bounded, and the distance between Σ1(s) and Σ2(s) is greater than a positive value independent of s. The main goal of this section is to show that if the interaction defining HΛ(s) is smooth and decays sufficiently fasts, then we can use the theory described in Sec. V to construct a quasilocal automorphism αs:AΛ→AΛ⁠, which we call the spectral flow, that maps the spectral projection of HΛ(s) onto Σ1(s) back to the spectral projection of HΛ(0) onto Σ1(0). In Sec. VII, we use the spectral flow to discuss the classification of gapped ground state phases. A second important application concerns models with a spectral gap above their ground states, for which s parameterizes a perturbation of the system; this is the main topic we analyze in Paper II.96 While both these applications are for ground states, the methods we introduce here are more general and work equally well for isolated bounded subsets anywhere in the spectrum.

Denoting by P(s) the spectral projection of HΛ(s) onto Σ1(s), the existence of an automorphism αs satisfying

(6.1)

is well-known. As shown by Kato in Ref. 67, under certain conditions which guarantee the smoothness of P(s), the unique strong solution of

(6.2)

is unitary and satisfies

αsK(P(s))≔UK(s)*P(s)UK(s)=P(0).

The automorphism studied by Kato was for a family of Hamiltonians defined on a general Hilbert space H⁠, and so his results do not take into account the locality structure of a quantum lattice system. As a result, the automorphism induced by UK(s) is not obviously quasilocal. Hastings and Wen were the first to introduce a technique for constructing an automorphism on a quantum lattice system that both satisfies (6.1) and is quasilocal.59 In that work, they referred to the quasilocal automorphism as the quasiadiabatic evolution (or continuation). It is this approach that we follow in this section. Neither name, spectral flow, or quasiadiabatic continuation, accurately and unambiguously captures the essence of this quasilocal automorphism. It suffices to say that it is a unitary dynamics with useful properties. In other works, Hastings introduced novel ways to combine particular instances of the spectral flow with quasilocality properties of quantum lattice systems, most notably in Ref. 54. This work inspired a string of new results in the theory of quantum lattice systems, and so it seems appropriate to refer to the generator of this spectral flow as the Hastings generator.

We first consider a family of parameter dependent Hamiltonians on a general complex Hilbert space H and later return to apply our results to the setting of quantum lattice systems. Specifically, we consider operators that depend on a parameter s ∈ [0, 1], and we note that the choice of interval [0, 1] is a matter of convenience. We begin with the following definition:

Definition 6.1.

Let H be a complex Hilbert space. We say that a map Φ:[0,1]→B(H) is strongly C1 if Φ(s) is strongly differentiable for all s ∈ [0, 1], and the derivative Φ′:[0,1]→B(H) is continuous in the strong operator topology.

We consider a family of parameter dependent Hamiltonians of the form
(6.3)
where H is a self-adjoint operator acting on some dense domain D⊂H⁠, and Φ:[0,1]→B(H) is strongly C1 and pointwise self-adjoint, i.e., Φ(s)* = Φ(s) for all 0 ≤ s ≤ 1. Since Φ is bounded and self-adjoint, for each s ∈ [0, 1] it is clear that H(s) corresponds to a well-defined, self-adjoint operator with the same dense domain D⊂H⁠. We will refer to {H(s)}s∈[0,1] as a smooth family of Hamiltonians on H⁠.
For each 0 ≤ s ≤ 1, let us denote by τt(s) the Heisenberg dynamics corresponding to H(s), i.e.,
(6.4)
It is clear that for each s, this dynamics is a one-parameter family of automorphisms of B(H)⁠, and so for any real-valued function W∈L1(R)⁠, the mapping D:[0,1]→B(H) given by
(6.5)
is well-defined, pointwise self-adjoint, bounded, and continuous in the strong operator topology. In this case, the methods of Sec. II B show that the unique strong solution of
(6.6)
is well-defined, unitary, and norm-continuous. In terms of these unitaries, we can define an automorphism αs:B(H)→B(H) for each 0 ≤ s ≤ 1 by
(6.7)
Note that here, D(s), U(s), and αs all depend on the choice of weight function W∈L1(R)⁠. We will use this construction to define the spectral flow of interest.
As described in the Introduction, we will consider the situation that the smooth family of Hamiltonians defined as in (6.3) has a spectrum which can be decomposed into two disjoint, nonempty sets. This decomposition will depend on 0 ≤ s ≤ 1, and we are particularly interested in cases where the gap between these sets has a uniform lower bound. To be precise, some additional notation will be convenient: for any two nonempty sets X,Y⊂R⁠, denote by d(X, Y) the distance between these sets,
(6.8)

Assumption 6.2.
For each 0 ≤ s ≤ 1, the spectrum of H(s) can be partitioned into two disjoint sets Σ1(s) and Σ2(s), i.e., spec(H(s)) = Σ1(s) ∪ Σ2(s), such that
(6.9)
and moreover, there are compact intervals I(s) with end-points depending smoothly on s, for which Σ1(s)⊂I(s)⊂(R\Σ2(s)) and μ(s) ≔ d(I(s), Σ2(s)) satisfies μ ≔ inf0≤s≤1μ(s) > 0.

In many concrete examples, one can pick the interval I(s) as the smallest interval containing Σ1(s), and in that case, μ = γ′.

Given a smooth family of Hamiltonians H(s) that satisfy Assumption 6.2, the spectral flow of interest depends on the choice of an auxiliary parameter 0 < γ ≤ γ′. For any such γ and 0 ≤ s ≤ 1, we define the spectral flow αsγ:B(H)→B(H) by
(6.10)
where Uγ(s) is the unitary solution to (6.6) for the self-adjoint operator
(6.11)
defined by a well-chosen γ-dependent, real-valued weight function Wγ∈L1(R)⁠. In Sec. VI B, we state the necessary conditions for choosing Wγ and give an explicit example of a weight function that satisfies these conditions. In fact, we will be able to define weight functions Wγ for any γ > 0. However, to obtain the spectral flow property, i.e., (6.1), one must choose Wγ with 0 < γ ≤ γ′. As discussed in the Introduction, the Hamiltonian Dγ(s) will be called a Hastings generator. We can now state the first main result of this section.

Theorem 6.3.
Let H be a complex Hilbert space, and H(s) be a smooth family of Hamiltonians as in (6.3) satisfying Assumption 6.2. For any 0 < γ < γ′, there is a real-valued function Wγ∈L1(R) such that the automorphism αsγ:B(H)→B(H) defined as in (6.10) satisfies
(6.12)
for all 0 ≤ s ≤ 1. Here, P(s) denotes the spectral projection associated with H(s) onto the isolated part of the spectrum Σ1(s).
In the context of a quantum lattice system, the novel feature of the Hastings generator is that it generates a quasilocal family of automorphisms. This is the second main result of this section. Recall that given a quantum lattice system (Γ, d) and AΓ⁠, the local Hamiltonians for a strongly continuous interaction Φ:P0(Γ)×[0,1]→AΓloc are given by
(6.13)
Note that if Φ(X, s) is strongly C1 for all X ⊂ Λ in the sense of Definition 6.1, then the Hamiltonian HΛ(s) is also strongly C1. In this case, for every Λ∈P0(Γ)⁠, we may define the finite volume Hastings generator by
(6.14)
where for each s ∈ [0, 1], τt(s) is the Heisenberg dynamics associated with HΛ(s). We may now state the quasilocality result.

Theorem 6.4.
Consider a quantum lattice system composed of ν-regular metric space (Γ, d) and quasilocal algebra AΓ. Suppose that Φ∈BF([0,1]) for an F-function of the form
(6.15)
If Φ(X, s) is strongly C1 for all X∈P0(Γ) and Φ1′∈BF([0,1]) where Φ1′(X,s)=|X|Φ′(X,s), then for any γ > 0 there is an F-function, F(γ), such that for any Λ∈P0(Γ)⁠,
(6.16)
for a strongly continuous interaction ΨΛ∈BF(γ)([0,1]). Moreover, there is an interaction Ψ∈BF(γ)([0,1]) such that ΨΛn converges locally in F-norm to Ψ with respect to F(γ) for any sequence of increasing and absorbing finite volumes Λn↑Γ.

We give some context for this result. Suppose that Φ∈BF([0,1]) is such that the local Hamiltonians HΛ(s) are strongly C1. Recall that for any γ > 0, the Hastings generator DΛγ(s)⁠, which is defined in terms of HΛ′(s) [see (6.14)], is strongly continuous and self-adjoint. The automorphism αsγ,Λ defined as in (6.10) can then be recognized as the Heisenberg dynamics associated with DΛ(s). If Theorem 6.4 holds, then DΛ(s) is itself a local Hamiltonian associated with a strongly continuous interaction ΨΛ∈BF(γ)([0,1])⁠. Applying the Lieb-Robinson bound, i.e., Theorem 3.1, to αsγ,Λ shows that the spectral flow is quasilocal as claimed. In the proof of Theorem 6.4, we show that the norm ‖ΨΛ‖F(γ) is bounded from above by a constant independent of Λ, from which local F-norm convergence will follow. The interaction Ψ then defines an infinite volume spectral flow automorphism αsγ:AΓ→AΓ that is also quasilocal.

Note that we do not require Assumption 6.2 for Theorem 6.4, and, in particular, the quasilocality result holds where the spectrum of HΛ(s) is or is not gapped. If, however, HΛ(s) satisfies Assumption 6.2 with gap γΛ′>0⁠, then the finite-volume automorphisms αsΛ,γ generated by DΛγ(s) for any 0<γ≤γΛ′ will both be quasilocal and satisfy (6.12). In applications to stability, one is interested in the situation that there is some sequence of finite volumes (Λn)n≥1 for which both Theorems 6.3 and 6.4 hold simultaneously and that the gaps γΛn′ as in (6.9) are uniformly bounded from below by a positive constant independent of n.

In what follows, we will typically work with a Hastings generator and spectral flow automorphism that depend on a fixed value of γ. As such, we will often suppress the dependence of γ from our notation.

The remainder of this section is organized as follows: In Sec. VI B, we define the explicit weight function Wγ used in our results and prove some basic decay estimates on this function. The reader can skip these details on first reading. Recall that the definition of the Hastings generator is given in terms of a specific weighted integral operator. In Sec. VI C, we define several general weighted integral operators in terms of appropriate L1 functions and prove some useful properties. We use the results from this section to give the proof of Theorem 6.3 in Sec. VI D. We consider quantum lattice systems in Sec. VI E where we show that, in this context, the weighted integral operators introduced in Sec. VI C are quasilocal when defined using the weight functions from Sec. VI B. We then use these results to prove Theorem 6.4 (which is restated as Theorem 6.14). We end the section by showing that there is a well-defined spectral flow automorphism in the thermodynamic limit when the conditions of Theorem 6.4 hold.

To write down the generator of the spectral flow dynamics [see (6.11)], requires a weight function with certain properties. In Sec. VI C, we will define a class of transformations on the algebra of observables of the form

I(A)=∫Rw(t)τt(A) dt,

where w∈L1(R)⁠. In fact, we will make increasingly detailed assumptions of w in order to prove useful properties of the map I. At some point, it becomes more efficient to work with a specific family of functions w for which the assumptions hold. Having such a family of functions will make it possible to state explicit decay estimates that are useful for applications. As such, in this section, we introduce this family of functions, for which interesting properties were already investigated in Ref. 60, and prove some basic estimates; these will be particularly relevant in Sec. VI E 2. It will be clear that other functions can be used to derive similar results. The details of this section can be skipped on first reading. Its main importance is to demonstrate the existence of functions with all the desired properties.

Consider the sequence (an)n≥1 defined by

(6.17)

In terms of this sequence, define a function w:R→R by setting

(6.18)

where c > 0 is chosen so that

(6.19)

It follows from Lemma 6.5 that w∈L1(R) and so this constant is well-defined.

It is clear that w is non-negative and even. Moreover, if we denote by ŵγ:R→R the unitary Fourier transform of w⁠, i.e., for each k∈R⁠,

(6.20)

then it is easy to check (see, e.g., Ref. 12) that supp(ŵ) ⊂ [−1, 1]. Lemma 6.5 provides a useful estimate on w⁠.

Lemma 6.5.
Let a > 0 and p ≥ 0 be an integer. For any x > 1 with ln(x)≥max9,p+1a, one has that
(6.21)
As a consequence, there is a number η ∈ (2/7, 1) for which if x ≥ e9, then
(6.22)

Proof.
To see (6.21), consider the change of variables: u = at/ln(t)2. Clearly,
(6.23)
It will be convenient to take x large enough so that ln(x)4 ≤ x. As one readily checks, this is the case if x ≥ e9; however, we note that this lower bound is not optimal. In any case, using this one also has that
(6.24)
Consequently,
(6.25)
For integers p ≥ 0, the above integral may be bounded using
(6.26)
where the final inequality is valid whenever k ≥ p + 1. With the further constraint that ln(x)≥p+1a⁠, the bound
(6.27)
follows, using again that ln(x)4 ≤ x. Now (6.21) follows from (6.25) and (6.26).
We now estimate w to establish (6.22). Note that for any N ≥ 1 and t ≠ 0,
(6.28)
Using Stirling’s formula, i.e., N!≤eNN+12e−N⁠, and choosing N=⌊a1tln(t)2⌋⁠, we find that
(6.29)
where, for the final inequality above, we used that t is large enough so that both 1 ≤ ln(t)2 and ln(a1t)2≤ln(t)2 hold. Since (6.17) implies that a1 < 1/2, both inequalities are true if t ≥ e. As
(6.30)
it is clear that a1 > 1/7. Now, setting η = 2a1, we have found that
(6.31)
Now (6.22) follows from (6.21).
For our estimates on the spectral flow, it will be convenient to rescale this weight function w⁠. For any γ > 0, define wγ:R→R by setting
(6.32)
It is clear that each such wγ is non-negative, even, L1-normalized, and moreover,
(6.33)
The function Wγ:R→R given by
(6.34)
where H(x) is the Heaviside function [for clarity, we take H(0) = 1] will also play a key role below. This may be rewritten as
(6.35)
Thus, Wγ is odd, and since wγ is normalized and even, one has that ‖Wγ‖∞≤1/2⁠. In fact, a short calculation shows that
(6.36)
It is clear from (6.34) that the distributional derivative of Wγ is
(6.37)
and thus, its (unitary) Fourier transform satisfies
(6.38)
In particular, we have
(6.39)
As we will see, a “well-chosen” weight function Wγ for defining the spectral flow as described following (6.11) is the one which satisfies (6.39) and has a decay estimate that is at least stretched exponential, similar to the next result.

Corollary 6.6.
Let γ > 0. If γx ≥ e9, then
(6.40)
with c as in (6.18) [see also (6.19)], and η ∈ (2/7, 1) as in Lemma 6.5. Moreover, if γx ≥ e9, then
(6.41)
again with c and η as above.

In this section, we briefly discuss some general facts about weighted integrals of a dynamics. Such operators arise as the generator of the spectral flow, and in this case, a number of their basic properties are relevant.

1. Some generalities

Let H be a densely defined self-adjoint operator on a Hilbert space H⁠. Denote by τt the associated Heisenberg dynamics, i.e., the one parameter family of automorphisms of B(H) given by

(6.42)

For any w∈L1(R)⁠, a bounded mapping I:B(H)→B(H) is defined by setting

(6.43)

In fact, Stone’s theorem guarantees that this integral is well-defined in both the weak and strong sense. We refer to the operator I above as the integral of the dynamics τt weighted by w⁠, or more briefly, as a weighted integral operator.

Our applications will mainly concern families of these weighted integral operators. In fact, suppose H(s) = H + Φ(s) is as described in (6.3) and for each 0 ≤ s ≤ 1, consider Is:B(H)→B(H) with

(6.44)

where τt(s) is the dynamics corresponding to H(s) [see (6.3) and (6.4)] and w∈L1(R) is real-valued. Lemma 6.7 is a useful observation.

Lemma 6.7.
Let H be a densely defined self-adjoint operator on a Hilbert space H, and for s ∈ [0, 1], let Φ(s)=Φ(s)*∈B(H) be continuous in s for the strong operator topology. Suppose w∈L1(R) is real-valued, and A:[0,1]→B(H) is pointwise self-adjoint and continuous in the strong operator topology. Then, the mapping D:[0,1]→B(H) given by
(6.45)
is pointwise self-adjoint and continuous in the strong operator topology.

