1 Introduction
We prove an adiabatic theorem for the automorphism group acting on the observable algebra of quasi-local operators that describes the dynamics of an interacting fermionic quantum system on the lattice 
$ \mathbb {Z} ^d$ with short range interactions. For this we assume that the generator of the automorphism group, the Liouvillian, has a unique ground state and that the Gelfand–Naimark–Segal (GNS) Hamiltonian constructed from it has a spectral gap above zero.
Such systems are usually obtained by taking the thermodynamic limit 
$\Lambda \nearrow \mathbb {Z} ^d$ for a sequence of systems defined on finite domains 
$\Lambda \subset \mathbb {Z} ^d$. In [Reference Henheik and Teufel15] we prove an analogous adiabatic theorem assuming a uniform gap for each finite system in such a sequence. A spectral gap above the ground state – that is, a minimal energy for excitations – is characteristic for insulating materials. However, on finite domains it is often the case that one-body states supported close to the boundary, so-called edge states, allow for excitations with arbitrarily small energy and thus the corresponding finite-volume Hamiltonians have no spectral gap. While in many models edge states can be suppressed by choosing appropriate boundary conditions, our result shows that this is not a necessary condition for the adiabatic theorem to hold in the thermodynamic limit. Moreover, we obtain an adiabatic theorem even for finite systems with edge states by ‘restricting’ the infinite-volume result to the bulk of the finite system.
While a precise statement of our result requires considerable preparation, let us at least indicate a few more details here. Let 
$\{ H_0^\Lambda (t)\}_{\Lambda \subset \mathbb {Z} ^d, \,t\in I}$ be a time-dependent family of Hamiltonians with local interactions describing the system on finite domains 
$\Lambda $. It is known that under quite general conditions (see, e.g., [Reference Bratteli and Robinson7, Reference Nachtergaele, Sims and Young24]) the Heisenberg time evolution generated by 
$H_0^\Lambda (t)$ at fixed 
$t\in I\subset \mathbb {R} $ induces in the thermodynamic limit 
$\Lambda \nearrow \mathbb {Z} ^d$ a one-parameter group 
$s\mapsto {\mathrm {e}}^{\mathrm {i} s \mathcal {L}_{H_0(t)}}$ of automorphisms on the observable algebra 
$\mathcal {A}$ of quasi-local operators for the infinite system. Here 
denotes the Liouvillian, a densely defined derivation on 
$\mathcal {A}$. We assume that for each 
$t\in I$ the Liouvillian 
$ \mathcal {L}_{H_0(t)}$ has a unique ground state 
$\rho _0(t)$. The most important innovation of our result is that we only assume that the ‘infinite-volume Hamiltonian’ associated with 
$\rho _0(t)$ via the GNS construction has a spectral gap above its ground state, whereas in all previous work on adiabatic theorems for extended interacting systemsFootnote 1 [Reference Bachmann, Bols, De Roeck and Fraas3, Reference Monaco and Teufel20, Reference Teufel28, Reference Henheik and Teufel15] the gap assumption was imposed on the finite-volume Hamiltonians 
$H_0^\Lambda (t)$.
Let 
$\mathfrak {U}_{t,t_0}^{ \eta }$ be the cocycle of automorphisms on 
$\mathcal {A}$ generated by the time-dependent Liouvillian 
$\frac {1}{\eta }\mathcal {L}_{H_0 (\cdot )} $; that is, the physical time evolution of the infinite system. The adiabatic parameter 
$\eta>0$ controls the separation of two timescales: the internal timescale defined by the spectral gap and the timescale on which the Hamiltonian varies; for example, due to time-dependent external fields. The asymptotic regime 
$\eta \ll 1$ is called the adiabatic limit. The following infinite-volume version of the standard super-adiabatic theorem is a special case of our result: There exist super-adiabatic states 
$\rho _0^\eta (t)$ on 
$\mathcal {A}$ that are close to the instantaneous ground states 
$\rho _0(t)$ in the sense that
$$ \begin{align*} |\rho_0^\eta(t)(A) - \rho_0(t)(A) | = {\mathcal{O}} (\eta) \quad\mbox{for all } \,A\in\mathcal{D}\subset \mathcal{A}, \end{align*} $$such that the evolution 
$\mathfrak {U}^{ \eta }_{t,t_0}$ intertwines the super-adiabatic states to all orders in the adiabatic parameter 
$\eta $; that is, for all 
$A\in \mathcal {D}\subset \mathcal {A}$,
The scope of the adiabatic theorem expands considerably when additive perturbations 
$\varepsilon V^\Lambda (t)$ of 
$H_0^\Lambda (t)$ are also taken into account; for example, when justifying linear response theory with the help of the adiabatic theorem [Reference Henheik and Teufel16]. Hence, our general results concern the dynamics generated by
$$ \begin{align*} H^{\Lambda,\varepsilon}(t) := H_0^\Lambda(t) + \varepsilon V^\Lambda(t) \end{align*} $$ in the thermodynamic limit. Here 
$V^\Lambda (t)$ can be a sum-of-local-terms (SLT) operator; that is, a local Hamiltonian like 
$H_0^\Lambda (t)$, or a Lipschitz potential or a sum of both. Note that for the perturbed Hamiltonian 
$H^{\Lambda ,\varepsilon }(t)$ we do not even assume a spectral gap in the bulk, since many interesting perturbations, like a linear potential across the sample, necessarily close the spectral gap of the unperturbed system when 
$\Lambda $ is sufficiently large. In this situation the ground states 
$\rho _0(t)$ turn into resonances 
$\Pi ^\varepsilon (t)$ with lifetime of order 
${\mathcal {O}}(\varepsilon ^{-\infty })$ for the dynamics 
$s\mapsto {\mathrm {e}}^{\mathrm {i} s \mathcal {L}_{H^\varepsilon (t)}}$; that is for all 
$n,m\in \mathbb {N} $ it holds that
These resonance states we call nonequilibrium almost-stationary states (NEASS) in this context. Our adiabatic theorem, Theorem 3.4, then establishes the existence of super-adiabatic NEASSs 
$\Pi ^{ \varepsilon ,\eta }(t)$ on 
$\mathcal {A}$ close to 
$\Pi ^{ \varepsilon }(t)$ such that the adiabatic evolution 
$\mathfrak {U}^{\varepsilon , \eta }_{t,t_0}$ generated by 
$\frac {1}{\eta } \mathcal {L}_{H^{\varepsilon }(\cdot )}$ approximately intertwines the super-adiabatic NEASSs in the following sense: for any 
$n>d$ and for all 
$A\in \mathcal {D}\subset \mathcal {A}$,
uniformly for t in compact sets. While for 
$\varepsilon =0$ this statement reduces to the standard adiabatic theorem (1.1), for 
$0<\varepsilon \ll 1$ the right-hand side of (1.2) is small if only if also 
$\eta $ is small, but not too small compared to 
$\varepsilon $; that is, 
$ \varepsilon ^{n/d}\ll \eta \ll 1$ for some 
$n\in \mathbb {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 already been observed in adiabatic theory for resonances before; see, for example, [Reference Abou Salem and Fröhlich1, Reference Elgart and Hagedorn11].
Our results build on a number of recent developments in adiabatic theory for extended lattice systems. Bachmann et al. [Reference Bachmann, Bols, De Roeck and Fraas3] showed how locality in the form of the Lieb–Robinson bounds [Reference Lieb, Robinson, Nachtergaele, Solovej and Yngvason18] and the SLT-inverse of the Liouvillian [Reference Hastings and Wen14, Reference Bachmann, Michalakis, Nachtergaele and Sims5] can be used to obtain uniform adiabatic approximations for extended lattice systems with spectral gaps in finite volumes. Motivated by space-adiabatic perturbation theory [Reference Panati, Spohn and Teufel27], in [Reference Teufel28] one of us generalised the results of [Reference Bachmann, Bols, De Roeck and Fraas3, Reference Monaco and Teufel20] to situations where the spectral gap is closed; for example, due to perturbations by external fields; that is, to NEASSs. In [Reference Henheik and Teufel15] we lifted the results of [Reference Teufel28] to states for the infinite system by taking a thermodynamic limit, using recent advances on the thermodynamic limit of quantum lattice systems from [Reference Nachtergaele, Sims and Young24]. Finally, the construction of the spectral flow in the infinite volume with a gap only in the bulk by Moon and Ogata [Reference Moon and Ogata22] provided the motivation and a technical basis for the adiabatic theorem proved in this article.
As mentioned above, the adiabatic theorem has profound implications for the validity of linear response theory in such systems and thus also for transport in topological insulators. We will not discuss these questions here but instead refer to the recent reviews [Reference Henheik and Teufel16, Reference Bachmann, De Roeck and Fraas2] and to [Reference Marcelli, Panati and Teufel19]. Note, however, that all previous results listed above assume a uniform spectral gap of the Hamiltonians describing the unperturbed finite size systems and are thus not applicable to topological insulators with boundaries. Instead, periodic boundary conditions must be assumed. While one expects that in the absence of phase transitions the effect of the boundaries is negligible, this must be proved, and we achieve this in this article as far as the adiabatic approximation is concerned.
From a technical point of view, the mentioned adiabatic theorems are all based on the same perturbative scheme, called adiabatic perturbation theory. The specific challenge for extended systems in the thermodynamic limit is to control that under all of the algebraic operations used, the corresponding operators remain in spaces of operators with good locality properties and with a good thermodynamic limit. The critical operation to make adiabatic perturbation theory work as intended in (1.2) is the inversion of the Liouvillian, the step at which the gap condition is needed. Good locality properties means, very roughly speaking, that each member 
$A^\Lambda $ of an operator family 
$\{A^\Lambda \}_{\Lambda \subset \mathbb {Z} ^d}$ can be written as a sum 
$A^\Lambda = \sum _{i\in \Lambda } \Phi _i$ of uniformly bounded and uniformly local terms 
$\Phi _i$ and only the number of terms grows with the volume 
$|\Lambda |$ of 
$\Lambda $. We call such operators SLT operators.Footnote 2 The main technical challenge is thus the definition of suitable normed spaces of SLT operators, within which the adiabatic perturbation scheme can be iterated an arbitrary number of times and, in particular, within which one can invert the Liouvillian 
$\mathcal {L}_{H_0(t)}$ under the assumption of a spectral gap in the bulk. Here we rely, up to small modifications, on spaces already used in [Reference Bachmann, Bols, De Roeck and Fraas3]. However, in order to control the thermodynamic limit in a quantitative way that allows finally also statements about the bulk behaviour of finite systems with boundary, we define a property called ‘having a rapid thermodynamic limit’ and need to show that this property is preserved under the relevant algebraic operations as well.
More explicitly, this property improves the convergence and quasi-locality estimates of the dynamics, the Liouvillian and the SLT-inverse of the Liouvillian from standard norm estimates to norms measuring the quality of localisation (later called ‘f-norms’), which were introduced in [Reference Moon and Ogata22]. Since the spectral gap only appears in the bulk, the Liouvillian can finally only be inverted in the thermodynamic limit (Proposition 3.3) and thus the perturbative scheme can only yield the adiabatic theorem after taking that limit. This requires the improved quantitative control of the dynamics, the Liouvillian and the SLT-inverse of the Liouvillian in the thermodynamic limit, which is guaranteed by assuming the above property (Proposition 3.2). Besides this concept, our proof of the adiabatic theorem in the bulk heavily relies on the idea of investigating the dynamics 
$\mathfrak {U}_{t,t_0}^{\varepsilon , \eta , \Lambda , \Lambda '}$ generated by 
$H_0^\Lambda $ and 
$V^{\Lambda '}$ for different system sizes 
$\Lambda ' \subset \Lambda $ and taking 
$\Lambda \nearrow \mathbb {Z} ^d$ and 
$\Lambda ' \nearrow \mathbb {Z} ^d$ separately.
We end the Introduction with a short plan of the article. In Section 2 we explain the mathematical setup. In Section 3 we state the generalised adiabatic theorem in the infinite volume as in (1.2). We also show that our general assumptions are indeed fulfilled for certain models of topological insulators. In Section 4 we state the generalised adiabatic theorem for finite systems with a gap only in the bulk and for observables A supported sufficiently far from the boundary. The latter is the result that applies to large but finite systems with edge states that close the spectral gap. Section 5 contains the proofs of our main results. Various technical details are collected in three appendices.
2 The mathematical framework
In this section, we briefly recall the mathematical framework relevant for formulating and proving our results. More details and explanations are provided in [Reference Henheik and Teufel15].
