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.

## I. INTRODUCTION

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 Haag^{48}) 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 II^{96} 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.

## II. SOME BASIC PROPERTIES OF QUANTUM DYNAMICS

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*), $s\u2264t\u2208R$, 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 $I\u2286R$ 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

where *H*_{0} is a time-independent self-adjoint operator with dense domain $D\u2282H$, and for *t* ∈ *I*, $\Phi (t)*=\Phi (t)\u2208B(H)$ and *t* ↦ Φ(*t*) is continuous in the strong operator topology. This means that for all $\psi \u2208H$, 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:

For bounded *H*(*t*), through a standard construction, we will see that there exists a family of unitaries $U(t,s)\u2208B(H)$, *s*, *t* ∈ *I*, that is jointly strongly continuous with *ψ*(*t*) = *U*(*t*, *t*_{0})*ψ*_{0} being the unique solution of (2.2) for all $\psi 0\u2208H$. 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*) = *H*_{0} + Φ(*t*), where *H*_{0} 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.

### A. Properties of continuity, measurability, and integration in $B(H)$

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 $I\u2282R$ be a finite or infinite interval and $A:I\u2192B(H)$ be strongly continuous, i.e., for all $\psi \u2208H$, *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

The strong continuity of *t* ↦ *A*(*t*) implies that *t* ↦ ∥*A*(*t*)*ψ*∥ is continuous for all $\psi \u2208H$, 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* $I\u2282R$ *be a finite or infinite interval and* $H$ *and* $K$ *be Hilbert spaces.*

*If*$A,B:I\u2192B(H)$*are strongly continuous*(*strongly differentiable*)*, then*(*t, s*) ↦*A*(*t*)*B*(*s*)*is jointly strongly continuous*(*separately strongly differentiable*)*.**If*$A:I\u2192B(H)$*and*$B:I\u2192B(K)$*are strongly continuous*(*strongly differentiable*)*, then*(*t, s*) ↦*A*(*t*) ⊗*B*(*s*)*is jointly strongly continuous*(*separately strongly differentiable*)*.**If*$A:I\u2192B(H)$*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 $A:I\u2192B(H)$ and $\psi :I\u2192H$ 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.

*t*

_{0},

*s*

_{0}∈

*I*be fixed. Given that

*A*and

*B*are strongly continuous, and hence locally bounded, it follows that

*A*(

*t*)

*B*(

*s*)

*ψ*→

*A*(

*t*

_{0})

*B*(

*s*

_{0})

*ψ*as (

*t*,

*s*) → (

*t*

_{0},

*s*

_{0}) since

*ψ*=

*∑*

_{n}

*ψ*

_{n}⊗

*ϕ*

_{n}, and $\u2211n\Vert \psi n\Vert 2=\Vert \psi \Vert 2$. Fix

*s*∈

*I*, and let

*J*⊆

*I*be a compact interval that contains a neighborhood of

*s*. Using the orthonormality of

*ϕ*

_{n}, we find that for all

*t*∈

*J*,

*ϵ*> 0, choose

*N*large enough so that $\u2211n>N\Vert \psi n\Vert 2\u2264\u03f5/8MJ2$, where

*M*

_{J}> 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 $\Vert (A(t)\u2212A(s))\psi n\Vert 2<\u03f5/2N$. Putting these together, if |

*t*−

*s*| <

*δ*, then

*A*(

*t*) ⊗

*B*(

*s*) = (

*A*(

*t*) ⊗ $1$)($1$ ⊗

*B*(

*s*)), by (i), the tensor product of two strongly continuous maps $t\u21a6A(t)\u2208B(H)$ and $s\u21a6B(s)\u2208B(K)$ is jointly strongly continuous.

*A*(

*t*), the function ∥

*A*(

*t*)∥ can be expressed as a supremum of continuous functions,

*f*

^{−1}((

*s*,

*∞*)) is an open subset of

*I*. Now, if

*f*is the supremum of a family of functions {

*f*

_{α}}, we have that $f\u22121((s,\u221e))=\u22c3\alpha f\alpha \u22121((s,\u221e))$. In our case, the

*f*

_{α}are indexed by a pair of unit vectors in $H$ and each

*f*

_{α}is continuous. Therefore, $f\alpha \u22121((s,\u221e))$ is open for all

*α*and so is $\u22c3\alpha f\alpha \u22121((s,\u221e))$. This shows the lower semicontinuity.

Since we have that *f*^{−1}((*s*, *∞*)) is open, this set is also Borel measurable, for all $s\u2208R$. 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.

*t*↦ ⟨

*ϕ*,

*A*(

*t*)

*ψ*⟩ is measurable, then for any compact

*J*⊂

*I*the integral of

*A*over

*J*is defined as the operator $BJ\u2208B(H)$ corresponding to the bounded sesquilinear form

*B*

_{J}=

*∫*

_{J}

*A*(

*t*)

*dt*to denote this operator. For strongly continuous functions

*A*(

*t*), the same integral can be interpreted in the strong sense

*J*⊂

*I*compact:

*J*if, e.g.,

*A*(

*t*) =

*B*(

*t*)$w$(

*t*), with

*B*(

*t*) strongly continuous and bounded and $w$ ∈

*L*

^{1}(

*J*). Finally, it is easy to see that if

*A*is strongly continuous, then $B(t)=\u222bt0tA(s)ds$ is norm continuous and strongly differentiable with $ddtB(t)=A(t)$. As such, the fundamental theorem of calculus also holds in the strong sense.

### B. Dynamical equations and the Dyson series

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, $I\u2282R$ be a finite or infinite interval, and $H:I\u2192B(H)$ 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 *t*_{0} ∈ *I* and every initial condition $\psi 0\u2208H$, the time-dependent Schrödinger equation

has a unique solution in the sense that there is a unique, strongly differentiable function $\psi :I\u2192H$ which satisfies (2.10). This solution can be characterized in terms of a two-parameter family of unitaries ${U(t,s)}s,t\u2208I\u2282B(H)$ such that

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 $\psi \u2208H$, the Hilbert space-valued series

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

which is to be understood in the sense of strong derivatives. Of course, under the stronger assumption that $H:I\u2192B(H)$ 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 $\varphi ,\psi \u2208H$ and any *s*, *t* ∈ *I*, *U*(*t*, *s*) satisfies

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

*Proposition 2.2.*

*Let*$A:I\u2192B(H)$

*be strongly continuous, and consider the differential equation*

*The following statements hold:*

*There is a unique strong solution*$V:I\u2192B(H)$*of*(2.15)*, and V is norm continuous.**Any locally norm-bounded, weak solution of*(2.15)*coincides with the strong solution.**Let*$D\u2282H$*be dense. Suppose*$V:I\u2192B(H)$*is strongly continuous and satisfies*

*for all*$\psi \u2208D$

*and t*∈

*I. Then, V is the unique strong solution.*

- (iv)
*If V*_{0}*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*

*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

*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

*U*(

*t*,

*s*) =

*V*(

*t*)

*V*(

*s*)

^{*}must coincide with the Dyson series constructed in (2.12).