Proof.
Self-adjointness of D(s), which uses that w is real-valued, is clear. Set A(s,t)∈B(H) by
(6.46)
With s0 ∈ [0, 1] fixed, for any 0 ≤ s ≤ 1, we have that
(6.47)
Stone’s theorem guarantees that for each 0 ≤ s ≤ 1, the mapping A(s,⋅):R→B(H) is continuous in the strong operator topology, and so the integrand above is clearly measurable. We now claim that for each t∈R⁠, A(⋅,t):[0,1]→B(H) is also continuous in the strong operator topology. Given this, the claimed continuity of D will follow from an application of dominated convergence. Here, we are using that strong continuity of A implies sup0≤s≤1∥A(s)∥ < ∞.
Due to the form of A(s, t) [see (6.46)], we need only to show that s ↦ eitH(s) is strongly continuous for each fixed t∈R⁠. To see this, note that for any ϕ∈D⁠, the common domain of all H(s),
(6.48)
from which the well-known Duhamel’s formula is proven. As a consequence,
(6.49)
is valid for all ψ∈H and t ≥ 0 (a similar bound holds for t < 0). Dominated convergence applied here, using the continuity assumption on Φ, shows that s ↦ eitH(s) is continuous in the strong operator topology for each fixed t∈R⁠. The proof is now completed as described above.

2. Two particular weighted integrals

For the applications that follow, two particular weighted integral operators play a key role. We introduce a notation for them here and discuss some basic properties.

Generally, the setup is as before. Let H be a densely defined self-adjoint operator on a Hilbert space H and denote by τt the corresponding dynamics [see, e.g., (6.42)].

For any fixed γ > 0, let wγ, Wγ∈L1(R) be any real-valued functions so that (6.33), (6.38), and (6.39) hold. Define two linear maps F,G:B(H)→B(H) by setting

(6.50)

As we will see, the properties of F and G depend crucially on the choice of γ > 0.

In the remainder of this subsection and Subsection VI D, we do not require the more detailed properties of wγ and Wγ that we have proved for the specific functions constructed in Sec. VI B [see (6.18), (6.32), and (6.34)]. These properties will become important later when we analyze the quasilocality properties of the spectral flow. In particular, in Lemma 6.8 and the proof of Theorem 6.3, the specific functions defined in Sec. VI B are not required.

Lemma 6.8.
Let H be a densely defined, self-adjoint operator on a Hilbert space H. Let γ > 0, wγ,Wγ∈L1(R) be real-valued and satisfy (6.33), (6.38), and (6.39), and F,G:B(H)→B(H) be as defined in (6.50). Suppose that the spectrum of H can be decomposed into two nonempty, disjoint sets Σ1 and Σ2,
(6.51)
with Σ1 contained in some compact set and d(Σ1, Σ2) ≥ γ. Denote by P the spectral projection associated with H onto Σ1. Then, for any A∈B(H)⁠,
(6.52)
and
(6.53)
Here, Eλ denotes the spectral family associated with H.

Proof.
We first prove (6.52). In fact, we will show that each F(A) is diagonal with respect to P in the sense that
(6.54)
Given this, one readily checks that
(6.55)
as claimed.
We now calculate the left-hand-side of (6.54). To do so, we will use results on double operator integrals [see, e.g., Ref. 17]. In fact, using Theorem 4.1 in Ref. 17, one sees that
(6.56)
Here, we have used Eλ to denote the spectral family associated with H. Moreover, wγ∈L1(R) is sufficient to guarantee the reordering of the integrals above; it is here that we apply Theorem 4.1(iii) of Ref. 17. The final equality is due to the fact that the Fourier transform of wγ is supported in [−γ, γ] [see (6.33)]. The other relation in (6.54) is proven similarly and (6.52) follows.
Arguing as above, we find that
(6.57)
The claim in (6.53) now follows from (6.39).

A useful observation for certain applications (see, e.g., Refs. 9 and 84) is that the map G is a (left-) inverse of the Liouvillean [H, ·] on the space of off-diagonal operators.

Proposition 6.9.
Let H be a densely defined, self-adjoint operator on a Hilbert space H, and let [H, ·] denote the generator of the Heisenberg dynamics generated by H. Let γ > 0, and F and G as defined in (6.50). Suppose that the spectrum of H can be decomposed into two nonempty, disjoint sets Σ1 and Σ2, with Σ1 compact and d(Σ1, Σ2) ≥ γ. Let P denote the spectral projection of H onto Σ1. Then, for all A∈B(H) such that G(A)∈dom[H,⋅], we have
(6.58)
If, in addition, A is off-diagonal with respect to P, meaning A∈PB(H)(1−P)⊕(1−P)B(H)P, we have F(A)=0 and
(6.59)

Proof.
For any u∈R⁠,
(6.60)
Since, by assumption, G(A)∈dom[H,⋅] and dom[H, ·] is τu-invariant, we then have
(6.61)
where the derivative of Wγ is taken in the distributional sense. Evaluation of (6.61) at u = 0 results in
(6.62)
If A∈PB(H)(1−P) [or A∈(1−P)B(H)P], then F(A)∈PB(H)(1−P) [or F(A)∈(1−P)B(H)P], and hence, in either case [by (6.54)], we have F(A)=0⁠. With this, (6.62) becomes
(6.63)

In applications to quantum spin systems, either finite or infinite, the domain condition on G(A) in this proposition is quite generally satisfied due to the quasilocality properties of both G and the generator of the Heisenberg dynamics. See, e.g., the discussion of the domain of the generator of the dynamics in the proof of Theorem 7.6.

The goal of this section is to complete the proof of Theorem 6.3. Let us recap our progress so far.

Let H(s) = H + Φ(s) be as defined in (6.3). For any γ > 0, a map D:[0,1]→B(H) is defined by

(6.64)

where τt(s) is the dynamics associated with H(s) as in (6.4) and Wγ is the particular weight function defined in (6.34). By Lemma 6.7, D(s) is pointwise self-adjoint and continuous in the strong operator topology. In this case, for any 0 ≤ s ≤ 1, an automorphism αs of B(H) is defined by setting

(6.65)

where the unitary U(s) is the unique strong solution of

(6.66)

The proof of Theorem 6.3 is completed by showing that if H(s) satisfies Assumption 6.2 for some γ > 0, then the automorphisms αs introduced above satisfy (6.12), i.e.,

(6.67)

Proof of Theorem 6.3.
As discussed above, we need only to verify (6.67). A formal calculation shows that
(6.68)
in the sense of strong derivatives. Since α0(P(0)) = P(0), we need only to prove that
(6.69)
It is well-known [see Ref. 67] that spectral projections can be determined through a contour integral of the resolvent, i.e.,
(6.70)
where R(z, s) = (H(s) − z)−1 is the resolvent of H(s) and η(s) is any contour in the complex plane that encircles the interval I(s), as described in Assumption 6.2. From this representation, it is clear that strong differentiability of P follows from strong differentiability of R(z, ·), and so the formal calculation in (6.68) is well-defined. Now, note that for any fixed s0 ∈ [0, 1], the gap assumption allows for a choice of contour η(s) which is independent of s in a neighborhood of s0. With such a contour, one checks that
(6.71)
As P(s) is a strongly differentiable family of orthogonal projections, one can also verify that
(6.72)
We conclude that
(6.73)
where we have set
(6.74)
To simplify the integrals on the right-hand-side of (6.73), we again appeal to the formalism of double operator integrals. In fact, let us denote by, Eλ(s)⁠, the spectral family associated with the self-adjoint operator H(s). One checks that
(6.75)
where again the reordering of the integrals appearing above is justified by Theorem 4.1(iii) in Ref. 17. Here, specifically, the required integrability condition on the contour is readily verified using Assumption 6.2. Applying similar arguments to the second term in (6.73), we find that
(6.76)
On the other hand, the right-hand-side of (6.69) is clearly given by
(6.77)
Here, we have used the notation G(s) for the weighted integral operator [see (6.50)] defined with respect to the parameter dependent dynamics, τt(s)⁠. Using Lemma 6.8, in particular, (6.53) with A = −iΦ′(s), the equality claimed in (6.69) is now clear. This completes the proof of Theorem 6.3.

For the remainder of this section, let us assume that (Γ, d) is a ν-regular metric space, in the sense of (5.35), and Hx is the complex Hilbert space of the quantum system at x ∈ Γ. We start by considering a finite system corresponding to Λ∈P0(Γ)⁠. Recall that for any X ⊂ Λ, we denote HX=⊗x∈XHx and AX=B(HX)⁠.

This section is divided into two parts. First, in Sec. VI E 1, we prove quasilocality estimates for the two weighted integral operators introduced in Sec. VI C 2. Then, in Sec. VI E 2, we establish quasilocality bounds for the spectral flow constructed in the proof of Theorem 6.3.

1. Quasilocality for two weighted integral operators

In Sec. VI C 2, we introduced two particular weighted integral operators that will appear frequently in our applications. We now demonstrate that, under certain additional conditions, each of these weighted integral operators satisfies an explicit quasilocality estimate in the sense of Sec. V.

Let us assume that there is a one-parameter family of automorphisms of AΛ⁠, which we denote by τt that satisfies a quasilocality estimate. More precisely, suppose that there are positive numbers C and v as well as a non-negative, nondecreasing function g for which given any X, Y ⊂ Λ,

(6.78)

for all A∈AX⁠, B∈AY⁠, and t∈R⁠. Here, d = d(X, Y) is the distance between the sets X and Y . As discussed in Sec. III, such a bound is known for the dynamics generated by a short range Hamiltonian; it is, e.g., a consequence of the Lieb-Robinson bounds in Theorem 3.1. In order to prove the quasilocal bounds below, we need only to know (6.78) and that g(d) becomes sufficiently large [see (6.90)]. In applications, we typically have (6.78) with

(6.79)

In terms of these automorphisms τt, for each γ > 0, define F,G:B(H)→B(H) by

(6.80)

for any A∈B(H) [compare with (6.50)]. Here again wγ and Wγ are the specific weight functions introduced in Sec. VI B.

Before we state our first result, recall that wγ(t) = γw(γt) and therefore ‖wγ‖∞≤γc with c being the L1-normalization of w [see (6.18)]. Moreover, Corollary 6.6 [see specifically (6.40)] demonstrates that there is an η ∈ (2/7, 1) for which given any x ≥ γ−1e9 the bound

(6.81)

holds. Here, for any b > 0, we have introduced the subadditive, nondecreasing function

(6.82)

See Subsection 2 a of the  Appendix for a discussion of the properties of fb.

Our quasilocality estimate on the weighted integral operator F follows.

Lemma 6.10.
Let τt be a family of automorphisms of B(H) satisfying (6.78) with (6.79). Let γ > 0 and take F to be the weighted integral operator defined in (6.80). For any 0 < ϵ < 1 and all X, Y ⊂ Λ, the bound
(6.83)
holds for all A∈AX and B∈AY. Here,
(6.84)
where dϵ* is the smallest value of d for which
(6.85)

It can be verified that the function GFϵ(d) given in (6.84) is monotone and strictly decreasing when |GFϵ(d)|<1⁠.

Proof.
Let X, Y ⊂ Λ. Since wγ is L1-normalized, it is clear that
(6.86)
In applications, this bound is best when d = d(X, Y) is small.
When d = d(X, Y) is sufficiently large, see below, a different estimate holds. In fact, let T ≥ 0 and estimate
(6.87)
For the first term above, we ignore the weight and use the locality bound for the dynamics, i.e., (6.78). For the second term, we ignore the dynamics and use the estimate on the weight [see (6.81)]. From these, we obtain the bound
(6.88)
It is important to note that Corollary 6.6 [summarized in (6.81)], has a constraint, and so (6.88) is only valid if γT ≥ e9. For any 0 < ϵ < 1, choose T=(1−ϵ)vg(d)⁠. In this case, we find that
(6.89)
whenever γϵg(d) ≥ e9 and γϵ is as in (6.85). Since limd→∞g(d) = ∞, it is clear that (6.89) can be estimated as in (6.84) for sufficiently large d. The relation
(6.90)
is used when defining dϵ* as above. This completes the proof.

Depending on the application one has in mind, more decay of the function governing the locality of the dynamics, specifically e−g(d), may be needed. For example, many applications require certain moments of the function GFϵ to be finite. Let us make two observations in this regard.

On polynomial decay: Let us consider a family of automorphisms τt with a locality estimate of the form (6.78). If the nondecreasing function g is of the form
(6.91)
then (for fixed t) the locality bound decays like a power-law. In this case, Lemma 6.10 holds; however, the resulting decay function [see (6.84)] has no finite moments. In fact, for any positive numbers a, b, c, and d, one readily checks that
(6.92)
where the functions fc and gd are as defined in (6.82) and (6.91), respectively. As we will see in Paper II,96 this lack of moments restricts the known proofs of stability of the spectral gap to perturbations that decay faster than any polynomial. We do not believe that arguments in Ref. 83 can be extended to obtain a uniform lower bound for the spectral gap in the case of perturbations with only power-law decay, contrary to the claim made in that work. To see why, note that the proof of Lemma 6.10 depends on the choice of T ≥ 0 [see, e.g., (6.87)]. This choice must be made in such a way that both terms on the right-hand-side of (6.88) decay. In order for the first term to decay, vT − g(d) < 0 and so one must take T < v−1g(d). As the function fγ(T) is increasing for large T, the most decay one can obtain from the second term is when T = v−1g(d). If g is logarithmic as discussed above, then even this choice has no finite moments.
On stretched-exponential decay: Let us consider a family of automorphisms τt with a locality estimate of the form (6.78). If the nondecreasing function g is of the form
(6.93)
then all moments of the decay function [see (6.84)] are finite. In fact, for any δ > 0, there is a number Cδ for which ln(x) ≤ Cδxδ whenever x ≥ 1. In this case, for any b > 0,
(6.94)
and therefore, for any δ < 1/2, the function in (6.84) decays at least as fast as a stretched exponential [see Subsection 2 a of the  Appendix more on this terminology].

Lemma 6.11 is the analog of Lemma 6.10 applicable to G⁠.

Lemma 6.11.
Let τt be a family of automorphisms of B(H) satisfying (6.78) with (6.79). Let γ > 0 and take G to be the weighted integral operator defined in (6.80). For any 0 < ϵ < 1 and all X, Y ⊂ Λ, the bound
(6.95)
holds for all A∈AX and B∈AY. Here,
(6.96)
and dϵ* is as defined in Lemma 6.10.

The proof of this lemma is almost identical to that of Lemma 6.10, except that one uses the estimate (6.41) from Corollary 6.6, instead of (6.40). We also use that ‖Wγ‖∞≤1/2⁠.

It can also be verified here that the function GGϵ(d) given in (6.96) is monotone and strictly decreasing when |GGϵ(d)|<‖Wγ‖1⁠.

2. Quasilocality of the spectral flow automorphism

We will now consider the spectral flow in the thermodynamic limit. In order to derive explicit estimates useful for applications, we work with F-functions of the ν-regular metric space (Γ, d) of the form F(r) = e−g(r)F0(r), where g is nondecreasing and subadditive, and

(6.97)

for a suitable ξ > 0. As is shown in the  Appendix (Subsection 1 a of the  Appendix), any choice of ξ > ν + 1 will define an F-function on a ν-regular (Γ, d). In the case Γ=Zν⁠, ξ > ν is sufficient. We will say that F is a weighted F-function on (Γ, d) with base F0.

Let us now introduce the models we consider through an assumption.

Assumption 6.12.

There is a collection {Hx}x∈Γ of densely defined, self-adjoint on-site Hamiltonians. For each 0 ≤ s ≤ 1, there is an interaction Φ(s) on AΓ for which

  • For each X∈P0(Γ)⁠, Φ(X,s)*=Φ(X,s)∈AX for all 0 ≤ s ≤ 1.

  • For each X∈P0(Γ)⁠, Φ(X,⋅):[0,1]→AX strongly C1 in the sense of Definition 6.1.

  • F is a weighted F-function on (Γ, d) with base F0 as in (6.97) and there is a bounded, measurable function ∥Φ∥1,1:[0, 1] → [0, ∞) for which given any x, y ∈ Γ, the estimate
    (6.98)
    holds for all 0 ≤ s ≤ 1.

Under Assumption 6.12, given any Λ∈P0(Γ)⁠,
(6.99)
is a well-defined self-adjoint operator on HΛ for each 0 ≤ s ≤ 1. If we denote by Dx⊂Hx the dense domain of the on-site Hamiltonian Hx, then each HΛ(s) has the same dense domain DΛ=⊗x∈ΛDx⊂HΛ⁠. We stress here that in our applications the number 0 ≤ s ≤ 1 plays the role of a parameter. In this case, the finite-volume Hamiltonians HΛ(s) do not depend on time t, and thus, using functional calculus, for each 0 ≤ s ≤ 1, the dynamics corresponding to these self-adjoint operators is simply given by
(6.100)
Interactions depending on the time t itself, as in the model (3.62) discussed in Sec. III B, can be accommodated without difficulty.
Assumption 6.12 implies that Φ(s)∈BF for each 0 ≤ s ≤ 1. Therefore, an application of Theorem 3.5 shows that there is an infinite volume dynamics defined by
(6.101)
In terms of this infinite volume dynamics, and any γ > 0, we now define a family of linear maps {Ksγ:AΓloc→AΓ}s∈[0,1] by setting
(6.102)
Here, Wγ is the weight function introduced in (6.34) of Sec. VI B. Often, we will regard γ as fixed and drop it from our notation.

