On adiabatic theory for extended fermionic lattice systems

Henheik, Joscha; Wessel, Tom

doi:10.1063/5.0123441

We review recent results on adiabatic theory for ground states of extended gapped fermionic lattice systems under several different assumptions. More precisely, we present generalized super-adiabatic theorems for extended but finite and infinite systems, assuming either a uniform gap or a gap in the bulk above the unperturbed ground state. The goal of this Review is to provide an overview of these adiabatic theorems and briefly outline the main ideas and techniques required in their proofs.

I. INTRODUCTION

We review four recent results on adiabatic theory for ground states of extended finite and infinite fermionic lattice systems at zero temperature.^1–3 These results are generalized super-adiabatic theorems (see Sec. I B) and concern Hamiltonians of the form

H^{ε} = H_{0} + ε V,

where the unperturbed Hamiltonian H₀ is a sum-of-local-terms (SLT) operator describing short-range interacting fermions and is assumed to have a spectral gap above its ground state. This gap might be closed by the (small) perturbation ɛV, which is given by a short-range Hamiltonian, a Lipschitz potential, or a sum of both. Consequently, the results presented in this Review are adiabatic theorems for resonances of H^ɛ (cf. Refs. 4 and 5).

The most important corollary and main motivation for proving such theorems in the context of extended fermionic lattice systems is the rigorous justification of linear response theory^1,6 and the Kubo formula⁷ for (topological) insulators,⁸ such as quantum Hall systems,⁹ where the prototypical relevant perturbation is a linear external potential modeling a constant electric field closing the gap of H₀ for every ɛ ≠ 0 (see Fig. 1).

FIG. 1.

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Let H₀ be a Hamiltonian with a gapped sector and a gap g. Perturbing with a Lipschitz potential v(x) = ɛ x, the gap gets closed (for large enough lattices). However, as indicated, a local gap persists and an electron at location x₀ would either need to overcome the gap (vertical arrow) or tunnel along the distance g/ɛ (horizontal arrow) in order to make a transition from the gapped sector.^1,6

In the remainder of this Introduction, we first briefly discuss the connection between linear response and adiabatic theory in Sec. I A (see also Refs. 1 and 6). Furthermore, we point out the key ingredients and developments which allowed to prove^1–3 the four adiabatic theorems presented in this Review. Afterward, in Sec. I B, we explain the notion of generalized super-adiabatic theorems and thereby introduce (super-adiabatic) non-equilibrium almost-stationary states (NEASSs)¹ as the above mentioned resonances of H^ɛ. A first brief but somewhat precise statement and overview of the results is given in Sec. I C.

A. Linear response and adiabatic theory

The formalism of linear response theory⁷ has been widely used in physics to calculate the response of a system in thermal equilibrium to external perturbations. Put briefly, linear response theory provides an answer to the following question: What is the response of a system described by a Hamiltonian H₀, which is initially in an equilibrium state ρ₀, to a small static perturbation ɛV? Or, in somewhat more mathematical terms, what is the change¹⁰

ρ_{ε} (A) - ρ_{0} (A) = ε σ_{A} + o (ε)

of the expectation value of an observable A induced by the perturbation ɛV to leading order in its strength 0 < ɛ ≪ 1? Here, ρ_ɛ denotes the state of the system after the perturbation has been (adiabatically) turned on and σ_A denotes the linear response coefficient.

The answer to this fundamental question of linear response clearly hinges on the problem of determining ρ_ɛ. Although in few particular situations one expects ρ_ɛ to remain an equilibrium state for the perturbed Hamiltonian H^ɛ = H₀ + ɛV, the original linear response theory⁷ was developed for situations where the system is driven out of equilibrium, i.e., ρ_ɛ being a resonance state. As prominently formulated by Simon¹¹ in his “Fifteen problems in mathematical physics” from 1984, the latter non-equilibrium situation causes the main challenges in a rigorous mathematical treatment. However, in either case, the linear response coefficient σ_A is customarily expected to be given by the celebrated Kubo formula,⁷ and rigorously justifying it was formulated as one of the problems by Simon.¹¹ For a more detailed recent review on the (mathematical) problem of proving Kubo’s formula and its relevance in the context of quantum Hall systems, we refer to Ref. 6.

In a nutshell, the problem of justifying linear response theory and proving Kubo’s formula is thus to verify that a system, initially in an equilibrium state ρ₀, is adiabatically driven by a small perturbation ɛV into a non-equilibrium state ρ_ɛ ≈ ρ₀. Since the perturbation acts over a very long (macroscopic) time, this problem clearly supersedes standard perturbation theory: The change of the state being small is not a trivial consequence of the smallness of the perturbation ɛV. Instead, verifying that the two states, ρ_ɛ and ρ₀, are close to each other requires an adiabatic-type theorem.

However, even in our rather simple setting (zero temperature, assuming that ρ₀ is the gapped ground state of H₀ describing an extended fermionic lattice system, the perturbation ɛV might close the gap), the problem of justifying the linear response formalism also goes beyond standard adiabatic theory. In fact, the applicability of the standard adiabatic theorem of quantum mechanics is rather restrictive for the following three reasons:

The standard adiabatic theorem requires the perturbation ɛV to not close the spectral gap. In that scenario, it asserts that ρ_ɛ is (close to) the gapped ground state of H^ɛ = H₀ + ɛV and, as such, a (nearly) equilibrium state.
Even if we neglect the first issue, the usual adiabatic theorem estimates the difference between ρ_ɛ and the ground state of the perturbed Hamiltonian H^ɛ in operator norm, leaving the translation to local differences in expectation values as an additional and potentially non-trivial step.
In general, extended systems are plagued by the orthogonality catastrophe: Whenever for single-particle states $ψ, \tilde{ψ}$ we have $‖ ψ - \tilde{ψ} ‖ \sim ε$ ⁠, the non-interacting many-particle states ⊗_x∈Λψ_x and $\otimes_{x \in Λ} {\tilde{ψ}}_{x}$ satisfy $‖ \otimes_{x \in Λ} ψ_{x} - \otimes_{x \in Λ} {\tilde{ψ}}_{x} ‖ \sim ε | Λ |$ ⁠, i.e., the norm-estimate deteriorates when |Λ| → ∞. This means that the approximation error in the standard adiabatic theorem grows with the systems size, and it is thus not applicable for macroscopic systems.

A major breakthrough in overcoming these obstacles has recently been achieved by Bachmann, De Roeck, and Fraas¹² (see also their introductory lecture notes¹³). They proved the first adiabatic theorem for extended (but finite) lattice systems with short-range interactions, thereby solving the second and third problem in the list above. More precisely, their result concerns differences in expectation values and provides error estimates, which are uniform in the system size.

For these lattice systems with short-range interactions, well-known Lieb–Robinson bounds^14–16 ensure a finite speed of correlation and prevent the build-up of long-range entanglement. Having Lieb–Robinson bounds at hand allowed Bachmann et al.¹⁸ to prove that the generator of the spectral flow, introduced by Hastings and Wen,¹⁷ is an SLT operator and thus preserves good locality properties. The general spectral flow technique can then be used to prove automorphic equivalence of two gapped ground states ρ₀ and ρ₁ of Hamiltonians H(0) and H(1), respectively: Given a smooth path s ↦ H(s) of (uniformly) gapped SLT Hamiltonians, their ground states are automorphically equivalent (equal up to a conjugation by unitaries) with the generator of the automorphism being an SLT operator.¹⁸ This automorphic equivalence allowed Bachmann et al.¹² to prove a super-adiabatic theorem (see Sec. I B for an explanation of this notion) for such systems, however, still requiring the spectral gap not only for H₀ but also for H^ɛ, i.e., the gap must remain open.¹⁹

The four theorems presented in this Review also solved the last remaining problem given under item (i) in the above list, i.e., they allow the perturbation ɛV to close the spectral gap of H₀. The main idea for establishing this generalization is that a spatially local gap should suffice for an adiabatic theorem to hold. This underlies the space-time adiabatic perturbation theory originally developed for non-interacting fermions by Panati, Spohn, and Teufel,^20,21 where one utilizes a gap that exists locally in space (and time) but does not exist globally. It also underlies the recent results by De Roeck, Elgart, and Fraas,²² where an adiabatic theorem holds even if the “spectral gap” is filled with eigenvalues, whose corresponding eigenvectors are spatially localized, leaving a gap (with smaller size) locally open. Finally, this is also the idea behind Theorems III and, especially, IV, where one still has an adiabatic-type theorem although the gap closes at the boundary of the lattices.

Combining the ideas from the space-time adiabatic perturbation theory with the methods invented in Ref. 12, the first of the four theorems presented in this Review was proven by Teufel.¹ It concerns extended but finite systems and requires a spectral gap for H₀, uniformly in the system size [see Assumption (GAP_unif)]. The precise statement is formulated in Theorem I. In order to extend this result from finite lattices to an infinite system, Henheik and Teufel² adapted ideas from Nachtergaele, Sims, and Young¹⁶ on controlling the thermodynamic limit of automorphisms with SLT generators. This result is formulated in Theorem II.

So far, all the mentioned results were obtained under the assumption of a (uniform) spectral gap for the finite systems (which also implies a gap for the infinite system). However, the recent result on automorphic equivalence with a gap only in the bulk (via the GNS construction) by Moon and Ogata²³ opened the door for a new class of adiabatic theorems, where the unperturbed Hamiltonian H₀ is no longer required to have a uniform spectral gap. Instead, Theorem III, originally proven by Henheik and Teufel,³ is a result for the infinite volume states and requires a gap in the bulk. This technically means a gap for the infinite system [cf. Assumption (GAP_bulk)] but can be understood as requiring a local gap in the interior of the finite lattices (cf. Remark 4).

Moreover, by employing strong locality estimates from Refs. 16 and 23, and assuming fast convergence of ground states, Theorem III can be traced back to extended but finite systems that only have a gap in the bulk. This was also proven in Ref. 3 and is formulated in Theorem IV.

B. Non-equilibrium almost-stationary states

For the results presented in this Review, we consider time-dependent families

H^{ε} (t) = H_{0} (t) + ε V (t), t \in I \subset R,

(1)

of many-body Hamiltonians for lattice fermions in $Γ \subset Z^{d}$ with short-range interactions. Here, Γ will either be a finite box Λ or the whole of $Z^{d}$ ⁠. For each t ∈ I, we denote by ρ₀(t) the instantaneous ground state of H₀(t) on the (quasi-local) algebra of observables $A_{Γ}$ ⁠. For simplicity of the presentation, we shall assume that the ground state is unique.²⁴ Moreover, we assume that the ground state is separated by a gap from the rest of the spectrum [see Assumptions (GAP_unif) and (GAP_bulk) in Sec. III for the precise formulation]. The perturbation V(t) can be a Hamiltonian with short-range interactions or a possibly unbounded external Lipschitz potential or a sum of both [see Sec. II and Assumptions (INT₁)–(INT₄) in Sec. III].