2.1 The lattice and algebras of local observables
We consider fermions with r spin or other internal degrees of freedom on the lattice 
$\mathbb {Z}^d$. Let 
$\mathcal {P}_0(\mathbb {Z}^d) := \{X \subset \mathbb {Z}^d : \vert X \vert < \infty \}$ denote the set of finite subsets of 
$\mathbb {Z}^d$, where 
$\vert X \vert $ is the number of elements in X, and let
$$ \begin{align*} \Lambda_k := \{-k,\ldots, +k\}^d \end{align*} $$be the centred box of size 
$2k$ with metric 
$d^{\Lambda _k}(\cdot ,\cdot )$. This metric may differ from the standard 
$\ell ^1$-distance 
$d(\cdot , \cdot )$ on 
$\mathbb {Z}^d$ restricted to 
$\Lambda _k$ if one considers discrete tube or torus geometries but satisfies the bulk-compatibility condition
$$ \begin{align*} \forall k \in \mathbb{N} \ \forall x,y \in \Lambda_k: d^{\Lambda_k}(x,y) \le d(x,y) \ \ \text{and} \ \ d^{\Lambda_k}(x,y) = d(x,y) \ \text{whenever} \ d(x,y) \le k. \end{align*} $$For each 
$X \in \mathcal {P}_0(\mathbb {Z}^d)$, let 
$\mathfrak {F}_X$ be the fermionic Fock space build up from the one-body space 
$\ell ^2(X,\mathbb {C}^r)$. The 
$C^*$-algebra of bounded operators 
$\mathcal {A}_{X} := \mathcal {L}(\mathfrak {F}_X)$ is generated by the identity element 
$\mathbf {1}_{\mathcal {A}_X}$ and the creation and annihilation operators 
$a_{x,i}^*, a_{x,i}$ for 
$x \in X$ and 
$1 \le i \le r$, which satisfy the canonical anti-commutation relations. Whenever 
$X\subset X'$, 
$\mathcal {A}_{X}$ is naturally embedded as a sub-algebra of 
$\mathcal {A}_{X'}$.
The algebra of local observables is defined as the inductive limit
$$ \begin{align*} \mathcal{A}_{\mathrm{loc}} := \bigcup_{X \in \mathcal{P}_0(\mathbb{Z}^d)} \mathcal{A}_{X}, \qquad\mbox{and its closure}\qquad \mathcal{A} := \overline{\mathcal{A}_{\mathrm{loc}}}^{\Vert \cdot \Vert} \end{align*} $$with respect to the operator norm 
$\Vert \cdot \Vert $ is a 
$C^*$-algebra, called the quasi-local algebra. The even elements 
$\mathcal {A}^+ \subset \mathcal {A}$ form a 
$C^*$-subalgebra.
Also, note that for any 
$X \in \mathcal {P}_0(\mathbb {Z}^d)$ the set of elements 
$\mathcal {A}_{X}^N$ commuting with the number operator
$$ \begin{align*} N_{X} := \sum_{x \in X} a^*_x \cdot a_x := \sum_{x \in X} \sum_{i=1}^{r}a^*_{x,i} a_{x,i} \end{align*} $$forms a subalgebra of the even subalgebra; that is, 
$\mathcal {A}_X^N \subset \mathcal {A}_X^+ \subset \mathcal {A}_X$.
For any bounded, nonincreasing function 
$f: [0,\infty ) \to (0, \infty )$ with 
${\lim _{t \to \infty }f(t) =0 }$ one defines the set 
$\mathcal {D}_f^+$ of all 
$A \in \mathcal {A}^+$ such that
See Appendix C of [Reference Henheik and Teufel15] for a definition and summary of the relevant properties of the conditional expectation 
$\mathbb {E}_{\Lambda _k}$. As shown in Lemma B.1 of [Reference Moon and Ogata22], 
$\mathcal {D}_f$ is a 
$*$-algebra and 
$(\mathcal {D}_f,\|\cdot \|_f)$ is a Banach space. Although 
$(\mathcal {D}_f,\|\cdot \|_f)$ is not a normed algebra, it follows from the proof of Lemma B.1 in [Reference Moon and Ogata22] that multiplication 
$\mathcal {D}_f\times \mathcal {D}_f\to \mathcal {D}_f$, 
$(A,B)\mapsto AB$, is continuous. Note that in (2.1) and in the following we write the argument of a linear map 
$\mathcal {A}\supset \mathcal {D}\to \mathcal {A}$ in double brackets 
. This notation helps to structure complicated compositions of maps with different types of arguments.
As only even observables will be relevant to our considerations, we will drop the superscript ‘
$^+$’ from now on and redefine 
$\mathcal {A} := \mathcal {A}^+$.
2.2 Interactions and associated operator families
An interaction on a domain 
$\Lambda _k$ is a map
$$ \begin{align*} \Phi^{\Lambda_k} : \left\{X \subset \Lambda_k\right\} \to \mathcal{A}_{\Lambda_k}^N,\; X \mapsto \Phi^{\Lambda_k} (X) \in \mathcal{A}_X^N \end{align*} $$with values in the self-adjoint operators. Note that the maps 
$\Phi ^{\Lambda _k}$ can be extended to maps on the whole 
$\mathcal {P}_0(\mathbb {Z}^d)$ or restricted to a smaller 
$\Lambda _l$, trivially.
In order to describe fermionic systems on the lattice 
$\mathbb {Z}^d$ in the thermodynamic limit, one considers sequences 
$\Phi = \left (\Phi ^{\Lambda _k}\right )_{k \in \mathbb {N}}$ of interactions on domains 
$\Lambda _k$ and calls the whole sequence an interaction. An infinite-volume interaction is a map
$$ \begin{align*} \Psi : \mathcal{P}_0(\mathbb{Z}^d) \to \mathcal{A}_{\mathrm{loc}}^N,\; X \mapsto \Psi (X)\in \mathcal{A}_{X}^N , \end{align*} $$again with values in the self-adjoint operators. Such an infinite-volume interaction defines a general interaction 
$\Psi = \big (\Psi ^{ \Lambda _k}\big )_{k \in \mathbb {N}}$ by restriction; that is, by setting 
$\Psi ^{ \Lambda _k} := \Psi |_{\mathcal {P}_0(\Lambda _k)}$.
With any interaction 
$\Phi $ one associates a sequence 
$A= (A^{\Lambda _k})_{k\in \mathbb {N} }$ of self-adjoint operators
$$ \begin{align*} A^{\Lambda_k} := A^{\Lambda_k}(\Phi) := \sum_{X \subset \Lambda_k} \Phi^{\Lambda_k} (X) \in \mathcal{A}_{\Lambda_k}^N. \end{align*} $$A so-called F-function for the lattice 
$\mathbb {Z}^d$ can be chosen as 
$F_1(r) =\frac {1}{ (1+r)^{d+1}}$, which is integrable and has a finite convolution constant. These properties remain valid for 
$F_{\zeta } = \zeta \cdot F_1$ if one multiplies by a bounded, nonincreasing and logarithmically superadditive weight function 
$\zeta :[0,\infty ) \to (0,\infty )$. The space of such weight functions is denoted by 
$\mathcal {W}$, while we write 
$\mathcal {S}$ for the space of weight functions decaying faster than any polynomial. An example for a function 
$\zeta \in \mathcal {S}$ is the exponential 
$\zeta (r) = \mathrm {e}^{-ar}$ for some 
$a>0$. The finite convolution constant associated with 
$F_{\zeta }$ is denoted by
$$ \begin{align*} C_{\zeta} := \sup_{k \in \mathbb{N}} \sup_{x,y\in \Lambda_k} \sum_{z \in \Lambda_k} \frac{F_{\zeta}(d^{\Lambda_k}(x,z)) \ F_{\zeta}(d^{\Lambda_k}(z,y))}{F_{\zeta}(d^{\Lambda_k}(x,y))} < \infty. \end{align*} $$For any 
$\zeta \in \mathcal {W}$ and 
$n\in \mathbb {N}_0$, a norm on the vector space of interactions is
$$ \begin{align} \Vert \Phi \Vert_{\zeta,n } := \sup_{k \in \mathbb{N}}\sup_{x,y \in \mathbb{Z}^d} \sum_{\substack{ X \in \mathcal{P}_0(\mathbb{Z}^d): \\x,y\in X}} {d^{\Lambda_k}}\mbox{-diam}(X)^n \frac{\Vert \Phi^{\Lambda_k}(X)\Vert}{F_{\zeta}(d^{\Lambda_k}(x,y))}. \end{align} $$In order to quantify the difference of interactions ‘in the bulk’, we also introduce for any interaction 
$\Phi ^{\Lambda _l}$ on the domain 
$\Lambda _l$ and any 
$\Lambda _M\subset \Lambda _l$ the quantity
$$ \begin{align*} \Vert \Phi^{\Lambda_l} \Vert_{\zeta, n,\Lambda_M} := \sup_{x,y \in \Lambda_M} \sum_{\substack{ X \subset \Lambda_M: \\x,y\in X}} \mbox{diam}(X)^n \frac{\Vert \Phi^{\Lambda_l}(X)\Vert}{F_{\zeta}(d (x,y))}, \end{align*} $$where d and diam now refer to the 
$\ell ^1$-distance on 
$\mathbb {Z}^d$. Note that these norms depend on the sequence of metrics 
$d^\Lambda _k$ on the cubes 
$\Lambda _k$; that is, on the boundary conditions. While this will in general not be made explicit in the notation, we add a superscript 
$^\circ $ to the norm and to the normed spaces defined below if we want to emphasise the use of open boundary conditions; that is, 
$d^\Lambda _k \equiv d$. The compatibility condition for the metrics 
$d^{\Lambda _k}$ implies that 
$\Vert \Psi \Vert _{\zeta , n } \le \Vert \Psi \Vert _{\zeta , n }^\circ $.
Let 
$\mathcal {B}_{\zeta , n }$ be the Banach space of interactions with finite 
$\Vert \cdot \Vert _{\zeta , n }$-norm. Clearly, 
$\mathcal {B}_{\zeta , n } \subset \mathcal {B}_{\zeta , m }$ whenever 
$n \ge m$. Not so obviously, for any pair 
$\zeta ,\xi \in \mathcal {S}$ it holds that either 
$\mathcal {B}_{\zeta , n } \subset \mathcal {B}_{\xi , n }$ or 
$\mathcal {B}_{\xi , n } \subset \mathcal {B}_{\zeta , n }$ (or both) for all 
$n\in \mathbb {N} _0$; see Lemma A.1 in [Reference Monaco and Teufel20]. If 
$\mathcal {B}_{\zeta , n } \subset \mathcal {B}_{\xi , n }$ for all 
$n\in \mathbb {N} _0$, then we say that 
$\xi $ dominates 
$\zeta $ and write 
$\zeta \prec \xi $.
An operator family A is called an SLT operator family if 
$\Phi _A \in \mathcal {B}_{1, 0 }$. Moreover, define
$$ \begin{align*} \mathcal{B}_{\zeta, \infty } := \bigcap_{n \in \mathbb{N}_0} \mathcal{B}_{\zeta, n }, \quad \mathcal{B}_{\mathcal{S},n } := \bigcup_{\zeta \in \mathcal{S}} \mathcal{B}_{\zeta, n }, \quad \mathcal{B}_{\mathcal{S},\infty } := \bigcap_{n \in \mathbb{N}_0} \mathcal{B}_{\mathcal{S}, n } \end{align*} $$as spaces of interactions used in the sequel. The corresponding spaces of operator families are denoted as 
$\mathcal {L}_{\zeta , n }$, 
$\mathcal {L}_{\zeta , \infty }$, 
$\mathcal {L}_{\mathcal {S},n }$ and 
$\mathcal {L}_{\mathcal {S},\infty }$. Lemma A.1 in [Reference Monaco and Teufel20] shows that the spaces 
$\mathcal {B}_{\mathcal {S},n }$ and therefore also 
$\mathcal {B}_{\mathcal {S},\infty }$ are indeed vector spaces.
Let 
$I \subset \mathbb {R}$ be an open interval. We say that a map 
$A: I \to \mathcal {L}_{\zeta , n }$ is smooth and bounded whenever it is term-wise and point-wise smooth in 
$t \in I$ and if for all 
$i\in \mathbb {N} _0$ there exists 
$\zeta ^{(i)}\in \mathcal {S}$ such that 
$\zeta ^{(0)}=\zeta $ and 
$\sup _{t\in I} \Vert \frac {\mathrm {d}^i}{\mathrm {d} t^i}\Phi _A(t){\Vert}_{\zeta ^{(i)}, n } < \infty $. The corresponding spaces of smooth and bounded time-dependent interactions and operator families are denoted by 
$\mathcal {B}_{I, \zeta , n }$ and 
$\mathcal {L}_{I, \zeta , n}$ and are equipped with the norm 
$\Vert \Phi \Vert _{I, \zeta , n } := \sup _{t \in I} \Vert \Phi (t)\Vert _{\zeta , n }$.
We say that 
$A: I \to \mathcal {L}_{\mathcal {S}, \infty }$ is smooth and bounded if for any 
$n \in \mathbb {N}_0$ there is a 
$\zeta _n \in \mathcal {S}$ such that 
$A: I \to \mathcal {L}_{\zeta _n,n }$ is smooth and bounded and we write 
$\mathcal {L}_{I, \mathcal {S}, \infty }$ for the corresponding space (similarly for 
$\mathcal {L}_{I, \zeta , \infty }$ and 
$\mathcal {L}_{I, \mathcal {S}, n }$).