Proposition 6.13 relates the quantities introduced above to the methods discussed in Sec. V D 3.

Proposition 6.13.

Consider a quantum lattice system composed of a ν-regular metric space (Γ, d) and AΓ. Suppose Assumption 6.12 holds with a weighted F-function F(r) = e−g(r)F0(r) for which limr→∞g(r) = ∞ and F0 is as in (6.97). Then, for any γ > 0, the family of maps {Ksγ}s∈[0,1] [as defined in (6.102)], satisfies the conditions of Assumption 5.15.

Proof.

As is clear from the statement, the results we prove below hold for any γ > 0. For convenience of presentation, we now fix such a value γ > 0 and then suppress it in our notation.

Let {Λn}n≥1 be any increasing, exhaustive sequence of (nonempty) finite subsets of Γ. For each n ≥ 1, define a family of maps {Ks(n)}s∈[0,1]⁠, with Ks(n):AΛn→AΛn for all 0 ≤ s ≤ 1, by setting
(6.103)
Here, we have used the finite volume dynamics [see (6.99) and (6.100)] with Λ = Λn. We will show that any such choice of {Λn}n≥1 determines a sequence of maps Ks(n)⁠, as defined in (6.103), which satisfies all four conditions in Assumption 5.15.

In the first part of Assumption 5.15, we show that for each n ≥ 1, the finite volume families of maps {Ks(n)}s∈[0,1] satisfy Assumption 5.11. Since Wγ∈L1(R) is real-valued, Assumption 5.11(i) is easily verified. We note that integrability of Wγ is a consequence of the estimate (6.41) in Corollary 6.6. To check the remaining parts of Assumption 5.11, we recall that the properties of maps with the form (6.103) were discussed in Example 4.11. Assumption 6.12 guarantees that the methods of Example 4.11 apply, and the remaining details are readily checked.

For our applications, the simple bound
(6.104)
suffices, and thus, Assumption 5.15(ii) is trivially satisfied with p = 0 and B(s)=‖Wγ‖1⁠.
The uniform quasilocality estimate in Assumption 5.15 (iii) can be seen as follows: By Assumption 6.12 (iii), Φ(s)∈BF for each 0 ≤ s ≤ 1. As a result, given any n ≥ 1, the model’s finite-volume dynamics, i.e., τtΛn,s⁠, satisfies a Lieb-Robinson bound as in Theorem 3.3. In fact, for any X, Y ⊂ Λn, with X ∩ Y ≠ ∅, and any A∈AX and B∈AY⁠, the bound
(6.105)
holds for all t∈R and s ∈ [0, 1]. Here, we have estimated the weighted F-function F(r) = e−g(r)F0(r) and set
(6.106)
In this case, Lemma 6.11 applies. We have that for any 0 < ϵ < 1 and A and B as above, the bound
(6.107)
holds with decay function Gϵ as in (6.96). In principle, here we have used that limr→∞g(r) = ∞, although one only needs that it becomes sufficiently large. Note further that the bound in (6.107) is uniform in n ≥ 1 as well as 0 ≤ s ≤ 1, and we have proven Assumption 5.15 (iii).
We now demonstrate that Assumption 5.15 (iv) holds. Fix X∈P0(Γ) and A∈AX⁠. For n ≥ 1 sufficiently large, X ⊂ Λn, and thus for any T > 0, the following estimate holds:
(6.108)
Appealing to the continuity result in Corollary 3.6 [see specifically (3.80)], for any t ∈ [−T, T],
(6.109)
where v is as in (6.106), and in this particular application,
(6.110)
[see, e.g., (3.21)]. Thus, if T > 0 is sufficiently large (e.g., γT ≥ e9), then by Corollary 6.6,
(6.111)
where fγ(t) is as defined in (6.82). Arguing now as in the proof of Lemma 6.10, let 0 < ϵ < 1, set dn = d(X, Γ\Λn), and take T to be defined by vT = (1 − ϵ)g(dn). We have proven that there are positive numbers C1 and C2 for which
(6.112)
for n ≥ 1 sufficiently large. For example, one may take
(6.113)
This completes Assumption 5.15 (iv), and so Proposition 6.13 is proven.

We can now introduce the spectral flow for the class of models under consideration. As above, all comments below are valid for any choice of γ > 0, which is suppressed in the notation. The spectral flow automorphism can be defined for any choice of γ > 0, and under the general assumptions above, the spectral flow is quasilocal with γ-dependent estimates. It is only for the special relation with the spectral projection P(s) as in (6.12) that γ needs to be a lower bound for the gap in the spectrum as described in Assumption 6.2.

Given Proposition 6.13, we know that the family of maps {Ks}s∈[0,1] satisfies Assumption 5.15. By Assumption 6.12, Φ′ is a well-defined interaction on AΓ⁠, and moreover, Φ′ is a suitable initial interaction in the sense of Assumption 5.16, in particular, Φ1′∈BF([0,1]) as is clear from (6.98). In this case, we are in a position to apply Theorem 5.17; we first introduce the relevant notation. Let {Λn}n≥1 be a sequence of (nonempty) increasing and exhaustive finite subsets of Γ. For each n ≥ 1 and any 0 ≤ s ≤ 1, consider the transformed (bounded) Hamiltonian
(6.114)
which acts on HΛn [compare with (5.95)]. As in Sec. V D 3, the unique strong solution of
(6.115)
can be used to define a family of automorphisms of AΛn by setting
(6.116)
[see specifically (5.98) and (5.97)]. We will refer to the automorphisms αs(n) as the finite-volume spectral flow dynamics. To estimate the quasilocality of αs(n)⁠, we fix a locally normal product state ρ on AΓ and proceed as in Sec. V D 3. Consider the finite-volume, s-dependent interaction Ψn with terms Ψn(Z, s), for Z ⊂ Λn and 0 ≤ s ≤ 1, defined by
(6.117)
and further the corresponding infinite-volume interaction Ψ given by
(6.118)
again, one should compare with (5.98) and (5.100).

The main result of this subsection is as follows:

Theorem 6.14.
Consider a quantum lattice system composed of a ν-regular metric space (Γ, d) and AΓ. Suppose that Assumption 6.12 holds with a weighted F-function of the form F(r) = e−g(r)F0(r), where F0(r) is as in (6.97) and
(6.119)
for some a > 0 and 0 < θ ≤ 1. Then, Ψn converges locally in F-norm to Ψ with respect to an F-function on (Γ, d). Here, Ψn and Ψ are as defined in (6.117) and (6.118).

We make some comments and point out two corollaries before proving the theorem.

First, in principle, one can do better than the growth assumption in (6.119); in fact, one needs only that there is some 0 < ϵ < 1 for which the decay function Gϵ [see (6.107)] satisfies the conditions of Theorem 5.17 (ii). As can be seen from our comments in Sec. VI E 1 after Lemma 6.10, any weight function g satisfying (6.119) corresponds to such a decay function Gϵ which satisfies the conditions of Theorem 5.17 (ii). However, no weight function g which grows proportional to a logarithm [see (6.91)] corresponds to a decay function Gϵ satisfying the conditions of Theorem 5.17 (ii).

Next, let us say some more about the F-function whose existence plays a crucial role in the proof of Theorem 6.14. Given the decay of the initial interaction Φ [see (6.98)], it is proven in Proposition 6.13 that the weighted integral operators used in defining the generator of the spectral flow [see (6.103) and subsequently (6.114)] satisfy the quasilocality estimate (6.107). The corresponding decay function G (take ϵ = 1/2 for convenience) has the following form: there exist positive numbers C1, C2, and d* for which
(6.120)
where we stress that the function g above corresponds to the weight in the F-function governing the decay of the initial interaction Φ. It should be clear that any F-function governing the decay of Ψ (and similarly Ψn) will decay no faster than this G. Our estimates show that there are positive numbers C1′⁠, C2′⁠, and d*′ for which Ψ,Ψn∈BF̃([0,1]) with
(6.121)
where η′ is any number strictly less that η and the function g̃ may be taken as g̃(d)=ãdθ with the same value of θ (as g) but, in general, a smaller value of a. As is discussed in Subsection 2 of the  Appendix, any function with the form (6.121) is an F-function on (Γ, d). To obtain the local convergence of Ψn to Ψ, we will need to modify the above F-function slightly, but all relevant estimates on Ψ and Ψn [see, for example, Corollaries 6.15 and 6.16] will be made with respect to the function F̃ described above. For more details on this, see the discussion following the statement of Theorem 5.17.
Finally, our proof of Theorem 6.14 guarantees that Theorem 3.8 applies and so
(6.122)
Here, the limit is in norm and the quantity, αs, is the well-defined, infinite-volume dynamics associated with Ψ∈BF̃([0,1])⁠, with terms as in (6.118), whose existence is guaranteed by Theorem 3.5.

For ease of later reference, we now state two corollaries providing explicit estimates on the quasilocality of the spectral flow.

Corollary 6.15.
Under the assumptions of Theorem 6.14, for any X,Y∈P0(Γ) with X ∩ Y = ∅, the bound
(6.123)
holds for any A∈AX⁠, B∈AY, and 0 ≤ s ≤ 1. Here, F̃ may be taken as in (6.121).
Since Ψ∈BF̃([0,1])⁠, the above is an immediate consequence of Corollary 3.6 (i). Our estimates actually show that sup0≤s≤1‖Ψ‖F̃(s)<∞ and, therefore, we have
(6.124)
Given (6.124), it is clear that the bound in Corollary 6.15 may be further estimated with a linear dependence on s. This observation is useful in some applications.

The following corollary is a direct application of Theorem 3.8 (i):

Corollary 6.16.
Under the assumptions of Theorem 6.14, for any X∈P0(Γ), the bound
(6.125)
holds for all A∈AX and 0 ≤ s ≤ 1. Here, F̃ may be taken as in (6.121).

Proof of Theorem 6.14.

It is clear that, under the decay assumptions (6.119), Proposition 6.13 holds. In this case, for each γ > 0, the family of maps {Ksγ}s∈[0,1] satisfies Assumption 5.15, and moreover, Φ′ is a suitable initial interaction in the sense of Assumption 5.16. As before, we will suppress the dependence on γ > 0 in what follows.

For any x, y ∈ Γ and n ≥ 1 large enough so that x, y ∈ Λn, an application of Theorem 5.13 shows
(6.126)
where Ψn is the finite-volume interaction in (6.117), F is the weighted F-function governing the decay of Φ as in Assumption 6.12(iii), and Gϵ is the decay function associated with the family {Ks(n)}s∈[0,1] as in (6.107) [see also (6.96)]. Our estimates show that one may take
(6.127)
and C2 = 8κ∥F∥sup0≤s≤1∥Φ∥1,1(s). As we have argued before, the analog of (6.126) also holds with the interaction Ψ replacing Ψn on the left-hand-side and the right-hand-side unchanged.
It is also clear that the decay assumption (6.119) guarantees that for any 0 < δ < η,
(6.128)
As such, for d = d(x, y) sufficiently large, we may further estimate the right-hand-side of (6.126) by
(6.129)
For large d, the second term above dominates. Using the facts provided in Subsection 3 of the  Appendix, it is clear that an F-function, F̃⁠, of the form in (6.121) bounds this quantity. For any such F̃⁠, our estimates are uniform with respect to s, and so we have proven that
(6.130)
In fact, the same uniform estimate also holds for the finite-volume interactions Ψn.

Since the form of the decay function Gϵ, as in (6.107), is explicit [see (6.96)], it is clear that Gϵ has a finite 2ν + 1 moment. Arguing as above, a similar, yet different F-function F^ can be produced which satisfies the assumptions of Theorem 5.17 (ii) with α = 1/2; in fact, any choice of 0 < α < 1 suffices. In this case, Ψn converges locally in F-norm to Ψ with respect to this function F^⁠. As indicated previously, for the estimates in Corollaries 6.15 and 6.16, one can use the original F-function F̃ having the form (6.121).

In this section, we use the spectral flow to study gapped ground state phases of a quantum lattice model (Γ, d) and AΓ⁠. As before, we will discuss both finite and infinite volume systems and take the thermodynamic limit along a sequence of increasing and absorbing finite volumes. To this end, we will consider the following setup for this section.

Throughout this section, let (Γ, d) be a fixed ν-regular metric space with a weighted F-function of the form F(r) = e−g(r)F0(r), where F0 is an F-function for (Γ, d) of the form (6.97) and g is a non-negative, nondecreasing, subadditive function bounded below by arθ for some θ ∈ (0, 1]. In addition, we consider a fixed sequence of increasing and absorbing finite volumes Λn↑Γ and with the convention that we always take the thermodynamic limit with respect to a subsequence of this sequence. We will use the notation BF1([0,1]) to denote the space of differentiable curves of interactions Φ(s)∈BF⁠, s ∈ [0, 1], satisfying Assumption 6.12. At each x ∈ Γ, we may have a densely defined self-adjoint Hx, but these we regard as fixed. Specifically, we only consider here relations between models with different interactions Φ(s) but with the same {Hx∣x ∈ Γ}.

For simplicity, we will assume that the finite-volume Hamiltonians for the models parametrized by s ∈ [0, 1], are defined by

(7.1)

Within the context described above, we now introduce the notion of a uniformly gapped curve of models or, equivalently, a curve of uniformly gapped interactions for which we use the notation EΛ(s) = inf specHΛ(s) to denote the ground state energy of HΛ(s).

Definition 7.1.
Let γ > 0. A curve of interactions Φ∈BF1([0,1]) is called uniformly gapped with gap γ, if there exists a non-negative sequence (δn)n⁠, with limnδn = 0, such that for all n ≥ 1 and s ∈ [0, 1],
(7.2)
where HΛn(s) is the finite volume Hamiltonian defined in (7.1).

We can leave γ unspecified and call the curve simply uniformly gapped if there exists γ > 0 such that it is uniformly gapped with gap γ.

It is well-known that the spectral gap generally depends on the boundary conditions. Our choice to define the path of Hamiltonians (7.1) with respect to a single interaction leads to boundary conditions that are not necessarily the most general ones of interest; studying all possible cases at once would lead to quite onerous notation, which we want to avoid in this discussion. Suffice it to note that everything in this section could be generalized to the situation where we have boundary conditions expressed by a sequence Φn∈BF1([0,1])⁠, by requiring that Φn converges locally (in a uniform version of Definition 3.7) in a suitable norm to some Φ∈BF1([0,1])⁠. In this case, Φn is then used to define the Hamiltonian (7.1) on the volume Λn for each n. For example, this can be used to study finite systems in Zν with periodic boundary conditions. Even without considering n-dependent interactions, the present setup allows one to study the effects of certain boundary conditions. For example, by replacing Γ by a subset Γ0 ⊂ Γ and different sequences of finite volumes Λn, models defined with the same interaction Φ may show different behavior. An example of this is discussed in detail for a class of so-called PVBS models in Refs. 11 and 18. There, Γ0 is the half-space in Γ=Zν defined by an arbitrary hyperplane. For these models, the spectral gap is shown to depend nontrivially on the orientation of the hyperplane.

We use the notion of a uniform gap to define a relation ∼ on BF as follows:

Definition 7.2.

For Φ0,Φ1∈BF⁠, we say that Φ0 and Φ1 are equivalent, denoted by Φ0 ∼ Φ1, if there exists a uniformly gapped curve Φ∈BF1([0,1]) such that Φ(0) = Φ0 and Φ(1) = Φ1.

In the physics literature, two models Φ0 and Φ1 are said to be in the same gapped ground state phase if Φ0 ∼ Φ1.32,33 Studying curves of models has proved to be fruitful also in mathematical studies.12–16 In this section, we explore some essential properties of models that belong to the same gapped ground state phase. First, however, we show that the relation ∼ used to define this notion is indeed an equivalence relation.

We found it convenient to state Definition 7.2 in terms of differentiable curves because we rely on the differentiability in the construction of the spectral flow automorphisms. Equivalently, however, one can just assume the existence of a piecewise differentiable continuous curve. As we show in the proof of the Proposition 7.3, a simple reparametrization of the concatenation of two differentiable curves yields a differentiable curve with the desired properties.

Proposition 7.3.

The relation ∼ defined in Definition 7.2 is an equivalence relation on BF.