As mentioned above, the main results presented in this Review are so-called generalized super-adiabatic theorems for ρ₀(t), which we briefly explain in the following. For ɛ = 0, the results are “standard” super-adiabatic theorems and establish the existence of super-adiabatic states $ρ_{0}^{η} (t)$ on $A_{Γ}$ close to ρ₀(t), i.e.,

|ρ_{0}^{η} (t) (A) - ρ_{0} (t) (A)| = O (η),

such that the adiabatic time-evolution $U_{t, t_{0}}^{η}$ on $A_{Γ}$ generated by $\frac{1}{η} H_{0} (\cdot)$ intertwines the super-adiabatic states to all orders in the adiabatic parameter η > 0, i.e.,

|ρ_{0}^{η} (t_{0}) (U_{t, t_{0}}^{η} [[A]]) - ρ_{0}^{η} (t) (A)| = O (η^{\infty})

(2)

for all A in a dense subspace $D \subset A_{Γ}$ ⁠. Throughout this entire Review, we shall study our system in the Heisenberg picture, meaning that the observable A evolves in time, not the state $ρ_{0}^{η} (t_{0})$ (see also Proposition 3). Note that the comparison state $ρ_{0}^{η} (t)$ does not involve any time evolution but simply depends on the Hamiltonian at time t (see Definition 1 for details). Here and in the following, we write the arguments of (densely defined) linear operators on $A_{Γ}$ inside the brackets [[⋅]] for better readability.

For ɛ > 0, the scope of the adiabatic theorem (2) extends considerably since the perturbation ɛV(t) might close the spectral gap and turn the ground state ρ₀(t) of H₀(t) into an instantaneous resonance state Π^ɛ(t) for H^ɛ(t). These states have a lifetime of order $O (ε^{- \infty})$ for the dynamics $s \mapsto e^{i s L_{H^{ε} (t)}}$ ⁠, with $L_{H^{ε} (t)} [[\cdot]] ≔ [H^{ε} (t), \cdot]$ (formally) denoting the derivation associated with H^ɛ(t). That is, for all $n \in N$ and fixed t, it holds that

|Π^{ε} (t) (e^{i s L_{H^{ε} (t)}} [[A]]) - Π^{ε} (t) (A)| = O (ε^{n} (1 + | s |^{d + 1})),

which is why they were called non-equilibrium almost-stationary states (NEASSs) in this context by Teufel.¹ The generalized super-adiabatic theorems then establish the existence of a super-adiabatic NEASS Π^ɛ,η(t) on $A_{Γ}$ close to Π^ɛ(t) such that the adiabatic time-evolution $U_{t, t_{0}}^{ε, η}$ generated by $\frac{1}{η} H^{ε} (\cdot)$ approximately intertwines the super-adiabatic NEASSs in the following sense: for any n > d and for all $A \in D \subset A_{Γ}$ ⁠, we have

|Π^{ε, η} (t_{0}) (U_{t, t_{0}}^{ε, η} [[A]]) - Π^{ε, η} (t) (A)| = O (η^{n - d} + \frac{ε^{n + 1}}{η^{d + 1}})

(3)

uniformly for t in compact sets, which we call a generalized super-adiabatic theorem.

In our setting of gapped Hamiltonians H₀ describing insulating materials, there is indeed a clear and simple physical picture suggesting the existence of NEASSs for H^ɛ, as observed in Refs. 1 and 6 (see Fig. 1). For simplicity, assume that H₀ is a periodic one-body operator in one spatial dimension and that the Fermi energy μ (chemical potential) lies in a gap of size g. For the perturbation, we consider the potential of a small constant electric field ɛ. In the initial state ρ₀, before the perturbation is turned on, all one-body states with energy smaller than μ are occupied. After the voltage has been applied, the energy of an electron located at position x₀ gets substantially shifted by ɛ x₀, but is only subject to small force of order ɛ. As indicated in Fig. 1, in order to make a transition, such an electron must either overcome the gap of size g or tunnel a macroscopic distance of order g/ɛ. Thus, although ρ₀ is neither close to the ground state nor any other equilibrium state of the perturbed Hamiltonian H^ɛ = H₀ + ɛV, it is still almost stationary for H^ɛ. This heuristic picture remains valid if short-range interactions between the electrons are taken into account.

While for ɛ = 0 the generalized super-adiabatic theorem (3) reduces to the standard one (2), for 0 < ɛ ≪ 1, the right-hand side of (3) is small if and only if also η is small, but not too small compared to ɛ, i.e., $ε^{\frac{n + 1}{d + 1}} ≪ η ≪ 1$ for some $n \in N$ ⁠. Physically, this simply means that the adiabatic approximation breaks down when the adiabatic switching occurs at times that exceed the lifetime of the NEASS, an effect that has been observed in adiabatic theory for resonances before; see, e.g., Refs. 4 and 5. It can also be heuristically understood from the tunneling picture given in Fig. 1.

Moreover, in view of the linear response problem discussed in Sec. I A, let us only mention here that a statement such as (3), in fact, yields a solution to this problem after expanding the state Π^ɛ,η(t) in powers of ɛ, where the linear term [eventually stemming from the first order operator A₁ given in (18)] does, in fact, constitute Kubo’s formula. See Refs. 1, 2, and 6 for details.

C. Brief statement of the results

We shall establish the existence of super-adiabatic NEASSs in four generally quite different situations; the main differences are also summarized in Table I:

On finite systems $Λ_{k} ⋐ Z^{d}$ with suitable boundary conditions, assuming that the unperturbed Hamiltonians $H_{0}^{Λ_{k}} (t)$ have a gapped ground state uniformly in Λ_k, there exist NEASSs on $A_{Λ_{k}}$ such that the constants in (3) are independent of Λ_k. See Theorem I and Ref. 1.
Additionally assuming convergence of the Hamiltonians (they have a thermodynamic limit; cf. Definition 2) and ground states, there also exists a super-adiabatic NEASS on $A_{Z^{d}}$ after taking the thermodynamic limit $Λ_{k} ↗ Z^{d}$ ⁠. See Theorem II and Ref. 2.
For the infinite system $Z^{d}$ ⁠, assuming that the unperturbed Hamiltonian H₀ has a unique gapped ground state (via the GNS construction), there exists a NEASS on $A_{Z^{d}}$ ⁠, while a (uniform) spectral gap for finite sub-systems is not required. See Theorem III and Ref. 3.
Additionally assuming a quantitative control on the convergence of the finite volume Hamiltonians $H^{Λ_{k}} (t)$ (they have a rapid thermodynamic limit; cf. Definition 5) and the unperturbed ground states in the thermodynamic limit, there also exist NEASSs on $A_{Λ_{k}}$ (again with a uniform constant) up to an error vanishing faster than any inverse polynomial in the distance to the boundary. See Theorem IV and Ref. 3.

TABLE I.

Overview of the adiabatic theorems and the original papers.

	Finite volume	Infinite volume
Uniform gap	Theorem I; see Ref. 1	Theorem II; see Ref. 2
Gap in the bulk	Theorem IV; see Ref. 3	Theorem III; see Ref. 3

	Finite volume	Infinite volume
Uniform gap	Theorem I; see Ref. 1	Theorem II; see Ref. 2
Gap in the bulk	Theorem IV; see Ref. 3	Theorem III; see Ref. 3

A typical example of a physically relevant class of Hamiltonians,^1,6,25 to which the above generalized super-adiabatic theorems apply, is given by

\begin{aligned} H_{0}^{Λ_{k}} = & \sum_{x, y \in Λ_{k}} a_{x}^{*} T (x - y) a_{y} + \sum_{x \in Λ_{k}} a_{x}^{*} ϕ (x) a_{x} + \sum_{x, y \in Λ_{k}} a_{x}^{*} a_{x} W () d^{Λ_{k}} (x, y) () a_{y}^{*} a_{y} - μ N_{Λ_{k}}, \end{aligned}

(4)

modeling Chern or topological insulators. In agreement with the precise locality assumptions (INT₁)–(INT₄) in Sec. III, we suppose that the kinetic term $T : Z^{d} \to L (C^{r})$ is an exponentially decaying function with T(−x) = T(x)*, the potential term $ϕ : Z^{d} \to L (C^{r})$ is a bounded function taking values in the self-adjoint matrices, and the two-body interaction $W : [0, \infty) \to L (C^{r})$ is exponentially decaying and also takes values in the self-adjoint matrices. Note that x – y in the kinetic term refers to the difference modulo the imposed boundary condition on Λ_k. In the first two terms of (4), a_x is the column vector of the annihilation operators a_x,i (i labels internal degrees of freedom, such as spin) and $a_{x}^{*}$ is the row vector of the creation operators $a_{x, i}^{*}$ (see Sec. II). In addition, with a slight abuse of notation in the third term of (4), we wrote $a_{x}^{*} a_{x}$ for the row vector with entries $a_{x, i}^{*} a_{x, i}$ and $a_{y}^{*} a_{y}$ for the column vector with entries $a_{y, i}^{*} a_{y, i}$ ⁠.

It is well known that non-interacting Hamiltonians H₀, i.e., with W ≡ 0, of type (4) on a torus (periodic boundary condition) have a uniform spectral gap [see Assumption(GAP_unif)] whenever the chemical potential μ multiplying the number operator lies in a gap of the spectrum of the corresponding one-body operator on the infinite domain. It was recently shown^26,27 that the spectral gap remains open when perturbing by sufficiently small short-range interactions W ≠ 0. On the other hand, the Hamiltonian H₀ on a cube with open boundary condition has, in general, no longer a spectral gap because of the appearance of edge states. However, away from the boundary, a gap in the bulk [see Assumption (GAP_bulk)] is still present. While also the uniqueness of the ground state is expected to hold for such models, to our knowledge, it has been shown only for certain types of quantum spin systems; cf. Refs. 28–32. For further details, we refer to the original papers.^1–3 Finally, it is an interesting program to extend Table I by further rows representing different notions of a spectral gap for H₀, e.g., a local gap as in Ref. 32 or even only a mobility gap (see Ref. 22 for a first result in this direction).

After a brief introduction to the relevant mathematical framework in Sec. II, we formulate the four main theorems in Sec. III. Ideas of their proofs are provided in Sec. IV.

II. MATHEMATICAL FRAMEWORK

In this section, we briefly introduce the (standard) mathematical framework used in the formulation of the adiabatic theorems. More explanations and details are provided in Refs. 1–3.

A. Algebra of observables

We consider fermions with r spin or other internal degrees of freedom on the lattice $Z^{d}$ ⁠. Let ${X ⋐ Z^{d}} ≔ {X \subset Z^{d} : | X | < \infty}$ denote the set of finite subsets of $Z^{d}$ ⁠, where |X| is the number of elements in X. For each $X ⋐ Z^{d}$ ⁠, let $F_{X}$ be the fermionic Fock space built up from the one-body space $ℓ^{2} (X, C^{r})$ ⁠. The C*-algebra of bounded operators $A_{X} ≔ L (F_{X})$ is generated by the identity element $1_{A_{X}}$ and the creation and annihilation operators $a_{x, i}^{*}$ ⁠, a_x,i for x ∈ X and 1 ≤ i ≤ r, which satisfy the canonical anti-commutation relations (CARs). Whenever X ⊂ X′, then $A_{X}$ is naturally embedded as a subalgebra of $A_{X^{'}}$ ⁠. For infinite systems, the algebra of local observables is defined as the inductive limit

A_{loc} ≔ ⋃_{X ⋐ Z^{d}} A_{X}, and its completion A_{Z^{d}} ≔ {\bar{A_{loc}}}^{‖ \cdot ‖}

with respect to the operator norm ‖ ⋅‖ is a C*-algebra, called the quasi-local algebra. The even elements $A_{Z^{d}}^{+} \subset A_{Z^{d}}$ form a C*-subalgebra. In addition, note that for any $X ⋐ Z^{d}$ ⁠, the set of elements $A_{X}^{N}$ commuting with the number operator $N_{X} ≔ \sum_{x \in X} a_{x}^{*} a_{x} ≔ \sum_{x \in X} \sum_{i = 1}^{r} a_{x, i}^{*} a_{x, i}$ forms a subalgebra of the even subalgebra, i.e., $A_{X}^{N} \subset A_{X}^{+} \subset A_{X}$ ⁠. As only even observables will be relevant to our considerations, we will drop the superscript ⁺ from now on and redefine $A_{Z^{d}} ≔ A_{Z^{d}}^{+}$ ⁠.