2.3 Lipschitz potentials
For the perturbation we will allow external potentials 
$ v = \left ( v^{\Lambda _k}: \Lambda _k \to \mathbb {R}\right )_{ k \in \mathbb {N}} $ that satisfy the Lipschitz condition
$$ \begin{align} C_v := \sup_{k \in \mathbb{N}} \sup_{\substack{x,y \in \Lambda_k}} \frac{\vert v^{\Lambda_k}(x)- v^{\Lambda_k}(y)\vert}{ d^{\Lambda_k}(x,y)} < \infty \end{align} $$and call them Lipschitz potentials. With a Lipschitz potential v we associate the corresponding operator sequence 
$V_v = (V_v^{ \Lambda _k})_{ k \in \mathbb {N}}$ defined by
$$ \begin{align*} V_v^{\Lambda_k} := \sum_{x \in \Lambda_k} v^{ \Lambda_k}(x) \hspace{0.5mm} a^*_x\hspace{-0.5mm} \cdot \hspace{-0.5mm}a_x. \end{align*} $$The space of Lipschitz potentials is denoted by 
$\mathcal {V}$. As above, we add a superscript 
$^\circ $ to the constant and to the space of Lipschitz potentials if we want to emphasise the use of open boundary conditions; that is, 
$d^\Lambda _k \equiv d$. The compatibility condition for the metrics 
$d^{\Lambda _k}$ implies that 
$C_v \ge C_v^\circ $. Note that a Lipschitz function 
$v_\infty :\mathbb {Z}^d\to \mathbb {R} $ defines, again by restriction, a Lipschitz potential in 
$\mathcal {V}^\circ $ and we write 
$v_\infty \in \mathcal {V}^\circ $ with a slight abuse of notation. We call 
$v_\infty $ an infinite-volume Lipschitz potential.
We say that the map 
$v: I \to \mathcal {V}$ is smooth and bounded whenever it is term-wise and point-wise smooth in 
$t \in I$ and satisfies 
$\sup _{t \in I} C_{ \frac {\mathrm {d}^i}{\mathrm {d} t^i}v (t)} < \infty $ for all 
$i \in \mathbb {N}_0$. The space of smooth and bounded Lipschitz potentials on the interval I is denoted by 
$\mathcal {V}_I$.
A crucial property of Lipschitz potentials is that their commutator with an SLT operator family is again an SLT operator family (see Lemma 2.1 in [Reference Teufel28]).
Lemma 2.1. Let 
$v \in \mathcal {V}_I$ and 
$A \in \mathcal {A}_X$. Then we have
$$ \begin{align*} \sup_{k \in \mathbb{N}} \sup_{s \in I} \left\Vert \left[V_v^{\Lambda_k}(s), A\right] \right\Vert \le \tfrac{1}{2}C_v r \,\mathrm{diam}(X)^{d+1} \Vert A \Vert. \end{align*} $$For 
$A \in \mathcal {L}_{I,\zeta ,n+d+1 }$ it holds that
$$ \begin{align*} \left\Vert \Phi_{[A,V_v]} \right\Vert{}_{I, \zeta, n } \le \tfrac{1}{2}C_v r \left\Vert \Phi_A \right\Vert{}_{I, \zeta, n+d+1 } \end{align*} $$and hence, in particular, 
$ [A,V_v] \in \mathcal {L}_{I, \zeta , n }. $
3 The adiabatic theorem for infinite systems with a gap in the bulk
The subject of our investigations is time-dependent Hamiltonians of the form
$$ \begin{align*} H^{\varepsilon}(t)= H_0( t) + \varepsilon V(t) \end{align*} $$for 
$t\in I\subset \mathbb {R} $. The unperturbed Hamiltonian 
$H_0(t)$, defined in terms of its short-range infinite-volume interaction 
$\Psi _{H_0}$, is assumed to have a spectral gap in the bulk uniformly in t, a notion to be made precise in the following. This gap may be closed by the perturbation 
$V(t)=V_{v_\infty }(t) + H_1(t)$ consisting of a Lipschitz potential and an additional short-range Hamiltonian, both defined on all of 
$\mathbb {Z}^d$ through an infinite-volume Lipschitz potential 
$ v_\infty $ (respectively an infinite-volume interaction 
$\Psi _{H_1}$).
Before we can formulate the precise assumptions and the statement of the theorem of this section, we first need to discuss the definition and the properties of the infinite-volume dynamics generated by 
$H^\varepsilon (t)$. With the infinite-volume interactions one naturally associates sequences of finite-volume systems, 
$\Psi _{H_0} = \big ( \Psi _{H_0}^{\Lambda _k}\big )_{k\in \mathbb {N} } $, 
$\Psi _{H_1} = \big ( \Psi _{H_1}^{\Lambda _k}\big )_{k\in \mathbb {N} } $, 
$v_\infty = ( v_\infty |_{\Lambda _k})_{k\in \mathbb {N} }$. The corresponding finite-volume Hamiltonians 
$H^{\varepsilon , \Lambda _k}(t)$ are self-adjoint and generate unique unitary evolution families 
$\mathfrak {U}^{\Lambda _k}_{t,s} $ (in the Heisenberg picture) on the finite-volume algebras 
$\mathcal {A}_{\Lambda _k}$. It is shown, for example, in [Reference Bratteli and Robinson7, Reference Bru and de Siqueira Pedra9, Reference Nachtergaele, Sims and Young24] that 
$\mathfrak {U}^{\Lambda _k}_{t,s} $ converges strongly on 
$\mathcal {A}_{\textrm {loc}}$ to a unique co-cycle of automorphisms 
$\mathfrak {U}_{t,s} $ on 
$\mathcal {A}$ whenever 
$H_0,H_1 \in \mathcal {L}_{I, \zeta , 0 }^\circ $. However, for our proofs we need norm convergence of 
$\mathfrak {U}^{\Lambda _k}_{t,s}$ to 
$\mathfrak {U}_{t,s} $ with respect to 
$\|\cdot \|_f$-norms for suitable decay functions 
$f\in \mathcal {S}$. With the next section in mind, where we consider finite systems with a gap in the bulk, we now introduce a new property for general interactions 
$\big (\Phi ^{\Lambda _k}\big )_{k\in \mathbb {N} }$ called rapid thermodynamic limit. This property guarantees norm convergence of the unitary evolution families generated by the corresponding operators. It extends the standard notion of having a thermodynamic limit as used, for example, in Definition 2.1 in [Reference Henheik and Teufel15], by a quantitative control on the rate of convergence. Definition 3.1 might seem unnecessarily complicated at first sight, but it balances two requirements: On the one hand, the condition is strong enough, such that for interactions with a rapid thermodynamic limit one can show that the induced finite-volume dynamics, derivations and the SLT inverse of the Liouvillian converge to their infinite-volume limits in the norm of bounded operators between normed spaces 
$\mathcal {D}_{f_1}, \mathcal {D}_{f_2} \subset \mathcal {A}$ of quasi-local observables with appropriate 
$f_1,f_2 \in \mathcal {S}$ (cf. Proposition 3.2). Moreover, one obtains sufficient control of the rate of convergence of these induced operations within the bulk (cf. Appendix B) in order to conclude an adiabatic theorem in the bulk of a finite-volume system as formulated in Theorem 4.1. On the other hand, the condition of having a rapid thermodynamic limit is sufficiently weak, so that it is preserved under all operations used in the perturbative construction of super-adiabatic states (cf. Appendix C). Finally, it is satisfied by all physically meaningful models known to us and, most important for the present section, it is satisfied by interactions arising from restricting an infinite-volume interaction.
Definition 3.1 Rapid thermodynamic limit of interactions and potentials
(a) A time-dependent interaction
$\Phi \in \mathcal {B}_{I,\zeta , n }$ is said to have a rapid thermodynamic limit with exponent $\gamma \in (0,1)$ if there exists an infinite-volume interaction $\Psi \in \mathcal {B}_{I,\zeta , n }^\circ $ such that (3.1)We write$$ \begin{align} \forall i \in \mathbb{N}_0 \ \ &\exists \lambda, C>0 \ \ \forall M\in \mathbb{N} \ \ \forall \, k \, \ge \, M + \lambda M^{\gamma} : \\ &\sup_{t \in I}\left\Vert \frac{\mathrm{d}^i}{\mathrm{d} t^i}\left( \Psi- \Phi^{ \Lambda_k}\right) (t) \, \right\Vert{}_{\zeta^{(i)}, n,\Lambda_M} \le C \, \zeta^{(i)}(M^{\gamma}) =: C \zeta^{(i)}_{\gamma}(M). \nonumber \end{align} $$$\Phi \stackrel {\textrm {r.t.d.}}{\rightarrow } \Psi $ in this case. Note that $\zeta \in \mathcal {S}$ implies $\zeta _{\gamma } \in \mathcal {S}$.A time-dependent interaction
$\Phi \in \mathcal {B}_{I,\mathcal {S}, n }$ is said to have a rapid thermodynamic limit with exponent $\gamma \in (0,1)$ if there exists $\zeta _n \in \mathcal {S}$ such that $\Phi \in \mathcal {B}_{I,\zeta _n, n }$ has a rapid thermodynamic limit with exponent $\gamma \in (0,1)$; a time-dependent interaction $\Phi \in \mathcal {B}_{I,\mathcal {S}, \infty }$ is said to have a rapid thermodynamic limit with exponent $\gamma \in (0,1)$ if for any $n \in \mathbb {N}$ there exists $\zeta _n \in \mathcal {S}$ such that $\Phi \in \mathcal {B}_{I,\zeta _n, n }$ has a rapid thermodynamic limit with exponent $\gamma \in (0,1)$ (similarly for $\mathcal {B}_{I, \zeta , \infty }$ and $\mathcal {B}_{I, \mathcal {S}, n }$).A family of operators is said to have a rapid thermodynamic limit if and only if the corresponding interaction does.
(b) A Lipschitz potential
$v \in \mathcal {V}_I$ is said to have a rapid thermodynamic limit with exponent $\gamma \in (0,1)$ if it is eventually independent of k; that is, if there exists an infinite-volume Lipschitz potential $ v_\infty \in \mathcal {V}_I^\circ $ such that We write$$ \begin{align*} \exists \lambda>0\ \ \forall M\in \mathbb{N} \ \ \forall \, k \, \ge \, M + \lambda M^{\gamma}, t \in I: \ \ v_{ \infty}(t, \cdot) |_{\Lambda_M} = v^{ \Lambda_k}(t, \cdot) |_{\Lambda_M} . \end{align*} $$$v\stackrel {\textrm {r.t.d.}}{\rightarrow } v_\infty $ in this case.
Trivially, an infinite-volume interaction 
$\Psi \in \mathcal {B}_{I,\zeta , n }^\circ $ has a rapid thermodynamic limit with any exponent 
$\gamma \in (0,1)$. Moreover, an interaction (respectively a Lipschitz potential) having a rapid thermodynamic limit with exponent 
$\gamma \in (0,1)$ clearly has a thermodynamic limit in the sense of Definition 2.1 from [Reference Henheik and Teufel15] (see also Definition 3.7 from [Reference Nachtergaele, Sims and Young24]) and thus the property of having a rapid thermodynamic limit for the interaction (respectively potential) guarantees also the existence of a thermodynamic limit for the associated evolution families and derivations as in Proposition 2.2 from [Reference Henheik and Teufel15] (see also Proposition B.1 and Proposition B.6).
Definition 3.1 also applies to time-independent interactions and potentials trivially and the interactions in 
$\mathcal {B}_{I,\zeta , n }$ and that the Lipschitz potentials in 
$\mathcal {V}_I$ that have a rapid thermodynamic limit with exponent 
$\gamma \in (0,1)$ form sub-vector spaces. We provide an equivalent characterisation of interactions having a rapid thermodynamic limit in terms of a Cauchy condition in Lemma C.1.
The improved quasi-locality estimates as required for the proof of the adiabatic theorem in the infinite volume, Theorem 3.4, are summarised in the following proposition. More detailed statements, which make explicit the rate of convergence and also apply to the SLT operator families that appear in the perturbative construction of the NEASS, are given in Appendix B. They are, however, only required for the proof of the adiabatic theorem in finite volume (Theorem 4.1).
Proposition 3.2 Rapid thermodynamic limit and quasi-locality estimates
Let 
$H_0 \in \mathcal {L}_{I, \exp (-a \,\cdot ), 0 }$ and 
$v \in \mathcal {V}_I$ both have a rapid thermodynamic limit with exponent 
$\gamma \in (0,1)$, set 
$H=H_0 + V_v$ and let 
$\mathfrak {U}_{t,s}$ denote the bulk dynamics generated by the derivation (the Liouvillian) 
$\mathcal {L}_{H (t)}: D(\mathcal {L}_{H (t)}) \to \mathcal {A}$ associated with 
$H(t)$ at 
$t \in I$. Moreover, let 
$\mathcal {I}_{H_0 (t)} $ be the SLT-inverse of the Liouvillian 
$\mathcal {L}_{H_0(t)}$ (see Appendix A). The finite-volume versions for the cubes 
$\Lambda _k$ are indicated by the superscript 
$\Lambda _k$.
Then the dynamics, the Liouvillian and the SLT-inverse of the Liouvillian preserve quasi-locality and the finite-volume versions converge rapidly to their bulk versions in the following sense: For suitable functions 
$f_1, f_2 \in \mathcal {S}$ (which may be different in every case below) it holds that
1.
$\mathfrak {U}_{t,s} :\mathcal {D}_{f_1} \to \mathcal {D}_{f_2}$ is a bounded operator and $\mathfrak {U}^{\Lambda _k}_{t,s} \xrightarrow {k \to \infty }\mathfrak {U}_{t,s}$ in the corresponding operator norm, both uniformly for $s,t$ in compacts.2.
$\mathcal {D}_{f_1} \subset D(\mathcal {L}_{H (t)})$, $\mathcal {L}_{H (t)} :\mathcal {D}_{f_1} \to \mathcal {D}_{f_2}$ is a bounded operator and $\mathcal {L}_{H (t)}^{\Lambda _k} \xrightarrow {k \to \infty }\mathcal {L}_{H_ (t)}$ in the corresponding operator norm, both uniformly for $t \in I$.3.