Proof.
The defining properties of reflexivity and symmetry of an equivalence relation follow by considering constant curves Φ(s) = Φ0, for all s ∈ [0, 1], and reversed curves Φ−1(s) = Φ(1 − s). For transitivity, consider two curves Φ(1)(s), Φ(2)(s)∈BF1([0,1]) such that Φ(1)(1) = Φ(2)(0), and define Φ(s)∈BF1([0,1]) by
Φ(s)=Φ(1)(sin(πs)) s≤1/2Φ(2)(1−sin(πs)) s>1/2.
Here, the reparameterization of s in the piecewise definition is chosen only to ensure the differentiability of Φ at s = 1/2. Other reparameterizations will also work. Transitivity follows from setting δn=max(δn(1),δn(2)) and γ = min(γ(1), γ(2)), where δn(i) and γ(i), i = 1, 2, refer to the sequences and the gap for the two curves.

Note that, without loss of generality, we can assume that the sequence (δn) in Definition 7.1 is nonincreasing. It is also easy to see that for uniformly gapped Φ, the spectral projection Pn(s) of HΛn(s) associated with the interval [EΛn(s),EΛn(s)+δn] becomes independent of the choice of sequence (δn) for large n in the sense that for any two sequences (δn) and (δn′) for which (7.2) holds, the spectral projections associated with the intervals [EΛn(s),EΛn(s)+δn] and [EΛn(s),EΛn(s)+δn′] coincide for sufficiently large n.

Let Φ be uniformly gapped. Then, consider the collection of states of AΛn supported on the spectral subspace of HΛn(s) associated with the intervals [EΛn(s),EΛn(s)+δn]⁠. More precisely, define
Sn(s)={ω∈S(AΛn)∣ω(Pn(s))=1}.
Here, for any complex C*-algebra A with unit 1⁠, S(A) denotes the state space of A that is the set of positive linear functionals on A with ω(1)=1⁠. The remarks in the previous paragraph show that Sn(s) becomes independent of the choice of the sequence (δn) for large n. Therefore, it is possible to define
S(s)={ω∈S(AΓ)∣∃(nk)increasing andωk∈Snk(s)s.t.limkωk(A)=ω(A),∀A∈AΓloc}.
In this sense, S(s) is the set of all weak −* limits of states in Sn(s)⁠. It can be shown that the elements ω∈S(s) are ground states of the infinite-volume model defined by the dynamics τt obtained as the thermodynamic limit of the model with Hamiltonians (7.1).96 The prime example to keep in mind is that the set Sn(s) also consists of ground states for a Hamiltonian HΛn(s) that has a uniform lower-bound, denoted by γ > 0, separating the ground state energy from the rest of the spectrum. However, it will be interesting to consider the slightly more general setup we have introduced above.

Let us now fix a uniformly gapped curve Φ∈BF1([0,1]) with gap γ. As an application of Theorem 6.14 and the comments following it, we have strongly continuous spectral flow automorphisms αs(n) for the curve of finite-volume on Λn and αs for the infinite system on Γ. Here, the uniform gap of the curve plays the role of the parameter γ in the construction of the spectral flow. Moreover, Theorem 6.14 [see specifically (6.122)] establishes that αs is the strong limit of αs(n)⁠, and the convergence of this limit is uniform for s ∈ [0, 1]. Moreover, we can use the spectral flow αs to construct a cocycle of automorphisms αt,s≔αs−1◦αt⁠, for all t, s ∈ [0, 1]. We can similarly define a collection of finite volume cocycles, αt,s(n)⁠.

Our next goal is to show that the spectral flow cocycle establishes a close relationship between the sets S(s) for different values of s, which we refer to as automorphic equivalence of gapped ground state phases. Using the definition of αt,s(n)⁠, Theorem 6.3 establishes the following relationships between the spectral projections Pn(s):
(7.3)
As an immediate consequence, we have that ωn◦αt,s(n)∈Sn(t) for any ωn∈Sn(s) as
ωn◦αt,s(n)(Pn(t))=ωn(Pn(s))=1.
Since αt,s(n) is an automorphism, it is invertible. In fact, its inverse is given by αs,t(n)⁠. As such, we see that composition with αt,s(n) defines a bijection between the sets Sn(s) and Sn(t)⁠. Explicitly,
(7.4)
Theorem 7.4 extends this bijection to the thermodynamic limit. The quasilocal properties of the spectral flow established in Sec. VI play an important role both at the technical and the conceptual level. Technically, they are the main ingredients in establishing the convergence of the thermodynamic limit. Conceptually, the fact that αs is a dynamics generated by a short-range interaction shows that the local properties of the states at different values of s are related by a “natural,” finite-time, unitary evolution.

Theorem 7.4.
For all s, t ∈ [0, 1], the spectral flow automorphism αs,t provides a bijection between the sets S(s) and S(t) by composition,
(7.5)

Proof.

This is a direct consequence of (7.4), Theorem 6.14, and Lemma 7.5.

Lemma 7.5.

Let (αn)n be a strongly convergent sequence of automorphisms of a C*-algebra A, converging to α, and let (ωn)n be a sequence of states on A. Then, the following are equivalent:

  • ωn converges to ω in the weak-* topology;

  • ωn ◦ α converges to ω ◦ α in the weak-* topology;

  • ωn ◦ αn converges to ω ◦ α in the weak-* topology.

Proof.
(i) ⇔ (ii) follows immediately from the fact that α and α−1 are automorphisms. Now if (ii) holds, the limit of the second term in the RHS of
|(ωn◦αn)(A)−(ω◦α)(A)|≤|ωn(αn(A)−α(A))|+|ωn(α(A))−ω(α(A))| 
vanishes. So does the limit of the first term since
|ωn(αn(A)−α(A))|≤‖ωn‖‖αn(A)−α(A)‖→0
as ωn is a state. Therefore, (iii) holds. A similar argument implies (iii) ⇒ (ii).
Recall the following essential property of the spectral flow automorphisms with parameter γ > 0 constructed in Sec. VI for a family of Hamiltonians HΛ(s). Suppose that P(s) is the spectral projection of HΛ(s) associated with a bounded interval [a(s), b(s)] [with a(s) and b(s) differentiable] that is gapped from the rest of the spectrum by γ > 0, i.e.,
(a(s)−γ,a(s))∩spec(HΛ(s))=(b(s),b(s)+γ)∩spec(HΛ(s))=∅,for all s∈[0,1].
Then, by Theorem 6.3 the spectral flow αt,s with parameter γ associated with HΛ(s) once again maps P(t) to P(s). In the discussion above, we focused on gapped ground state phases, for which the relevant part of the spectrum is at the bottom. We will describe examples in some detail in Sec. VII C. There is no reason, however, why a similar strategy could not be employed to study states supported in spectral subspaces associated with an isolated part elsewhere in the spectrum. For example, an isolated band of excited states could also be studied with the help of spectral automorphisms. This new and largely unexplored territory seems promising to us.

We conclude this section with the following result regarding the continuity of the spectral gap above the ground state energy of the GNS Hamiltonian Hωs of an infinite volume ground state ωs∈S(s) for the case of quantum spin systems, i.e., the single-site Hilbert spaces Hx are finite-dimensional and the dynamics is generated by an interaction Φ∈BF1([0,1]) for a suitable F-function F. The restriction to the case of quantum spin systems is because we rely on some well-known properties of the dynamics and, in particular, its generator in that case. The finite-dimensionality of the single-site Hilbert spaces is not essential, but the boundedness of the interactions is, in addition to the general setup described at the beginning of this section, including Assumption 6.12.

Theorem 7.6.
Consider a quantum spin model defined by a uniformly gapped curve of interactions Φ∈BF1([0,1]), with gap γ > 0. Fix s0 ∈ [0, 1] and let {Hωs}s∈[0,1] denote the set of GNS Hamiltonians associated with the states ωs∈S(s) of the form ωs=ωs0◦αs,s0 for some ωs0∈S(s0) and with the spectral flow αs,s0 corresponding to the parameter γ. If for all s ∈ [0, 1], kerHωs is one-dimensional, then
γ(s)≔sup{δ>0 : spec(Hωs)∩(0,δ)=∅}
is a upper-semicontinuous function of s.

Proof.

Recall that for each s, the infinite volume dynamics τt(s) is a strongly continuous group of automorphisms of AΓ generated by a closed operator δ(s), i.e., τt(s)=eitδ(s)⁠, and that AΓloc is a core for δ(s).

First, we show that for all t, s, s0 ∈ I, αt,s(AΓloc) is a core for δ(s0)⁠. Since AΓloc is a core and αt,s is an automorphism, we only need to show that αt,s(A)∈dom(δ(s0))⁠, for all A∈AΓloc⁠. Let X = supp(A) and ΠX(n) be as in (4.11). Then by (4.13),
αt,s(A)=limn→∞ΠX(n)(αt,s(A)).
The result follows from showing that δ(s0)(ΠX(n)(αt,s(A))) is convergent. Using the telescopic property of ΔX(n) and linearity of δ(s0)⁠, it follows that for any n ≥ 0,
δ(s0)(ΠX(n)(αt,s(A)))=∑m=0nδ(s0)(ΔX(m)(αt,s(A))),
which is absolutely convergent by Proposition 5.9.
Now, let γ(s) denote the spectral gap of Hωs and pick any s0 ∈ [0, 1]. By the variational principle,
(7.6)
where C is any core for δ(s). Using that ωs=ωs0◦αs,s0 and Cs=αs0,s(AΓloc) is a core for δ(s), we have the following identity:
(7.7)
For any A∈AΓloc⁠, define the function fA(s)=ωs0(A*αs,s0◦δ(s)◦αs0,s(A)) and consider the family of functions,
F={ fA∣A∈AΓloc,ωs0(A)=0,ωs0(A*A)=1}.
Using (7.7), the expression for the gap in (7.6) can be rewritten in terms of the family F as
(7.8)
The result follows from showing that all f∈F are continuous. This can be seen by expressing the operator αs,s0◦δ(s)◦αs0,s as the generator of the dynamics for a new s-dependent interaction Ψ∈BF̃(I)⁠. Using the continuity of automorphisms and αs,s0◦αs0,s=id⁠, we have
(7.9)
Theorem 5.13 and Corollary 6.15 imply the existence of an F-function F̃ [see (5.89)] and a strongly continuous Ψ∈BF̃(I) such that the RHS is the generator determined by Ψ(s), which is again locally bounded and quasilocal. It follows that the map s↦αs,s0◦δ(s)◦αs0,s(A) is continuous for each finite X ⊂ Γ and A∈AX⁠. Therefore, f(s)=ωs0(A*αs,s0◦δ(s)◦αs0,s(A)) defines a continuous function.

As far as we are aware, for all models that satisfy the conditions of the theorem, the gap appears to be continuous in the parameter, not just semicontinuous. In particular, the gap is continuous when perturbation theory applies. This raises the question whether one indeed has continuity of the spectral gap as long as it is strictly positive, or whether additional assumptions are needed for continuity. Needless to say, the gap is not always stable and so should not be expected to be continuous, in general, on a domain where it vanishes at some points.

In Sec. VII A, we introduced the classification of gapped ground state phases through equivalence classes of interactions for which there exists an interpolation by a uniformly gapped curve. We showed that within each equivalence class, the sets of ground states are mapped into each other by an automorphism with good quasilocality properties (the spectral flow derived from the uniformly gapped curve of interactions interpolating between the models). Implicit in this description is the idea that any curve of interactions interpolating between two models in distinct phases (different equivalence classes) must contain at least one point where the gap vanishes. Such points are called quantum critical points and one says that a quantum phase transition occurs in the system.115 

Physical systems often have symmetries that play an important role. In the description of certain phenomena, it may be essential that a certain symmetry be present in the model. This led to the concept of symmetry protected gapped phases31,39 due to the observation that if one only allows curves of interactions that all possess a given symmetry, a finer classification of gapped ground state phases may arise. A nice example of this are the Z2×Z2 protected phases of the spin-1 chain.33,102,104,122 In general, the equivalence classes break up into subclasses if a restricted set of uniformly gapped curves of interactions is used to define the equivalence relation. This prompts us to revisit the notion of automorphic equivalence in the presence of symmetry.

A symmetry is usually specified by the action of a group G (as automorphisms) on the algebra of observables of the system. Although there are interesting symmetries that do not fit the framework of group representations by automorphisms, such as dualities and quantum group symmetries, we limit the discussion here to that setting. In general, we use the label G to specify the presence of a certain symmetry. So, we will consider spaces of interactions BFG⊂BF and of curves of interactions BF1,G([0,1])⊂BF1([0,1])⁠. To be clear, in this context, G stands for the full specification of the symmetry including its action on the system, not just the abstract group.

Here are four important classes of symmetries:

  • Local symmetries are described by automorphisms β of AΓ with the property that they leave the single-site algebras, A{x}=B(Hx)⁠, invariant. Specifically, we assume that the restrictions of β to A{x}⁠, x ∈ Γ, are inner automorphisms given in terms of a unitary Ux∈B(Hx)⁠: β(A)=Ux*AUx for A∈B(Hx)⁠. These types of symmetries are sometimes called gauge symmetries because gauge symmetries are of this form. Thus, any local symmetry β is determined by a family of unitaries {Ux∈B(Hx)∣x∈Γ}⁠. We say that β is a symmetry of Φ if β(Φ(X)) = Φ(X), for all X∈P0(Γ)⁠. It is easy to see that this implies that β commutes with the dynamics τt generated by Φ: β ◦ τt = τt ◦ β. If Φ depends on a parameter s or on the time t, the symmetry condition is assumed to hold pointwise in s and/or t. The set of all local symmetries form a group under the law of composition of automorphisms. It is often useful to consider the (projective) representations of this group, G, given by the local unitaries Ux(g), g ∈ G.

  • Lattice symmetries are, in general, described by a bijection R: Γ → Γ. It is usually important that R preserves the local structure of (Γ, d), e.g., one requires that R is isometric: d(R(x), R(y)) = d(x, y), x, y ∈ Γ. Examples include translations of lattices such as Zν and reflection symmetries satisfying R2 = id. If we assume that HR(x)≅Hx⁠, R can be lifted to an automorphism of AΓ as follows. Denote by ix:B(Hx)→A{x} the natural isomorphism (or a well-chosen one) and define the automorphism βR of AΓ⁠, by putting

(7.10)

The symmetry of the interaction is expressed by the property Φ(R(X)) = βR(Φ(X)). In the case of lattice translations, this yields a representation of (Zm,+) on AΓ⁠, i.e., for a∈Zm,R(x)=x+a denotes the action of translations on Γ, and X + a = {x + a∣x ∈ X}. Correspondingly, Φ is called translation invariant if βa(Φ(X)) = Φ(X + a), for all a∈Zm⁠.

  • (iii)

    Time-reversal symmetry is expressed as a local symmetry [discussed in (i)] given by an antiautomorphism, implemented on each site by an antiunitary transformation. The latter are, in general, the composition of a unitary transformation and a complex conjugation. Besides taking into account the antilinearity, time reversal symmetry can be treated in the same way as linear local symmetries.

  • (iv)

    Chiral symmetry is described by a unitary, say C, that anticommutes with the Hamiltonian. So, at each point in the curve of Hamiltonians, we have C*H(s)C = −H(s). For the dynamics, this implies that for all s ∈ [0, 1], t∈R⁠, and A∈A⁠,

(7.11)

It should be noted that the basic types of symmetries can be combined. For example, some models are invariant under a combined lattice reflection and time-reversal transformation, without possessing either of these symmetries separately.

We assume the same setup as described in the beginning of Sec. VII of a fixed ν-regular metric space (Γ, d) with a specified weighted F-function F. Let G denote the symmetries under consideration. The fixed family of on-site Hamiltonians Hx, x ∈ Γ is assumed to have the symmetry G as if it were a zero-range interaction. For example, if Ux∈B(Hx) describes a local unitary symmetry, we assume that the domain of Hx is invariant under Ux and that Hx and Ux commute. Or, as another example, if Γ=Zν and the symmetry is the full translation invariance of the lattice, Hx is assumed to be the same self-adjoint operator at each site x.

For the interactions, let BFG⊂BF denote the space of interactions with finite F-norm that possess the symmetry G, and

(7.12)

Definitions 7.1 and 7.2 of uniformly gapped curves and the equivalence relation now carry over the situation with a symmetry G in the obvious way, as does the proof of the analog of Proposition 7.3. The resulting equivalence classes are called symmetry protected phases. Since the uniformly gapped curves with symmetry are a special case of the general situation, Theorem 7.4 applies and the spectral flow automorphism establishes a bijection between the sets of states S(s) along the curve.

For the study of the stability of gapped ground state phases with symmetry breaking we present in Paper II,96 it will be important that the automorphisms αt,s commute with the automorphisms βg, g ∈ G, representing the symmetry on AΓ⁠. Moreover, it will be desirable that the interaction Ψ(s) generating αt,s and its finite-system analogs all have the symmetry. There are a few subtleties that merit further discussion concerning the construction of a spectral flow with the desired symmetry properties.