*Proof.*

*t*∈

*I*, set

*V*is the unique strong solution of (2.15).

*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*(

*t*

_{1})⋯

*A*(

*t*

_{n}) is jointly strongly continuous in the variables

*t*

_{1}, …,

*t*

_{n}, and thus, the integrands are locally integrable. Next, for any

*t*≥

*t*

_{0}, the bound

*A*∥(

*t*) for ∥

*A*(

*t*)∥. As it is clear that a similar argument holds for

*t*<

*t*

_{0}, we see that

*V*is well-defined as an absolutely convergent (in norm) series.

*V*is a strong solution of (2.15). To see this, define recursively a sequence of operators ${Vn}n\u22651$, $Vn:I\u2192B(H)$ by setting

*h*≠ 0,

*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.

*V*

_{1}and

*V*

_{2}be two strong solutions of (2.15). For any

*t*∈

*I*, set

*t*

_{+}= max{

*t*,

*t*

_{0}} and

*t*

_{−}= min{

*t*,

*t*

_{0}}. Given $\psi \u2208H$, we have that

*V*is norm continuous. In fact, let

*t*,

*t*

_{0}∈

*I*and $\psi \u2208H$. Clearly,

*V*follows.

- (ii)
The uniqueness statement in (ii) is proven similarly. In fact, let

*V*_{1}and*V*_{2}be two locally norm-bounded weak solutions of (2.15). In this case, for any $\varphi ,\psi \u2208H$ and each*t*∈*I*,

*t*

_{±}as above. Taking the supremum of (2.26) over all normalized $\varphi ,\psi \u2208H$ gives

- (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 $\varphi \u2208H$, (2.16) implies

*t*∈

*I*. Let $\psi \u2208H$ and take any sequence ${\psi n}n\u22651$ in $D$ with

*ψ*

_{n}converging to

*ψ*. Consider the sequence of functions $fn:I\u2192C$ defined by

*f*(

*t*) = lim

_{n→∞}

*f*

_{n}(

*t*). Note that these pointwise limits exist as

*ψ*

_{n}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 $fn\u2032$(

*t*) = ⟨

*ϕ*,

*A*(

*t*)

*V*(

*t*)

*ψ*

_{n}⟩. Since

*AV*is strongly continuous, and hence locally bounded, the same argument shows that $g:I\u2192C$ with

*g*(

*t*) = lim

_{n→∞}$fn\u2032$(

*t*) = ⟨

*ϕ*,

*A*(

*t*)

*V*(

*t*)

*ψ*⟩ is well-defined. Observing further that $fn\u2032$ 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 $W:I\u2192B(H)$ defined by setting

*V*being the strong solution of (2.15), consider the function $Y:I\u2192B(H)$ given by

*Y*(

*t*) =

*V*(

*t*)

*W*(

*t*). One checks that

*W*is a right inverse of

*V*for all

*t*∈

*I*. Noting that the function $Z:I\u2192B(H)$ defined by

*Z*(

*t*) =

*W*(

*t*)

*V*(

*t*) satisfies the trivial initial value problem,

*W*(

*t*) =

*V*(

*t*)

^{−1}as claimed. In fact, the uniqueness of the strong solution of (2.31) follows.

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,*$I\u2282R$

*be a finite or infinite interval, and*$A,B:I\u2192B(H)$

*be strongly continuous with A pointwise self-adjoint, i.e., A*(

*t*)

^{*}=

*A*(

*t*)

*for all t*∈

*I. For each t*

_{0}∈

*I and*$V0\u2208B(H)$

*, the initial value problem*

*has a unique strong solution. In particular,*

*where t*

_{+}= max{

*t*,

*t*

_{0}}

*, t*

_{−}= min{

*t*,

*t*

_{0}}

*. Moreover, any locally bounded weak solution of*(2.34)

*coincides with the strong solution and, therefore, satisfies the estimate in*(2.35).

*Proof.*

*t*∈

*I*. As a product of strongly differentiable maps, $V:I\u2192B(H)$ given by

*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.

### C. The dynamics for a class of unbounded Hamiltonians

#### 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* = *H*_{0} + 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*) = *H*_{0} + Φ(*t*), where *H*_{0} 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 H*_{0} *be a self-adjoint operator with dense domain* $D\u2282H$*. Let* $I\u2282R$ *be a finite or infinite interval and* $\Phi :I\u2192B(H)$ *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*) = *H*_{0} + 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 t*_{0}∈*I and*$\psi 0\u2208H$*,*$\psi :I\u2192H$*given by ψ*(*t*) =*U*(*t, t*_{0})*ψ*_{0}*satisfies*(2.38)$ddt\u27e8\varphi ,\psi (t)\u27e9=\u2212i\u27e8H(t)\varphi ,\psi (t)\u27e9\u2003with\u2003\psi (t0)=\psi 0,$*for all*$\varphi \u2208D$*and t*∈*I.*

*Proof.*

*H*

_{0}is self-adjoint, Stone’s theorem implies that ${eitH0}t\u2208R$ is a strongly continuous, one-parameter unitary group. In this case, the map $H\u0303:I\u2192B(H)$ given by

*Ũ*(

*t*,

*s*)}

_{s,t∈I}which satisfy the cocycle property (2.19). In terms of this family, we define $U:I\xd7I\u2192B(H)$ by setting

*U*(

*t*,

*s*)}

_{s,t∈I}is a two-parameter family of unitaries satisfying (i) and (ii) above.

*t*

_{0}∈

*I*and $\psi 0\u2208H$. Define $\psi :I\u2192H$ by setting

*ψ*(

*t*) =

*U*(

*t*,

*t*

_{0})

*ψ*

_{0}. Observe that for any $\varphi \u2208D$ and each

*t*∈

*I*,

*t*. One calculates that

We need to justify only the uniqueness of the locally bounded weak solutions. Let *t*_{0} ∈ *I*, let $\psi 0\u2208H$, and suppose *ψ*_{1} and *ψ*_{2} are two locally bounded solutions of the initial value problem (2.38). Consider the functions $\psi \u03031(t)=eitH0\psi 1(t)$ and $\psi \u03032(t)=eitH0\psi 2(t)$. It is easy to check that these functions are locally bounded weak solutions of the Schrödinger equation associated with the bounded Hamiltonian $H\u0303(t)$ 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 *ψ*_{1} and *ψ*_{2}.