$\mathcal {I}_{H_0 (t)} :\mathcal {D}_{f_1} \to \mathcal {D}_{f_2}$ is a bounded operator and $\mathcal {I}_{H_0 (t)}^{\Lambda _k} \xrightarrow {k \to \infty } \mathcal {I}_{H_0 (t)}$ in the corresponding operator norm, both uniformly for $t \in I$.
The gap assumption for 
$H_0$ in the bulk can be formulated via the GNS construction. Let 
$\Psi _{H_0}\in \mathcal {B}_{\zeta ,0 }^\circ $ be an infinite-volume interaction and let 
$\mathcal {L}_{H_0}$ denote the induced derivation on 
$\mathcal {A}$. A state 
$\omega $ on 
$\mathcal {A}$ is called a 
$ \mathcal {L}_{H_0}$-ground state, iff 
$\omega (A^*\mathcal {L}_{H_0}(A)) \ge 0$ for all 
$A \in D(\mathcal {L}_{H_0})$. Let 
$\omega $ be a 
$ \mathcal {L}_{H_0}$-ground state and 
$(\mathcal {H}_{\omega }, \pi _{\omega }, \Omega _{\omega })$ be the corresponding GNS triple (
$\mathcal {H}_{\omega }$ a Hilbert space, 
$\pi _{\omega }: \mathcal {A} \to \mathcal {B}(\mathcal {H}_{\omega })$ a representation and 
$\Omega _{\omega } \in \mathcal {H}_{\omega }$ a cyclic vector). Then there exists a unique densely defined, self-adjoint positive operator 
$H_{0,\omega } \ge 0$ on 
$\mathcal {H}_{\omega }$ satisfying
for all 
$A \in \mathcal {A}$ and 
$t \in \mathbb {R}$. We call this 
$H_{0,\omega }$ the bulk Hamiltonian associated with 
$\Psi _{H_0}$ and 
$\omega $. See [Reference Bratteli and Robinson7] for the general theory.
We can now formulate the assumptions of the main theorem. Recall that 
$I \subseteq \mathbb {R}$ denotes an open time interval.
Assumptions on the interactions.
(I1) Let
$\Psi _{H_0}, \Psi _{H_1} \in \mathcal {B}_{I, \exp (-a \, \cdot ),\infty }^\circ $ be time-dependent infinite-volume interactions and $v_\infty \in \mathcal {V}^\circ _I$ be a time-dependent infinite-volume Lipschitz potential.(I2) Assume that the map
$I\to \mathcal {B}_{\exp (-a \, \cdot ),\infty }^\circ $, $t\mapsto \Psi _{H_0(t)}$, is continuously differentiable.Footnote 3
Moreover, we assume that 
$\Psi _{H_0}$ has a unique gapped ground state in the following sense.
Assumptions on the ground state of 
$\Psi _{H_0}$.
(G1) Uniqueness. For each
$t \in I$, there exists a unique $\mathcal {L}_{H_0(t)}$-ground state $\rho _0(t)$.(G2) Gap. There exists
$g>0$ such that $\sigma (H_{0,\rho _0(t)}(t)) \setminus \{0\} \subset [2g,\infty )$ for all $t \in I$.(G3) Regularity. For any
$f \in \mathcal {S}$ and $A \in \mathcal {D}_f$, $t\mapsto \rho _0(t)(A)$ is differentiable and there exists a constant $C_f$ such that for all $A \in \mathcal {D}_f$, $$ \begin{align*} \sup_{t \in I} \vert \dot{\rho_0}(t)(A)\vert \le C_f \Vert A\Vert_f. \end{align*} $$
The assumptions (I1) and (I2) on the interactions are quite explicit and easily checked for concrete models. Moreover, they are very general and hold for standard tight-binding models with short-range interactions. The assumptions (G1)–(G3) on the ground state of 
$\Psi _{H_0}$ are much harder to verify in concrete models. Assuming uniqueness – that is, (G1) – the gap assumption (G2) holds, in particular, if the finite-volume Hamiltonians have a uniform gap: that the gap persists in the thermodynamic limit was shown, for example, in Proposition 5.4 in [Reference Bachmann, Dybalski and Naaijkens4]. And the smoothness of expectation values of (almost) exponentially localised observables as under item (G3) is a consequence of a uniform gap in finite volumes (see Remark 4.15 in [Reference Moon and Ogata22] and Lemma 6.0.1 in [Reference Moon21]). The uniqueness of the ground state as required under (G1) has been shown, to our present knowledge, only in very specific quantum spin systems. We will discuss the validity of our assumptions in more detail below.
The key role of the spectral gap condition is that it allows constructing an SLT-inverse of the Liouvillian 
$\mathcal {L}_{H_0(t)}$. Assuming a gap only in the bulk, as we do, means that the action of the Liouvillian can only be inverted in the bulk. The following proposition shows that this is indeed a meaningful concept. Its proof is given in Appendix A.
Proposition 3.3 Inverting the Liouvillian in the bulk
Let 
$\Psi _{H_0} \in \mathcal {B}_{ \exp (-a \cdot ), 0 }^\circ $ be an infinite-volume interaction and 
$\rho _0$ a 
$\mathcal {L}_{H_0}$-ground state that satisfies the gap assumption (G2). Then for all 
$A \in \mathcal {A}$ with 
$\mathcal {I}_{H_0}(A) \in D(\mathcal {L}_{H_0})$ and all 
$B \in \mathcal {A}$, we have
We now state our main theorem. Its proof is given in Section 5 and based on the statements summarised in Proposition 3.2 and Proposition 3.3.
Theorem 3.4 Adiabatic theorem for infinite systems with a gap in the bulk
Let 
$\Psi _{H_0}$, 
$\Psi _{H_1}$ and 
$v_\infty $ satisfy (I1), (I2), (G1), (G2) and (G3). Denote by 
$\mathfrak {U}_{t,t_0}^{\varepsilon , \eta }$ the Heisenberg time evolution on 
$\mathcal {A}$ generated by 
$\frac {1}{\eta }\Psi _{H^{\varepsilon }}$ with
$$ \begin{align*} \Psi_{H^{\varepsilon}}:= \Psi_{H_0} + \varepsilon (V_{v_\infty} + \Psi_{H_1}) \end{align*} $$and adiabatic parameter 
$\eta \in (0,1]$ (cf. Corollary B.2). Then for any 
$\varepsilon , \eta \in (0,1]$ and 
$t \in I$ there exists a near-identity automorphism 
$\beta ^{\varepsilon , \eta }(t)= \mathrm {e}^{\mathrm {i} \varepsilon \mathcal {L}_{S^{\varepsilon , \eta }(t)}}$ with 
$\Psi _{ S^{\varepsilon , \eta }}\in \mathcal {B}^\circ _{I,\mathcal {S},\infty } $ such that the super-adiabatic NEASS defined by
$$ \begin{align*} \Pi^{\varepsilon, \eta}(t) := \rho_0(t)\circ \beta^{\varepsilon, \eta}(t) \end{align*} $$has the following properties:
1.
$\Pi ^{\varepsilon , \eta }$ almost intertwines the time evolution: For any $n \in \mathbb {N}$, there exists a constant $C_n$ such that for any $ t \in I$ and for all finite $X \subset \mathbb {Z}^d$ and $A \in \mathcal {A}_X \subset \mathcal {A}$, (3.3)For any$f \in \mathcal {S}$ the same statement (with a different constant $C_n$) holds also for all $A \in \mathcal {D}_f$ after replacing $\Vert A \Vert \vert X \vert ^2$ by $ \Vert A \Vert _f$.2.
$\Pi ^{\varepsilon , \eta }$ is local in time: $\beta ^{\varepsilon , \eta }(t)$ depends only on $\Psi _{H^{\varepsilon }}$ and its time derivatives at time t.3.
$\Pi ^{\varepsilon , \eta }$ is stationary whenever the Hamiltonian is stationary: if for some $t\in I$ all time derivatives of $\Psi _{H^{\varepsilon } }$ vanish at time t, then $\Pi ^{\varepsilon , \eta }({t}) = \Pi ^{\varepsilon , 0}({t})$.4.
$\Pi ^{\varepsilon , \eta } $ equals the ground state $\rho _0$ of $\mathcal {L}_{H_0 }$ whenever the perturbation vanishes and the Hamiltonian is stationary: if for some $t\in I$ all time derivatives of $\Psi _{H^{\varepsilon } }$ vanish at time t and $\Psi _{H_1(t)} = 0$ and $v_\infty (t) =0$, then $\Pi ^{\varepsilon , \eta }(t) = \Pi ^{\varepsilon , 0}(t) = \rho _0(t)$.
Let us discuss a class of Hamiltonians that are used to model Chern or topological insulators and for which our assumptions can be shown or are expected to hold. Consider the unperturbed (time-independent) Hamiltonian
$$ \begin{align} H_0^{\Lambda_k} \;= \;\sum_{x,y \in \Lambda_k} a_x^*T(x -y)a_y \;+\; \sum_{x \in \Lambda_k} a_x^*\phi(x)a_x + \sum_{x,y \in \Lambda_k} a_x^* a_x W(d^{\Lambda_k}(x,y))a_y^*a_y\; -\; \mu N_{\Lambda_k}. \end{align} $$Assume that the kinetic term 
$T : \mathbb {Z}^d \to \mathcal {L}(\mathbb {C}^r)$ is an exponentially decaying function with 
$T(-x) = T(x)^*$, the potential term 
$\phi : \mathbb {Z}^d \to \mathcal {L}(\mathbb {C}^r)$ is a bounded function taking values in the self-adjoint matrices and the two-body interaction 
$W: [0,\infty ) \to \mathcal {L}(\mathbb {C}^r)$ 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 
$\Lambda _k$ if 
$\Lambda _k$ is supposed to have a torus geometry.
It is well known that noninteracting Hamiltonians 
$H_0$ – that is, with 
$W \equiv 0$ – of the type (3.4) on a torus (periodic boundary condition) have a spectral gap whenever the chemical potential 
$\mu $ lies in a gap of the spectrum of the corresponding one-body operator on the infinite domain. It was recently shown [Reference Hastings13, Reference De Roeck and Salmhofer10] that the spectral gap remains open when perturbing by sufficiently small short-range interactions 
$W \neq 0$. On the other hand, the Hamiltonian 
$H_0$ on a cube with open boundary condition in general, no longer has a spectral gap because of the appearance of edge states.
Note that for either boundary condition the corresponding interactions have a rapid thermodynamic limit and converge to the same infinite-volume interaction 
$\Psi _{H_0}$ defined by T, 
$\phi $ and W on all of 
$\mathbb {Z}^d$. Thus, in the thermodynamic limit the different boundary conditions lead to the same infinite-volume dynamics on the quasi-local algebra 
$\mathcal {A}$. More precisely, by application of, for example, Proposition 2.2 from [Reference Henheik and Teufel15], both sequences of dynamics are strongly convergent; that is,
for any observable 
$A\in \mathcal {A}_{\mathrm {loc}}$.
The spectral gap for the finite-volume ground states (which we have for the torus geometry) implies a spectral gap also for the GNS Hamiltonian associated to any limiting 
$ \mathcal {L}_{H_0}$-ground state, as it cannot close abruptly in the thermodynamic limit (see Proposition 5.4 in [Reference Bachmann, Dybalski and Naaijkens4]). Therefore, the Hamiltonian (3.4) with open boundary conditions has a rapid thermodynamic limit and a gap only in the bulk. While uniqueness of the ground state is expected to hold for such models, to our knowledge it has been shown only for certain types of spin systems (cf. [Reference Yarotsky29, Reference Fröhlich and Pizzo12, Reference Nachtergaele, Sims and Young25, Reference Nachtergaele, Sims and Young26, Reference Henheik, Teufel and Wessel17]).
A typical perturbation that is physically relevant is the adiabatic switching of a small constant electric field. Theorem 4.1 then shows that the response in the bulk is local and independent of the boundary condition. For more details on the implications for linear response, we refer to [Reference Henheik and Teufel16].
4 The adiabatic theorem for finite domains
Let 
$\Phi _{H_0}, \Phi _{H_1} \in \mathcal {B}_{I, \exp (-a \, \cdot ),\infty } $ be interactions and 
$v\in \mathcal {V}_I$ a Lipschitz potential, all with a rapid thermodynamic limit; that is, 
$\Phi _{H_0} \stackrel {\textrm {r.t.d.}}{\rightarrow } \Psi _{H_0}$, 
$\Phi _{H_1} \stackrel {\textrm {r.t.d.}}{\rightarrow } \Psi _{H_1}$ and 
$v \stackrel {\textrm {r.t.d.}}{\rightarrow } v_\infty $. For the infinite-volume interaction 
$\Psi _{H_0}$ we require again existence of a unique gapped ground state as made precise in assumptions (G1)–(G3). However, in this section we prove an adiabatic theorem for the finite-volume dynamics of states 
$\rho _0^{\Lambda _k}(t)$ that are close to the infinite-volume ground state 
$\rho _0(t)$ in the bulk. More precisely, we require
(S) The sequence
$\big (\rho _0^{\Lambda _k}(t)\big )_{k\in \mathbb {N} }$ of states on $\mathcal {A}_{\Lambda _k}$ converges rapidly to $\rho _0(t)$ in the bulk: there exist $C\in \mathbb {R} $, $m \in \mathbb {N}$ and $h \in \mathcal {S}$ such that for any finite $X \subset \mathbb {Z}^d$, $A \in \mathcal {A}_X$ and $\Lambda _k\supset X$, $$ \begin{align*} \sup_{t \in I} \left\vert \rho_0(t)(A) - \rho_0^{\Lambda_k}(t)(A) \right\vert \le C \Vert A \Vert \,\mathrm{diam}(X)^m \, h\big(\mathrm{dist}(X,\mathbb{Z}^d \setminus \Lambda_k)\big). \end{align*} $$
While the sequence 
$\rho _0(t)|_{\mathcal {A}_{\Lambda _k}}$ obtained by just restricting the infinite-volume ground state trivially satisfies conditions (S), the adiabatic theorem below holds for super-adiabatic NEASS constructed from any sequence satisfying (S). Most interesting for application would be a sequence of ground states 
$\rho _0^{\Lambda _k}(t)$ of the finite-volume Hamiltonians 
$H_0^{\Lambda _k}(t)$. While (S) is expected to hold for any sequence of finite-volume ground states for the models discussed in the previous section, the only result we are aware of indeed proving such a statement is again for weakly interacting spin systems [Reference Yarotsky29].