As mentioned above, it is important that both the spectral flow αt,s and its generating interaction Ψ(s) respect the symmetry of the initial interaction Φ(s). Recall that the conditional expectations ΠX from (4.11) play a crucial role in the quasilocality properties of αt,s and the definition of Ψ(s) [see Corollary 6.15, (6.118) and Sec. IV B]. In the presence of a local symmetry β, it is useful to choose the locally normal product state in the definition of the conditional expectations ΠX that is β-invariant, meaning ρx(A)=ρx(β(A)),A∈A{x} or, equivalently, β(ρx) = ρx [see (4.8)]. This requirement guarantees that if A is invariant under β, then so is ΠX(A), i.e.,

(7.13)

If dimHx<∞⁠, then a β-invariant locally normal product state always exists. For example, setting ρx to be the tracial state will produce a β-invariant state. Given any Φ(s) with a local symmetry and any symmetric ρ, it is easy to see using (7.13) that the Hastings interactions Ψn defined in (6.117) and, consequently, the spectral flow αs(n) derived from Φ(s) both inherit this symmetry. The same holds true for the corresponding infinite volume objects, Ψ and αs. In particular, αs commutes with any local symmetry automorphism β that leaves Φ′(s) invariant for all s ∈ [0, 1].

For infinite-dimensional Hx⁠, a symmetric normal state on Hx may or may not exist. One may have to relax either the normality or the symmetry requirement. Which of the two is more relevant would depend on the situation at hand but for the type of applications we are considering here, it is important to use normal states. In the case of a gauge symmetry described by a compact Lie group, constructing symmetric normal states is not a problem. However, even when such a state does not exist, the symmetry of the spectral flow is restored in the thermodynamic limit. This follows from the observation that although the infinite-volume interaction Ψ(s) depends on the choice of the locally normal state ρ used in its construction, the infinite-volume flow is the thermodynamic limit of automorphisms generated by self-adjoint operators that commute with the symmetry. This is apparent, e.g., from expression (5.71) in which HΛΦ is to be replaced by HΛ′(s) and K is Gs defined in (6.50).

In the presence of a lattice symmetry such as translation invariance, it makes sense to pick a translation invariant product state ρ to define the conditional expectations ΠX. This is obviously always possible and yields a covariant family of conditional expectations, meaning that βa ◦ ΠX = ΠX+a ◦ βa, where for a∈Zm⁠, X + a denotes the action of the translations on Γ and βa denotes the corresponding action on AΓ⁠. Finite subsystems defined on quotient lattices, e.g., Zn/(⁠LZm⁠) with n ≥ m, can have the corresponding quotient symmetry (ZmmodL)⁠, which is equivalent to considering the system with periodic boundary conditions. In general, finite systems will not have an exact translation symmetry but, again, the symmetry is recovered in the thermodynamic limit.

The case of time-reversal symmetry can be treated in the same way as local unitary symmetries. Due to the oddness of the function Wγ in (6.35), the Hastings interaction Ψ(s) changes sign under time-reversal. Since the time-reversal automorphism is antilinear, however, this is exactly the requirement for αs(n) and αs to commute with it.

The case of a fixed chiral symmetry C along the curve of Hamiltonians H(s) = H + Φ(s) implies that Φ′(s) anticommute with C. Using again the oddness of the function Wγ, and the property (7.11), it is straightforward to check that C then commutes with the generator of the spectral flow, i.e., it is a symmetry of the spectral flow.

Theorem 7.7.
Let {βg∣g ∈ G} be the automorphisms on AΓ representing symmetries of the system of the type described in (i–iv) above. Then, for any uniformly gapped curve of interactions Φ∈BF1,G([0,1]), there exists a strongly continuous cocycle of spectral flow automorphisms αt,s, s, t ∈ [0, 1] such that
(7.14)
and
βg◦αt,s=αt,s◦βg, for all s,t∈[0,1],g∈G.

The list of types of symmetries we have discussed here is not exhaustive. For example, another type of symmetry relevant for applications is duality symmetries. We postpone the discussion of those to Paper II,96 where we will study the stability of gapped ground state phases.

The construction of the automorphisms αt,s assumes the existence of a uniform lower bound for the spectral gap above the ground states along the curve of models in the sense of Definition 7.1. Establishing a uniform bound for the gap is generally a very hard problem. Fortunately, there are a good number of interesting examples where the existence of a positive uniform lower bound can be proved.

The largest variety of examples is found as a result of perturbing models for which the ground state and the existence of a spectral gap above it are known. We will review the state of the art of perturbative results of this type in Paper II.96 For this reason, we limit ourselves here to citing a few works that illustrate the broad range of examples that exist in the literature: some exactly solvable models such as the anisotropic XY chain,74 quantum perturbations of classical spin models,69,80 perturbations of the AKLT chain2,128 and similar models,121 perturbations of simple models with topological order in the ground state such as the Toric code model,21 general perturbations of frustrationfree models satisfying a local topological order condition,83 and perturbations of quasifree fermion systems.37 

Other interesting examples for which explicit lower bounds for the gap can be obtained and classes of models for which the equivalence classes can be explicitly determined are the frustrationfree spin chains with finitely correlated ground states, also known as matrix product states.13,15,40,87,99,100,101 Allowing for general perturbations of such models typically leads to splitting of the degenerate ground states found in the frustrationfree model. The so-called Kennedy triplet of “excited’”states of the spin-1 Heisenberg antiferromagnetic chain of even length can be regarded as an example of this phenomenon.68 In general, sufficiently small perturbations of one-dimensional frustrationfree models with a gap above the ground state will have a group of eigenvalues near the bottom of the spectrum separated by a gap (uniform in the size of the system) from the rest of the spectrum. The associated eigenstates all converge to ground states in the thermodynamic limit. Both statements are proved in Ref. 85.

We postpone a more comprehensive discussion of examples of models with distinct gapped ground state phases until after the presentation of the stability results of gapped phases in Paper II.96 

We would like to thank Valentin Zagrebnov for illuminating discussions about the nonautonomous Cauchy problems that arise in quantum dynamical systems. We also thank Martin Gebert for reading an early version of this paper and asking good questions and Derek Robinson for several useful remarks and informative comments about the history of the subject. All three authors wish to thank the Department of Mathematics of the University of Arizona and the University of California, Davis, for extending their kind hospitality to us and for the stimulating atmosphere they offered during several visits back and forth over the years it took to complete this project. B.N. also acknowledges the support of a CRM-Simons Professorship for a stay at the Centre de Recherches Mathématiques (Montréal) during Fall 2018, which created the perfect circumstances to complete this paper. B.N. was supported by the National Science Foundation under Grant Nos. DMS-1515850 and DMS-1813149.

This section collects a number of facts about the decay bounds used throughout this paper. In general, we will assume that Γ is a countable set equipped with a metric, and we denote this metric by d. A good example to keep in mind is Γ=Zν with the ℓ1-metric. When necessary, we will also assume Γ is ν-regular [see (A13)].

When considering the Heisenberg dynamics associated with a Hamiltonian, our quasilocality estimates require a short-range assumption on the corresponding interaction. For general sets Γ, which need not have the structure of a lattice, a sufficient condition for the existence of a dynamics in the thermodynamic limit can be expressed in terms of a norm on the interaction. We have found it convenient to express the decay of interactions with distance by a so-called F-function, which we discuss below. Depending on the application one has in mind, more explicit forms of decay, again expressed in terms of a family of F-functions, is convenient. These are by no means the only ways to express decay assumptions for interaction. If generality is not the concern, one can easily re-express decay into a more suitable form for the case at hand, say, e.g., for systems with pair interactions only. In this appendix, our goal is to briefly summarize various notions of decay which occur frequently in the main text.

1. On F-functions

Let (Γ, d) be a countable metric space. When Γ is finite, most notions introduced below are trivial, and for that reason, we will mainly consider the situation where Γ has infinite cardinality. We will say that Γ is equipped with an F-function if there is a nonincreasing function F:[0, ∞) → (0, ∞) for which

  • F is uniformly integrable:

(A1)
  • (ii)

    F satisfies a convolution condition:

(A2)

Any function F satisfying (A1) and (A2) will be called an F-function on Γ. We note that an immediate consequence of (A2) is that for any pair x, y ∈ Γ, we have the bound

(A3)

The constant CF enters into a number of our estimates. We say that an F-function on Γ is normalized if CF = 1. Of course, for any F-function F, the function F̃=CF−1F defines a new F-function on Γ for which CF̃=1⁠.

Note that if Γ is equipped with an F-function F, then

(A4)

where the left-hand-side above is a uniform estimate on the cardinality of the ball of radius n centered at x ∈ Γ. The above follows immediately from the estimate

(A5)

This estimate also demonstrates that the existence of an F-function guarantees that Γ is uniformly, locally finite.

Moreover, if Γ is infinite, the existence of an F-function implies that the diameter of Γ is infinite. In this situation, if {Λn}n≥1 is an increasing, exhaustive sequence of finite subsets of Γ (i.e., Λn ⊂ Λn+1 for all n ≥ 1 and Λn↑Γ), then for any finite X ⊂ Γ,

(A6)

This follows by observing that for any m ≥ 1,

(A7)

is a finite subset of Γ. Since {Λn}n≥1 is absorbing, there is an N ≥ 1 for which X(m) ⊂ ΛN. Since Γ has infinite cardinality, the set Γ\ΛN is nonempty. It immediately follows that d(x, y) ≥ m for all x ∈ X and y ∈Γ\ΛN, from which (A6) follows.

a. Two common examples of F-functions

First, many well-studied quantum spin models are defined on the hypercubic lattice Γ=Zν for some integer ν ≥ 1. For concreteness, consider Zν equipped with the ℓ1-metric

(A8)

Other translation invariant metrics can be treated similarly. For any ϵ > 0, the function

(A9)

is an F-function on Γ=Zν⁠. Integrability follows from

(A10)

Moreover, for any metric space (Γ, d): if p ≥ 1, the bound

(A11)

holds for all x, y, z ∈ Γ, since the function t ↦ tp is (midpoint) convex. In this case, the function defined in (A9) satisfies (A2) with

(A12)

Next, we note that for many of our results, it is not necessary that Γ has the structure of a lattice. We will say that a metric space (Γ, d) is ν-regular if there exist ν > 0 and κ < ∞ for which

(A13)

Here, for any x ∈ Γ and n ≥ 0, bx(n) is the ball of radius n centered at x and | · | denotes cardinality. From (A4), we see that if Γ has an F-function for which F(r) ≤ Cr−ν, then Γ is ν-regular.

If (Γ, d) is ν-regular, then for any ϵ > 0, the function

(A14)

is an F-function on Γ.

To see that this is the case, we need only to check uniform integrability, i.e., (A1), as an argument using (A11) shows that this F satisfies (A2). Fix x ∈ Γ. Set Bx(1) = bx(1) and Bx(n) = bx(n)\bx(n − 1) for any n ≥ 2. It is then clear that (A13) implies

(A15)

uniformly in x; hence, (A1) holds. A computation similar to (A11) shows that CF < ∞.

We can combine the above discussion with (A5) to prove the following result:

Proposition A.1.

Let (Γ, d) be a countable metric space.

  • If (Γ, d) is ν-regular then F(r) = (1 + r)−(ν+1+ϵ) is an F function of (Γ, d) for all ϵ > 0.

  • If F(r) = (1 + r)−(ν+ϵ) is an F-function of (Γ, d) for all ϵ > 0, then Γ is ν-regular.

Proof.
Part (i) follows immediately from the discussion after (A14). For part (ii), suppose that F(r) = (1 + r)−(ν+ϵ) is an F-function of (Γ, d) for all ϵ > 0. Fix ϵ > 0. Then, by (A4), for any n ≥ 1 and x ∈ Γ,
(A16)
Taking the infimum over ϵ > 0 shows that (Γ, d) is ν-regular with κ = 2ν∥F∥.

2. On weighted F-functions

For certain applications, it is convenient to consider families of F-functions of a specific form, which we call weighted F-functions.

Let (Γ, d) be a metric space equipped with an F-function F as described in Subsection 1 of the  Appendix. Let g:[0, ∞) → [0, ∞) be a non-negative, nondecreasing, sub-additive function, i.e.,

(A17)

Corresponding to any such g, the function

(A18)

is an F-function on Γ. In fact, since g is non-negative, Fg satisfies (A1) with ∥Fg∥ ≤ ∥F∥. Moreover, since g is nondecreasing and sub-additive, one also has that

(A19)

Thus, (A2) holds with CFg≤CF⁠.

We may refer to F as the base F-function associated with Fg; note that F0 = F for g = 0. The function g induces a factor r ↦ e−g(r) which is often referred to as a weight. We may also loosely refer to g as a weight and similarly Fg as a weighted F-function. One readily checks that sums, non-negative scalar multiples, and compositions of weights are also weights; in the sense that if g1 and g2 are both non-negative, nondecreasing, sub-additive functions, then so too are g1 + g2, ag1 (for a ≥ 0), and g1 ◦ g2.

In certain applications, it is useful to introduce a one-parameter family of weighted F-functions by taking a base F-function F on Γ, fixing a weight g, and associating to any a ≥ 0, the function ga(r) = ag(r), for which Fga(r)=e−ag(r)F(r)⁠. When g is understood, we often just write Fa≔Fga to describe this family of weighted F-functions. For example, if F(r) = (1 + r)−p and g(r) = r, then Fa(r) = e−ar(1 + r)−p is the family of weighted F-functions defined in Sec. III A 1.

As a motivation for introducing these weights, consider again Γ=Zν⁠. As discussed in Subsection A 1 a of the  Appendix, the polynomially decaying function F in (A9) is an F-function associated with Γ=Zν⁠. Such an F-function is appropriate for interactions Φ with terms that decay polynomially with the diameter of their support: ∥Φ(X)∥ ≤ C(1 + diam(X))−ν+ϵ. However, we are typically interested in interactions whose terms decay much faster, in particular, exponentially fast. One readily checks that for any a > 0, the exponential function g(r) = e−ar fails to satisfy (A2) and as such is not an F-function on Γ=Zν⁠, but does satisfy the criterion to be a weight. Since exponential functions often govern the decay of our interactions, it is convenient that one can obtain an exponentially decaying F-function on Γ=Zν by making an appropriate choice of weight.

Before moving on to discussing several useful weights, we point out one added benefit of these functions. In the situation where we do not assume to have a weighted F-function and given X, Y∈P0(Γ)⁠, we will often use the simple bound

(A20)

when applying a Lieb-Robinson bound or a quasilocality bound [see (3.24) and (5.2)]. For weighted F-functions, however, the following is also frequently used:

(A21)

Here, one typically is considering a quantum lattice system defined on a (large) finite volume Λ, and in the situation that Y = Λ\X(n), then the RHS of (A21) decays as e−g(n).

a. Three common weights

With an eye toward our specific applications, we now introduce three particular classes of weights.

First, let μ ∈ [0, 1]. The function g:[0, ∞) → [0, ∞) given by g(r) = rμ is non-negative, nondecreasing, and sub-additive in the sense of (A17). The constant function g(r) = 1 corresponding to μ = 0 is of minor interest; however, the choice of μ = 1 generates exponentially decaying weights. When 0 < μ < 1, the function e−rμ is often called a stretched exponential.

Next, we provide an example between exponential and stretched exponential decay. As we will show, for any p > 0, the function

(A22)

is non-negative, nondecreasing, and sub-additive. In our applications of the spectral flow (see, e.g., Sec. VI), the choice of p = 2 is particularly relevant.

Note that at r = ep, the nonconstant part of g has a zero derivative, and for r > ep, g is strictly increasing. That motivates this particular choice of cutoff. Also, it is easy to see that this function is subadditive by taking cases: Let r, s ≥ 0. Consider (i) r + s ≤ ep and (ii) r + s > ep. Both cases are easy to see. For the second case, use

(A23)

Note that

(A24)

the latter fact using that r + s > ep.

Finally, the function g:[0, ∞) → [0, ∞) given by

(A25)

is clearly non-negative and nondecreasing. Since for any r, s ≥ 0, we have that

(A26)

and subadditivity of g readily follows. Starting with a base F-function, as in (A9) or (A14), a proper scaling of F by this weight allows for arbitrary power-law decay.

3. Simple transformations of F-functions

In certain applications, it is convenient to know that various decaying functions are in fact F-functions. Quantities of interest can be estimated in terms of translations or rescalings of known F-functions. For ν-regular Γ, these modifications preserve the basic properties of an F-function. The following two propositions show that suitably defined truncations, shifts, and dilations of F-functions are again F-functions.

Proposition A.2.
Let (Γ, d) be a ν-regular metric space with an F-function F. For any a ≥ 0 and any choice of c ≥ F(a), the function F̃:[0,∞)→(0,∞) defined by setting
(A27)
is an F-function on (Γ, d). In fact,
(A28)

Proof.
Fix x ∈ Γ. Note that
(A29)
and the first bound in (A28) follows from ν-regularity.
To see the second bound, note that
(A30)
for all sites x, y ∈ Γ. By considering the cases d(x, y) ≥ a and d(x, y) < a separately, one can show
cF(d(x,y))≤max{c, F(0)}F̃(d(x,y)),
from which the second bound in (A28) follows.