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 *H*_{0} 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 *H*_{0} 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.

*Let H*

_{0}

*be a self-adjoint operator with dense domain*$D\u2282H$

*and suppose H*

_{0}≥ 0

*. Suppose*$\Phi :R\u2192B(H)$

*is pointwise self-adjoint and*“

*Lipschitz*”

*continuous in the sense that for any bounded interval*$I\u2282R$

*, there exists a constant C such that for all s, t*∈

*I, we have*

*Then, there exists a strongly continuous propagator U*(

*t, s*)

*, such that*$U(t,s)D\u2282D$

*, for all s*≤

*t*∈

*I, and such that t*↦

*U*(

*t, t*

_{0})

*ψ*

_{0}

*is the unique strong solution of*

*for all*$\psi 0\u2208D$.

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 Kato^{65,66} and Kisyński.^{70}

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

In this section, we consider families of Hamiltonians *H*_{λ}(*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 *H*_{0} be a densely defined, self-adjoint operator on a Hilbert space $H$ and denote by $D\u2282H$ the corresponding dense domain. Let $I,J\u2282R$ be intervals and consider the family of Hamiltonians *H*_{λ}(*t*), *t* ∈ *I* and *λ* ∈ *J*, acting on $D\u2282H$ given by

where for each *t* ∈ *I* and *λ* ∈ *J*, $\Phi \lambda (t)*=\Phi \lambda (t)\u2208B(H)$. The self-adjointness of *H*_{λ}(*t*) on the common domain, $D$, is clear. We will assume that (*t*, *λ*) ↦ Φ_{λ}(*t*) is jointly strongly continuous. We will also assume that for each fixed *t* ∈ *I*, the mapping *λ* ↦ Φ_{λ}(*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 ${U\lambda (t,s)}s,t\u2208I$ which generates the weak solutions of the Schrödinger equation associated with *H*_{λ}(*t*) [see (2.38)]. Our goal here is to show that for fixed *s*, *t* ∈ *I*, the map *λ* ↦ *U*_{λ}(*t*, *s*) is strongly differentiable, and moreover,

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*

*Let H*

_{λ}(

*t*)

*be a family of self-adjoint operators as in*(2.46)

*, and let U*

_{λ}(

*t, s*)

*denote the corresponding unitary propagator. Then, for all s, t*∈

*I with s*≤

*t, we have that*

*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*) ↦ Φ_{λ}(*t*) is jointly norm continuous,for each

*t*∈*I*, the map*λ*↦ Φ_{λ}(*t*) is norm differentiable, with the derivative denoted by Φ′_{λ}(*t*), andthe map $(t,\lambda )\u21a6\Phi \lambda \u2032(t)$ is jointly norm continuous,

*λ*↦

*U*

_{λ}(

*t*,

*s*) is norm differentiable and its derivative satisfies (2.48).

*Proof.*

*U*

_{λ}(

*t*,

*s*), as defined in the proof of Proposition 2.4, is

*Ũ*

_{λ}(

*t*,

*s*) is the unique strong solution of

*Ũ*

_{λ}(

*t*,

*s*), i.e.,

*λ*-derivative of (2.49) is easily seen to satisfy (2.48).

*n*≥ 1, define a map $\Psi \lambda :In\u2192B(H)$ by setting

*t*

_{1}, …,

*t*

_{n}) ∈

*I*

^{n}fixed, our assumptions imply that

*λ*↦ Ψ

_{λ}(

*t*

_{1}, …,

*t*

_{n}) is strongly differentiable, and moreover,

The proof of (2.51) is now completed by demonstrating that upon inserting the Dyson series for *Ũ*_{λ}(*t*, *r*) and *Ũ*_{λ}(*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).

*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

*t*

_{1}, …,

*t*

_{p}and

*t*

_{p+2}, …,

*t*

_{p+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

*λ*∈

*J*, the Heisenberg dynamics $\tau t,s\lambda $,

*s*,

*t*∈

*I*, associated with the family of Hamiltonians in (2.46) is the cocycle of automorphisms of $B(H)$ given by

*Proposition 2.7.*

*Let H*_{λ}(*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*$A\u2208B(H)$*, the map*$(s,t)\u21a6\tau t,s\lambda (A)$*is jointly strongly continuous.**For each s, t*∈*I and*$A\u2208B(H)$*, the map*$\lambda \u21a6\tau t,s\lambda (A)$*is strongly differentiable*(*and hence strongly continuous*)*. Moreover, one has the estimate*

- (iii)
*For fixed s, t*∈*I and λ*∈*J, the map*$\tau t,s\lambda (\u22c5):B(H)\u2192B(H)$*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 $\tau t,s\lambda (A)$ [see (2.58)] is the product of jointly strongly continuous mappings.

*s*≤

*t*, then

*s*,

*t*∈

*I*and let [

*a*,

*b*] ⊂

*J*. Without loss of generality, assume that

*s*≤

*t*. Let

*ϵ*> 0. Since $(r,\lambda )\u21a6\Phi \lambda \u2032(r)$ is jointly strongly continuous,

*δ*> 0 so that

*N*≥ 1 and numbers

*λ*

_{1}, …,

*λ*

_{N}∈ [

*a*,

*b*] for which the balls of radius

*δ*centered at

*λ*

_{i}, 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

*B*<

*∞*be such that sup

_{n≥1}∥

*A*

_{n}∥ ≤

*B*. Using (2.58) and the strong convergence of

*A*

_{n}to

*A*, it is easy to verify that for any $\psi \u2208H$ and any 1 ≤

*i*≤

*N*, the sequence ${\tau t,s\lambda i(An)\psi}n\u22651$ converges to $\tau t,s\lambda i(A)\psi $ in $H$. Pick

*n*

_{0}≥ 1 so that for all

*n*≥

*n*

_{0}and each 1 ≤

*i*≤

*N*,

*λ*∈ [

*a*,

*b*], there is an

*i*for which

*n*≥

*n*

_{0}. This proves that the strong convergence is uniform for

*λ*∈ [

*a*,

*b*], or in other words, that the family of maps ${\tau t,s\lambda (\u22c5)\u2223\lambda \u2208[a,b]}$, for

*s*,

*t*∈

*I*fixed, is equicontinuous on bounded sets in $B(H)$ with respect to the strong operator topology.