Since a very similar construction is common for quantum spin systems (see, e.g., Ref. 16), all the results immediately translate to this setting.

B. Interactions and operator families

We shall consider sequences of Hamiltonians defined on centered boxes Λ_k ≔ {−k, …,+k}^d of size 2k with metric $d^{Λ_{k}} (\cdot, \cdot)$ ⁠. This metric may differ from the standard ℓ¹-distance d(⋅, ⋅) on $Z^{d}$ restricted to Λ_k if one considers discrete tube or torus geometries, but satisfies the bulk-compatibility condition,

\forall k \in N \forall x, y \in Λ_{k} : d^{Λ_{k}} (x, y) \leq d (x, y) and d^{Λ_{k}} (x, y) = d (x, y) whenever d (x, y) \leq k .

An interaction on a domain Λ_k is a map

Φ^{Λ_{k}} : \{X \subset Λ_{k}\} \to A_{Λ_{k}}^{N}, X \mapsto Φ^{Λ_{k}} (X) \in A_{X}^{N}

with values in the self-adjoint operators. Note that the maps $Φ^{Λ_{k}}$ can be extended to maps on the whole ${X ⋐ Z^{d}}$ or restricted to a smaller Λ_l, trivially. In order to describe fermionic systems on the lattice $Z^{d}$ in the thermodynamic limit, one considers sequences $Φ = {(Φ^{Λ_{k}})}_{k \in N}$ of interactions on domains Λ_k and calls the whole sequence an interaction.

An infinite volume interaction is a map,

Ψ : {X ⋐ Z^{d}} \to A_{loc}^{N}, X \mapsto Ψ (X) \in A_{X}^{N},

again with values in the self-adjoint operators. Such an infinite volume interaction defines a general interaction $Ψ = {(Ψ^{Λ_{k}})}_{k \in N}$ by restriction, i.e., by setting $Ψ^{Λ_{k}} ≔ Ψ |_{{X \subset Λ_{k}}}$ ⁠.³³ With any interaction Φ, one associates an operator family, which is a sequence $A = {(A^{Λ_{k}})}_{k \in N}$ of self-adjoint operators,

A^{Λ_{k}} ≔ A^{Λ_{k}} (Φ) ≔ \sum_{X \subset Λ_{k}} Φ^{Λ_{k}} (X) \in A_{Λ_{k}}^{N} .

For any a > 0 and $n \in N_{0}$ ⁠, we define the norm

‖ Φ ‖_{a, n} ≔ \sup_{k \in N} \sup_{x, y \in Z^{d}} \sum_{\begin{array}{c} X \subset Λ_{k} : \\ x, y \in X \end{array}} d^{Λ_{k}} -diam {(X)}^{n} e^{a \cdot d^{Λ_{k}} (x, y)} ‖ Φ^{Λ_{k}} (X) ‖

(5)

on the space of interactions.³⁴ Note that these norms depend on the sequence of metrics $d^{Λ_{k}}$ on the cubes Λ_k, i.e., on the boundary conditions.

Similar constructions for interactions and interaction norms are long known. More commonly, the norms are independent of the particular lattice Λ_k and the interaction ${(Φ^{Λ_{k}})}_{k \in N}$ is given by restrictions of a single infinite volume interaction. Moreover, in earlier works,^35,36 the authors did not require additional decay properties, which were only added later (see, e.g., Refs. 16, 37, and 38). The use of interactions and corresponding norms, which are not simply restrictions of an infinite volume interaction, originates in Ref. 25 to incorporate non-trivial boundary conditions. In order to control commutators with Lipschitz potentials (see Sec. II C), the dependence on the diameter $d^{Λ_{k}} -diam (X)$ was added in Ref. 1. Finally, to ensure the existence of the thermodynamic limit, it is necessary to require the bulk-compatibility condition.^2,3 Yet another variant of defining interaction norms is to replace dist(x, y) with diam(X) in (5) (see, e.g., Refs. 38 and 39).

In order to quantify the difference of interactions in the bulk (see Sec. III B), we also introduce for any interaction $Φ^{Λ_{l}}$ on the domain Λ_l and any Λ_M ⊂ Λ_l the quantity

‖ Φ^{Λ_{l}} ‖_{a, n, Λ_{M}} ≔ \sup_{x, y \in Λ_{M}} \sum_{\begin{array}{c} X \subset Λ_{M} : \\ x, y \in X \end{array}} diam {(X)}^{n} e^{a \cdot d (x, y)} ‖ Φ^{Λ_{l}} (X) ‖,

where d and diam now refer to the ℓ¹-distance on $Z^{d}$ ⁠.

Let $B_{a, n}$ be the Banach space of interactions with finite ‖ ⋅‖_a,n-norm, and define the space of exponentially localized interactions as the intersection $B_{a, \infty} ≔ ⋂_{n \in N_{0}} B_{a, n}$ ⁠. In the literature, the vector spaces of operator families, which can be written in terms of such interactions, are denoted by $L_{a, n}$ and $L_{a, \infty}$ ⁠. Moreover, we will be a bit sloppy in the following terminology and call the elements $A^{Λ_{k}}$ of an operator sequence A sum-of-local-terms (SLT) operators whenever its interaction Φ_A has a finite interaction norm similar to (5), but with the exponential replaced by a function growing faster than any polynomial. This will allow us to formulate the results and the ideas of the proofs without too many details. For the precise conditions, see, e.g., see Ref. 2, Sec. 2.2.

Now, let $I \subset R$ be an open interval. We say that a map $Φ : I \to B_{a, n}$ is smooth and bounded whenever it is (i) term- and point-wise smooth in t ∈ I, i.e., $t \mapsto Φ^{Λ_{k}} (t, X)$ are C^∞-functions for all $k \in N$ and X ⊂ Λ_k, and (ii) $\sup_{t \in I} ‖ \frac{d^{i}}{d t^{i}} Φ (t) ‖_{a, n} < \infty$ for all $i \in N_{0}$ ⁠. The corresponding spaces of smooth and bounded time-dependent interactions and operator families are denoted by $B_{I, a, n}$ and $L_{I, a, n}$ and are equipped with the norm ‖Φ‖_I,a,n ≔ sup_t∈I‖Φ(t)‖_a,n. We say that $Φ : I \to B_{a, \infty}$ is smooth and bounded if $Φ : I \to B_{a, n}$ is smooth and bounded for all $n \in N_{0}$ ⁠, and we write $B_{I, a, \infty}$ and $L_{I, a, \infty}$ for the corresponding spaces of time-dependent exponentially localized interactions and operator families, respectively.

For (time-dependent) infinite volume interactions Ψ, we add a superscript ° to the norms and to the normed spaces defined above, emphasizing, in particular, the use of open boundary conditions, i.e., $d^{Λ_{k}} \equiv d$ ⁠. Note that the compatibility condition for the metrics $d^{Λ_{k}}$ implies that ‖Ψ‖_a,n ≤ ‖Ψ‖°_a,n.

C. Lipschitz potentials

For the perturbation, we will allow external potentials $v = {(v^{Λ_{k}} : Λ_{k} \to R)}_{k \in N}$ that satisfy the Lipschitz condition,

C_{v} ≔ \sup_{k \in N} \sup_{\begin{array}{c} x, y \in Λ_{k} : \\ x \neq y \end{array}} \frac{|v^{Λ_{k}} (x) - v^{Λ_{k}} (y)|}{d^{Λ_{k}} (x, y)} < \infty,

(6)

and call them for short Lipschitz potentials.⁴⁰ With a Lipschitz potential v, we associate the corresponding operator-sequence $V_{v} = {(V_{v}^{Λ_{k}})}_{k \in N}$ defined by

V_{v}^{Λ_{k}} ≔ \sum_{x \in Λ_{k}} v^{Λ_{k}} (x) a_{x}^{*} a_{x}

and denote the space of Lipschitz potentials by $V$ ⁠. We emphasize that, since $\sup_{k \in N} \sup_{\begin{array}{c} x \in Λ_{k} \end{array}} | v^{Λ_{k}} (x) |$ might be infinite, V_v is, in general, no SLT operator. However, this is still more restrictive than general onsite potentials because it only varies slowly in space. Moreover, we say that the map $v : I \to V$ is smooth and bounded whenever (i) $v^{Λ_{k}} (x, \cdot)$ are C^∞-functions for all $k \in N$ and x ∈ Λ_k and (ii) it satisfies $\sup_{t \in I} C_{\frac{d^{i}}{d t^{i}} v (t)} < \infty$ for all $i \in N_{0}$ ⁠. The space of smooth and bounded time-dependent Lipschitz potentials is denoted by $V_{I}$ ⁠.

As above, we also introduce infinite volume Lipschitz potentials $v_{\infty} : Z^{d} \to R$ ⁠, which, again by restriction and invoking the compatibility condition for the metrics $d^{Λ_{k}}$ ⁠, can be viewed as a Lipschitz potential with $d^{Λ_{k}} \equiv d$ in (6). In addition, analogously to Sec. II B, for (time-dependent) infinite volume Lipschitz potentials, we add a superscript ° to the constant from (6) and to the spaces, emphasizing the use of open boundary conditions. Note that the compatibility condition for the metrics $d^{Λ_{k}}$ implies that $C_{v} \geq {C °}_{v}$ ⁠.

III. ADIABATIC THEOREMS FOR GAPPED QUANTUM SYSTEMS

As mentioned in the Introduction, we shall distinguish two generally quite different settings regarding the presence of a spectral gap of the unperturbed Hamiltonian H₀ grouped as Theorem I and Theorem II in Sec. III A and Theorem III and Theorem IV in Sec. III B. First, in Sec. III A, we will work under the assumption that there exists a sequence of subsystems ${(Λ_{k})}_{k \in N}$ equipped with an appropriate metric (reflecting, e.g., periodic boundary conditions), ensuring that all (but finitely many) $H_{0}^{Λ_{k}}$ have a uniform gap above their ground state, which is made precise in Assumption (GAP_unif). Then, in Sec. III B, however, we drop this assumption and solely assume that H₀ has a gap in the bulk, meaning that the GNS Hamiltonian, describing the system in the thermodynamic limit, has a spectral gap above its ground state eigenvalue zero [see Assumption (GAP_bulk)]. Note that the second group of results is more general than the first group with regard to the gap condition since a uniform gap for finite systems guarantees a spectral gap for the GNS Hamiltonian describing the infinite system (see Proposition 5.4 in Ref. 41). Therefore, the second row in Table I somewhat improves the results in the first row since finding a suitable geometry for which one already has a spectral gap for finite systems is no longer necessary.

In the precise formulation of the adiabatic theorems, we shall frequently use the abbreviating phrase that a state Π^ɛ,η(t) is a super-adiabatic NEASS (see Sec. I B), which we generally define as follows: reminiscent of Refs. 1–3.

Definition 1

(Super-adiabatic non-equilibrium almost-stationary states).