The following theorem asserts that by assuming a spectral gap in the bulk for the infinite system one obtains similar adiabatic bounds also for states of finite systems (without a spectral gap!) that are close to the infinite-volume ground state in the bulk. As adiabaticity need not hold at the boundaries of the finite systems, nonadiabatic effects arising at the boundary may propagate into the bulk. For this reason, an additional error term appears, but it decays faster than any polynomial in the size of the finite system for any fixed 
$\eta $.
Theorem 4.1 Adiabatic theorem for finite systems with gap in the bulk
Let 
$\Phi _{H_0}, \Phi _{H_1} \in \mathcal {B}_{I, \exp (-a \, \cdot ),\infty } $ be interactions and 
$v\in \mathcal {V}_I$ a Lipschitz potential, all with a rapid thermodynamic limit; that is, 
$\Phi _{H_0} \stackrel {\textrm {r.t.d.}}{\rightarrow } \Psi _{H_0}$, 
$\Phi _{H_1} \stackrel {\textrm {r.t.d.}}{\rightarrow } \Psi _{H_1}$ and 
$v \stackrel {\textrm {r.t.d.}}{\rightarrow } v_\infty $. Let 
$\Psi _{H_0}$, 
$\Psi _{H_1}$ and 
$ v_\infty $ satisfy (I1), (I2), (G1), (G2) and (G3). Moreover, let 
$\big (\rho _0^{\Lambda _k}(t)\big )_{k\in \mathbb {N} }$ satisfy (S). For all 
$k\in \mathbb {N} $ let 
$\mathfrak {U}_{t,t_0}^{\varepsilon , \eta , \Lambda _k}$ be the Heisenberg time evolution on 
$\mathcal {A}_{\Lambda _k}$ generated by
$$ \begin{align*} \tfrac{1}{\eta}H^{\varepsilon,\Lambda_k}(t) := \tfrac{1}{\eta} \left( H^{ \Lambda_k}_0(t) + \varepsilon \left( V_v^{\Lambda_k} (t) + H^{\Lambda_k}_1(t) \right)\right) \end{align*} $$with adiabatic parameter 
$\eta \in (0,1]$. Then for any 
$\varepsilon , \eta \in (0,1]$ and 
$t \in I$ there exists a near-identity automorphism 
$\beta ^{\varepsilon , \eta ,\Lambda _k}(t)$ of 
$\mathcal {A}_{\Lambda _k}$ such that the super-adiabatic NEASS defined by
$$ \begin{align*} \Pi^{\varepsilon, \eta,\Lambda_k}(t) := \rho_0^{\Lambda_k}(t)\circ \beta^{\varepsilon, \eta,\Lambda_k}(t) \end{align*} $$has the following properties:
1. It almost intertwines the time evolution of observables in the bulk: There exists
$\lambda>0$ such that for any $n \in \mathbb {N}$ there exists a constant $C_n$ and for any compact $K\subset I$ and $m\in \mathbb {N} $ there exists a constant $\tilde {C}_{n,m,K}$ such that for all $k\in \mathbb {N} $, all finite $X \subset \Lambda _k$, all $A \in \mathcal {A}_X$ and all $t,t_0\in K$, (4.1)2. It is local in time:
$\beta ^{\varepsilon , \eta ,\Lambda _k}(t)$ depends only on $H^{\varepsilon ,\Lambda _k}$ and its time derivatives at time $t $.3. It is stationary whenever the Hamiltonian is stationary: if for some
$t\in I$ all time derivatives of $H^{\varepsilon ,\Lambda _k} $ vanish at time t, then $\Pi ^{\varepsilon , \eta ,\Lambda _k}({t}) = \Pi ^{\varepsilon , 0,\Lambda _k}({t})$.4. It equals
$ \rho ^{\Lambda _k}_0(t)$ whenever the perturbation vanishes and the Hamiltonian is stationary: if for some $t\in I$ all time derivatives of $H^{\varepsilon , \Lambda _k} $ vanish at time t and $V^{\Lambda _k}(t)=0$, then $\Pi ^{\varepsilon , \eta ,\Lambda _k}(t) = \Pi ^{\varepsilon , 0,\Lambda _k}(t) = \rho ^{\Lambda _k}_0(t)$.
As mentioned before, the second term in the bound (4.1) arises when nonadiabatic transitions that can occur in the boundary region propagate into the bulk. While we do not prove this explicitly, our proof suggests that if the perturbations v, 
$H_1$ and 
$\dot H_0$ are supported in the bulk and if 
$\rho _0^{\Lambda _k}(t)$ is the ground state of 
$H_0(t)$, then this second part of the bound can be replaced by a term that is uniform in time and 
$\eta $. Finally, for the topological insulators discussed in the previous section, one expects that excitations at the boundary cannot propagate into the bulk, since the chiral edge states do not couple to states supported in the bulk. That is, in such systems we expect (4.1) to hold with the second term replaced by a bound that is uniform in time and 
$\eta $ even if the adiabatic perturbation acts also near the boundary.
5 Proofs of the main results
5.1 Proof of the adiabatic theorem in the bulk
Proof of Theorem 3.4. The proof is an adaption of the strategies for proving the analogous theorem for finite domains (see Theorem 5.1 in [Reference Teufel28]) and its thermodynamic limit version (see Theorem 3.2 in [Reference Henheik and Teufel15]). In order to cope with the new situation of ‘a gap only in the bulk’, we use and adapt techniques originally developed in [Reference Moon and Ogata22] and summarised in a modified way in Appendix B.
Before going into details, let us briefly outline the structure of the proof. We first construct a sequence of densely defined derivations 
$(\mathcal {L}_{A_j^{\varepsilon , \eta }(t)})_{j\in \mathbb {N} }$ such that for each 
$n\in \mathbb {N} $ the state 
$\Pi _n^{\varepsilon ,\eta }(t) := \rho _0(t)\circ \beta _n^{\varepsilon , \eta }(t)$ constructed from the automorphism
generated by
$$ \begin{align} \varepsilon \mathcal{L}_{S_n^{\varepsilon, \eta}(t)} := \sum_{j = 1}^{n} \varepsilon^{j} \mathcal{L}_{A_j^{\varepsilon, \eta}(t)} \end{align} $$satisfies (3.3). The n-independent states 
$\Pi ^{\varepsilon ,\eta }(t)$ are then obtained from a suitable resummation of the asymptotic series (5.1) for 
$n\to \infty $.
In [Reference Teufel28] the corresponding SLT operators 
$A_j^{\varepsilon , \eta , \Lambda _l}(t)$ on finite domains 
$\Lambda _l$ are constructed from an algebraic iterative scheme. Basically, we would like to apply the same scheme to construct the associated derivations in the infinite volume with good control of the remainders that add up to the right-hand side of (3.3). To do so, we first construct the coefficients 
$A_j^{\varepsilon , \eta , \Lambda _l,\Lambda _k}(t)$ in finite volumes 
$\Lambda _l$ exactly as in [Reference Teufel28], where the perturbation is restricted to an even smaller volume 
$\Lambda _k$. Because of the absence of a spectral gap in finite volume, however, the remainders are not small; that is, a bound like (3.3) does not hold. The two main steps are now
(1) to show that the SLT operators
$A_j^{\varepsilon , \eta , \Lambda _l, \Lambda _k}(t)$ have a rapid thermodynamic limit.Footnote 4(2) to show that (3.3) holds in the thermodynamic limit.
Part (1) is rather technical: The iterative construction of the operators 
$A_j^{\varepsilon , \eta , \Lambda _l,\Lambda _k}(t)$ takes as an initial input the instantaneous Hamiltonian 
$H^{ \Lambda _l}_0(t)$, its time derivative 
$\dot H^{ \Lambda _k}_0(t)$ and the perturbation 
$V^{\Lambda _k}(t)$. Then it proceeds (see (5.16) and (5.17)) by taking sums of commutators, further time derivatives and applying the map 
$\mathcal {I}$ defined in Appendix B.3 and called, for simplicity, the inverse Liouvillian, although it does not have this property in finite volumes. In Appendix C we show that all of these operations preserve the property of having a rapid thermodynamic limit (see Lemma C.2, Lemma C.3 and Lemma C.4) and thus also all 
$A_j^{\varepsilon , \eta , \Lambda _l,\Lambda _k}(t)$ have a rapid thermodynamic limit.
Part (2) is now discussed in detail and we divide it into four steps.
(a) Decomposition of 
As a first step we construct a two-parameter family of automorphisms 
$\beta _n^{\varepsilon , \eta , \Lambda _l, \Lambda _k}(t)$ where the unperturbed Hamiltonian 
$H_0(t)$ and the ‘perturbations’ 
$V_v$, 
$H_1$ and 
$\dot {H}_0$ enter on different scales 
$\Lambda _l$ and 
$\Lambda _k$ with 
$ l\geq k$.
To do so, we extend each restriction 
$v_{\infty }^{\Lambda _k}:\Lambda _k\to \mathbb {R} $ of the infinite-volume Lipschitz potential 
$v_\infty $ to a function 
$\tilde v^{\Lambda _k}:\mathbb {Z}^d\to \mathbb {R} $ in such a way that 
$\mathrm {supp}( \tilde v^{\Lambda _k}) \subset \Lambda _{2k}$ and 
$\tilde v^{\Lambda _k}$ satisfies the Lipschitz condition (2.3) on all of 
$\mathbb {Z}^d$, possibly with a different but k-independent constant 
${C}^\circ _{\tilde v}$. Hence, 
$\tilde v^{\Lambda _k}$ defines a Lipschitz potential on any 
$\Lambda _l$ with 
$l\geq 2k$. Moreover, the corresponding operator satisfies 
$V_{\tilde v}^{\Lambda _k} \in \mathcal {A}_{\Lambda _{l}} \subset \mathcal {A}_{\textrm {loc}} \subset \mathcal {D}_f$ for any 
$l\geq 2k$ and all 
$f \in \mathcal {S}$.
By 
$\mathfrak {U}_{t,t_0}^{\varepsilon , \eta ,\Lambda _l, \Lambda _k}$ we denote the dynamics generated by 
$H_0^{\Lambda _l} + \varepsilon (V^{\Lambda _k}_{ \tilde v} + H_1^{\Lambda _k}) =: H_0^{\Lambda _l} + \varepsilon V^{\Lambda _k}$ on 
$\mathcal {A}_{\Lambda _l}$ with adiabatic parameter 
$\eta>0$. Similarly, let 
$\beta _n^{\varepsilon , \eta , \Lambda _l, \Lambda _k}(t)$ denote the automorphism on 
$\mathcal {A}_{\Lambda _l}$ constructed from the inputs 
$H_0^{\Lambda _l}$, 
$ V_{\tilde v}^{\Lambda _k}$, 
$H_1^{\Lambda _k}$ and 
$\dot {H}_0^{\Lambda _k}$ (cf. (5.16)) and define the state
$$ \begin{align*} \Pi_n^{\varepsilon, \eta, \Lambda_l, \Lambda_k}(t) := \rho_0(t) \circ \beta_n^{\varepsilon, \eta, \Lambda_l, \Lambda_k}(t) \end{align*} $$on 
$\mathcal {A}_{\Lambda _l}$. In the following we will consider 
$H^{\varepsilon ,k} := \{ H^{\Lambda _l}_0 + \varepsilon H_1^{\Lambda _k} \}_{l\in \mathbb {N} }$ for each fixed 
$k\in \mathbb {N} $ as an SLT operator and use, often implicitly, that the 
$\Vert \cdot \Vert _{I, \zeta , n }$-norms of the corresponding interactions are bounded uniformly in k.
From now on, we drop the superscripts 
$\varepsilon $ and 
$\eta $ and fix 
$X\in \mathcal {P}_0(\mathbb {Z}^d)$ and 
$A\in \mathcal {A}_X$. Then for any 
$l,k\in \mathbb {N} $ with 
$X\subset \Lambda _l$ and 
$l\geq 2k$ we can split the left-hand side of (3.3) as
$$ \begin{align} & \hphantom{a\,\left\vert \Pi_n^{}-\Pi_n^{}(t)(A)\right\vert}\quad + \ \left\vert \Pi^{\Lambda_l, \Lambda_k}_n( t)(A) -\Pi_n^{}( t)(A)\right\vert . \end{align} $$We will now show that (5.2) and (5.4) can be made arbitrarily small by choosing 
$l,k$ large enough. To be precise, by this phrase we mean that for any 
$\delta>0$ there exists 
$N_\delta \in \mathbb {N} $ such that the corresponding term is smaller than 
$\delta $ for all 
$k\geq N_\delta $ and all 
$l\geq 2k$. For (5.3) we show that it satisfies the bound (3.3), up to terms that vanish in the limit 
$l\to \infty $ for any 
$k\in \mathbb {N} $. Together these statements then imply (3.3).