## III. LIEB-ROBINSON BOUNDS AND INFINITE VOLUME DYNAMICS OF LATTICE SYSTEMS

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 quadratic^{34} 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 {Λ_{n}}, i.e., Λ_{n}*↑* Γ. Often, the goal is to obtain estimates for the finite systems defined on Λ_{n} that are uniform in *n*. The definitions below prepare for this goal. It is possible to consider a finite set $\Gamma $ 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 Radin^{107} and, in particular, by Robinson^{112} 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.

### A. Lieb-Robinson estimates for bounded time-dependent interactions

#### 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 $P0(\Gamma )$ be the collection of all finite subsets of Γ. For any $\Lambda \u2208P0(\Gamma )$, the Hilbert space of states and algebra of local observables over Λ are denoted by

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*^{*}-norm^{116}). For any two finite sets Λ_{0} ⊂ Λ ⊂ Γ, each $A\u2208A\Lambda 0$ can be naturally identified with $A\u22971\Lambda \\Lambda 0\u2208A\Lambda $. With respect to this identification, the algebra of *local observables* is then defined as the inductive limit

and the *C*^{*}-algebra of *quasilocal observables*, which we denote by $A\Gamma $, is the completion of $A\Gamma loc$ 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 $A\Gamma $.

A model on a quantum lattice system is given in terms of an *interaction* Φ. In the time-independent case, this is a map $\Phi :P0(\Gamma )\u2192A\Gamma loc$ such that $\Phi (Z)*=\Phi (Z)\u2208AZ$ for all $Z\u2208P0(\Gamma ).$ The *quantum lattice model* associated with Φ is the collection of all local Hamiltonians of the form

We will also consider time-dependent interactions. Let $I\u2282R$ be an interval. A map $\Phi :P0(\Gamma )\xd7I\u2192A\Gamma loc$ is said to be a *strongly continuous interaction* if

To each

*t*∈*I*, the map $\Phi (\u22c5,t):P0(\Gamma )\u2192A\Gamma loc$ is an interaction.For each $Z\u2208P0(\Gamma )$, $\Phi (Z,\u22c5):I\u2192AZ$ 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

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*, *H*_{Λ}(*t*) is a bounded, self-adjoint operator on $H\Lambda $. Moreover, by Proposition 2.1, $H\Lambda :I\u2192A\Lambda $ is strongly continuous. In this case, Proposition 2.2 demonstrates that there exists a two-parameter family of unitaries ${U\Lambda (t,s)}s,t\u2208I\u2282A\Lambda $, defined as the unique strong solution of the initial value problem

In terms of these unitary propagators, we define a Heisenberg dynamics $\tau t,s\Lambda :A\Lambda \u2192A\Lambda $ by setting

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

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.,

- (ii)
*F*satisfies the convolution condition

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

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

also satisfies (i) and (ii) with ∥*F*_{g}∥ ≤ ∥*F*∥ and $CFg\u2264CF$. We call any *F*-function of this form a *weighted F-function*.

It is easy to produce examples of these *F*-functions when $\Gamma =Z\nu $ for some *ν* ≥ 1 and *d*(*x*, *y*) = |*x* − *y*| is the *ℓ*^{1}-distance. In fact, for any *ϵ* > 0, the function

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

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 *F*_{a}, given by *F*_{a}(*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 $\Phi :P0(\Gamma )\u2192A\Gamma loc$ be an interaction. The *F-norm* of Φ is defined by

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

Note that for any $Z\u2208P0(\Gamma )$, 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

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

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 $A\Gamma $. Let $I\u2282R$ be an interval and $\Phi :P0(\Gamma )\xd7I\u2192A\Gamma loc$ 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

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,

#### 2. Lieb-Robinson estimates for bounded interactions

In Theorem 3.1, we demonstrate that the finite volume Heisenberg dynamics $\tau t,s\Lambda $, as defined in (3.6), associated with any $\Phi \u2208BF(I)$ 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 $X\u2208P0(\Gamma )$, we denote by $\u2202\Phi IX\u2282X$ the Φ-boundary of *X*,

In some estimates, it may be useful to restrict the time interval used to define the Φ-boundary. For instance, given $\Phi \u2208BF(R)$ one could find that $\u2202\Phi RX=X$ for some *X*, while $\u2202\Phi IX$ is strictly smaller for a subinterval $I\u2282R$. From now on, we will drop the time-interval *I* from the notation and simply write *∂*_{Φ}*X*. We note also that in many situations, not much is lost by using *X* instead of *∂*_{Φ}*X* in the following estimates.

Second, for $\Phi \u2208BF(I)$, and *s*, *t* ∈ *I*, the quantity *I*_{t,s}(Φ) defined by

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

with

*Let*$\Phi \u2208BF(I)$

*and*$X,Y\u2208P0(\Gamma )$

*with X ∩ Y*= ∅

*. For any*$\Lambda \u2208P0(\Gamma )$

*with X*∪

*Y*⊂ Λ

*and any*$A\u2208AX$

*and*$B\u2208AY$

*, we have*

*for all t, s*∈

*I. Here, C*

_{F}

*is the constant in*(3.9)

*, and the quantity D*(

*X, Y*)

*is given by*

It is easy to see that with the definition $F1(r)=CF\u22121F(r)$, *F*_{1} 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 *C*_{F} = 1 for each of the *F*-functions, we note that the final result can be expressed with a renormalized *F*-function such that *C*_{F} = 1.

*F*-function

*F*

_{g}(

*r*) =

*e*

^{−g(r)}

*F*(

*r*), we can further estimate

*d*(

*X*,

*Y*) is the distance between

*X*and

*Y*. When

*g*(

*r*) =

*ar*for some

*a*> 0 [i.e., $\Phi \u2208BFa(I)$ with

*F*

_{a}(

*r*) =

*e*

^{−ar}

*F*(

*r*)], it makes sense to define the quantity $va=2a\u22121CFa\Phi Fa$ 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

*X*in the volume Λ, denoted

*S*

_{Λ}(

*X*), as follows:

*X*and Λ$\$

*X*. We will also use the following notation for $X,Y\u2208P0(\Gamma )$:

*Lemma 3.2.*

*Let*$\Phi \u2208BF(I)$

*. Fix*$Y\u2208P0(\Gamma )$

*,*$B\u2208AY$

*, and*$\Lambda \u2208P0(\Gamma )$

*with Y*⊂ Λ

*. For any X*⊂ Λ

*, the family mappings*$gt,sX,B:AX\u2192A\Lambda $

*for t, s*∈

*I, defined by*

*are norm-continuous; more precisely,*$(s,t)\u21a6gt,sX,B$

*is jointly continuous in the norm on*$B(AX,A\Lambda )$

*. Moreover, for fixed t and s, the mapping*$gt,sX,B$

*satisfies*

*s*,

*t*) ↦

*U*

_{Λ}(

*t*,

*s*) as proven in Proposition 2.2(v), see also statements following. In fact, for any $A\u2208AX$, one has the simple estimate