We assume to be in the following general setting, which is made precise in concrete situations: For (small) ɛ > 0, define the time-dependent Hamiltonian,

H^{ε} (t) = H_{0} (t) + ε V (t), t \in I, on Γ \subset Z^{d},

and let ρ₀(t) be (close to)⁴² the ground state of H₀(t). Moreover, denote the Heisenberg time-evolution on the algebra of (quasi-local) observables

A_{Γ}

generated by

\frac{1}{η} H^{ε} (t)

as

U_{t, t_{0}}^{ε, η}

⁠, where t, t₀ ∈ I for some open interval

I \subset R

and η > 0 is a (small) adiabatic parameter.

Then, we say that a state Π^ɛ,η(t) on $A_{Γ}$ is a super-adiabatic non-equilibrium almost-stationary state for the state ρ₀(t) and the time-evolution $U_{t, t_{0}}^{ε, η}$ on $A_{Γ}$ if it satisfies the following properties:

Π^ɛ,η almost intertwines the time evolution: For any $n \in N$ ⁠, there exists a constant C_n such that for any t, t₀ ∈ I and for all X ⋐ Γ and $A \in A_{X} \subset A_{Γ}$ ⁠, we have
$|Π^{ε, η} (t_{0}) (U_{t, t_{0}}^{ε, η} [[A]]) - Π^{ε, η} (t) (A)| \leq C_{n} \frac{ε^{n + 1} + η^{n + 1}}{η^{d + 1}} (1 + | t - t_{0} |^{d + 1}) ‖ A ‖ | X |^{2} .$
(7)
Π^ɛ,η is local in time: Π^ɛ,η(t) only depends on H₀ and V and their time derivatives at time t.
Π^ɛ,η is stationary whenever the Hamiltonian is stationary: If for some fixed t ∈ I all time-derivatives of H₀ and V vanish at time t, then Π^ɛ,η(t) equals the NEASS⁴³ Π^ɛ(t) for the instantaneous ground state ρ₀(t) and the time-evolution $s \mapsto e^{i s L_{H^{ε} (t)}}$ generated by the time-independent Hamiltonian H^ɛ(t).
Π^ɛ,η equals the (approximate) ground state ρ₀ of H₀ whenever the perturbation vanishes and the Hamiltonian is stationary: If for some t ∈ I all time-derivatives of H₀ and V vanish at time t and V(t) = 0, then Π^ɛ,η(t) = Π^ɛ,0(t) = ρ₀(t).

We could have written bound (7) in a more general form as indicated by (3). For example, we could allow (1 + |t − t₀|^d+1) to be replaced by a constant C_K < ∞, depending only on a compact subset K ⊂ I of times or, similarly, |X|² to be replaced by a constant C_X < ∞, depending only on the support $X ⋐ Z^{d}$ of the observable A. In addition, the power of η in the denominator could be allowed to be more general, e.g., some constant C_d < ∞ instead of d + 1. However, the concrete form of (7) indeed matches the precise bounds of the results in Sec. III.

A. Systems with a uniform gap

Throughout this section, we assume that H₀ has a uniformly gapped unique ground state in the following sense.

(GAP_unif) Assumptions on the ground state of H₀.

Let $Φ_{H_{0}} = {(Φ_{H_{0}}^{Λ_{k}})}_{k \in N}$ be an interaction. There exists $L \in N$ such that for all t ∈ I, k ≥ L, and corresponding Λ_k, the operator $H_{0}^{Λ_{k}} (t)$ has a simple gapped ground state eigenvalue $E_{0}^{Λ_{k}} (t) = \inf σ (H_{0}^{Λ_{k}} (t))$ ⁠, i.e., there exists g > 0 such that $d i s t (E_{0}^{Λ_{k}} (t), σ (H_{0}^{Λ_{k}} (t)) \ {E_{0}^{Λ_{k}} (t)}) \geq g$ for all t ∈ I, k ≥ L. We denote the spectral projection of $H_{0}^{Λ_{k}} (t)$ corresponding to $E_{0}^{Λ_{k}} (t)$ by $P_{0}^{Λ_{k}} (t)$ and write $ρ_{0}^{Λ_{k}} (t) (\cdot) ≔ t r (P_{0}^{Λ_{k}} (t) \cdot)$ for the canonically associated state on $A_{Λ_{k}}$ ⁠.

A physically relevant class of Hamiltonians satisfying this assumption (possibly up to the uniqueness, which we require for simplicity of the presentation) was given in (4) in Sec. I C. In the following, we shall present adiabatic theorems for extended but finite systems (Theorem I) and for infinite systems (Theorem II) under Assumption (GAP_unif).

1. Extended but finite systems

The basic assumption on the Hamiltonian says that it is composed of exponentially localized interactions and/or a Lipschitz potential.

(INT₁) Assumptions on the interactions.

Let H₀, H₁ be the Hamiltonians of two time-dependent exponentially localized interactions, i.e., $Φ_{H_{0}}, Φ_{H_{1}} \in B_{I, a, \infty}$ for some a > 0, and $v \in V_{I}$ be a time-dependent Lipschitz potential.

The following results due to Teufel¹ marks the starting point for generalized super-adiabatic theorems for extended fermionic lattice systems.

Theorem I

[Adiabatic theorem for finite systems with a uniform gap (see Ref. 1, Theorem 5.1)].

Under Assumptions (GAP_unif) and (INT₁), there exists a sequence of near-identity⁴⁴ automorphisms

β^{ε, η, Λ_{k}} (t) = e^{i ε L_{S^{ε, η} (t)}^{Λ_{k}}}

with SLT generators S^ɛ,η for any ɛ, η ∈ (0, 1] and t ∈ I such that the states

Π^{ε, η, Λ_{k}} (t) ≔ ρ_{0}^{Λ_{k}} (t) ◦ β^{ε, η, Λ_{k}} (t)

(8)

are super-adiabatic NEASSs for the Heisenberg time-evolution

U_{t, t_{0}}^{ε, η, Λ_{k}}

on

A_{Λ_{k}}

generated by

\frac{1}{η} H^{ε, Λ_{k}} (\cdot)

with

\frac{1}{η} H^{ε, Λ_{k}} (t) ≔ \frac{1}{η} (H_{0}^{Λ_{k}} (t) + ε (V_{v}^{Λ_{k}} (t) + H_{1}^{Λ_{k}} (t)))

uniformly in k ≥ L. That is, for every

n \in N

, there exists a constant C_n, such that for any

A \in A_{X}

, ɛ, η ∈ (0, 1], and all t, t₀ ∈ I, it holds that

\sup_{k \geq L} |Π^{ε, η, Λ_{k}} (t_{0}) (U_{t, t_{0}}^{ε, η, Λ_{k}} [[A]]) - Π^{ε, η, Λ_{k}} (t) (A)| \leq C_{n} \frac{ε^{n + 1} + η^{n + 1}}{η^{d + 1}} (1 + | t - t_{0} |^{d + 1}) ‖ A ‖ | X |^{2} .

The proof of this result fundamentally builds on space-time adiabatic perturbation theory^20,21 and technical estimates originally derived in Ref. 12. The latter show that the operations necessary for the construction of the generator of the near-identity automorphism in the definition of the NEASS in (8) (almost) preserve exponential localization required for the Hamiltonian (see Sec. IV). As already mentioned in the Introduction, although the adiabatic theorem in Ref. 12 is at first sight quite similar to the one above, it requires the perturbation to not close the spectral gap of the Hamiltonian H₀ and is thus not generalized in the sense explained in Sec. I B.

2. Infinite systems

The next result is obtained from Theorem I by taking $Λ_{k} ↗ Z^{d}$ ⁠. This requires the interactions and the Lipschitz potential composing the Hamiltonian (1) to have a thermodynamic limit² in the following sense.

Definition 2

(Thermodynamic limit of interactions and potentials).

An exponentially localized time-dependent interaction $Φ \in B_{I, a, \infty}$ is said to have a thermodynamic limit (have a TDL) if there exists an infinite volume interaction $Ψ \in {B °}_{I, a, \infty}$ such that
$\forall n \in N, i \in N_{0}, M \in N : \lim_{k \to \infty} \sup_{t \in I} {‖\frac{d^{i}}{d t^{i}} (Ψ - Φ^{Λ_{k}}) (t)‖}_{a, n, Λ_{M}} = 0,$
and we write $Φ \overset{td}{\to} Ψ$ in this case.
An operator family is said to have a TDL if and only if the corresponding interaction does.
For more general (non-exponentially localized) SLT operators, the definition is completely analogous.
A Lipschitz potential $v \in V_{I}$ is said to have a TDL if there exists an infinite volume Lipschitz potential $v_{\infty} \in {V °}_{I}$ such that
$\forall M \in N, \exists K \geq M \forall k \geq K, t \in I : v^{Λ_{k}} (t, \cdot) |_{Λ_{M}} = v_{\infty} (t, \cdot) |_{Λ_{M}} .$
Again, we write $v \overset{td}{\to} v_{\infty}$ in this case.

Note that whenever Φ = Ψ for some infinite-volume interaction Ψ or v = v_∞ for some infinite volume Lipschitz potential v_∞, both Φ and v trivially have a TDL.

The following proposition is a standard consequence of Lieb–Robinson bounds and shows that the property of having a TDL for interactions and Lipschitz potentials guarantees the existence of the thermodynamic limit for the corresponding evolution operators.^16,45 We remark that it remains true under less restrictive assumptions on the localization quality of the interaction (see, e.g., Proposition 2.2 in Ref. 2).

Proposition 3

(Thermodynamic limit of evolution operators).

Let

K_{0} \in L_{I, a, \infty}

and

w \in V_{I}

both have a thermodynamic limit, i.e.,

Φ_{K_{0}} \overset{td}{\to} Ψ_{K_{0}}

and

w \overset{td}{\to} w_{\infty}

for some

Ψ_{K_{0}} \in {B °}_{I, a, \infty}

and

w_{\infty} \in {V °}_{I}

. Set K = K₀ + V_w, and let

U^{η, Λ_{k}} (t, t_{0})

denote the evolution family generated by

K^{Λ_{k}} (t)

in scaled time with η > 0, i.e., the solution to the Schrödinger equation

i η \frac{d}{d t} U^{η, Λ_{k}} (t, t_{0}) = K^{Λ_{k}} (t) U^{η, Λ_{k}} (t, t_{0})

with

U^{η, Λ_{k}} (t_{0}, t_{0}) = i d

. Then, there exists a co-cycle of automorphisms

U_{t, t_{0}}^{η} : A_{Z^{d}} \to A_{Z^{d}}

such that for all

A \in A_{l o c}

⁠,

U_{t, t_{0}}^{η} [[A]] = \lim_{k \to \infty} U_{t, t_{0}}^{η, Λ_{k}} [[A]] ≔ \lim_{k \to \infty} U^{η, Λ_{k}} {(t, t_{0})}^{*} A U^{η, Λ_{k}} (t, t_{0}) .

The co-cycle $U_{t, t_{0}}^{η}$ only depends on $Ψ_{K_{0}}$ and w_∞ and is generated by the time-dependent (closed) derivation $(L_{K (t)}, D (L_{K (t)}))$ associated with K(t).

As mentioned above, since Theorem II is deduced from Theorem I by taking $Λ_{k} ↗ Z^{d}$ ⁠, we will need to assume the existence of a thermodynamic limit for the building blocks of the Hamiltonian (1).

(INT₂) Assumptions on the interactions.