Proposition A.3.
Let (Γ, d) be a ν-regular metric space, and p ≥ 1 be such that
(A31)
is an F-function on (Γ, d).

  • If F(r) = e−g(r)F0(r) is a weighted F-function on (Γ, d), then for any ϵ > 0 the function defined by F̃(r)=F(ϵr) is an F-function on (Γ, d). Moreover,

(A32)
  • (ii)
    If F(r) = e−g(r)F0(r) is a weighted F-function on (Γ, d), then for any a > 0 the function defined by
    (A33)
    is an F-function on (Γ, d). In fact,
(A34)

Proof.
To see (i), first note that r ↦ g(ϵr) is non-negative, nondecreasing, and subadditive. In this case, we need only to verify that r ↦ (1 + ϵr)−p is an F-function. For any ϵ > 0, one has that
(A35)
holds for all r ≥ 0. The first bound in (A32) is then clear, and the second bound follows as in (A11).
To see (ii), a short calculation shows that
(A36)
The first inequality above is trivial since F is nonincreasing. The second follows from sub-additivity of g, namely, g(r) ≤ g(a) + g(r − a) for any r > a, as well as the fact that F0(r − a) ≤ F0(r)/F0(a). The bounds in (A34) readily follow.
A simple consequence of Proposition A.3 is the following. Under the assumptions of Proposition A.3, let F be a weighted F-function with base F0. For any r > 1, the bound
(A37)
is clear, and this bound extends to all r ≥ 0 using the function in Proposition A.3(ii). Thus, F(⌊r⌋) is an F-function as well.

4. Basic interaction bounds

In the main text, we frequently use a number of basic estimates concerning interactions that are expressed using F-functions. Here, we collect a few results for later reference.

We begin by recalling some of the basic notation associated with interactions. Let (Γ, d) be a metric space equipped with an F-function F as discussed in Subsection 1 of the  Appendix. Let P0(Γ) denote the set of finite subsets of Γ. We say that a mapping Φ is an interaction on Γ if Φ:P0(Γ)→AΓloc with the property that Φ(X)*=Φ(X)∈AX for every X∈P0(Γ)⁠. If Φ is an interaction on Γ, we write that Φ∈BF if and only if

(A38)

A basic consequence of (A38) is that for all x, y ∈ Γ,

(A39)
a. Estimates based on distance

We first provide a basic F-norm estimate based on summing interaction terms whose distance from a specific set is given. Recall that if X ⊂ Γ and n ≥ 0, the set X(n) ⊂ Γ is defined as

(A40)

Proposition A.4.
Let (Γ, d) be a metric space equipped with an F-function F. Let Φ be an interaction on Γ with Φ∈BF. For any X∈P0(Γ) and each R ≥ 0,
(A41)
Moreover,
(A42)

Proof.
Given the uniform integrability of F, i.e., (A1), the second estimate in (A41) is clear given the first. To see the first, note that by over-counting,
(A43)
The estimate in (A41) now follows from (A39) and (A1).
To prove (A42), note that again by over-counting,
(A44)
Thus, (A42) follows using the convolution condition on F.
A simple corollary of these bounds follows. To state, it requires that we introduce two notions. First, we describe compatible F-functions. Let (Γ, d) be a metric space equipped with two F-functions denoted by F1 and F2. We will say that F1 and F2 are compatible if there is a positive number C1,2 and a nonincreasing function F1,2:[0, ∞) → (0, ∞) for which given any x, y ∈ Γ,
(A45)
Next we briefly describe time-dependent interactions (see Sec. III A 1 for more details). Let I⊂R be an interval. We say that Φ:P0(Γ)×I→AΓloc is a strongly continuous interaction on Γ if for each X∈P0(Γ)⁠,

  • Φ(X,t)*=Φ(X,t)∈AX⁠, for all t ∈ I;

  • The map Φ(X,⋅):I→AX is continuous in the strong operator topology.

Moreover, we will write Φ∈BF(I) if Φ(⋅,t)∈BF⁠, for all t ∈ I, and ∥Φ(t)∥F, which we sometimes write as ∥Φ∥F(t), is a locally bounded function of t.

We now state a corollary of Proposition A.4.

Corollary A.5.
Let (Γ, d) be a metric space equipped with two compatible F-functions F1 and F2. For an interval I⊂R, suppose there is a cocycle of automorphisms {αt,s}t,s∈I of AΓ, which satisfy a Lieb-Robinson bound, i.e., for any disjoint X,Y∈P0(Γ), each A∈AX,B∈AY, and s, t ∈ I,
(A46)
with a prefactor Dα(t, s) that increases as |t − s| increases. Let Ψ be a time-dependent interaction with Ψ∈BF2(I)⁠. Given R ≥ 0 and s, t ∈ I with s ≤ t, one has that for any A∈AX with X∈P0(Γ)⁠,
(A47)
and
(A48)

Proof.
To prove (A47), note that for any s ≤ r ≤ t, the simple bound
(A49)
holds. Estimating as in (A41) yields (A47) as claimed.
To see (A48), take some s ≤ r ≤ t and note that (A46) applies. In fact, applying the Lieb-Robinson bound, we find
(A50)
Here, we use that, by our assumptions, Dα(t, r) ≤ Dα(t, s). Then, arguing as in the proof of (A41) [see, in particular, (A44)], we find
(A51)
Using compatibility, i.e., (A45), the bound in (A48) now follows upon integration.

b. Estimates based on diameter

In some situations, it is more convenient to form decay arguments based on the diameters of sets, rather than the distance between sets. For these cases, we will further assume that (Γ, d) is ν-regular so that (A13) holds.

Before we state our first result, let us introduce some convenient notation. Our estimates will often be in terms of moments of certain decay functions. To this end, let G : [0, ∞) → (0, ∞) be a decay function. For any p ≥ 0 and each m ≥ 0, set

(A52)

whenever the sum on the right-hand-side above is finite. We will refer to MpG(0) as the p-th moment of G. The notation

(A53)

will be used for iterated moments. A rough estimate involving exchanging the order of the summations shows

(A54)

We now state our first result, compare with Proposition A.4.

Proposition A.6.
Let (Γ, d) be a ν-regular metric space equipped with an F-function F, and let Φ be an interaction on Γ with Φ∈BF. Then, for any R ≥ 0,
(A55)
If, in addition, F has a finite 2ν-th moment, i.e., M2νF(0)<∞, then
(A56)

Proof.
For any fixed x ∈ Γ, note that
(A57)
Using this bound, (A55) follows from uniform integrability of F.
To see (A56), again fix x ∈ Γ and see that
(A58)
where, for the last line above, we used that F is nonincreasing and over-estimated using (A13). This proves (A56).
In some arguments, we encounter moments of interactions. More precisely, let Φ be an interaction on Γ. For any p ≥ 0, the mapping Φp:P0(Γ)→AΓloc given by
(A59)
also defines an interaction on Γ. We refer to Φp as the p-th moment of Φ. Lemma A.7 provides a basic estimate for interactions of this type.

Lemma A.7.
Let (Γ, d) be a ν-regular metric space equipped with an F-function F, and let Φ be an interaction on Γ with Φ∈BF. If p ≥ 0 and M(p+2)νF(0)<∞, then the p-th moment of Φ satisfies
(A60)

Proof.
Fix x, y ∈ Γ. Set m = ⌊d(x, y)⌋ and note that
(A61)
Now, if x ∈ X and diam(X) < n + 1, then (A13) guarantees that
(A62)
Moreover, by over-counting, it is clear that
(A63)
where * represents a non-negative quantity. We conclude that
(A64)
which proves (A60).

When considering a weighted F-function F(r) = e−g(r)F0(r), one can often use the weight with (A60) to prove that the pth moment of Φ has a finite F-norm. This allows us to apply the Lieb-Robinson bound theory to Φp.

Corollary A.8.
Let p ≥ 0 and let F(r) = e−g(r)F0(r) be a weighted F-function on a ν-regular metric space (Γ, d). If M(p+2)ν−ag(0)<∞ for some 0 < a ≤ 1, then Φp∈BF̃ for any Φ∈BF, where F̃ is the F-function,
(A65)

Proof.
This is an immediate consequence of (A60), and the bound
M(p+2)νF(k)=∑n=⌊k⌋∞(1+n)(p+2)νe−g(n)F0(n)≤e(a−1)g(⌊k⌋)F0(⌊k⌋)∑n=0∞(1+n)(p+2)νe−ag(n)=M(p+2)ν−ag(0)F̃(k),
for all k ≥ 0.

c. An estimate on weighted sums

Lemma A.9.
Let (Γ, d) be a ν-regular metric space equipped with an F-function F, and let Φ be an interaction on Γ with Φ∈BF. If G: [0, ∞) → (0, ∞) satisfies M2ν+1G(0)<∞, then for any x, y ∈ Γ,
(A66)

Proof.
For each X∈P0(Γ)⁠, the sets X(n) [see, e.g., (A40)] are increasing, and therefore, if x, y ∈ X(n) for some n ≥ 1, then x, y ∈ X(m) for all m ≥ n. With this in mind, we write
(A67)
where we have inserted the characteristic function χm+1 defined by
(A68)
Note that x, y ∈ X(m + 1) means there exist w⁠, z ∈ X such that w ∈ bx(m + 1) and z ∈ by(m + 1). Using this, the definition of M0G(m) [see (A52)], and the F-norm of Φ, the estimate
(A69)
readily follows.
We now divide the final series on the right-hand-side of (A69) into sums of large and small m. More precisely, for any fixed 0 < ϵ < 1, set m0 = m0(ϵ, x, y) to be the largest integer m ≥ −1 satisfying
(A70)
For any 0 ≤ m ≤ m0, w ∈ bx(m + 1), and z ∈ by(m + 1), one has ϵd(x, y) ≤ d(⁠w⁠, z) as
(A71)
In this case, the first few terms may be estimated as
(A72)
where in the last equality we have used (A54).
For the remaining terms, uniform integrability of F implies
(A73)
Using the definition of m0 and again applying (A54), the bound claimed in (A66) follows from choosing ϵ = 1/3.

1.
Abrikosov
,
A. A.
,
Methods of Quantum Field Theory in Statistical Physics
(
Dover
,
1975
).
Google Scholar
 
2.
Affleck
,
I.
,
Kennedy
,
T.
,
Lieb
,
E. H.
, and
Tasaki
,
H.
, “
Valence bond ground states in isotropic quantum antiferromagnets
,”
Commun. Math. Phys.
 
115
(
3
),
477
–
528
(
1988
).
https://doi.org/10.1007/bf01218021
Google Scholar
Crossref
Search ADS
 
3.
Altland
,
A.
 and
Simons
,
B. D.
,
Condensed Matter Field Theory
(
Cambridge University Press
,
2010
).
Google Scholar
Crossref
Search ADS
 
4.
Amour
,
L.
,
Levy-Bruhl
,
P.
, and
Nourrigat
,
J.
, “
Dynamics and Lieb-Robinson estimates for lattices of interacting anharmonic oscillators
,”
Colloq. Math.
 
118
,
609
–
648
(
2010
).
https://doi.org/10.4064/cm118-2-17
Google Scholar
Crossref
Search ADS
 
5.
Bachmann
,
S.
 and
Bluhm
,
A.
, “
Local factorisation of the dynamics of quantum spin systems
,”
J. Math. Phys.
 
58
,
071901
(
2017
).
https://doi.org/10.1063/1.4991949
Google Scholar
Crossref
Search ADS
 
6.
Bachmann
,
S.
,
Bols
,
A.
,
De Roeck
,
W.
, and
Fraas
,
M.
, “
Quantization of conductance in gapped interacting systems
,”
Ann. Henri Poincaré
 
19
(
3
),
695
–
708
(
2018
).
https://doi.org/10.1007/s00023-018-0651-0
Google Scholar
Crossref
Search ADS
 
7.
Bachmann
,
S.
,
De Roeck
,
W.
, and
Fraas
,
M.
, “
Adiabatic theorem for quantum spin systems
,”
Phys. Rev. Lett.
 
119
,
060201
(
2017
); e-print arXiv:1612.01505.
https://doi.org/10.1103/physrevlett.119.060201
Google Scholar
Crossref
Search ADS
PubMed
 
8.
Bachmann
,
S.
,
De Roeck
,
W.
, and
Fraas
,
M.
, “
The adiabatic theorem and linear response theory for extended quantum systems
,”
Commun. Math. Phys.
 
361
,
997
–
1027
(
2018
); e-print arXiv:1705.02838.
https://doi.org/10.1007/s00220-018-3117-9
Google Scholar
Crossref
Search ADS
 
9.
Bachmann
,
S.
,
De Roeck
,
W.
, and
Fraas
,
M.
, “
The adiabatic theorem in a quantum many-body setting
,” e-print arXiv:1808.09985 (
2018
).
Google Scholar
 
10.
Bachmann
,
S.
,
Dybalski
,
W.
, and
Naaijkens
,
P.
, “
Lieb–Robinson bounds, Arveson spectrum and Haag–Ruelle scattering theory for gapped quantum spin systems
,”
Ann. Henri Poincare
 
17
,
1737
–
1791
(
2016
).
https://doi.org/10.1007/s00023-015-0440-y
Google Scholar
Crossref
Search ADS
 
11.
Bachmann
,
S.
,
Hamza
,
E.
,
Nachtergaele
,
B.
, and
Young
,
A.
, “
Product vacua and boundary state models in d dimensions
,”
J. Stat. Phys.
 
160
,
636
–
658
(
2015
).
https://doi.org/10.1007/s10955-015-1260-7
Google Scholar
Crossref
Search ADS
 
12.
Bachmann
,
S.
,
Michalakis
,
S.
,
Nachtergaele
,
B.
, and
Sims
,
R.
, “
Automorphic equivalence within gapped phases of quantum lattice systems
,”
Commun. Math. Phys.
 
309
,
835
–
871
(
2012
).
https://doi.org/10.1007/s00220-011-1380-0
Google Scholar
Crossref
Search ADS
 
13.
Bachmann
,
S.
 and
Nachtergaele
,
B.
, “
Product vacua with boundary states
,”
Phys. Rev. B
 
86
(
3
),
035149
(
2012
).
https://doi.org/10.1103/physrevb.86.035149
Google Scholar
Crossref
Search ADS
 
14.
Bachmann
,
S.
 and
Nachtergaele
,
B.
, “
On gapped phases with a continuous symmetry and boundary operators
,”
J. Stat. Phys.
 
154
,
91
–
112
(
2014
).
https://doi.org/10.1007/s10955-013-0850-5
Google Scholar
Crossref
Search ADS
 
15.
Bachmann
,
S.
 and
Nachtergaele
,
B.
, “
Product vacua with boundary states and the classification of gapped phases
,”
Commun. Math. Phys.
 
329
,
509
–
544
(
2014
).
https://doi.org/10.1007/s00220-014-2025-x
Google Scholar
Crossref
Search ADS
 
16.
Bachmann
,
S.
 and
Ogata
,
Y.
, “
C1-classification of gapped parent Hamiltonians of quantum spin chains
,”
Commun. Math. Phys.
 
338
,
1011
–
1042
(
2015
).
https://doi.org/10.1007/s00220-015-2350-8
Google Scholar
Crossref
Search ADS
 
17.
Birman
,
M. S.
 and
Solomyak
,
M.
, “
Double operator integrals in a Hilbert space
,”
Integr. Equations Oper. Theory
 
47
,
131
–
168
(
2003
).
https://doi.org/10.1007/s00020-003-1157-8
Google Scholar
Crossref
Search ADS
 
18.
Bishop
,
M.
,
Nachtergaele
,
B.
, and
Young
,
A.
, “
Spectral gap and edge excitations of d-dimensional PVBS models on half-spaces
,”
J. Stat. Phys.
 
162
,
1485
–
1521
(
2016
).
https://doi.org/10.1007/s10955-016-1457-4
Google Scholar
Crossref
Search ADS
 
19.
Brandao
,
F. G. S. L.
,
Cubitt
,
T. S.
,
Lucia
,
A.
,
Michalakis
,
S.
, and
Perez-Garcia
,
D.
, “
Area law for fixed points of rapidly mixing dissipative quantum systems
,”
J. Math. Phys.
 
56
,
102202
(
2015
).
https://doi.org/10.1063/1.4932612
Google Scholar
Crossref
Search ADS
 
20.
Bratteli
,
O.
 and
Robinson
,
D. W.
,
Operator Algebras and Quantum Statistical Mechanics
, 2nd ed. (
Springer Verlag
,
1997
), Vol. 2.
Google Scholar
Crossref
Search ADS
 
21.
Bravyi
,
S.
,
Hastings
,
M.
, and
Michalakis
,
S.
, “
Topological quantum order: Stability under local perturbations
,”
J. Math. Phys.
 