We also note that the map $gt,sX,B$ equals the restriction of $gt,s\Lambda ,B$ 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 *S*_{Λ}(*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.*

*X*⊂ Λ, $A\u2208AX$, and

*s*∈

*I*. Recall that the inverse dynamics is given by

*U*

_{X}(

*t*,

*s*) are defined as in (3.5) [see also (3.4)] with Λ =

*X*. Consider the function $fs:I\u2192A\Lambda $ given by $fs(t)=gt,sX,B(\tau ^t,sXs(A))$. It follows that $fs(t)=[\tau t,s\Lambda \u25e6\tau ^t,sX(A),B]$ is strongly differentiable in

*t*and a short calculation shows that

*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

*t*

_{0}=

*s*. Thus, we have

*Proof of Theorem 3.1.*

*D*(

*X*,

*Y*) defined to be the minimum in (3.24) is also clear.

*X*,

*Y*, Λ,

*A*, and

*B*be as in Theorem 3.1. An application of Lemma 3.2 demonstrates that

*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,

*R*

_{N+1}(

*t*) is estimated as follows: First, we observe that

*Z*

_{1},

*Z*

_{2}, …,

*Z*

_{N+1}) which satisfy

*Z*

_{1}∩

*∂*

_{Φ}

*X*≠ ∅ and

*Z*

_{j}∩

*Z*

_{j−1}≠ ∅ for 2 ≤

*j*≤

*N*+ 1. As such, there are points $w1$, $w2$, …, $wN+1$ ∈ Λ such that $w1$ ∈

*Z*

_{1}∩

*∂*

_{Φ}

*X*and $wj$ ∈

*Z*

_{j}∩

*Z*

_{j−1}for all 2 ≤

*j*≤

*N*+ 1. A simple upper bound on these sums is then obtained by overcounting,

*Z*

_{N+1}must contain more than one point since

*Z*

_{N+1}∈

*S*

_{Λ}(

*Z*

_{N}). As $\Phi \u2208BF(I)$, (3.19) implies that

*k*≤

*N*+ 1. Using this bound as well as (3.8) and (3.9), we conclude that

_{F}is locally integrable on

*I*, this remainder clearly goes to 0 as

*N*→

*∞*.

*a*

_{n}(

*t*). In fact, these terms are also sums over chains; however, there is a restriction: only those chains whose final link

*Z*

_{n}satisfies

*Z*

_{n}∩

*Y*≠ ∅ contribute to the sum. Recalling that $It,s(\Phi )=CF\u222bst\Vert \Phi \Vert F(r)dr$, the bound

*δ*

_{Y}(

*X*) = 0 and

*n*≥ 1, the bound in (3.23) is now clear.

### B. A class of unbounded Hamiltonians

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 $A\Gamma $. Let $I\u2282R$ be an interval, *F* be an *F*-function on (Γ, *d*), and $\Phi \u2208BF(I)$ be a time-dependent interaction. To each *z* ∈ Γ, fix a self-adjoint operator *H*_{z} with dense domain $Dz\u2282Hz$. For any $\Lambda \u2208P0(\Gamma )$ and *t* ∈ *I*, consider the finite-volume Hamiltonian

The noninteracting Hamiltonian

is essentially self-adjoint with domain

[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*, *H*_{Λ}(*t*) is essentially self-adjoint on $H\Lambda $ with domain $D\Lambda $. We proceed by using the notations $H\Lambda (0)$ and *H*_{Λ}(*t*) for the corresponding self-adjoint closures.

As $\Phi \u2208BF(I)$, it is a strongly continuous interaction, and so for any $\Lambda \u2208P0(\Gamma )$, Proposition 2.4 guarantees the existence of a finite volume unitary propagator corresponding to *H*_{Λ}(*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 $H\Lambda (0)$ generates a free-dynamics

in terms of a group of strongly continuous unitaries $U\Lambda (0)(t,0)=e\u2212itH\Lambda (0)$. In this case,

is pointwise self-adjoint with $H\u0303\Lambda :I\u2192A\Lambda $ strongly continuous. By Proposition 2.2(v), there is a unique strong solution of the initial value problem

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

for any *s*, *t* ∈ *I*. As is demonstrated in the proof of Proposition 2.4, the operators ${U\Lambda (t,s)}s,t\u2208I$ 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 *H*_{Λ}(*t*). We use these unitaries to define a dynamics associated with *H*_{Λ}(*t*), namely, for any *s*, *t* ∈ *I*, we take $\tau t,s\Lambda :A\Lambda \u2192A\Lambda $ as

One readily checks that the family ${\tau t,s\Lambda \u2223t,s\u2208I}$ of automorphisms on $A\Lambda $ satisfies the cocycle property and that the following analog of Theorem 3.1 holds for this dynamics.

*Let*${Hz}z\u2208\Gamma $

*be a collection of densely defined self-adjoint operators,*$\Phi \u2208BF(I)$

*, and*$\tau t,s\Lambda $

*be the dynamics given in*(3.56)

*. Let*$X,Y\u2208P0(\Gamma )$

*be disjoint sets. For any*$\Lambda \u2208P0(\Gamma )$

*with X*∪

*Y*⊂ Λ

*and any*$A\u2208AX$

*and*$B\u2208AY$

*, the bound*

*holds for all t, s*∈

*I. Here, C*

_{F}

*is the constant in*(3.9)

*, and the quantities I*

_{t,s}(Φ)

*and D*(

*X, Y*)

*are as discussed earlier [see*(3.21) and (3.24)

*, respectively].*

*Proof.*

*t*∈

*I*, $\tau t(0)(A)\u2208AX$ and $\tau t(0)(B)\u2208AY$. Moreover, the interaction-picture dynamics is generated by the strongly continuous interaction $\Phi \u0303$ with terms

*t*∈

*I*. Since $\Phi \u0303(Z,t)$ and Φ(

*Z*,

*t*) have the same support and the same norm, it is clear that $\Vert \Phi \u0303\Vert F(t)=\Vert \Phi \Vert F(t)$ for all

*t*∈

*I*. In this case, the bound in (3.57) follows from Theorem 3.1 applied to the interaction-picture dynamics $\Phi \u0303$.