For $Ψ_{H_{0}}, Ψ_{H_{1}} \in {B °}_{I, a, \infty}$ for some a > 0 and $v_{\infty} \in {V °}_{I}$ ⁠, there exist $Φ_{H_{0}}, Φ_{H_{1}} \in B_{I, a, \infty}$ and $v \in V_{I}$ with appropriate boundary conditions (encoded in the definition of the norms defining the spaces $L$ and the Lipschitz condition) all having a TDL with the respective object as the limit, i.e., $Φ_{H_{0}} \overset{td}{\to} Ψ_{H_{0}}$ ⁠, $Φ_{H_{1}} \overset{td}{\to} Ψ_{H_{1}}$ ⁠, and $v \overset{td}{\to} v_{\infty}$ ⁠.

We also assume the convergence of ground states by means of the Banach–Alaoglu theorem (the unit sphere in $A_{Z^{d}}^{*}$ is weak*-compact), essentially only in order to avoid the extraction of a subsequence.

(S_unif) Assumptions on the convergence of states.

Assume that for every t ∈ I, the sequence ${(ρ_{0}^{Λ_{k}} (t))}_{k \in N}$ of ground states (naturally extended to the whole of $A_{Z^{d}}$ ⁠) converges in the weak*-topology to a state ρ₀(t) on $A_{Z^{d}}$ ⁠, which we call the gapped limit ground state at t ∈ I.

We can now formulate the second generalized super-adiabatic theorem concerning infinite systems with a uniform gap.²

Theorem II

[Adiabatic theorem for infinite systems with a uniform gap (see Ref. 2, Theorems 3.2 and 3.5)].

Under Assumptions (GAP_unif), (INT₂), and (S_unif), there exists a near-identity automorphism

β^{ε, η} (t) = e^{i ε L_{S^{ε, η} (t)}}

with SLT generators S^ɛ,η for any ɛ, η ∈ (0, 1] and t ∈ I such that the state

Π^{ε, η} (t) ≔ ρ_{0} (t) ◦ β^{ε, η} (t)

is a super-adiabatic NEASS for the Heisenberg time-evolution on

A_{Z^{d}}

generated by

\frac{1}{η} Ψ_{H^{ε} (\cdot)}

with

Ψ_{H^{ε}} ≔ Ψ_{H_{0}} + ε (V_{v_{\infty}} + Ψ_{H_{1}}) .

The crucial point in the Proof of Theorem II in Ref. 2 is to show that the property of having a TDL is designed in such a way that it is preserved under all necessary operations for the construction of the NEASS (see Sec. IV). Therefore, also the near-identity automorphism from (8) converges as $Λ_{k} ↗ Z^{d}$ by means of Proposition 3.

B. Systems with a gap in the bulk

In this section, we drop Assumption (GAP_unif) of a uniform gap for finite systems, but merely work under the condition of a gap in the bulk, which is formulated via the Gelfand–Naimark–Segal (GNS) construction in Assumption (GAP_bulk): Let $Ψ_{H_{0}} \in {B °}_{a, 0}$ be an infinite volume interaction and $L_{H_{0}}$ denote the induced derivation on (a dense subset of) $A_{Z^{d}}$ ⁠. A state ω on $A_{Z^{d}}$ is called an $L_{H_{0}}$ -ground state if and only if $ω (A^{*} L_{H_{0}} (A)) \geq 0$ for all $A \in D (L_{H_{0}})$ ⁠. Let ω be an $L_{H_{0}}$ -ground state and $(H_{ω}, π_{ω}, Ω_{ω})$ be the corresponding GNS triple [i.e., $H_{ω}$ be a Hilbert space, i.e., $π_{ω} : A \to L (H_{ω})$ be a representation, and, i.e., $Ω_{ω} \in H_{ω}$ be a cyclic vector]. Then, there exists a unique densely defined, self-adjoint positive operator H_0,ω ≥ 0 on $H_{ω}$ satisfying

π_{ω} (e^{i t L_{H_{0}}} [[A]]) = e^{i t H_{0, ω}} π_{ω} (A) e^{- i t H_{0, ω}} and e^{- i t H_{0, ω}} Ω_{ω} = Ω_{ω}

(9)

for all $A \in A$ and $t \in R$ ⁠. We call this H_0,ω the bulk Hamiltonian (or GNS Hamiltonian) associated with $Ψ_{H_{0}}$ and ω. See Ref. 46 for the general theory.

We assume that $Ψ_{H_{0}}$ has a unique gapped ground state in the following sense (cf. Refs. 3 and 23):

(GAP_bulk) Assumptions on the ground state of $Ψ_{H_{0}}$ .

Uniqueness. For each t ∈ I, there exists a unique $L_{H_{0} (t)}$ -ground state ρ₀(t).
Gap. There exists g > 0 such that $σ (H_{0, ρ_{0} (t)} (t)) \ {0} \subset [g, \infty)$ for all t ∈ I.
Regularity. For any strictly positive $f \in S (R)$ (Schwarz functions), define $D_{f}$ as the set of observables $A \in A_{Z^{d}}$ for which $‖ A ‖_{f} ≔ ‖ A ‖ + \sup_{k \in N} (‖ (1 - E_{Λ_{k}}) [[A]] ‖ / f (k)) < \infty$ ⁠, where $E_{Λ_{k}} [[\cdot]]$ denotes the conditional expectation (see Ref. 2, Appendix C). Then, for any $A \in D_{f}$ ⁠, t ↦ ρ₀(t)(A) is differentiable and there exists a constant C_f such that
$\sup_{t \in I} | \dot{ρ_{0}} (t) (A) | \leq C_{f} ‖ A ‖_{f} .$

The smoothness of expectation values of (almost) exponentially localized observables as under item (iii) is a rather technical condition and a consequence of a uniform gap as in Assumption (GAP_unif) (see Remark 4.15 in Ref. 23 and Lemma 6.0.1 in Ref. 47). Although the uniqueness of the ground state in item (i), which we required throughout this Review, is expected to hold for the physically relevant type of Hamiltonian (4), it has been shown, to our present knowledge, only in very specific quantum spin systems. These include (a) weak perturbations of non-interacting gapped frustration-free systems^28,32 and (b) short-range interacting frustration-free models fulfilling local topological quantum order (LTQO).^31,39

Remark 4.

As mentioned in the beginning of Sec. III, item (ii) holds, in particular, if one has a uniform gap for finite systems as spelled out in Assumption (GAP_unif) since it cannot close abruptly in the thermodynamic limit for the GNS Hamiltonian (see Proposition 5.4 in Ref. 41). However, we observe that a considerably weaker sufficient condition for having a gap for the GNS Hamiltonian as in Assumption (GAP_bulk) (ii) is to have a gap in the bulk for the finite systems Λ_k in the following sense: There exists g > 0 such that for all

k \in N

⁠, there exists some

l = l (k) \in N

with l(k) → ∞ as k → ∞, and we have

ρ_{0}^{Λ_{k}} (t) () A^{*} L_{H_{0} (t)}^{Λ_{k}} [[A]] () \geq g (ρ_{0}^{Λ_{k}} (t) (A^{*} A) - {|ρ_{0}^{Λ_{k}} (t) (A)|}^{2})

(10)

for all

A \in A_{Λ_{l}}

and all t ∈ I, where

ρ_{0}^{Λ_{k}} (t)

denotes a suitable ground state of

H_{0}^{Λ_{k}} (t)

⁠. Indeed, assuming that

ρ_{0}^{Λ_{k}} (t) ⇀ ρ_{0} (t)

for every t ∈ I,⁴⁸ this simply follows after taking the limit k → ∞ on both sides of () and realizing that, as k → ∞, the set of admissible observables

A \in A_{Λ_{l (k)}}

exhausts

A_{l o c}

⁠, which is dense in

A_{Z^{d}}

by definition. The resulting inequality immediately yields the desired spectral gap for the GNS Hamiltonian (cf. Ref. 46, Proposition 5.3.19, and Ref. 30, Sec. 7).

In the following, we shall present adiabatic theorems for infinite systems (Theorem III) and for extended but finite systems (Theorem IV) under Assumption (GAP_bulk).

1. Infinite systems

Analogously to Sec. III A, the basic assumptions on the Hamiltonian say that it is composed of exponentially localized interactions and/or a Lipschitz potential. In addition, the Hamiltonian H₀ satisfies a technical regularity assumption in t, for which we recall that $I \subset R$ denotes an open time interval.

(INT₃) Assumptions on the interactions.

Let $Ψ_{H_{0}}, Ψ_{H_{1}} \in {B °}_{I, a, \infty}$ be time-dependent infinite volume interactions and $v_{\infty} \in {V °}_{I}$ be a time-dependent infinite volume Lipschitz potential.
Assume that the map $I \to {B °}_{a, \infty}$ ⁠, $t \mapsto Ψ_{H_{0} (t)}$ is continuously differentiable.⁴⁹

We can now formulate the third generalized super-adiabatic theorem concerning infinite systems with a gap in the bulk.³

Theorem III

[Adiabatic theorem for infinite systems with a gap in the bulk (see Ref. 3, Theorem 3.4)].

Under Assumptions (GAP_bulk) and (INT₃), there exists a near-identity automorphism

β^{ε, η} (t) = e^{i ε L_{S^{ε, η} (t)}}

on

A_{Z^{d}}

with SLT generators S^ɛ,η for any ɛ, η ∈ (0, 1] and t ∈ I such that the state

Π^{ε, η} (t) ≔ ρ_{0} (t) ◦ β^{ε, η} (t)

is a super-adiabatic NEASS for ρ₀(t) and the Heisenberg time-evolution on

A_{Z^{d}}

generated by

\frac{1}{η} Ψ_{H^{ε} (\cdot)}

with

Ψ_{H^{ε}} ≔ Ψ_{H_{0}} + ε (V_{v_{\infty}} + Ψ_{H_{1}}) .

The key role of the spectral gap condition is that it allows us to construct an inverse of the Liouvillian $L_{H_{0} (t)}$ ⁠, appearing in the construction of the NEASS, which maps SLT operators to SLT operators with slightly deteriorated locality properties. Hence, the inverse of $L_{H_{0} (t)}$ is called the quasi-local inverse of the Liouvillian.⁵⁰ Assuming a gap only in the bulk, as done in (GAP_bulk), means that the action of the Liouvillian can only be inverted in the bulk (see Sec. IV).

2. Extended but finite systems

Contrary to the results in Sec. III A, the adiabatic theorem describing an infinite system with a gap in the bulk did not require any notion of having a TDL in its formulation. Instead, in order to derive a finite-volume analog from Theorem III (with qualitative additional error terms; see Theorem IV), we need to introduce the stronger notion of having a rapid thermodynamic limit for the exponentially localized interactions and the Lipschitz potential. We refer to Ref. 3 for a detailed discussion of this property.

Definition 5

(Rapid thermodynamic limit of interactions and potentials).

An exponentially localized time-dependent interaction $Φ \in B_{I, a, \infty}$ is said to have a rapid thermodynamic limit with exponent γ ∈ (0, 1) (have a RTDL_γ) if there exists an infinite volume interaction $Ψ \in {B °}_{I, a, \infty}$ such that
$\begin{aligned} \forall n \in N, i \in N_{0} \exists λ, C > 0 & \forall M \in N \forall k \geq M + λ M^{γ} : \\ \sup_{t \in I} {‖\frac{d^{i}}{d t^{i}} (Ψ - Φ^{Λ_{k}}) (t)‖}_{a, n, Λ_{M}} \leq C e^{- a M^{γ}}, \end{aligned}$
(11)
and we write $Φ \overset{rtd}{\to} Ψ$ in this case.
A family of operators is said to have a RTDL if and only if the corresponding interaction does.
For more general (non-exponentially localized) SLT operators, the definition is completely analogous.
A Lipschitz potential $v \in V_{I}$ is said to have a RTDL_γ if it is eventually independent of k, i.e., if there exists an infinite volume Lipschitz potential $v_{\infty} \in {V °}_{I}$ such that
$\exists λ > 0 \forall M \in N \forall k \geq M + λ M^{γ}, t \in I : v_{\infty} (t, \cdot) |_{Λ_{M}} = v^{Λ_{k}} (t, \cdot) |_{Λ_{M}} .$
Again, we write $v \overset{rtd}{\to} v_{\infty}$ in this case.