51
,
093512
(
2010
).
https://doi.org/10.1063/1.3490195
Google Scholar
Crossref
Search ADS
 
22.
Bravyi
,
S.
 and
Hastings
,
M. B.
, “
A short proof of stability of topological order under local perturbations
,”
Commun. Math. Phys.
 
307
,
609
(
2011
).
https://doi.org/10.1007/s00220-011-1346-2
Google Scholar
Crossref
Search ADS
 
23.
Bravyi
,
S.
,
Hastings
,
M. B.
, and
Verstraete
,
F.
, “
Lieb-Robinson bounds and the generation of correlations and topological quantum order
,”
Phys. Rev. Lett.
 
97
,
050401
(
2006
).
https://doi.org/10.1103/physrevlett.97.050401
Google Scholar
Crossref
Search ADS
PubMed
 
24.
Bru
,
J.-B.
 and
de Siquiera Pedra
,
W.
,
Lieb-Robinson Bounds for Multi-Commutators and Applications to Response Theory
, Springer Briefs in Mathematical Physics Vol. 13 (
Springer Nature
,
2017
).
Google Scholar
Crossref
Search ADS
 
25.
Buchholz
,
D.
, “
The resolvent algebra for oscillating lattice systems: Dynamics, ground and equilibrium states
,”
Commun. Math. Phys.
 
353
,
691
–
716
(
2017
); e-print arXiv:1605.05259.
https://doi.org/10.1007/s00220-017-2869-y
Google Scholar
Crossref
Search ADS
 
26.
Buerschaper
,
O.
, “
Dynamics of correlations and quantum phase transitions in bosonic lattice systems
,” M.Sc. thesis,
Ludwig-Maximilians University
,
Munich
,
2007
.
Google Scholar
 
27.
Burrell
,
C. K.
 and
Osborne
,
T. J.
, “
Bounds on the information propagation in disordered quantum spin chains
,”
Phys. Rev. Lett.
 
99
,
167201
(
2007
).
https://doi.org/10.1103/physrevlett.99.167201
Google Scholar
Crossref
Search ADS
PubMed
 
28.
Buttá
,
P.
,
Caglioti
,
E.
,
Di Ruzza
,
S.
, and
Marchioro
,
C.
, “
On the propagation of a perturbation in an anharmonic system
,”
J. Stat. Phys.
 
127
,
313
–
325
(
2007
).
https://doi.org/10.1007/s10955-007-9278-0
Google Scholar
Crossref
Search ADS
 
29.
Cha
,
M.
,
Naaijkens
,
P.
, and
Nachtergaele
,
B.
, “
The complete set of infinite volume ground states for Kitaev’s abelian quantum double models
,”
Commun. Math. Phys.
 
357
,
125
–
157
(
2018
).
https://doi.org/10.1007/s00220-017-2989-4
Google Scholar
Crossref
Search ADS
 
30.
Cha
,
M.
,
Naaijkens
,
P.
, and
Nachtergaele
,
B.
, “
On the stability of charges in infinite quantum spin systems
,” e-print arXiv:1804.03203 (
2018
).
Google Scholar
 
31.
Chen
,
X.
,
Gu
,
Z.-C.
,
Liu
,
Z.-X.
, and
Wen
,
X.-G.
, “
Symmetry protected topological orders and the group cohomology of their symmetry group
,”
Phys. Rev. B
 
87
,
155114
(
2013
).
https://doi.org/10.1103/physrevb.87.155114
Google Scholar
Crossref
Search ADS
 
32.
Chen
,
X.
,
Gu
,
Z.-C.
, and
Wen
,
X.-G.
, “
Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order
,”
Phys. Rev. B
 
82
(
15
),
155138
(
2010
).
https://doi.org/10.1103/physrevb.82.155138
Google Scholar
Crossref
Search ADS
 
33.
Chen
,
X.
,
Gu
,
Z.-C.
, and
Wen
,
X.-G.
, “
Classification of gapped symmetric phases in one-dimensional spin systems
,”
Phys. Rev. B
 
83
(
3
),
035107
(
2011
).
https://doi.org/10.1103/physrevb.83.035107
Google Scholar
Crossref
Search ADS
 
34.
Cramer
,
M.
,
Serafini
,
A.
, and
Eisert
,
J.
, “
Locality of dynamics in general harmonic quantum systems
,” in
Quantum Information and Many Body Quantum Systems
, CRM Series, No. 8, edited by
Ericsson
,
M.
 and
Montangero
,
S.
 (
Edizioni della Normale
,
Pisa
,
2008
), pp.
51
–
72
.
Google Scholar
 
35.
Damanik
,
D.
,
Lemm
,
M.
,
Lukic
,
M.
, and
Yessen
,
W.
, “
New anomalous Lieb-Robinson bounds in quasiperiodic XY chains
,”
Phys. Rev. Lett.
 
113
,
127202
(
2014
).
https://doi.org/10.1103/physrevlett.113.127202
Google Scholar
Crossref
Search ADS
PubMed
 
36.
Damanik
,
D.
,
Lukic
,
M.
, and
Yessen
,
W.
, “
Quantum dynamics of periodic and limit-periodic Jacobi and block Jacobi matrices with applications to some quantum many body problems
,”
Commun. Math. Phys.
 
337
,
1535
–
1561
(
2015
).
https://doi.org/10.1007/s00220-015-2346-4
Google Scholar
Crossref
Search ADS
 
37.
De Roeck
,
W.
 and
Salmhofer
,
M.
, “
Persistence of exponential decay and spectral gaps for interacting fermions
,”
Commun. Math. Phys.
 
365
,
773
–
796
(
2018
).
https://doi.org/10.1007/s00220-018-3211-z
Google Scholar
Crossref
Search ADS
 
38.
De Roeck
,
W.
 and
Schütz
,
M.
, “
Local perturbations perturb exponentially–locally
,”
J. Math. Phys.
 
56
,
061901
(
2015
).
https://doi.org/10.1063/1.4922507
Google Scholar
Crossref
Search ADS
 
39.
Duivenvoorden
,
K.
 and
Quella
,
T.
, “
From symmetry-protected topological order to Landau order
,”
Phys. Rev. B
 
88
,
125115
(
2013
).
https://doi.org/10.1103/physrevb.88.125115
Google Scholar
Crossref
Search ADS
 
40.
Fannes
,
M.
,
Nachtergaele
,
B.
, and
Werner
,
R. F.
, “
Finitely correlated states of quantum spin chains
,”
Commun. Math. Phys.
 
144
,
443
–
490
(
1992
).
https://doi.org/10.1007/bf02099178
Google Scholar
Crossref
Search ADS
 
41.
Folland
,
G. B.
,
Real Analysis: Modern Techniques and Their Applications
(
John Wiley & Sons
,
1999
).
Google Scholar
 
42.
Fradkin
,
E.
,
Field Theories of Condensed Matter Physics
, 2nd ed. (
Cambridge University Press
,
2013
).
Google Scholar
Crossref
Search ADS
 
43.
Fredenhagen
,
K.
, “
A remark on the cluster theorem
,”
Commun. Math. Phys.
 
97
,
461
–
463
(
1985
).
https://doi.org/10.1007/bf01213409
Google Scholar
Crossref
Search ADS
 
44.
Giuliani
,
A.
,
Mastropietro
,
V.
, and
Porta
,
M.
, “
Universality of the Hall conductivity in interacting electron systems
,”
Commun. Math. Phys.
 
349
,
1107
–
1161
(
2017
).
https://doi.org/10.1007/s00220-016-2714-8
Google Scholar
Crossref
Search ADS
 
45.
Gong
,
Z.-K.
,
Foss-Feig
,
M.
,
Michalakis
,
S.
, and
Gorshkov
,
A. V.
, “
Persistence of locality in systems with power-law interactions
,”
Phys. Rev. Lett.
 
113
,
030602
(
2014
).
https://doi.org/10.1103/physrevlett.113.030602
Google Scholar
Crossref
Search ADS
PubMed
 
46.
Gong
,
Z.-X.
,
Foss-Feig
,
M.
,
Brandão
,
F. G. S. L.
, and
Gorshkov
,
A. V.
, “
Entanglement area laws for long-range interacting systems
,”
Phys. Rev. Lett.
 
119
,
050501
(
2017
).
https://doi.org/10.1103/physrevlett.119.050501
Google Scholar
Crossref
Search ADS
PubMed
 
47.
Grundling
,
H.
 and
Rudolph
,
G.
, “
Dynamics for QCD on an infinite lattice
,”
Commun. Math. Phys.
 
349
,
1163
–
1202
(
2017
).
https://doi.org/10.1007/s00220-016-2733-5
Google Scholar
Crossref
Search ADS
 
48.
Haag
,
R.
,
Local Quantum Physics
, 2nd ed., Texts and Monographs in Physics (
Springer-Verlag
,
Berlin
,
1996
).
Google Scholar
Crossref
Search ADS
 
49.
Haah
,
J.
,
Hastings
,
M. B.
,
Kothari
,
R.
, and
Low
,
G. H.
, “
Quantum algorithm for simulating real time evolution of lattice Hamiltonians
,” in
2018 IEEE 59th Annual Symposium on Foundations of Computer Science
(
IEEE
,
2018
), pp.
350
–
360
.
Google Scholar
 
50.
Haegeman
,
J.
,
Michalakis
,
S.
,
Nachtergaele
,
B.
,
Osborne
,
T. J.
,
Schuch
,
N.
, and
Verstraete
,
F.
, “
Elementary excitations in gapped quantum spin systems
,”
Phys. Rev. Lett.
 
111
,
080401
(
2013
).
https://doi.org/10.1103/physrevlett.111.080401
Google Scholar
Crossref
Search ADS
PubMed
 
51.
Hainzl
,
C.
 and
Seiringer
,
R.
, “
The Bardeen–Cooper–Schrieffer functional of superconductivity and its mathematical properties
,”
J. Math. Phys.
 
57
,
021101
(
2016
).
https://doi.org/10.1063/1.4941723
Google Scholar
Crossref
Search ADS
 
52.
Hamza
,
E.
,
Sims
,
R.
, and
Stolz
,
G.
, “
Dynamical localization in disordered quantum spin systems
,”
Commun. Math. Phys.
 
315
,
215
–
239
(
2012
).
https://doi.org/10.1007/s00220-012-1544-6
Google Scholar
Crossref
Search ADS
 
53.
Han
,
X.
 and
Hartnoll
,
S. A.
, “
Locality bound for dissipative quantum transport
,”
Phys. Rev. Lett.
 
121
,
170601
(
2018
).
https://doi.org/10.1103/physrevlett.121.170601
Google Scholar
Crossref
Search ADS
PubMed
 
54.
Hastings
,
M. B.
, “
Lieb-Schultz-Mattis in higher dimensions
,”
Phys. Rev. B
 
69
,
104431
(
2004
).
https://doi.org/10.1103/physrevb.69.104431
Google Scholar
Crossref
Search ADS
 
55.
Hastings
,
M. B.
, “
An area law for one dimensional quantum systems
,”
J. Stat. Mech.: Theory Exp.
 
2007
,
P08024
.
https://doi.org/10.1088/1742-5468/2007/08/p08024
Crossref
Search ADS
 
56.
Hastings
,
M. B.
, “
The stability of free Fermi Hamiltonians
,”
J. Math. Phys.
 
60
,
042201
(
2019
).
https://doi.org/10.1063/1.5053869
Google Scholar
Crossref
Search ADS
 
57.
Hastings
,
M. B.
 and
Koma
,
T.
, “
Spectral gap and exponential decay of correlations
,”
Commun. Math. Phys.
 
265
,
781
–
804
(
2006
).
https://doi.org/10.1007/s00220-006-0030-4
Google Scholar
Crossref
Search ADS
 
58.
Hastings
,
M. B.
 and
Michalakis
,
S.
, “
Quantization of Hall conductance for interacting electrons on a torus
,”
Commun. Math. Phys.
 
334
,
433
–
471
(
2015
).
https://doi.org/10.1007/s00220-014-2167-x
Google Scholar
Crossref
Search ADS
 
59.
Hastings
,
M. B.
 and
Wen
,
X. G.
, “
Quasi-adiabatic continuation of quantum states: The stability of topological ground-state degeneracy and emergent gauge invariance
,”
Phys. Rev. B
 
72
,
045141
(
2005
).
https://doi.org/10.1103/physrevb.72.045141
Google Scholar
Crossref
Search ADS
 
60.
Ingham
,
A. E.
, “
A note on Fourier transforms
,”
J. London Math. Soc.
 
s1-9
,
29
–
32
(
1934
).
https://doi.org/10.1112/jlms/s1-9.1.29
Google Scholar
Crossref
Search ADS
 
61.
Islambekov
,
U.
,
Sims
,
R.
, and
Teschl
,
G.
, “
Lieb-Robinson bounds for the Toda lattice
,”
J. Stat. Phys.
 
148
,
440
–
479
(
2012
).
https://doi.org/10.1007/s10955-012-0554-2
Google Scholar
Crossref
Search ADS
 
62.
Itzykson
,
C.
 and
Drouffe
,
J.-M.
,
Statistical Field Theory
(
Cambridge University Press
,
1989
), Vols. 1 and 2.
Google Scholar
 
63.
Kadison
,
V. R.
 and
Ringrose
,
J. R.
,
Fundamentals of the Theory of Operator Algebras. Volume II: Advanced Theory
(
American Mathematical Society
,
1997
).
Google Scholar
Crossref
Search ADS
 
64.
Kanda
,
T.
 and
Matsui
,
T.
, “
Regular KMS states of weakly coupled anharmonic crystals and the resolvent CCR algebra
,” in
Analysis and Operator Theory
, edited by
Rassias
,
T.
 and
Zagrebnov
,
V.
 (Springer Optimization and Its Applications, 2016), Vol. 146.
65.
Kato
,
T.
, “
Integration of the equation of evolution in a Banach space
,”
J. Math. Soc. Jpn.
 
5
,
208
–
234
(
1953
).
https://doi.org/10.2969/jmsj/00520208
Google Scholar
Crossref
Search ADS
 
66.
Kato
,
T.
, “
On the linear differential equations in Banach spaces
,”
Commun. Pure Appl. Math.
 
9
,
479
–
486
(
1956
).
https://doi.org/10.1002/cpa.3160090319
Google Scholar
Crossref
Search ADS
 
67.
Kato
,
T.
,
Perturbation Theory for Linear Operators
(
Springer-Verlag
,
1995
).
Google Scholar
Crossref
Search ADS
 
68.
Kennedy
,
T.
, “
Exact diagonalisations of open spin-1 chains
,”
J. Phys.: Condens. Matter
 
2
(
26
),
5737
(
1999
).
https://doi.org/10.1088/0953-8984/2/26/010
Google Scholar
Crossref
Search ADS
 
69.
Kennedy
,
T.
 and
Tasaki
,
H.
, “
Hidden symmetry breaking and the Haldane phase in S = 1 quantum spin chains
,”
Commun. Math. Phys.
 
147
,
431
–
484
(
1992
).
https://doi.org/10.1007/bf02097239
Google Scholar
Crossref
Search ADS
 
70.
Kisyński
,
J.
, “
Sur les opérateurs de Green des problèmes de Cauchy abstraits
,”
Stud. Math.
 
XXIII
,
285
–
328
(
1964
).
https://doi.org/10.4064/sm-23-3-285-328
Google Scholar
Crossref
Search ADS
 
71.
Kitaev
,
A.
, “
Anyons in an exactly solved model and beyond
,”
Ann. Phys.
 
321
,
2
–
111
(
2006
).
https://doi.org/10.1016/j.aop.2005.10.005
Google Scholar
Crossref
Search ADS
 
72.
Klein
,
A.
 and
Perez
,
J. F.
, “
Localization in the ground state of a disordered array of quantum rotators
,”
Commun. Math. Phys.
 
147
,
241
–
252
(
1992
).
https://doi.org/10.1007/bf02096586
Google Scholar
Crossref
Search ADS
 
73.
Landau
,
L. J.
,
Perez
,
J. F.
, and
Wreszinski
,
W. F.
, “
Energy gap, clustering, and the Goldstone theorem in statistical mechanics
,”
J. Stat. Phys.
 