*λ*,

*t*) ↦ Φ

_{λ}(

*X*,

*t*) is jointly strongly continuous and strongly differentiable with respect to

*λ*, Proposition 2.7 applies to the corresponding finite-volume dynamics

*λ*-derivative of $\tau t,s\Lambda ,\lambda (A)$ [see (2.59)] will generally depend on the volume Λ. However, under the additional assumption that $\Phi \lambda ,\Phi \lambda \u2032\u2208BF(I)$, 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

### C. The infinite-volume dynamics

In this section, we will prove several convergence and continuity results for the Heisenberg dynamics associated with interactions $\Phi \u2208BF(I)$ 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 $\Phi \u2208BF(I)$ 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 $\Phi ,\Psi \u2208BF$ 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 *I*_{t,s}(Φ), which is defined in (3.21).

*Consider a quantum lattice system composed of*(Γ

*, d*)

*and*$A\Gamma $

*. Let*$I\u2282R$

*be an interval, F be an F-function on*(Γ

*, d*)

*, and*$\Phi ,\Psi \u2208BF(I)$

*be time-dependent interactions. Fix a collection of densely defined, self-adjoint on-site Hamiltonians*${Hz}z\u2208\Gamma $

*, and for any*$\Lambda \u2208P0(\Gamma )$

*, define Hamiltonians*

*as well as their corresponding dynamics,*$\tau t,s\Lambda $

*and*$\alpha t,s\Lambda $

*, for each s, t*∈

*I, respectively*.

*For any*$X,\Lambda \u2208P0(\Gamma )$*with X*⊂ Λ*, the bound*(3.68)$\Vert \tau t,s\Lambda (A)\u2212\alpha t,s\Lambda (A)\Vert \u22642\Vert A\Vert CFe2min(It,s(\Phi ),It,s(\Psi ))It,s(\Phi \u2212\Psi )\u2211x\u2208X\u2211y\u2208\Lambda F(d(x,y))$*holds for all*$A\u2208AX$*and s, t*∈*I*.*For any*$X,\Lambda 0,\Lambda \u2208P0(\Gamma )$*with X*⊂ Λ_{0}⊂ Λ*, the bound*(3.69)$\Vert \tau t,s\Lambda (A)\u2212\tau t,s\Lambda 0(A)\Vert \u22642\Vert A\Vert CFe2It,s(\Phi )It,s(\Phi )\u2211x\u2208X\u2211y\u2208\Lambda \\Lambda 0F(d(x,y))$*holds for all*$A\u2208AX$*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 $A\Lambda $, that are uniform in Λ but grow linearly in |*X*|. Since the dynamics are, of course, automorphisms, the bounds of Theorem 3.4 are only nontrivial if the RHS of (3.68) and (3.69) are smaller than 2∥*A*∥. As is well known, this will be true for both cases if |*t* − *s*| is sufficiently small. Additionally, the bound in (3.68) will be nontrivial if ∥Φ − Ψ∥_{F} is small, and the bound in (3.69) will be nontrivial if *d*(*X*, Λ$\$Λ_{0}) is small. Note that even if a map is bounded on all of $A\Gamma loc$, as is the case in Theorem 3.4, norm bounds for their local restriction can be very useful.

*Proof.*

*H*

_{z}= 0 for all

*z*∈ Γ. For any $\Lambda \u2208P0(\Gamma )$,

*X*⊂ Λ,

*s*,

*t*∈

*I*, and $A\u2208AX$, we write the corresponding difference as

*s*≤

*t*, a simple norm bound then shows that

*R*= 0 then gives

*H*

_{z}, we argue with the interaction picture dynamics as was done in the proof of Theorem 3.3. Using (3.58), it is clear that

*H*

_{z}= 0 for all

*z*∈ Γ, the analog of (3.72) is

A first application of Theorem 3.4 is a proof that given any collection of self-adjoint on-sites ${Hz}z\u2208\Gamma $ and an interaction $\Phi \u2208BF(I)$, there is a corresponding infinite-volume dynamics on $A\Gamma $. We obtain this infinite-volume dynamics as a limit of finite-volume dynamics. With this in mind, we will say that a sequence ${\Lambda n}n\u22651\u2282P0(\Gamma )$ is increasing and exhaustive if Λ_{n} ⊂ Λ_{n+1} for all *n* ≥ 1 and given any $X\u2208P0(\Gamma )$, there is an *N* ≥ 1 for which *X* ⊂ Λ_{N}.

*Under the assumptions of Theorem*3

*.*4

*, for each*$A\u2208A\Gamma loc$

*and any s, t*∈

*I,*

*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, τ*

_{t,s}

*is a cocycle of automorphisms of*$A\Gamma $

*. If there exists M*≥ 0

*such that*∥

*H*

_{z}∥ ≤

*M for all z*∈ Γ,

*then*(

*t, s*) ↦

*τ*

_{t,s}(

*A*)

*is norm continuous for all*$A\u2208A\Gamma $.

For unbounded *H*_{z}, the continuity of *τ*_{t,s} is limited by the continuity of the on-site dynamics $\tau t(0)$. In a suitable representation of $A\Gamma $ on a Hilbert space, one can retrieve weak continuity of the dynamics. See Ref. 90 for an example. Continuity properties of *τ*_{t,s} and other families of maps will be further discussed in Sec. IV B 1.

*Proof.*

*N*≥ 1 for which

*X*⊂ Λ

_{n}for all

*n*≥

*N*. For any integers

*n*,

*m*with

*N*≤

*m*≤

*n*, Theorem 3.4(ii) implies

*I*

_{t,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 ${\tau t,s\Lambda n(A)}n\u22650$ 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*$X,\u2009Y\u2208P0(\Gamma )$*such that X ∩ Y*= ∅*, the bound*$\Vert [\tau t,s(A),B]\Vert \u22642\Vert A\Vert \Vert B\Vert CF(e2It,s(\Phi )\u22121)D(X,Y)$*holds for all*$A\u2208AX$, $B\u2208AY$*, and t, s*∈*I*.*For any*$X\u2208P0(\Gamma )$*, the bound*(3.79)$\Vert \tau t,s(A)\u2212\alpha t,s(A)\Vert \u22642\Vert A\Vert CFe2min(It,s(\Phi ),It,s(\Psi ))It,s(\Phi \u2212\Psi )\u2211x\u2208X\u2211y\u2208\Gamma F(d(x,y))$*holds for all*$A\u2208AX$*and s, t*∈*I*.*For any*$X,\Lambda \u2208P0(\Gamma )$*with X*⊂ Λ*, the bound*(3.80)$\Vert \tau t,s(A)\u2212\tau t,s\Lambda (A)\Vert \u22642\Vert A\Vert CFe2It,s(\Phi )It,s(\Phi )\u2211x\u2208X\u2211y\u2208\Gamma \\Lambda F(d(x,y))$*holds for all*$A\u2208AX$*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.