In a nutshell, having a RTDL_γ means that the interaction (or the Lipschitz potential) essentially agrees with a corresponding infinite volume object, up to terms located on a thin shell with relative size of order k^γ−1 right at the boundary of Λ_k. Note that whenever Φ = Ψ for some infinite-volume interaction Ψ or v = v_∞ for some infinite volume Lipschitz potential v_∞, both Φ and v trivially have a RTDL_γ [with any exponent γ ∈ (0, 1)].

Theorem IV is deduced from Theorem III by comparing the time evolution $U_{t, t_{0}}^{ε, η}$ and the near identity automorphism β^ɛ,η in the definition of the NEASS on the infinite system $Z^{d}$ with the same objects for large (but finite) systems Λ_k. Therefore, we will need to assume the existence of a rapid thermodynamic limit for the building blocks of Hamiltonian (1).

(INT₄) Assumptions on the interactions.

The interactions $Φ_{H_{0}}, Φ_{H_{1}} \in B_{I, a, \infty}$ and the Lipschitz potential $v \in V_{I}$ all have a RTDL, i.e., $Φ_{H_{0}} \overset{rtd}{\to} Ψ_{H_{0}}$ ⁠, $Φ_{H_{1}} \overset{rtd}{\to} Ψ_{H_{1}}$ ⁠, and $v \overset{rtd}{\to} v_{\infty}$ ⁠. The limiting objects $Ψ_{H_{0}}$ ⁠, $Ψ_{H_{1}}$ ⁠, and v_∞ satisfy Assumption (INT₃).

In Theorem IV, we shall consider finite volume states $ρ_{0}^{Λ_{k}} (t)$ ⁠, which are close to the infinite volume ground state ρ₀(t) away from the boundary in following sense.

(S_bulk) Assumption on the convergence of states.

The sequence ${(ρ_{0}^{Λ_{k}} (t))}_{k \in N}$ of states on $A_{Λ_{k}}$ converges rapidly to ρ₀(t) in the bulk: there exist $C \in R$ ⁠, $m \in N$ ⁠, and $h \in S$ such that for any finite $X \subset Z^{d}$ ⁠, $A \in A_{X}$ ⁠, and Λ_k ⊃ X,

\sup_{t \in I} |ρ_{0} (t) (A) - ρ_{0}^{Λ_{k}} (t) (A)| \leq C ‖ A ‖ diam {(X)}^{m} h () d i s t (X, Z^{d} \ Λ_{k}) () .

While the sequence $ρ_{0}^{Λ_{k}} (t) \equiv ρ_{0} (t) |_{A_{Λ_{k}}}$ of simple restrictions satisfies Assumption (S_bulk) trivially, the adiabatic theorem ensures the existence of a super-adiabatic NEASS constructed for any such sequence.⁵¹ Most interesting for physical application would be a sequence of ground states $ρ_{0}^{Λ_{k}} (t)$ of the finite volume Hamiltonians $H_{0}^{Λ_{k}} (t)$ ⁠. While the above assumption is expected to hold for any sequence of finite volume ground states for Hamiltonians modeling Chern or topological insulators such as in (4), the only result we are aware of indeed proving such a statement is again [see the discussion below Assumption (GAP_bulk)] for weakly interacting spin systems.²⁸ In spirit, assuming is very similar to supposing that the system satisfies local topological quantum order (LTQO)^31,52 or a strong local perturbations perturb locally (LPPL) principle for perturbations acting at the boundary of the system.^32,39

We can now formulate the fourth and last generalized super-adiabatic theorem concerning finite systems with a gap in the bulk.³

Theorem IV

[Adiabatic theorem for finite systems with gap in the bulk (see Ref. 3, Theorem 4.1)].

Under Assumptions (GAP_bulk), (INT₄), and (S_bulk), there exists a sequence of near-identity automorphisms

β^{ε, η, Λ_{k}} (t) = e^{i ε L_{S^{ε, η} (t)}^{Λ_{k}}}

with SLT generators S^ɛ,η for any ɛ, η ∈ (0, 1] and t ∈ I such that the states

Π^{ε, η, Λ_{k}} (t) ≔ ρ_{0}^{Λ_{k}} (t) ◦ β^{ε, η, Λ_{k}} (t)

are super-adiabatic NEASSs for the Heisenberg time-evolution

U_{t, t_{0}}^{ε, η, Λ_{k}}

on

A_{Λ_{k}}

generated by

\frac{1}{η} H^{ε, Λ_{k}} (\cdot)

with

\frac{1}{η} H^{ε, Λ_{k}} (t) ≔ \frac{1}{η} () H_{0}^{Λ_{k}} (t) + ε () V_{v}^{Λ_{k}} (t) + H_{1}^{Λ_{k}} (t) () (),

up to an error vanishing faster than any inverse polynomial in the distance to the boundary. That is, for any

n \in N

⁠, there exists a constant C_n, and for any compact K ⊂ I and

m \in N

⁠, there exists a constant

{\tilde{C}}_{n, m, K}

such that for all

k \in N

, all X ⊂ Λ_k, all

A \in A_{X}

⁠, and all t, t₀ ∈ K,

\begin{align} |Π^{ε, η, Λ_{k}} (t_{0}) (U_{t, t_{0}}^{ε, η, Λ_{k}} [[A]]) - Π^{ε, η, Λ_{k}} (t) (A)| & \leq C_{n} \frac{ε^{n + 1} + η^{n + 1}}{η^{d + 1}} (1 + | t - t_{0} |^{d + 1}) ‖ A ‖ | X |^{2} \\ + {\tilde{C}}_{n, m, K} {(1 + η d i s t (X, Z^{d} \ Λ_{k}))}^{- m} ‖ A ‖ diam {(X)}^{2 d} . \end{align}

(12)

The above theorem asserts that by assuming (GAP_bulk), one obtains similar adiabatic bounds also for states of finite systems (without a spectral gap), which are close to the infinite volume ground state in the bulk as formulated in Assumption (S_bulk). Since the adiabaticity potentially breaks at the boundaries of the finite systems, non-adiabatic effects arising close to the boundary may propagate into the bulk. Therefore, an additional error term appears, but it decays faster than any polynomial in the size of the finite system for any fixed η. The actual form of the additional error term in the last line of (12) coming out of the proof in (see Ref. 3, Sec. 5) is slightly better but more complicated, which is why we refrain from stating it here.

The main points in the Proof of Theorem IV, which we discuss in Sec. IV, are to show that (i) the property of having a RTDL_γ is preserved under all necessary operations for the construction of the NEASS (similarly as for Theorem II) and (ii) having a RTDL_γ for an interaction provides an explicit rate of convergence for the associated evolution family as in Proposition 3.

IV. IDEA OF THE PROOFS

The goal of the present section is to convey the main ideas relevant for proving the individual theorems from Sec. III, where we already glimpsed the key steps required in their proofs. For many technical details, we refer the reader to the original works.^1–3

A. Systems with a uniform gap

The fundamental conceptual idea behind the proof for all four variants of the generalized super-adiabatic theorems is a perturbative scheme, which was called space-time adiabatic perturbation theory in Refs. 20 and 21. The basic structure of this computation is most easily presented for finite systems, where no further technical difficulties arise since all appearing operators are in fact matrices and thus bounded. However, it is still necessary to show that all estimates are uniform in the size of the system Λ_k.

1. Extended but finite systems: Proof of Theorem I

The form in which we presented Theorem I differs slightly from the original result (see Ref. 1, Theorem 5.1). The original statement concerns a sequence $Π_{n}^{ε, η, Λ_{k}} (t) ≔ ρ_{0}^{Λ_{k}} (t) ◦ β_{n}^{ε, η, Λ_{k}} (t)$ of states on Λ_k (indexed by $n \in N$ ⁠), where

β_{n}^{ε, η, Λ_{k}} (t) [[A]] ≔ e^{- i ε L_{S_{n}^{ε, η} (t)}^{Λ_{k}}} [[A]] and ε L_{S_{n}^{ε, η} (t)}^{Λ_{k}} ≔ \sum_{j = 1}^{n} ε^{j} L_{A_{j}^{ε, η} (t)}^{Λ_{k}} .

From this, Theorem I (and similarly all other three theorems) follows by a simple resummation of $ε^{j} L_{A_{j}^{ε, η} (t)}^{Λ_{k}}$ ⁠, which will be discussed in Sec. IV C.

The main idea of the proof is to choose each operator $A_{j}^{ε, η, Λ_{k}} (t)$ ⁠, j = 1, …, n, in such a way that the jth-order term in the perturbative scheme vanishes. For the n-dependent result (i.e., prior to resummation), we apply the fundamental theorem of calculus to get

Π_{n}^{ε, η, Λ_{k}} (t_{0}) (U_{t, t_{0}}^{ε, η, Λ_{k}} [[A]]) - Π_{n}^{ε, η, Λ_{k}} (t) (A) = - \int_{t_{0}}^{t} d s \frac{d}{d s} ρ_{0}^{Λ_{k}} (s) (β_{n}^{ε, η, Λ_{k}} (s) ◦ U_{t, s}^{ε, η, Λ_{k}} [[A]])

(13)

and then aim to bound the integrand. Calculating the derivative by using the chain rule and Duhamel’s formula leaves us with

\frac{d}{d s} ρ_{0}^{Λ_{k}} (s) (β_{n}^{ε, η, Λ_{k}} (s) ◦ U_{t, s}^{ε, η, Λ_{k}} [[A]]) = - \frac{i}{η} ρ_{0}^{Λ_{k}} (s) ([Q_{n}^{ε, η, Λ_{k}} (s), β_{n}^{ε, η, Λ_{k}} (s) ◦ U_{t, s}^{ε, η, Λ_{k}} [[A]]]),

(14)

where $Q_{n}^{ε, η, Λ_{k}} (s)$ is a shorthand notation for

\begin{align} η I_{s}^{Λ_{k}} () {\dot{H}}_{0}^{Λ_{k}} (s) () + η \int_{0}^{1} d λ e^{- i λ ε S_{n}^{ε, η, Λ_{k}} (s)} ε {\dot{S}}_{n}^{ε, η, Λ_{k}} (s) e^{i λ ε S_{n}^{ε, η, Λ_{k}} (s)} + e^{- i ε S_{n}^{ε, η, Λ_{k}} (s)} () H_{0}^{Λ_{k}} (s) + ε V^{Λ_{k}} (s) () e^{i ε S_{n}^{ε, η, Λ_{k}} (s)} \\ = : H_{0}^{Λ_{k}} (s) + \sum_{j = 1}^{n} ε^{j} R_{j}^{ε, η, Λ_{k}} (s) + ε^{n + 1} R_{n + 1}^{ε, η, Λ_{k}} (s), \end{align}

(15)

and $V^{Λ_{k}} = V_{v}^{Λ_{k}} + H_{1}^{Λ_{k}}$ ⁠. Here, $I_{s}^{Λ_{k}} ({\dot{H}}_{0}^{Λ_{k}} (s))$ is the SLT generator of the parallel transport within the vector-bundle $Ξ_{0, I}^{Λ_{k}}$ over I defined by $t \mapsto ρ_{0}^{Λ_{k}} (t)$ ⁠.⁵³ This parallel transport is also known as the spectral flow, which plays a fundamental role in proving automorphic equivalence of gapped ground state phases (see, e.g., Refs. 12 and 18). Moreover, the operator $I_{s}^{Λ_{k}} : A_{Λ_{k}} \to A_{Λ_{k}}$ is called the quasi-local inverse of the Liouvillian³⁸ $L_{H_{0} (s)}^{Λ_{k}}$ since it satisfies^1,12

ρ_{0}^{Λ_{k}} (s) () [) L_{H_{0} (s)}^{Λ_{k}} ◦ I_{s}^{Λ_{k}} [[B_{1}]] - i B_{1}, B_{2} (] () = 0 for all B_{1}, B_{2} \in A_{Λ_{k}}, s \in I,

(16)

and also preserves good localization of its argument (in particular, it maps SLT operators to SLT operators). This combined property of $I_{s}^{Λ_{k}}$ heavily relies on the ground state $ρ_{0}^{Λ_{k}} (s)$ being gapped^1,12,16 and will be of fundamental importance in the following.