26
(
4
),
755
–
766
(
1981
).
https://doi.org/10.1007/bf01010937
Google Scholar
Crossref
Search ADS
 
74.
Lieb
,
E.
,
Schultz
,
T.
, and
Mattis
,
D.
, “
Two soluble models of an antiferromagnetic chain
,”
Ann. Phys.
 
16
,
407
–
466
(
1961
).
https://doi.org/10.1016/0003-4916(61)90115-4
Google Scholar
Crossref
Search ADS
 
75.
Lieb
,
E. H.
 and
Robinson
,
D. W.
, “
The finite group velocity of quantum spin systems
,”
Commun. Math. Phys.
 
28
,
251
–
257
(
1972
).
https://doi.org/10.1007/bf01645779
Google Scholar
Crossref
Search ADS
 
76.
Lucia
,
A.
,
Cubitt
,
T. S.
,
Michalakis
,
S.
, and
Pérez-García
,
D.
, “
Rapid mixing and stability of quantum dissipative systems
,”
Phys. Rev. A
 
91
,
040302
(
2015
).
https://doi.org/10.1103/physreva.91.040302
Google Scholar
Crossref
Search ADS
 
77.
Ma
,
M.
,
Halperin
,
B.
, and
Lee
,
P.
, “
Strongly disordered superfluids; quantum fluctuations and critical behavior
,”
Phys. Rev. B
 
34
,
3136
–
3143
(
1986
).
https://doi.org/10.1103/physrevb.34.3136
Google Scholar
Crossref
Search ADS
 
78.
Mariën
,
M.
,
Audenaert
,
K. M. R.
,
Van Acoleyen
,
K.
, and
Verstraete
,
F.
, “
Entanglement rates and the stability of the area law for the entanglement entropy
,”
Commun. Math. Phys.
 
346
,
35
–
73
(
2016
).
https://doi.org/10.1007/s00220-016-2709-5
Google Scholar
Crossref
Search ADS
 
79.
Martin
,
Ph. A.
 and
Rothen
,
F.
,
Many-Body Problems and Quantum Field Theory
(
Springer-Verlag
,
2002
).
Google Scholar
Crossref
Search ADS
 
80.
Matsui
,
T.
, “
Uniqueness of the translationally invariant ground state in quantum spin systems
,”
Commun. Math. Phys.
 
126
,
453
–
467
(
1990
).
https://doi.org/10.1007/bf02125695
Google Scholar
Crossref
Search ADS
 
81.
Matsui
,
T.
, “
On spectral gap and split property in quantum spin chains
,”
J. Math. Phys.
 
51
,
015216
(
2010
).
https://doi.org/10.1063/1.3285046
Google Scholar
Crossref
Search ADS
 
82.
Matsui
,
T.
, “
Boundedness of entanglement entropy, and split property of quantum spin chains
,”
Rev. Math. Phys.
 
25
,
1350017
(
2013
).
https://doi.org/10.1142/s0129055x13500177
Google Scholar
Crossref
Search ADS
 
83.
Michalakis
,
S.
 and
Zwolak
,
J. P.
, “
Stability of frustration-free Hamiltonians
,”
Commun. Math. Phys.
 
322
,
277
–
302
(
2013
).
https://doi.org/10.1007/s00220-013-1762-6
Google Scholar
Crossref
Search ADS
 
84.
Monaco
,
D.
 and
Teufel
,
S.
, “
Adiabatic currents for interacting electrons on a lattice
,”
Rev. Math. Phys.
 
31
,
1950009
(
2019
).
https://doi.org/10.1142/s0129055x19500090
Google Scholar
Crossref
Search ADS
 
85.
Moon
,
A.
 and
Nachtergaele
,
B.
, “
Stability of gapped ground state phases of spins and fermions in one dimension
,”
J. Math. Phys.
 
59
,
091415
(
2018
).
https://doi.org/10.1063/1.5036751
Google Scholar
Crossref
Search ADS
 
86.
Mussardo
,
G.
,
Statistical Field Theory
(
Oxford University Press
,
2010
).
Google Scholar
 
87.
Nachtergaele
,
B.
, “
The spectral gap for some quantum spin chains with discrete symmetry breaking
,”
Commun. Math. Phys.
 
175
,
565
–
606
(
1996
).
https://doi.org/10.1007/bf02099509
Google Scholar
Crossref
Search ADS
 
88.
Nachtergaele
,
B.
,
Ogata
,
Y.
, and
Sims
,
R.
, “
Propagation of correlations in quantum lattice systems
,”
J. Stat. Phys.
 
124
,
1
–
13
(
2006
).
https://doi.org/10.1007/s10955-006-9143-6
Google Scholar
Crossref
Search ADS
 
89.
Nachtergaele
,
B.
,
Raz
,
H.
,
Schlein
,
B.
, and
Sims
,
R.
, “
Lieb-Robinson bounds for harmonic and anharmonic lattice systems
,”
Commun. Math. Phys.
 
286
,
1073
–
1098
(
2009
).
https://doi.org/10.1007/s00220-008-0630-2
Google Scholar
Crossref
Search ADS
 
90.
Nachtergaele
,
B.
,
Schlein
,
B.
,
Sims
,
R.
,
Starr
,
S.
, and
Zagrebnov
,
V.
, “
On the existence of the dynamics for anharmonic quantum oscillator systems
,”
Rev. Math. Phys.
 
22
,
207
–
231
(
2010
).
https://doi.org/10.1142/s0129055x1000393x
Google Scholar
Crossref
Search ADS
 
91.
Nachtergaele
,
B.
,
Scholz
,
V.
, and
Werner
,
R. F.
, “
Local approximation of observables and commutator bounds
,” in
Operator Methods in Mathematical Physics
, Operator Theory: Advances and Applications Vol. 227, edited by
Janas
,
J.
,
Kurasov
,
P.
,
Laptev
,
A.
, and
Naboko
,
S.
 (
Birkäuser Verlag
,
2013
), pp.
143
–
149
.
Google Scholar
Crossref
Search ADS
 
92.
Nachtergaele
,
B.
 and
Sims
,
R.
, “
Lieb-Robinson bounds and the exponential clustering theorem
,”
Commun. Math. Phys.
 
265
,
119
–
130
(
2006
).
https://doi.org/10.1007/s00220-006-1556-1
Google Scholar
Crossref
Search ADS
 
93.
Nachtergaele
,
B.
 and
Sims
,
R.
, “
A multi-dimensional Lieb-Schultz-Mattis theorem
,”
Commun. Math. Phys.
 
276
,
437
–
472
(
2007
).
https://doi.org/10.1007/s00220-007-0342-z
Google Scholar
Crossref
Search ADS
 
94.
Nachtergaele
,
B.
 and
Sims
,
R.
, “
Locality estimates for quantum spin systems
,” in
New Trends in Mathematical Physics. Selected Contributions of the XVth International Congress on Mathematical Physics
, edited by
Sidovaricius
,
V.
 (
Springer Verlag
,
2009
), pp.
591
–
614
.
Google Scholar
Crossref
Search ADS
 
95.
Nachtergaele
,
B.
 and
Sims
,
R.
, “
On the dynamics of lattice systems with unbounded on-site terms in the Hamiltonian
,” e-print arXiv:1410.8174 (
2014
).
Google Scholar
 
96.
Nachtergaele
,
B.
,
Sims
,
R.
, and
Young
,
A.
, “
Quasi-locality bounds for quantum lattice systems and perturbations of gapped ground states. Part II
” (to be published).
97.
Nachtergaele
,
B.
,
Sims
,
R.
, and
Young
,
A.
, “
Lieb-Robinson bounds, the spectral flow, and stability for lattice Fermion systems
,” in
Mathematical Results in Quantum Physics
, Contemporary Mathematics Vol. 717, edited by
Bonetto
,
F.
,
Borthwick
,
D.
,
Harrell
,
E.
, and
Loss
,
M.
 (
American Mathematical Society
,
2018
), pp.
93
–
115
.
Google Scholar
Crossref
Search ADS
 
98.
Nachtergaele
,
B.
,
Vershynina
,
A.
, and
Zagrebnov
,
V. A.
, “
Lieb-Robinson bounds and existence of the thermodynamic limit for a class of irreversible quantum dynamics
,” in
Entropy and the Quantum II: Arizona School of Analysis with Applications
, Contemporary Mathematics Vol. 522, edited by
Sims
,
R.
 and
Ueltschi
,
D.
 (
American Mathematical Society
,
2011
), pp.
161
–
175
.
Google Scholar
Crossref
Search ADS
 
99.
Ogata
,
Y.
, “
A class of asymmetric gapped Hamiltonians on quantum spin chains and its characterization I
,”
Commun. Math. Phys.
 
348
(
3
),
847
–
895
(
2016
), MR 3555356.
https://doi.org/10.1007/s00220-016-2696-6
Google Scholar
Crossref
Search ADS
 
100.
Ogata
,
Y.
, “
A class of asymmetric gapped Hamiltonians on quantum spin chains and its characterization II
,”
Commun. Math. Phys.
 
348
(
3
),
897
–
957
(
2016
), MR 3555357.
https://doi.org/10.1007/s00220-016-2697-5
Google Scholar
Crossref
Search ADS
 
101.
Ogata
,
Y.
, “
A class of asymmetric gapped Hamiltonians on quantum spin chains and its characterization III
,”
Commun. Math. Phys.
 
352
(
3
),
1205
–
1263
(
2017
).
https://doi.org/10.1007/s00220-016-2810-9
Google Scholar
Crossref
Search ADS
 
102.
Ogata
,
Y.
 and
Tasaki
,
H.
, “
Lieb-Schultz-Mattis type theorems for quantum spin chains without continuous symmetry
,”
Commun. Math. Phys.
(published online
2019
); e-print arXiv:1808.08740.
https://doi.org/10.1007/s00220-019-03343-5
Google Scholar
 
103.
Petz
,
D.
,
Quantum Information Theory and Quantum Statistics
(
Springer Verlag
,
2008
).
Google Scholar
 
104.
Pollmann
,
F.
,
Turner
,
A. M.
,
Berg
,
E.
, and
Oshikawa
,
M.
, “
Entanglement spectrum of a topological phase in one dimension
,”
Phys. Rev. B
 
81
,
064439
(
2010
).
https://doi.org/10.1103/physrevb.81.064439
Google Scholar
Crossref
Search ADS
 
105.
Poulin
,
D.
, “
Lieb-Robinson bound and locality for general Markovian quantum dynamics
,”
Phys. Rev. Lett.
 
104
,
190401
(
2010
).
https://doi.org/10.1103/physrevlett.104.190401
Google Scholar
Crossref
Search ADS
PubMed
 
106.
Prémont-Schwarz
,
I.
,
Hamma
,
A.
,
Klich
,
I.
, and
Markopoulou-Kalamara
,
F.
, “
Lieb-Robinson bounds for commutator-bounded operators
,”
Phys. Rev. A
 
81
,
040102(R)
(
2010
).
https://doi.org/10.1103/physreva.81.040102
Google Scholar
Crossref
Search ADS
 
107.
Radin
,
C.
, “
Signal propagation in lattice models of quantum many-body systems
,”
Commun. Math. Phys.
 
62
,
159
–
166
(
1978
).
https://doi.org/10.1007/bf01248669
Google Scholar
Crossref
Search ADS
 
108.
Raz
,
H.
 and
Sims
,
R.
, “
Estimating the Lieb-Robinson velocity for classical anharmonic lattice systems
,”
J. Stat. Phys.
 
137
,
79
–
108
(
2009
).
https://doi.org/10.1007/s10955-009-9839-5
Google Scholar
Crossref
Search ADS
 
109.
Reed
,
M.
 and
Simon
,
B.
,
Fourier Analysis, Self-Adjointness
, Methods of Modern Mathematical Physics Vol. 2 (
Academic Press
,
1975
).
Google Scholar
 
110.
Reed
,
M.
 and
Simon
,
B.
,
Functional Analysis. Methods of Modern Mathematical Physics
, Revised and Enlarged Edition Vol. 1 (
Academic Press
,
1980
).
Google Scholar
 
111.
Richerme
,
P.
,
Gong
,
Z.-X.
,
Lee
,
A.
,
Senko
,
C.
,
Smith
,
J.
,
Foss-Feig
,
M.
,
Michalakis
,
S.
,
Gorshkov
,
A. V.
, and
Monroe
,
C.
, “
Non-local propagation of correlations in long-range interacting quantum systems
,”
Nature
 
511
,
198
(
2014
).
https://doi.org/10.1038/nature13450
Google Scholar
Crossref
Search ADS
PubMed
 
112.
Robinson
,
D. W.
, “
Properties of propagation of quantum spin systems
,”
J. Aust. Math. Soc., Ser. B: Appl. Math.
 
19
,
387
–
399
(
1976
).
https://doi.org/10.1017/s0334270000001260
Google Scholar
Crossref
Search ADS
 
113.
Rudin
,
W.
,
Principles of Mathematical Analysis
(
McGraw-Hill
,
1976
).
Google Scholar
 
114.
Ruelle
,
D.
, “
On the asymptotic condition in quantum field theory
,”
Helv. Phys. Acta
 
35
,
147
(
1962
).
https://doi.org/10.5169/seals-113272
Google Scholar
 
115.
Sachdev
,
S.
,
Quantum Phase Transitions
(
Cambridge University Press
,
1999
).
Google Scholar
 
116.
Sakai
,
S.
,
C*-Algebras and W*-Algebras
, Classics in Mathematics (
Springer-Verlag
,
1998
), Reprint of the 1971 Edition.
Google Scholar
Crossref
Search ADS
 
117.
Seiringer
,
R.
, “
Bose gases, Bose–Einstein condensation, and the Bogoliubov approximation
,”
J. Math. Phys.
 
55
,
075209
(
2014
).
https://doi.org/10.1063/1.4881536
Google Scholar
Crossref
Search ADS
 
118.
Shiraishi
,
N.
 and
Tajima
,
H.
, “
Efficiency versus speed in quantum heat engines: Rigorous constraint from Lieb-Robinson bound
,”
Phys. Rev. E
 
96
,
022138
(
2017
).
https://doi.org/10.1103/physreve.96.022138
Google Scholar
Crossref
Search ADS
PubMed
 
119.
Simon
,
B.
,
Quantum Mechanics for Hamiltonians Defined as Quadratic Forms
(
Princeton University Press
,
1971
).
Google Scholar
 
120.
Simon
,
B.
,
The Statistical Mechanics of Lattice Gases
(
Princeton University Press
,
1993
), Vol. 1.
Google Scholar
 
121.
Szehr
,
O.
 and
Wolf
,
M. M.
, “
Perturbation theory for parent Hamiltonians of matrix product states
,”
J. Stat. Phys.
 
159
,
752
–
771
(
2015
); e-print arXiv:1402.4175.
https://doi.org/10.1007/s10955-015-1204-2
Google Scholar
Crossref
Search ADS
 
122.
Tasaki
,
H.
, “
Topological phase transition and Z2-index for S = 1 quantum spin chains
,”
Phys. Rev. Lett.
 
121
,
140604
(
2018
); e-print arXiv:1804.04337.
https://doi.org/10.1103/physrevlett.121.140604
Google Scholar
Crossref
Search ADS
PubMed
 
123.
Teufel
,
S.
, “
Non-equilibrium almost-stationary states for interacting electrons on a lattice
,”
Commun. Math. Phys.
(published online, 2019).
https://doi.org/10.1007/s00220-019-03407-6
124.
Tran
,
M. C.
,
Guo
,
A.
,
Su
,
Y.
,
Garrison
,
J. R.
,
Eldredge
,
Z.
,
Foss-Feig
,
M.
,
Childs
,
A. M.
, and
Gorshkov
,
A. V.
, “
Locality and digital simulation of power-law interactions
,” e-print arXiv:1808:05225 (
2018
).
Google Scholar
 
125.
Tsvelik
,
A. M.
,
Quantum Field Theory in Condensed Matter Physics
(
Cambridge University Press
,
1995
).
Google Scholar
 
126.
Weidmann
,
J.
,
Linear Operators in Hilbert Spaces
, Graduate Texts in Mathematics Vol. 68 (
Springer
,
1980
).
Google Scholar
Crossref
Search ADS
 
127.
Wreszinski
,
W. F.
, “
Charges and symmetries in quantum theories without locality
,”
Fortschr. Phys.
 
35
,
379
–
413
(
1987
).
https://doi.org/10.1002/prop.2190350502
Google Scholar
Crossref
Search ADS
 
128.
Yarotsky
,
D. A.
, “
Ground states in relatively bounded quantum perturbations of classical lattice systems
,”
Commun. Math. Phys.
 
261
,
799
–
819
(
2006
).
https://doi.org/10.1007/s00220-005-1456-9
Google Scholar
Crossref
Search ADS
 
129.
Yosida
,
K.
,
Functional Analysis
, 6th ed., Grundlehren der Mathematishen Wissenschaften (
Springer-Verlag
,
1980
).
Google Scholar
 
130.
Young
,
A.
, “
Spectral properties of multi-dimensional quantum spin systems
,” Ph.D. thesis,
University of California
,
Davis
,
2016
.
Google Scholar
 
© 2019 Author(s).
2019
Author(s)