_{Λ}can be extended to an interaction on all of Γ by declaring that Φ

_{Λ}(

*Z*,

*t*) = 0 for any $Z\u2208P0(\Gamma )$ with

*Z*∩ (Γ$\$Λ) ≠ ∅. We call this new mapping the extension of Φ

_{Λ}to Γ and continue to denote it by Φ

_{Λ}. Similarly, given Φ

_{Λ}and Λ

_{0}⊂ Λ, we define $\Phi \Lambda \u21be\Lambda 0:P0(\Lambda 0)\xd7I\u2192A\Lambda 0loc$ by

_{Λ}to Λ

_{0}. If the dynamics associated with $\Phi \Lambda \u21be\Lambda 0$ exists, we will denote it by $\tau t,s\Phi \Lambda ,\Lambda 0$. Of course, if Λ

_{0}is finite, such a dynamics always exists and it is generated by the time-dependent Hamiltonian

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

*Definition 3.7.*

Let (Γ, *d*) and $A\Gamma $ be a quantum lattice system, and let $I\u2282R$ be an interval. We say that a sequence of interactions ${\Phi n}n\u22651$ *converges locally in F-norm* to Φ if there exists an *F*-function, *F*, such that

$\Phi n\u2208BF(I)$ for all

*n*≥ 1,$\Phi \u2208BF(I)$,

for any $\Lambda \u2208P0(\Gamma )$ and each [

*a*,*b*] ⊂*I*,

*F*is an

*F*-function for which (i)–(iii) are satisfied, we say that Φ

_{n}

*converges locally in F-norm to*Φ

*with respect to F*.

_{F}:

*I*→ [0,

*∞*) given by

*H*

_{z}= 0 for all

*z*∈ Γ (see comments following the proof of Theorem 3.8).

*Let* ${\Phi n}n\u22651$ *be a sequence of time-dependent interactions on* Γ *with* Φ_{n} *converging locally in the F-norm to* Φ *with respect to F.*

*If for every*[*a, b*] ⊂*I,*(3.85)$supn\u22651\u222bab\Vert \Phi n\Vert F(t)\u2009dt<\u221e\u2009,$*then, for any*$X\u2208P0(\Gamma )$, $A\u2208AX$,*and s, t*∈*I, s*≤*t, we have convergence of the dynamics*(3.86)$limn\u2192\u221e\Vert \tau t,s\Phi n(A)\u2212\tau t,s\Phi (A)\Vert =0.$*Moreover, the convergence is uniform for s, t in compact intervals, and the dynamics is continuous,*(3.87)$\Vert \tau t,s\Phi (A)\u2212A\Vert \u22642|X|\Vert A\Vert \Vert F\Vert \u222bst\Vert \Phi \Vert F(r)dr.$*If, in addition, for all*$\Lambda \u2208P0(\Gamma )$*, and r*∈*I, we also have pointwise local convergence,*(3.88)$limn\u2192\u221e\Vert (\Phi n\u2212\Phi )\u21be\Lambda \Vert F(r)=0,$*and uniform boundedness of the interactions on compact intervals I*_{0}⊂*I,*(3.89)$supnsupt\u2208I0\Vert \Phi n\Vert F(t)<\u221e,$*then the generators converge uniformly for t in compacts: for all compact I*_{0}⊂*I and*$A\u2208A\Gamma loc$*, we have*(3.90)$limnsupt\u2208I0\Vert \delta t(n)(A)\u2212\delta t(A)\Vert =0,$*where*$\delta r(n)(A)=\u2211Z\u2208P0(\Gamma )[\Phi n(Z,r),A],\u2003\delta r(A)=\u2211Z\u2208P0(\Gamma )[\Phi (Z,r),A].$

*Proof.*

*a*,

*b*] ⊂

*I*be fixed. For any $\Lambda \u2208P0(\Gamma )$ with

*X*⊂ Λ, define the restricted interactions Φ

_{n}↾

_{Λ}and Φ↾

_{Λ}. By Theorem 3.5, a dynamics may be associated with each of these interactions and the estimate

*s*,

*t*∈ [

*a*,

*b*].

By assumption (3.85), it is clear that sup_{n}*I*_{t,s}(Φ_{n}) 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.

*F*-norm and the first term gives the desired estimate.

- (ii)
The interactions Φ

_{n}(*t*) and Φ(*t*) have finite*F*-norm. Hence, the corresponding derivations, $\delta t(n)$ and*δ*_{t}, are well-defined on $A\Gamma loc$. For any $\Lambda \u2208P0(\Gamma )$, we then have

*R*= 0 to the first term and using a similar argument for the second term, we get

*n*→

*∞*. Therefore,

*↑*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 *H*_{z} = 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 $\tau t,s\Phi n$ to the cocycle $\tau t,s\Phi $, as well as the convergence of the generators $\delta t(n)$ to densely defined derivations *δ*_{t}. 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 *δ*_{t}. Theorem 5.9 addresses this question.

*For all t in an interval I, let*$\Phi (t)\u2208BF$

*be interactions such that t*↦ Φ(

*Z, t*)

*is norm-continuous for all*$Z\u2208P0(\Gamma )$

*and such that*∥Φ(

*t*)∥

_{F}

*is bounded on all compact intervals I*

_{0}⊂

*I. Let τ*

_{t,s}

*denote the strongly continuous dynamics generated by*Φ(

*t*)

*. Define, for all t*∈

*I,*

*Then, t*→

*δ*

_{t}(

*A*)

*is norm-continuous for all*$A\u2208A\Gamma loc$

*, and for all s, t*∈

*I, s*<

*t*,

*Proof.*

_{n}. Therefore, we have

*δ*

_{t}(

*A*) as a function of

*t*, note that for $s,t\u2208I,X\u2208P0(\Gamma ),A\u2208AX$, and Λ ⊂ Γ, we have

*t*=

*s*.