In the last line of (15), we expanded in powers of ɛ and η in the sense that $R_{j}^{ε, η, Λ_{k}} (s)$ ⁠, for j ≤ n, are polynomials in η/ɛ of order (at most) j with ɛ- and η-independent SLT operators as coefficients. A more detailed step-by-step calculation can be found in the Proof of Proposition 5.1 in Ref. 1. Let us here only report the general structure

R_{j}^{ε, η, Λ_{k}} (s) = - i L_{H_{0} (s)}^{Λ_{k}} () A_{j}^{ε, η, Λ_{k}} (s) () + {\tilde{R}}_{j}^{ε, η, Λ_{k}} (s),

(17)

where the first remainder term is given by

{\tilde{R}}_{1}^{ε, η, Λ_{k}} (s) = \frac{η}{ε} I_{s}^{Λ_{k}} () {\dot{H}}_{0}^{Λ_{k}} (s) () - V^{Λ_{k}} (s)

and all other ${\tilde{R}}_{j}^{ε, η, Λ_{k}} (s)$ are composed of iterated commutators of the operators $A_{i}^{ε, η, Λ_{k}} (s)$ and ${\dot{A}}_{i}^{ε, η, Λ_{k}} (s)$ ⁠, for i < j ≤ n, with $H_{0}^{Λ_{k}} (s)$ and $V^{Λ_{k}} (s)$ ⁠. In contrast to general onsite potentials, the commutator of a Lipschitz potential with an SLT operator is an SLT operator itself (see Ref. 1, Lemma 2.1). For the commutator of SLT operators, this is easy to see.

We now consider individual terms from (15) when plugged into (14). The zero-order term vanishes because $ρ_{0}^{Λ_{k}} (s)$ is the ground state of $H_{0}^{Λ_{k}} (s)$ ⁠. By application of (16), we can iteratively choose

A_{j}^{ε, η, Λ_{k}} (s) = - I_{s}^{Λ_{k}} ({\tilde{R}}_{j}^{ε, η, Λ_{k}} (s))

(18)

such that (14) vanishes up to

- i \frac{ε^{n + 1}}{η} ρ_{0}^{Λ_{k}} (s) ([R_{n + 1}^{ε, η, Λ_{k}} (s), β_{n}^{ε, η, Λ_{k}} (s) ◦ U_{t, s}^{ε, η, Λ_{k}} [[A]]]) .

(19)

Moreover, all the operations involved in calculating the $A_{j}^{ε, η, Λ_{k}} (s)$ ⁠, i.e., taking commutators and applying the quasi-local inverse of the Liouvillian, preserve the locality properties of the operators, as shown in the appendices of Refs. 1 and 25, which are heavily based on Ref. 12. Hence, also all $A_{j}^{ε, η, Λ_{k}}$ are SLT operators.

It turns out that also $R_{n + 1}^{ε, η, Λ_{k}}$ is a polynomial in η/ɛ of order at most n + 1 and its coefficients, as we just explained, are SLT operators (see Ref. 1, Proof of Proposition 6.1). Thus, the absolute value of (13) is bounded by

\begin{align} \frac{ε^{n + 1}}{η} |\int_{t_{0}}^{t} d s ρ_{0}^{Λ_{k}} (s) ([R_{n + 1}^{ε, η, Λ_{k}} (s), β_{n}^{ε, η, Λ_{k}} (s) ◦ U_{t, s}^{ε, η, Λ_{k}} [[A]]])| \\ \leq \frac{ε^{n + 1}}{η} | t - t_{0} | \sup_{s \in [t_{0}, t]} ‖[{() β_{n}^{ε, η, Λ_{k}} ()}^{- 1} (s) () R_{n + 1}^{ε, η, Λ_{k}} (s) (), U_{t, s}^{ε, η, Λ_{k}} [[A]]]‖ \\ \leq C_{n} \frac{ε^{n + 1}}{η} | t - t_{0} | (1 + {(\frac{η}{ε})}^{n + 1}) (1 + {(\frac{| t - t_{0} |}{η})}^{d}) ‖ A ‖ | X |^{2} \\ \leq C_{n} \frac{ε^{n + 1} + η^{n + 1}}{η^{d + 1}} | t - t_{0} | (1 + | t - t_{0} |^{d}) ‖ A ‖ | X |^{2}, \end{align}

(20)

where we essentially used a generalized Lieb–Robinson bound (see Ref. 1, Lemma B.5) to estimate the commutator. Note that the $(1 + {(| t - t_{0} | / η)}^{d})$ -factor comes from the Lieb–Robinson bound and the adiabatic 1/η-scaling of the time evolution $U_{t, s}$ ⁠. The (1 + (η/ɛ)ⁿ⁺¹)-factor comes from bounding the interaction norm of $R_{n + 1}^{ε, η, Λ_{k}} (s)$ by separating the polynomial dependence on η/ɛ such that C_n is independent of Λ_k, ɛ, and η. We have thus shown that the NEASS almost intertwines the time evolution, i.e., item 1 of Definition 1.

We are left with discussing the remaining three characterizing properties of the NEASS given in Definition 1: By construction, all $A_{j}^{ε, η, Λ_{k}} (t)$ depend only on $H_{0}^{Λ_{k}} (t)$ and $V^{Λ_{k}} (t)$ and their jth derivatives at time t. This shows that the NEASS is local in time, i.e., item 2. Moreover, if all time derivatives of H₀ and V vanish for some t ∈ I, then all non-constant (i.e., in front of some positive power of η/ɛ) coefficients in $R_{j}^{ε, η, Λ_{k}}$ vanish and $Π_{n}^{ε, η, Λ_{k}} (t) = Π_{n}^{ε, 0, Λ_{k}} (t)$ ⁠. This shows that the NEASS is stationary whenever the Hamiltonian is stationary, i.e., item 3. If, for some t ∈ I, ${\dot{H}}_{0}^{Λ_{k}} (t)$ and $V^{Λ_{k}} (t)$ vanish, then ${\tilde{R}}_{1}^{ε, η, Λ_{k}}$ and thus $A_{1}^{ε, η, Λ_{k}}$ also vanish. If additionally all derivatives of $H_{0}^{Λ_{k}}$ and $V^{Λ_{k}}$ at t vanish, also ${\tilde{R}}_{j}^{Λ_{k}} (t)$ and thus $A_{j}^{ε, η, Λ_{k}} (t)$ vanish. Hence, $β_{n}^{ε, η, Λ_{k}} (t) = 1_{Λ_{k}}$ and the NEASS equals the ground state, i.e., item 4 holds.

The above listed general arguments immediately translate to the other three theorems.

2. Infinite systems: Proof of Theorem II

Without any further assumptions, the sequence Hamiltonian $H^{ε, Λ_{k}}$ and its constituents $H_{0}^{Λ_{k}}$ and $V^{Λ_{k}}$ could have nothing in common for different lattice sizes k (they might even describe different physical systems), so taking the limit $Λ_{k} ↗ Z^{d}$ might not be well-defined. In order to avoid this somewhat meaningless situation, we assumed that the building blocks of the Hamiltonian have a TDL [see Definition 2 and Assumption (INT₂)] and also the sequence of ground states ${(ρ_{0}^{Λ_{k}} (t))}_{k \in N}$ converges [Assumption (S_unif)]. Since the property of having a TDL guarantees the existence of the thermodynamic limit for the corresponding evolution operators (see Proposition 3 and Ref. 16), it remains to show that the operator sequences ${() A_{j}^{ε, η, Λ_{k}} (t) ()}_{k \in N}$ ⁠, j = 1, …, n, constructed in Sec. IV A 1 also have a TDL. More precisely, one needs to show that taking time-derivatives, sums of commutators with the building blocks of H^ɛ (and ${\dot{H}}_{0}$ ⁠), and the inverse of the Liouvillian [see (17)] leaves the property of having a TDL for SLT operators invariant, which is in fact the main point of the proof in Ref. 2. It is then straightforward to show that compositions of states and automorphisms, all having a thermodynamic limit, converge as $Λ_{k} ↗ Z^{d}$ ⁠. Since the constant C_n from (20) is uniformly bounded in k, the (sketch of a) Proof of Theorem II is complete.

B. Systems with a gap in the bulk

For systems having a spectral gap only in the bulk (i.e., for the GNS Hamiltonian), the characteristic (16) of $I_{s}^{Λ_{k}}$ ⁠, that it essentially inverts the Liouvillian $L_{H_{0} (s)}^{Λ_{k}}$ (and still maps SLT operators to SLT operators), is now only fulfilled for certain B₁ and B₂ in a dense domain $D \subset A_{Z^{d}}$ after taking the limit $Λ_{k} ↗ Z^{d}$ (see Ref. 3, Proposition 3.3). Presuming that the limit actually exists, this point is the main challenge in proving an adiabatic theorem under the less restrictive gap assumption (GAP_bulk).