(b) Thermodynamic limits of the automorphisms
To see that (5.2) and (5.4) are small for 
$l,k$ large enough, we merely use that the involved automorphisms have a thermodynamic limit. We only discuss (5.2) in some detail and estimate
The first term (5.5) can be made arbitrarily small by choosing l large enough. The precise argument was carried out in the proof of Theorem 3.5 in [Reference Henheik and Teufel15] and is not repeated here. For the second term, one combines Theorem 3.4 and Theorem 3.8 from [Reference Nachtergaele, Sims and Young24] (see also Theorem B.2 from [Reference Henheik and Teufel15]) in the same manner as in Proposition B.1. Using these statements and the construction procedure (see (5.16) and (5.17)) of 
$\varepsilon {S_n^{}} = \sum _{j = 1}^{n} \varepsilon ^{j} {A_j^{}}$ generating the automorphism 
$\beta _n^{\Lambda _l, \Lambda _k}$, one obtains that (5.6) can be made arbitrarily small by choosing 
$l,k$ large enough. Note that the composition of the automorphisms can be estimated with the aid of Lemma 4.1 from [Reference Henheik and Teufel15]. One can deal with (5.4) similarly.
(c) Thermodynamic limit of the adiabatic approximation
We are now left to study the term (5.3) that compares the full time evolution with the adiabatic evolution on the cube 
$\Lambda _l$ with the perturbations restricted to the smaller cube 
$\Lambda _k$. For this difference one obtains an expansion in powers of 
$\varepsilon $ exactly like in the proof of Proposition 5.1 in [Reference Teufel28]. However, in the absence of a spectral gap of the finite-volume Hamiltonians, the coefficients up to order n are not identically zero anymore. Instead, we need to show that they vanish as 
$l \to \infty $ for arbitrary 
$k \in \mathbb {N}$. To do so, we will apply Proposition 3.3 and the technical lemmata in Appendix B, which have been summarised in Proposition 3.2.
To begin, observe that for 
$A \in \mathcal {A}_{\mathrm {loc}}$,
According to Lemma B.3 and Lemma B.5, there exists a positive constant 
$C_\eta $ such that
uniformly for 
$s, t$ in compacts and for suitable 
$f_1$ and 
$f_2$. The bound in (5.8) is uniform in l and k, since the constants on the right-hand side of the relevant estimates in Lemma B.3 and Lemma B.5 only depend on the interaction norm, 
$\Vert \cdot \Vert _{I, \zeta , n }$, and the constant, 
$ C^\circ _{\tilde v}$, of the Lipschitz potential, which are both independent of l and k. However, the constant 
$C_\eta $ diverges as 
$\eta $ goes to zero. But this will be of no concern for us, since we will use (5.8) only to show that certain quantities vanish in the thermodynamic limit at a fixed value of 
$\eta>0$.
At this point, a technical digression is necessary. In (5.8) and many times in the following, we will consider (sequences of) automorphisms or derivations on the algebra 
$\mathcal {A}$ as (sequences of) bounded operators between normed spaces 
$(\mathcal {D}_{f_1}, \|\cdot \|_{f_1})$ and 
$(\mathcal {D}_{f_2}, \|\cdot \|_{f_2})$ for ‘suitable’ 
$f_1,f_2\in \mathcal {S}$, where 
$f_2$ dominates 
$f_1$; that is, 
$f_1\prec f_2$. The precise connection between 
$f_1$ and 
$f_2$ is quantified by the corresponding lemmata in Appendix B and also depends on the weight function 
$\zeta $ of the generator of the automorphism (respectively derivation). In Lemma B.11 we show that relative to any fixed weight function 
$\zeta \in \mathcal {S}$ there exists a sequence 
$(f_j)_{j\in \mathbb {N} }$ of weight functions 
$f_j\in \mathcal {S}$ such that the pair 
$f_i,f_{i+1}$ is ‘suitable’ for all 
$i\in \mathbb {N} $; that is, that it can be used in all of the lemmata of Appendix B together with any decay function 
$\xi \prec \zeta $. Now in this proof we fix such a sequence 
$(f_j)_{j\in \mathbb {N} }$ relative to the ‘worst’ weight function 
$\zeta _n$ appearing in the construction at level n; that is, all (finitely many) SLT operators are elements of 
$\mathcal {L}^\circ _{I, \zeta _n, \infty }$. More precisely, let 
$(f_j)_{j\in \mathbb {N} }$ be the sequence in 
$\mathcal {S}^R_\zeta $ constructed in Lemma B.11 for 
$\zeta (N) = \zeta _{n}\left (\lfloor \frac {N}{2} \rfloor ^\gamma \right )^{\frac {1}{2}}$ and 
$R = C \cdot 2^n$ with some 
$C < \infty $ (a polynomial growth would also suffice, as follows from the discussion in the proof of Lemma B.10). Hence, in the finitely many norm estimates like (5.8) that will appear in the proof, we can just pick successive pairs 
$f_i,f_{i+1}$ from this sequence 
$f_1\prec f_2\prec f_3 \cdots \prec f_i \prec f_{i+1}$. As the precise weight function of the final target space plays no role in the statement of the theorem, we do not keep track of this explicitly.
We now return to (5.7). Assuming (G3), the derivative can be evaluated by using the linearity and continuity of the involved maps. We find
The first term (5.9) can be evaluated with the help of the following lemma (cf. Equation (2.27) in [Reference Moon and Ogata22]). Its proof is given after the proof of the theorem.
Lemma 5.1 Derivative of the ground state
Let 
$f \in \mathcal {S}$ and 
$A \in \mathcal {D}_f$. Then
Proceeding with (5.9) we obtain
For (5.10) we get
and for (5.11)
(i) Approximating (5.12): We now replace (5.12) by an expression involving the map 
$\mathcal {I}$ defined in Appendix A. Using that the function w appearing in its definition is even and that 
$\rho _0(s)$ is a 
$ \mathcal {L}_{H_0(s)}$-ground state, we have
Using (5.8) together with Lemma B.9 and Lemma B.4, we see that this difference can be made arbitrarily small for 
$l,k$ large enough. More precisely, by the uniform bound in (5.8) (see also Lemma B.3 and Lemma B.5), we have the convergence of the dynamics, Lemma B.4, as well as the convergence of the derivation, Lemma B.9, both in suitable f-norms. Hence, we may replace (5.12) by the negative of (5.15) up to an arbitrarily small error for 
$l,k$ large enough.
(ii) Terms up to order n : In the second step, we expand the first entry of the commutator in (5.13) and (5.14) together with the negative of (5.15) as in Proposition 5.1 of [Reference Teufel28]. Having 
$-\mathrm {i}/\eta $ as a common prefactor, we get
$$ \begin{align*} \eta \int_{0}^{1} \mathrm{d} \lambda \,& \mathrm{e}^{-\mathrm{i} \lambda \varepsilon S_n^{ \Lambda_l, \Lambda_k}} \varepsilon \dot{S}_n^{ \Lambda_l, \Lambda_k} \, \mathrm{e}^{ \mathrm{i} \lambda \varepsilon S_n^{ \Lambda_l, \Lambda_k}} +\ \mathrm{e}^{-\mathrm{i}\varepsilon S_n^{ \Lambda_l, \Lambda_k}} \left( H_0^{\Lambda_l} + \varepsilon V^{\Lambda_k} \right)\mathrm{e}^{\mathrm{i}\varepsilon S_n^{\Lambda_l, \Lambda_k}} + \ \eta\,\mathcal{I}^{\Lambda_l}(\dot{H}_0^{\Lambda_k}) \\ =: \; &H_0^{\Lambda_l} + \sum_{j=1}^{n} \varepsilon^j R_j^{\Lambda_l, \Lambda_k} + \varepsilon^{n+1} R_{n+1}^{\Lambda_l, \Lambda_k}. \end{align*} $$Inserting the leading order term 
$H_0^{\Lambda _l}$ back into the commutator in (5.14), we find that
for any 
$k \in \mathbb {N}$ and uniformly for s and t in compacts. Here we used (5.8) and the fact that Lemma B.9 and continuity of 
$\rho _0(s)$ imply that
$$ \begin{align*} \lim_{l \to \infty} \rho_0(s) \circ \mathcal{L}_{H_0^{\Lambda_l}(s)} = \rho_0(s) \circ \mathcal{L}_{H_0 (s)} \equiv 0 \end{align*} $$uniformly on bounded subsets of 
$\mathcal {D}_f$.
The first-order term has the form
$$ \begin{align*} R_1^{\Lambda_l, \Lambda_k}= -\mathrm{i} \mathcal{L}_{H_0}^{\Lambda_l} (A_1^{\Lambda_l, \Lambda_k}) + \left(\tfrac{\eta}{\varepsilon}\,\mathcal{I}^{\Lambda_l}(\dot{H}_0^{\Lambda_k}) - V^{\Lambda_k} \right) =: -\mathrm{i} \mathcal{L}_{H_0}^{\Lambda_l} (A_1^{\Lambda_l, \Lambda_k}) +\tilde{R}_1^{\Lambda_l, \Lambda_k} \end{align*} $$with
$$ \begin{align} A_1^{\Lambda_l, \Lambda_k} := \mathcal{I}^{\Lambda_l}\left(V^{\Lambda_k} - \tfrac{\eta}{\varepsilon}\,\mathcal{I}^{\Lambda_l}(\dot{H}_0^{\Lambda_k})\right). \end{align} $$With the aid of Lemma B.10, it follows that, for any 
$k \in \mathbb {N}$, 
$A_1^{\Lambda _l, \Lambda _k}$ is convergent in a suitable f-norm as 
$l \to \infty $. So, using Lemma B.9 and Lemma B.10, for any 
$k \in \mathbb {N}$,
$R_1^{\Lambda _l, \Lambda _k}$ is also convergent in a suitable f-norm and its limit is given by
$$ \begin{align*} - \mathrm{i} \mathcal{L}_{H_0} \left( \mathcal{I}\left( \tfrac{\eta}{\varepsilon}\,\mathcal{I}(\dot{H}_0^{\Lambda_k})- V^{\Lambda_k}\right)\right) + \left(\tfrac{\eta}{\varepsilon}\,\mathcal{I}(\dot{H}_0^{\Lambda_k}) - V^{\Lambda_k} \right). \end{align*} $$Thus, we get by using (5.8) and Proposition 3.3 that
for any 
$k \in \mathbb {N}$ and uniformly for s and t in compacts.
The remainder terms 
$R_j^{\Lambda _l, \Lambda _k}(s)$ for 
$j=2,\ldots ,n$ have the form
$$ \begin{align*} R_j^{\Lambda_l, \Lambda_k} = -\mathrm{i} \mathcal{L}_{H_0}^{\Lambda_l} (A_j^{\Lambda_l, \Lambda_k}) + \tilde{R}_j^{\Lambda_l, \Lambda_k} \end{align*} $$where the operators 
$A_j^{\Lambda _l, \Lambda _k}$ are determined inductively as
$$ \begin{align} A_j^{\Lambda_l, \Lambda_k} := - \mathcal{I}^{\Lambda_l}(\tilde{R}_j^{\Lambda_l, \Lambda_k}) \end{align} $$and 
$\tilde {R}_j^{\Lambda _l, \Lambda _k}$ is composed of a finite number of iterated commutators of the operators 
$A_i^{\Lambda _l, \Lambda _k}$ and 
$\dot {A}_i^{\Lambda _l, \Lambda _k}$, 
$i=1,\ldots , j-1$, with 
$H_0^{\Lambda _l}$ and 
$V^{\Lambda _k}$.Footnote 5 By this structure, we have with the aid of Lemma B.9, Lemma B.10 and the continuity of multiplication in 
$(\mathcal {D}_f,\|\cdot \|_f)$ that 
$R_j^{\Lambda _l, \Lambda _k}$ is convergent in a suitable f-norm and we get
for 
$j=2,\ldots ,n$, any 
$k \in \mathbb {N}$ and uniformly for s and t in compacts by Proposition 3.3 and (5.8). So, the coefficients up to order 
$\varepsilon ^n$ vanish in the limit 
$l \to \infty $.
(iii) Remainder term: In the last step, the remainder term involving 
$R_{n+1}^{\Lambda _l, \Lambda _k}$ can be estimated as in the proof of Proposition 5.1 and Theorem 5.1 in [Reference Teufel28] by
where the constant 
$C_n$ is independent of 
$l,k \in \mathbb {N}$ and all other parameters (cf. [Reference Bachmann, Bols, De Roeck and Fraas3, Reference Monaco and Teufel20, Reference Teufel28]). More precisely, the estimate follows from Lemma C.5 from [Reference Monaco and Teufel20]. Note that the bound in this Lemma only depends on the the Lieb–Robinson velocity (B.1), the relevant interactions norms and the Lipschitz constant 
$ C^\circ _{\tilde {v}}$.