*τ*

_{t,s}, and we estimate the second term using Corollary 3.6(iii). This produces

*t*and

*n*, the RHS vanishes as

*n*→

*∞*, uniformly for

*s*,

*t*in any compact interval

*I*

_{0}. This shows the uniform convergence of the derivative, i.e., for all $X\u2208P0(\Gamma )$ and $A\u2208AX$,

*τ*

_{t,s}(

*δ*

_{t}(

*A*)) in

*t*and the usual argument using the fundamental theorem of calculus, it now follows that

*τ*

_{t,s}(

*A*) is differentiable in

*t*with the derivative given by

*iτ*

_{t,s}(

*δ*

_{t}(

*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 $A\Gamma $. With this understanding, one sees that Theorem 3.8 implies Theorem 3.5. In fact, let $\Phi \u2208BF(I)$ and take ${\Lambda n}n\u22651$ to be an increasing, exhaustive sequence of finite subsets of Γ. Define the sequence of interactions $\Phi n=\Phi \u21be\Lambda n$ and extend Φ_{n} to all of Γ as indicated above. In this case, it is clear that ${\Phi n}n\u22651$ is locally Cauchy in the *F*-norm defined by *F*, and moreover, (3.85) holds. Thus, the corresponding dynamics converge. Since the sequence ${\Phi n}n\u22651$ 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.

## IV. LOCAL APPROXIMATIONS

In Sec. III A 2, we proved a Lieb-Robinson bound for the finite volume dynamics generated by an interaction $\Phi \u2208BF(I)$. 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.

### A. Local approximations of observables

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 $B(H1)\u22971$ in $B(H1\u2297H2)$ is given by $1\u2297B(H2)$ (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 $\Lambda \u2208P0(\Gamma )$ and *X* ⊂ Λ, if $A\u2208A\Lambda $ satisfies [*A*, *B*] = 0 for all $B\u2208A\Lambda \X$, then $A\u2208AX$. 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 $A\u2032\u2208AX$.

*Lemma 4.1.*

*Let* $H1$ *and* $H2$ *be complex Hilbert spaces. There is a completely positive linear map* $E:B(H1\u2297H2)\u2192B(H1)$ *with the following properties:*

*For all*$A\u2208B(H1)$*,*$E(A\u22971)=A$*.**Whenever*$A\u2208B(H1\u2297H2)$*satisfies the commutator bound*$E(A)$$[A,1\u2297B]\u2264\u03f5\Vert B\Vert \u2009\u2009for all\u2009B\u2208B(H2),$*satisfies the estimate*

- (iii)
*For all*$C,D\u2208B(H1)$*and*$A\u2208B(H1\u2297H2)$*, we have*

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 $E=id\u2297tr$, 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 Π*

_{ρ}= id ⊗

*ρ satisfies properties*(

*i*)

*and*(

*iii*)

*of Lemma*4

*.*1

*. Moreover, if*$A\u2208B(H1\u2297H2)$

*is such that there is an ϵ*≥ 0

*for which*

*then*

*Proof.*

_{ρ}satisfies properties (i) and (iii) in Lemma 4.1. Moreover, since ∥Π

_{ρ}∥ = 1, we also have that for any $A\u2208B(H1\u2297H2)$,

*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 Π

_{ρ}(

*A*) in terms of its matrix elements. Since

*ρ*is given by a density matrix, there is a countable set of orthonormal vectors $\xi n\u2208H2$ and positive numbers

*ρ*

_{n}with

*∑*

_{n}

*ρ*

_{n}= 1 so that

*ρ*(

*A*′) =

*∑*

_{n≥1}

*ρ*

_{n}⟨

*ξ*

_{n},

*A*′

*ξ*

_{n}⟩ for all $A\u2032\u2208B(H2)$. The matrix elements of Π

_{ρ}(

*A*) are then given by

A number of further comments are in order. First, although the mapping Π_{ρ} 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 Π_{ρ} therefore appears to be nonoptimal. The map $E$ from Lemma 4.1 is only known to be bounded (specifically, $\Vert E\Vert =1$) 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 Π_{ρ} that is continuous on bounded subsets of the domain when both $B(H1\u2297H2)$ 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 $K:B(H1)\u2192B(H2)$ is continuous on bounded subsets with both its domain and codomain considered with the strong operator topology if given any bounded net $A\alpha \u2208B(H1)$ that converges strongly to $A\u2208B(H1)$, the net $K(A\alpha )\u2208B(H2)$ converges strongly to $K(A)\u2208B(H2)$.

*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:*

$\Pi \rho =id\u2297\rho :B(H1\u2297H2)\u2192B(H1)$,

$\Pi \u0303\rho :B(H1\u2297H2)\u2192B(H1\u2297H2),A\u21a6\Pi \rho (A)\u22971$

*.*

*Proof.*

_{ρ}is continuous on bounded sets, without loss of generality (WLOG), let {

*A*

_{α}∣

*α*∈

*I*} be a net in the unit ball of $B(H1\u2297H2)$ that converges to $A\u2208B(H1\u2297H2)$ in the strong operator topology, i.e., for all $\psi \u2208H1\u2297H2$, the net

*A*

_{α}

*ψ*converges to

*Aψ*with respect to the Hilbert space norm. Since ∥

*A*

_{α}∥ ≤ 1 for all

*α*∈

*I*, we necessarily have that ∥

*A*∥ ≤ 1. Let {

*ξ*

_{n},

*n*≥ 1} denote the orthonormal set of eigenvectors of

*ρ*corresponding to its nonzero eigenvalues

*ρ*

_{n}. Let $\varphi \u2208H1$ with ∥

*ϕ*∥ = 1. We use (4.6) to show that Π

_{ρ}(

*A*

_{α})

*ϕ*→ Π

_{ρ}(

*A*)

*ϕ*. Note that

*ϵ*> 0, choose

*N*so that

*∑*

_{n>N}

*ρ*

_{n}<

*ϵ*/4 and pick

*α*

_{0}∈

*I*so that for all

*α*≥

*α*

_{0}and any

*n*= 1, …,

*N*,

*α*≥

*α*

_{0},

_{ρ}.

- (ii)
Let {

*A*_{α}|*α*∈*I*} be a net in the unit ball of $B(H1\u2297H2)$ that converges to $A\u2208B(H1\u2297H2)$. By (i), we know that*B*_{α}= Π_{ρ}(*A*_{α}) converges to*B*= Π_{ρ}(*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 {*B*_{α}⊗ 1l∣*α*∈*I*} strongly converges to*B*⊗ 1l in $B(H1\u2297H2)$.

### B. Application to quantum lattice models

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 $A\Lambda $, for all $\Lambda \u2208P0(\Gamma )$. To this end, we consider a *locally normal product state ρ*, i.e., for each site *x* ∈ Γ, we fix a normal state *ρ*_{x} on $Ax=B(Hx)$ and take the unique state *ρ* on $A\Gamma $ such that $\rho \u21beA\Lambda =\u2297x\u2208\Lambda \rho x$ for all finite Λ ⊂ Γ. Then, given $X\u2282\Lambda \u2208P0(\Gamma )$, we define conditional expectations $\Pi \rho X,\Lambda :A\Lambda \u2192AX$