1. Infinite systems: Proof of Theorem III

As just explained, the main difficulty in proving Theorem III is that (16) only holds if $H_{0}^{Λ_{k}}$ is gapped. On top of that, we cannot handle the limit $Λ_{k} ↗ Z^{d}$ of ${\tilde{R}}_{j}^{ε, η, Λ_{k}}$ directly nor could they be used in the infinite volume version of (16) because it only holds for $B_{1}, B_{2} \in D \subset A_{Z^{d}}$ ⁠. However, the rest of the construction from Sec. IV A 1 is still valid, but the lower order terms in (13) have a non-vanishing contribution in finite domains. We thus repeat this construction but take coefficients $A_{j}^{ε, η, Λ_{k}, Λ_{l}} (t)$ ⁠, which are built up from $H_{0}^{Λ_{k}} (t)$ but restricting the perturbations ${\dot{H}}_{0} (t)$ and V(t) to Λ_l with l < k. In this way, one can take the limit $Λ_{k} ↗ Z^{d}$ in (16) with $B_{1} = \lim_{k \to \infty} {\tilde{R}}_{j}^{ε, η, Λ_{k}, Λ_{l}} \in A_{Z^{d}}$ [see (22) and (23) and the comment thereafter for technical obstructions in taking the limit]. Using this notational convention, we introduce the states

Π_{n}^{ε, η, Λ_{k}, Λ_{l}} (t) = ρ_{0} (t) ◦ β^{ε, η, Λ_{k}, Λ_{l}} (t),

where ρ₀(t) is the infinite volume ground state, and compare them to the actual objects in infinite volume while estimating

\begin{align} |Π_{n}^{ε, η} (t_{0}) (U_{t, t_{0}}^{ε, η} [[A]]) - Π_{n}^{ε, η} (t) (A)| & \leq |Π_{n}^{ε, η} (t_{0}) (U_{t, t_{0}}^{ε, η} [[A]]) - Π_{n}^{ε, η, Λ_{k}, Λ_{l}} (t_{0}) (U_{t, t_{0}}^{ε, η, Λ_{k}, Λ_{l}} [[A]])| \\ + |Π_{n}^{ε, η, Λ_{k}, Λ_{l}} (t_{0}) (U_{t, t_{0}}^{ε, η, Λ_{k}, Λ_{l}} [[A]]) - Π_{n}^{ε, η, Λ_{k}, Λ_{l}} (t) (A)| \\ + |Π_{n}^{ε, η, Λ_{k}, Λ_{l}} (t) (A) - Π_{n} (t) (A)| \end{align}

(21)

by means of the triangle inequality. Since all the interactions (and the Lipschitz potential) have a TDL, one can prove [see Ref. 3, Sec. 5.1(b)] that the first and last summand in (21) can be made arbitrarily small for $k, l \in N$ large enough, and we can thus focus on the second summand. However, since (16) only holds in the limit $Λ_{k} ↗ Z^{d}$ and also ρ₀(t) is not necessarily a ground state of $H_{0}^{Λ_{k}} (t)$ ⁠, the lower order terms in the analog of (14) and (15) do not vanish for finite k and l. Instead, only

\lim_{k \to \infty} ρ_{0} (s) ([H_{0}^{Λ_{k}} (s), β_{n}^{ε, η, Λ_{k}, Λ_{l}} (s) ◦ U_{t, s}^{ε, η, Λ_{k}, Λ_{l}} [[A]]]) = 0

(22)

and

\lim_{k \to \infty} ρ_{0} (s) ([R_{j}^{ε, η, Λ_{k}, Λ_{l}} (s), β_{n}^{ε, η, Λ_{k}, Λ_{l}} (s) ◦ U_{t, s}^{ε, η, Λ_{k}, Λ_{l}} [[A]]]) = 0

(23)

for all $l \in N$ and uniformly for s and t in compacts. These statements require a careful analysis of deteriorating localization properties along the expansion and convergence estimates in norms measuring the quality of localization [cf. the norm ‖ ⋅‖_f introduced in Assumption (GAP_bulk) (iii)] such that the limits really converge to the infinite volume version of (16) with B₁ and B₂ in a dense domain $D \subset A_{Z^{d}}$ ⁠. For further details, we refer to Proposition 3.2 and the statements in Appendix B of Ref. 3, which are adaptions of technical estimates that were originally established for the proof of automorphic equivalence with a gap only in the bulk.²³ Now, combining (22) and (23) with the estimates on the first and third summand in (21), we conclude that all the lower order terms vanish in the limit k → ∞ followed by l → ∞, which finishes our sketch of the Proof of Theorem III.

2. Extended but finite systems: Proof of Theorem IV

Let us briefly explain the strategy to prove Theorem IV. In order to show (12), we first estimate

\begin{align} |Π_{n}^{ε, η, Λ_{k}} (t_{0}) (U_{t, t_{0}}^{ε, η, Λ_{k}} [[A]]) - Π_{n}^{ε, η, Λ_{k}} (t) (A)| & \leq |Π_{n}^{ε, η, Λ_{k}} (t_{0}) (U_{t, t_{0}}^{ε, η, Λ_{k}} [[A]]) - Π_{n}^{ε, η} (t_{0}) (U_{t, t_{0}}^{ε, η} [[A]])| \\ + |Π_{n}^{ε, η} (t_{0}) (U_{t, t_{0}}^{ε, η} [[A]]) - Π_{n}^{ε, η} (t) (A)| \\ + |Π_{n}^{ε, η, Λ_{k}} (t) (A) - Π_{n}^{ε, η} (t) (A)| \end{align}

(24)

and treat the three summands separately. The second summand corresponds to the infinite system and can be estimated by means of Theorem III such that it accounts for the first contribution on the RHS of (12). We are left with bounding the remaining two summands in (24). These contribute the additional error term on the RHS of (12). To estimate them, we need explicit control on the speed of convergence (it must be faster than any inverse polynomial) for the states [see Assumption (S_bulk)] and automorphisms $β_{n}^{ε, η, Λ_{k}}$ and $U_{t, t_{0}}^{ε, η, Λ_{k}}$ ⁠. For the time evolution $U_{t, t_{0}}^{ε, η, Λ_{k}}$ ⁠, the rapid convergence to $U_{t, t_{0}}^{ε, η}$ is ensured by supposing that the building blocks of H^ɛ have a RTDL [see Definition 5 and Assumption (INT₄)]. This was carried out in see Ref. 3, Appendix B, building on estimates from see Ref. 16, Sec. 3. We remark that the adiabatic 1/η-scaling of the time evolution is responsible for the factor η appearing in the additional error term in (12). In order to show that also $β_{n}^{ε, η, Λ_{k}} \to β_{n}^{ε, η}$ converges sufficiently fast, we need to show that all $A_{j}^{ε, η}$ have a RTDL, i.e., the operations involved in constructing the generator of $β_{n}^{ε, η}$ leave the property of having a RTDL (essentially) invariant (see Ref. 3, Appendix C). This finishes the sketch of the Proof of Theorem IV and we refer to see Ref. 3, Sec. V B for further details.

C. Resummation of the NEASS

As mentioned in the beginning of Sec. IV A 1, the statements formulated in Sec. III require a resummation, which we explain in the following. First, note that the generator $ε S_{n}^{ε, η}$ of $β_{n}^{ε, η}$ constructed above can be rewritten as $ε S_{n}^{ε, η} = \sum_{j = 1}^{n} \sum_{i = 0}^{j} ε^{i} η^{j - 1} A_{j, i}$ ⁠, where the coefficients A_j,i are time-dependent SLT operators and independent of ɛ and η. Now, it is easy to show (see Ref. 2, Lemma E.1) that there exist a sequence δ_j → 0 and constants C_n such that the resummed generator

ε S^{ε, η} = \sum_{j = 1}^{\infty} χ_{[0, 1]} (ε / δ_{j}) χ_{[0, 1]} (η / δ_{j}) \sum_{i = 0}^{j} ε^{i} η^{j - i} A_{j, i}

(25)

satisfies

{‖ε S^{ε, η} - ε S_{n}^{ε, η}‖}_{SLT} \leq C_{n} (ε^{n} + η^{n}),

where ‖ ⋅‖_SLT denotes an interaction norm similar to (5). Resummations of this type are standard, e.g., in microlocal analysis,⁵⁴ and the above estimate immediately leads to the bounds (cf. Ref. 2, Lemmata E.3, E.4),

\sup_{t \in I} |Π^{ε, η, Λ_{k}} (t) (A) - Π_{n}^{ε, η, Λ_{k}} (t) (A)| \leq C_{n}^{'} (ε^{n} + η^{n}) ‖ A ‖ | X |^{2}

(26)

and

|Π^{ε, η, Λ_{k}} (t_{0}) (U_{t, t_{0}}^{ε, η, Λ_{k}} [[A]]) - Π_{n}^{ε, η, Λ_{k}} (t_{0}) (U_{t, t_{0}}^{ε, η, Λ_{k}} [[A]])| \leq C_{n}^{″} \frac{ε^{n} + η^{n}}{η^{d + 1}} {(1 + | t - t_{0} |)}^{d + 1} ‖ A ‖ | X |^{2},

(27)

uniformly in the size of the system Λ_k. In the context of Theorems II and III, corresponding estimates hold in infinite volume, i.e., without the subscript Λ_k.

Next, since the sum in (25) is finite for every fixed ɛ > 0, also the resummed generator $S^{ε, η, Λ_{k}}$ has a TDL as soon as $S_{n}^{ε, η, Λ_{k}}$ has a TDL. Therefore, the states $Π^{ε, η, Λ_{k}}$ constructed using $S^{ε, η, Λ_{k}}$ instead of $S_{n}^{ε, η, Λ_{k}}$ have a well-defined thermodynamic limit Π^ɛ,η (see Ref. 2, Lemma E.2), and since the bounds (26) and (27) are independent of Λ_k, they also hold for the respective objects in the thermodynamic limit. Hence, the results formulated in Sec. III can be concluded by combining the n-dependent statements discussed earlier in this section with the bounds (26) and (27) (or their infinite volume correspondents).

ACKNOWLEDGMENTS

It is a pleasure to thank Stefan Teufel for numerous interesting discussions, fruitful collaboration, and many helpful comments on an earlier version of the manuscript. J.H. acknowledges partial financial support from the ERC Advanced Grant No. 101020331 “Random matrices beyond Wigner-Dyson-Mehta.” T.W. acknowledges financial support from the DFG research unit FOR 5413 “Long-range interacting quantum spin systems out of equilibrium: Experiment, Theory and Mathematics.”

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

Joscha Henheik: Writing – original draft (equal); Writing – review & editing (equal). Tom Wessel: Writing – original draft (equal); Writing – review & editing (equal).

DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

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Author(s)

On adiabatic theory for extended fermionic lattice systems

I. INTRODUCTION

A. Linear response and adiabatic theory

B. Non-equilibrium almost-stationary states

C. Brief statement of the results

II. MATHEMATICAL FRAMEWORK

A. Algebra of observables

B. Interactions and operator families

C. Lipschitz potentials

III. ADIABATIC THEOREMS FOR GAPPED QUANTUM SYSTEMS

A. Systems with a uniform gap

1. Extended but finite systems

2. Infinite systems

B. Systems with a gap in the bulk

1. Infinite systems

2. Extended but finite systems

IV. IDEA OF THE PROOFS

A. Systems with a uniform gap

1. Extended but finite systems: Proof of Theorem I

2. Infinite systems: Proof of Theorem II

B. Systems with a gap in the bulk

1. Infinite systems: Proof of Theorem III

2. Extended but finite systems: Proof of Theorem IV

C. Resummation of the NEASS

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

REFERENCES

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On adiabatic theory for extended fermionic lattice systems

I. INTRODUCTION

A. Linear response and adiabatic theory

B. Non-equilibrium almost-stationary states

C. Brief statement of the results

II. MATHEMATICAL FRAMEWORK

A. Algebra of observables

B. Interactions and operator families

C. Lipschitz potentials

III. ADIABATIC THEOREMS FOR GAPPED QUANTUM SYSTEMS

A. Systems with a uniform gap

1. Extended but finite systems

2. Infinite systems

B. Systems with a gap in the bulk

1. Infinite systems

2. Extended but finite systems

IV. IDEA OF THE PROOFS

A. Systems with a uniform gap

1. Extended but finite systems: Proof of Theorem I

2. Infinite systems: Proof of Theorem II

B. Systems with a gap in the bulk

1. Infinite systems: Proof of Theorem III

2. Extended but finite systems: Proof of Theorem IV

C. Resummation of the NEASS

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

REFERENCES

Citing articles via

Submit your article

Sign up for alerts

Related Content

Resources

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