(d) Conclusion and resummation
Summarising our considerations, we have shown that for any 
$n\in \mathbb {N} $ there exists a positive constant 
$C_n$ such that
The n-independent states 
$\Pi ^{\varepsilon , \eta }$ that satisfy the estimate (3.3) can be constructed using Lemma E.3 and Lemma E.4 from [Reference Henheik and Teufel15]. All other statements on 
$\Pi _n^{\varepsilon , \eta }(t)$ are clear by construction (as in [Reference Teufel28]).
Proof of Lemma 5.1. The proof of this identity in [Reference Moon and Ogata22] uses the technical Lemmata 4.4, 4.5, 4.6, 4.12, 4.13, which we have adapted to our notion of interactions in Appendix B, Lemmata B.4, B.3, B.10, B.8 and B.9, respectively. Therefore, we only sketch the main arguments. Again, at each step, one chooses suitable f-norms.
Let 
$f \in \mathcal {S}$ and 
$A \in \mathcal {D}_f$. First, using the spectral gap of the bulk Hamiltonian, one can show that
where 
$\mathcal {J}_s$ is defined in Appendix A. By application of the Duhamel formula, one arrives at
Using (I2) in the form of Lemma B.8, (G3) and that
, since 
$\rho _0(s)$ is the 
$ \mathcal {L}_{H_0(s)}$-ground state, we have
5.2 Proof of the adiabatic theorem for finite domains
Proof of Theorem 4.1. We show that both terms on the left-hand side of (4.1) converge to their infinite-volume limits with a rate given by the second term on the right-hand side of (4.1). Together with Theorem 3.4, this implies the claim. Again, we do this first for fixed n and comment on the resummation afterwards.
Fix 
$X\in \mathcal {P}_0(\mathbb {Z}^d)$ and 
$A\in \mathcal {A}_X$. Then for any 
$k_2 \le k_1 \le k$ such that 
$X \subset \Lambda _{k_2}$ we obtain for the first term in (4.1) that
For (5.18) and (5.22) we combine the estimates provided by Proposition B.1 and Corollary B.2 with the trivial bound by 
$2 \Vert A \Vert $ using the following trivial inequality with 
$\alpha = \eta $:
$$ \begin{align} 0 \leq c \leq \min(a,b) \qquad\Rightarrow\qquad c \le a^{\alpha} \cdot b^{1-\alpha} \quad \mbox{for all} \quad \alpha \in (0,1). \end{align} $$This yields
$$ \begin{align*} (5.17) + (5.21)\; \le\; C(t,t_0) \, \Vert A \Vert \, \mathrm{diam}(X)^{(d+1)\eta} \exp\left(-a\, \eta \, \mathrm{dist}(X, \mathbb{Z}^d \setminus \Lambda_{\lceil k_2 - \lambda k_2^{\gamma} \rceil})^{\gamma}\right), \end{align*} $$where 
$C(t,t_0)$ depends only on t, 
$t_0$ and 
$H_0$.
Since 
$\varepsilon S^{\varepsilon , \eta }_n \in \mathcal {L}_{I,\mathcal {S}, \infty , \infty }$ has a rapid thermodynamic limit with exponent 
$\gamma \in (0,1)$ by application of Lemma C.2, Lemma C.3 and Lemma C.4, we have that for some 
$\zeta _n \in \mathcal {S}$ in particular 
$\varepsilon S^{\varepsilon , \eta }_n \in \mathcal {L}_{I,\zeta _n, 0, \infty }$ has a rapid thermodynamic limit with exponent 
$\gamma \in (0,1)$. Hence, (5.19) and (5.21) can be estimated using the ‘local decomposition technique’, (5.23), Proposition B.1 and Corollary B.2 as follows: For 
$Z \subset Z'$, let 
$\mathbb {E}^{Z'}_Z: \mathcal {A}_{Z'} \to \mathcal {A}_Z $ be the conditional expectation (on even observables!). Moreover, for a set 
$Y \in \mathcal {P}_0(\mathbb {Z}^d)$ and 
$\delta \ge 0$, let 
$ Y_{\delta } = \{z \in \mathbb {Z}^d : \mathrm {dist}(z,Y) \le \delta \}$ be the ‘fattening’ of the set Y by 
$\delta $. Defining
where v is the Lieb–Robinson velocity for 
$\min (a,a')$ as in (B.1) and for 
$j\ge 1$,
we can write
, where the sum is always finite, since eventually 
$X_{\frac {v}{\eta }\vert t-t_0 \vert + j }\cap \Lambda _{k_2} = \Lambda _{k_2}$. Clearly, 
$A^{(j)} \in \mathcal {A}_{X_{\frac {v}{\eta }\vert t-t_0 \vert + j }}$ and we bound
$$ \begin{align*} \mathrm{diam} (X_{\frac{v}{\eta}\vert t-t_0 \vert + j } ) \le C\, \mathrm{diam}(X)\, (j+1) \left(1+\eta^{-1}\vert t-t_0\vert\right). \end{align*} $$According to the properties of the conditional expectation on even observables (see Lemma C.2 from [Reference Henheik and Teufel15]) and using the Lieb–Robinson bound (Proposition B.1) in combination with (5.23), we have
$$ \begin{align*} \Vert A^{(j)} \Vert \le \min\left(C \Vert A \Vert \vert X\vert \mathrm{e}^{-\min(a,a')j}, \, 2 \Vert A\Vert\right) \le C_{\alpha_1} \Vert A \Vert \vert X\vert^{\alpha_1} \mathrm{e}^{-\alpha_1 \min(a,a')j} \end{align*} $$with 
$\alpha _1 \in (0,1)$. Thus, with the aid of Corollary B.2 and using again (5.23), we get
where 
$\alpha _1, \alpha _2 \in (0,1)$, since the sum
$$ \begin{align*} \sum_{j=0}^{\infty} \mathrm{e}^{-\alpha_1 \min(a,a')j} (j+1)^{(d+1)\alpha_2} < \infty \end{align*} $$is finite for every choice of 
$\alpha _1, \alpha _2 \in (0,1)$. Therefore, we arrive at
$$ \begin{align*} (5.18) + (5.20) \le C_{\alpha_1, \alpha_2} \Vert A \Vert \, \mathrm{diam}(X)^{(d+1)(\alpha_1+ \alpha_2)}(1+\eta^{-1}\vert t-t_0\vert )^{(d+1)\alpha_2} \, \zeta_{\gamma}^{\alpha_2}(\lceil k_1 - \lambda k_1^{\gamma}\rceil - k_2). \end{align*} $$For (5.20), we perform the local decomposition twice. Using the notation from above, we define
and for 
$i \ge 1$
Hence, we can write
and the sums are always finite. Using this form, it can be estimated with the same methods as above by
$$ \begin{align*} (5.19) \;\le\; C_{\alpha_1, \alpha_2, \alpha_3}\Vert A \Vert \, \mathrm{diam}(X)^{m\alpha_1+ d(\alpha_2 + \alpha_3)} \, (1+\eta^{-1}\vert t-t_0\vert )^{ d\alpha_2} h^{\alpha_1}(k-k_1), \end{align*} $$where 
$\alpha _1, \alpha _2, \alpha _3 \in (0,1)$.
Analogously, we estimate the second term on the left-hand side of (4.1) by
for 
$k_3 \le k$ to be chosen and 
$A \in \mathcal {A}_X$ such that 
$X \subset \Lambda _{k_3}$. As above, we get
$$ \begin{align*} (5.23) + (5.25) \;\le\; C_{\alpha} \Vert A \Vert \,\mathrm{diam}(X)^{(d+1)\alpha}\, \zeta_{\gamma}^{\alpha}\,(\mathrm{dist}(X, \mathbb{Z}^d \setminus \Lambda_{\lceil k_3 - \lambda k_3^{\gamma}\rceil})) \end{align*} $$with 
$\alpha \in (0,1)$ and
$$ \begin{align*} (5.24)\; \le\; C_{\alpha_1, \alpha_2}\Vert A \Vert \, \mathrm{diam}(X)^{m\alpha_1+ (d+1)\alpha_2} \, h^{\alpha_1}(k-k_3), \end{align*} $$with 
$\alpha _1, \alpha _2 \in (0,1)$. Now, we choose
$$ \begin{align*} k_1 &= \left\lfloor k-\frac{\mathrm{dist}\left(X,\mathbb{Z}^d \setminus \Lambda\left(k\right)\right) - \lambda k^{\gamma}}{3} \right\rfloor \\ k_2 &= \left\lfloor k-\frac{2 \left(\mathrm{dist}\left(X,\mathbb{Z}^d \setminus \Lambda\left(k\right)\right) - \lambda k^{\gamma}\right)}{3} \right\rfloor\\ k_3 &= \left\lfloor k-\frac{\mathrm{dist}\left(X,\mathbb{Z}^d \setminus \Lambda\left(k\right)\right) - \lambda k^{\gamma}}{2} \right\rfloor, \end{align*} $$which satisfy 
$k \ge k_1 \ge k_2$ and 
$k \ge k_3 $, as well as 
$k_1 \ge \frac {k}{2}$, 
$k_2 \ge \frac { k}{4}$ and 
$k_3 \ge \frac {k}{3}$ for k large enough.
Putting everything together and choosing the 
$\alpha $-parameters appropriately, we have shown that
for all 
$m \in \mathbb {N}$ and compact 
$K \subset I$, which is a valid estimate for all 
$A \in \mathcal {A}_X$ with 
$X \subset \Lambda _k$ after a possible adjustment of 
$\tilde {C}$. The n-independent states 
$\Pi ^{\varepsilon , \eta , \Lambda _k}(t)$ that satisfy the estimate (4.1) can again be constructed using Lemma E.3 and Lemma E.4 from [Reference Henheik and Teufel15]. All other statements on 
$\Pi _n^{\varepsilon , \eta , \Lambda _k}(t)$ are clear by construction (as in [Reference Teufel28]).
A Inverting the Liouvillian in the bulk
In this Appendix, we prove Proposition 3.3. First note that there exists a nonnegative, even function 
$w \in L^1(\mathbb {R})$ with 
$\Vert w\Vert _1=1$ and
$$ \begin{align*} \sup_{s \in \mathbb{R} } \vert s \vert^n \vert w(s) \vert < \infty, \quad \forall n \in \mathbb{N}_0, \end{align*} $$such that its Fourier transform
$$ \begin{align*} \hat{w}(k) = \frac{1}{\sqrt{2\pi}} \int_{\mathbb{R}} \mathrm{e}^{-\mathrm{i}ks} \,w(s) \, \mathrm{d}s \end{align*} $$satisfies 
$\mathrm {supp}(\hat {w}) \subset [-1,1]$. An explicit function w having all of these properties is constructed in [Reference Bachmann, Michalakis, Nachtergaele and Sims5]. For any 
$g>0$ (spectral gap of 
$H_0$), set
$$ \begin{align*} w_g(s) = g \,w(gs). \end{align*} $$It is clear that 
$w_g$ is nonnegative, even, 
$L^1$-normalised and, moreover,
$$ \begin{align*} \mathrm{supp}(\hat{w}_g) \subset [-g,g]. \end{align*} $$We drop the subscript g from now on. Let 
$H_0 \in \mathcal {L}_{I, \exp (-a\,\cdot ), 0}$ have a rapid thermodynamic limit and define bounded linear maps
for every 
$s \in I$ and 
$k \in \mathbb {N}$. The corresponding ‘bulk’ versions are defined as
The integrals can be understood as Bochner integrals on 
$(\mathcal {A}, \Vert \cdot \Vert )$ by the continuity of
for any 
$A \in \mathcal {A}$ (see Proposition B.1). The map 
$\mathcal {I}_t$ is called the SLT-inverse of the Liouvillian, which is justified by Proposition 3.3 and Lemma C.4. See Appendix D of [Reference Henheik and Teufel15] for a definition of the SLT-inverse of the Liouvillian on interactions.
Proof of Proposition 3.3. We drop the subscript 
$H_0$ of 
$\mathcal {L}$ and 
$\mathcal {I}$ from Proposition 3.3 to simplify notation. We prove this proposition using the GNS representation of 
$\rho _0$. Let 
$P_{\Omega _{\rho _0}} =\ \mid\kern-3pt\Omega _{\rho _0}\rangle\kern2pt \langle\Omega _{\rho _0}\kern-3pt\mid$ be the projection on the GNS state and let 
$\mathcal {I}_{\rho _0}$ be the GNS version of the inverse Liouvillian; that is,
$$ \begin{align*} \mathcal{I}_{\rho_0}(A) = \int_{\mathbb{R}} \mathrm{d}s \, w(s) \int_{0}^{s}\mathrm{d}u \, \mathrm{e}^{\mathrm{i} u H_{0,\rho_0}} A \mathrm{e}^{- \mathrm{i} uH_{0,\rho_0}}, \quad A \in \mathcal{B}(\mathcal{H}_{\rho_0}), \end{align*} $$and 
$\mathcal {J}_{\rho _0}$ analogously, where 
$H_{0,\rho _0}$ is the GNS Hamiltonian. Then we have
Since
, we have by Theorem 4 from [Reference Bratteli and Robinson6] that for all 
$B \in D(\mathcal {L})$,
By application of Proposition 6.9 from [Reference Nachtergaele, Sims and Young24], we get
$$ \begin{align*} \mathrm{i} [H_{0,\rho_0}, \mathcal{I}_{\rho_0}(\pi_{\rho_0}(A))] = \mathcal{J}_{\rho_0}(\pi_{\rho_0}(A)) - \pi_{\rho_0}(A); \end{align*} $$that is, we inverted the Liouvillian up to 
$\mathcal {J}_{\rho _0}(\pi _{\rho _0}(A))$. But, again with the aid of Proposition 6.9 from [Reference Nachtergaele, Sims and Young24] and using the cyclicity of the trace, we have
$$ \begin{align*} \langle \Omega_{\rho_0},[ \mathcal{J}_{\rho_0}(\pi_{\rho_0}(A)),\pi_{\rho_0}(B)] \Omega_{\rho_0} \rangle& \;=\; \mathrm{tr}(P_{\Omega_{\rho_0}}[ \mathcal{J}_{\rho_0}(\pi_{\rho_0}(A)),\pi_{\rho_0}(B)])\\ & \;=\;\mathrm{tr}( \pi_{\rho_0}(B)\underbrace{ [P_{\Omega_{\rho_0}}, \mathcal{J}_{\rho_0}(\pi_{\rho_0}(A))]}_{=0}) \;=\; 0. \end{align*} $$By using the GNS representation again, we also have
$$ \begin{align*} \langle \Omega_{\rho_0},[\pi_{\rho_0}(A) ,\pi_{\rho_0}(B)]\, \Omega_{\rho_0} \rangle = \rho_0([A,B]). \end{align*} $$We thus showed that for all 
$B\in D(\mathcal {L})$,
By density of 
$D(\mathcal {L})$ in 
$\mathcal {A}$ and continuity of 
$\rho _0:\mathcal {A}\to \mathbb {C} $, the equality holds for all 
$B\in \mathcal {A}$.
B Quasi-locality estimates
In this appendix we show how to control the actions of automorphisms, derivations and inverse Liouvillians generated by SLT operators with a rapid thermodynamic limit on spaces 
$\mathcal {D}_f\subset \mathcal {A}$ of quasi-local observables of the infinite system.
Let us briefly summarise the structure of and the motivation for the following results. To control the adiabatic approximation in the thermodynamic limit we need to consider the actions of automorphisms, derivations and inverse Liouvillians also on the spaces 
$\mathcal {D}_f$ introduced in [Reference Moon and Ogata22]. Parts of the statements of the required lemmata, Lemmata B.3, B.4, B.5, B.8, B.9, and B.10, were established already in [Reference Moon and Ogata22], however, without explicit uniformity and only for SLT operators defined by restrictions of a fixed interaction on 
$\mathcal {P}_0(\mathbb {Z}^d)$. The SLT operators 
$A_j$ appearing in the adiabatic expansion are not of this form, even if we would assume this form for our original Hamiltonian. Thus, we generalise the results of [Reference Moon and Ogata22] to SLT operators having a thermodynamic limit and make the uniformity explicit. In addition, we prove norm convergence of automorphisms and derivations in spaces of bounded operators from 
$(\mathcal {D}_{f_1},\|\cdot \|_{f_1})$ to 
$(\mathcal {D}_{f_2},\|\cdot \|_{f_2})$ in the thermodynamic limit. For this, however, the quantitative notion of rapid thermodynamic limit is required.
As starting points, we first establish the convergence of automorphisms in the thermodynamic limit as in [Reference Nachtergaele, Sims and Young24] but with additional quantitative control on the rate of convergence implied by the condition that the generator has a rapid thermodynamic limit (Proposition B.1 and Corollary B.2). Similarly, we prove quantitative estimates on the rate of convergence of derivations in the thermodynamic limit (Proposition B.6 and Corollary B.7).
B.1 Dynamics
Since we refer to the Lieb–Robinson bounds several times in this work, we restate them for convenience of the reader in the following proposition. Its only novel content is, however, the implications on the rate of convergence of automorphism groups for generators with a rapid thermodynamic limit.
Proposition B.1 Lieb–Robinson bound and convergence of dynamics
Let 
$H_0 \in \mathcal {L}_{I, \zeta , 0 }$ and 
$v \in \mathcal {V}_I$ both have a thermodynamic limit (see Definition 2.1 in [Reference Henheik and Teufel15]) and set 
$H:= H_0 + V_v$. For 
$A\in \mathcal {A}_{\Lambda _k}$ define the local dynamics as
where 
$U^{\Lambda _k}(t,s)$ is the solution to the Schrödinger equation
$$ \begin{align*} \mathrm{i}\tfrac{\mathrm{d}}{\mathrm{d}t} U^{\Lambda_k}(t,s) = H^{\Lambda_k} (t) U^{ \Lambda_k}(t,s) \end{align*} $$with 
$U^{\Lambda _k}(s,s) = \mathrm {id}$. There exists 
$C<\infty $ such that for all 
$A \in \mathcal {A}_X$ and 
$B \in \mathcal {A}_Y$ with 
$X, Y \subset \Lambda _k$ and all 
$k\in \mathbb {N}$,
If 
$H_0 \in \mathcal {L}_{I, \exp (-a \, \cdot ), 0 }$ for some 
$a>0$, one defines the Lieb–Robinson velocity via
$$ \begin{align} v_a := 2 a^{-1}C_{\exp(-a \, \cdot)}\Vert\Phi_{H_0} \Vert_{I,\exp(-a \, \cdot ), 0 } \end{align} $$and obtains the more transparent bound
If 
$H_0$ and v have a rapid thermodynamic limit with exponent 
$\gamma \in (0,1)$, then there exist 
$\lambda _1>0$, 
$\lambda _2 \in (0,1)$ and 
$C<\infty $ such that for all 
$l,k \in \mathbb {N}$ with 
$l \ge k$, 
$X \subset \Lambda _k$ and 
$A \in \mathcal {A}_X$,
In every case above, the constant C depends only on 
$\zeta $, 
$\Vert \Phi _{H_0} \Vert _{I, \zeta , 0 }$ and the Lipschitz constant 
$C_v$.
Proof. The first part is the standard Lieb–Robinson bound (see [Reference Lieb, Robinson, Nachtergaele, Solovej and Yngvason18] for the first proof; for fermionic systems, see [Reference Bru and de Siqueira Pedra9], [Reference Nachtergaele, Sims and Young23]). Invoking the estimate
for any 
$A \in \mathcal {A}_X$ (see Theorem 3.8 in [Reference Nachtergaele, Sims and Young24] and Lemma 2.1), the estimate (B.2) with the first alternative in the maximum is a consequence of Theorem 3.4 and Theorem 3.8 in [Reference Nachtergaele, Sims and Young24] by choosing 
$\Lambda _M = \Lambda _{\lceil k - \lambda _1 k^{\gamma }\rceil }$ with 
$\lambda _1$ from Definition 3.1 (respectively the alternative characterisation in Lemma C.1). The second alternative can easily be concluded from the first and is used in the proofs of the lemmata below.
As a consequence of (B.2), 
is a Cauchy sequence in 
$(\mathcal {A},\|\cdot \|) $ for every 
$A \in \mathcal {A}_{\mathrm {loc}}$. By density, the limits 
define a co-cycle 
$\mathfrak {U}_{t,s}$ of automorphisms on 
$\mathcal {A}$ (the bulk dynamics) and the map 
$I\times I \to \mathcal {A}$, 
is continuous for every 
$A \in \mathcal {A}$.Footnote 6
As the finite-volume metrics 
$d^{\Lambda _k}(\cdot , \cdot )$ are compatible in the bulk, we obtain the corresponding statements for the bulk dynamics.
Corollary B.2 Infinite-volume dynamics
Under the conditions of Proposition B.1 there exists 
$C<\infty $, such that for all 
$X,Y\in \mathcal {P}_0(\mathbb {Z}^d)$, 
$A \in \mathcal {A}_X$ and 
$B \in \mathcal {A}_Y$
If 
$H_0$ and v have a rapid thermodynamic limit, there exist 
$\lambda _1>0$, 
$\lambda _2 \in (0,1)$ such that
In both cases above, the constant C depends only on 
$\zeta $, 
$\Vert \Phi _{H_0} \Vert _{I, \zeta , 0 }$ and the Lipschitz constant 
$C_v$.
We are now ready to prove generalisations of Lemma 4.4 and Lemma 4.5 from [Reference Moon and Ogata22].
Lemma B.3 Quasi-locality of dynamics I
Let 
$H_0 \in \mathcal {L}_{I, \exp (-a \cdot ), 0 }$ and 
$v \in \mathcal {V}_I$ both have a thermodynamic limit, set 
$H=H_0 + V_v$ and let 
$\mathfrak {U}_{t,s}$ denote the infinite-volume dynamics generated by H. Let 
$f_1,f_2 :[0,\infty ) \to (0,\infty )$ be bounded, nonincreasing functions with 
$\lim _{s\to \infty } f_i(s) = 0$ for 
$i = 1,2$, such thatFootnote 7
$$ \begin{align*} \int_{ 0}^\infty \mathrm{d}s \, w_g(s) \frac{2s }{f_2(4v_a s)} < \infty, \end{align*} $$
$$ \begin{align*} \sup_{N \in \mathbb{N}} \frac{f_1( N- \lfloor \frac{N}{2} \rfloor)}{f_2(N)} < \infty,\qquad \sup_{N \in \mathbb{N}} \frac{N^{d+1} \, \mathrm{e}^{-a \frac{\left\lfloor \frac{N}{2} \right\rfloor}{2}}}{f_2(N)} < \infty, \end{align*} $$where 
$w_g$ is defined in Appendix A.
Then 
$\mathfrak {U}_{t,s} : \mathcal {D}_{f_1} \to \mathcal {D}_{f_2}$ is a bounded operator and the sequence 
$\mathfrak {U}^{\Lambda _N}_{t,s} : \mathcal {D}_{f_1} \to \mathcal {D}_{f_2}$ of operators is uniformly bounded, both uniformly for s and t in compacts. More precisely, there is a nonnegative, nondecreasing function 
$b_{f_1, f_2}: [0,\infty ) \to [0,\infty )$ such that
for all 
$A \in \mathcal {D}_{f_1}$. Moreover, we have that
$$ \begin{align*} \int_{0}^\infty \mathrm{d}t \, s \,w_g(s) \, b_{f_1, f_2} (s) < \infty. \end{align*} $$Besides the indicated dependence on 
$f_1$ and 
$f_2$, the function 
$b_{f_1,f_2}$ only depends on a, 
${\Vert \Phi _{H_0} \Vert _{I,\exp (-a \, \cdot ), 0 }}$ and the Lipschitz constant 
$C_v$.
Proof. The proof is analogous to the one of Lemma 4.5 from [Reference Moon and Ogata22]. Let 
$A \in \mathcal {D}_{f_1}$. Then, as 
$\mathfrak {U}_{t,s}$ is an automorphism,
From Corollary B.2 in combination with Lemma C.2 from [Reference Henheik and Teufel15], for 
$N,k \in \mathbb {N}$ with 
$k<N$, we obtain
For 
$N \in \mathbb {N}$ with 
$4v_a \vert t-s \vert \le N$, we use this bound with 
$k = N- \left \lfloor \tfrac {N}{2}\right \rfloor $ to estimate
On the other hand, for 
$N \in \mathbb {N}$ with 
$4v\vert t-s \vert> N$, we simply have
Hence, summarising our investigations, we obtain
The claim follows by the requirements on 
$f_1$ and 
$f_2$. The uniform boundedness of 
$\mathfrak {U}^{\Lambda _N}_{t,s}$ can be proven in the same way.
Lemma B.4 Quasi-locality of dynamics II
Let 
$H_0 \in \mathcal {L}_{I, \exp (-a\, \cdot ), 0 }$ and 
$v \in \mathcal {V}_I$ both have a rapid thermodynamic limit with exponent 
$\gamma \in (0,1)$, set 
$H=H_0 + V_v$ and let 
$\mathfrak {U}_{t,s}$ denote the dynamics generated by H. Let 
$f_1,f_2 :[0,\infty ) \to (0,\infty )$ be bounded, nonincreasing functions with 
$\lim _{s\to \infty } f_i(s) = 0$, for 
$i = 1,2$, such that
$$ \begin{align*} \lim_{N \to \infty} \frac{f_1(\lfloor \frac{N}{2} \rfloor)}{f_2(N)} = 0 \quad\mbox{and}\quad \lim_{N \to \infty} \frac{N^{d+1} \, \mathrm{e}^{-a (N-\lfloor \frac{N}{2} \rfloor)^{\gamma}}}{f_2(N)} = 0. \end{align*} $$Then 
$\mathfrak {U}_{t,s} : \mathcal {D}_{f_1} \to \mathcal {D}_{f_2}$ and 
$\mathfrak {U}^{\Lambda _N}_{t,s} : \mathcal {D}_{f_1} \to \mathcal {D}_{f_2}$ are bounded operators and 
$\mathfrak {U}_{t,s}^{\Lambda _N} \xrightarrow {N \to \infty } \mathfrak {U}_{t,s}$ in operator norm, uniformly for 
$s,t$ in compacts. In particular, for each 
$A \in \mathcal {D}_{f_1}$, 
is continuous with respect to the norm 
$\Vert \cdot \Vert _{f_2}$. This also holds for 
$(t,s) \mapsto \mathrm {e}^{\mathrm {i}s\mathcal {L}_{H(t)}}$.
Proof. The proof is inspired by the one of Lemma 4.4 in [Reference Moon and Ogata22]. Let 
$A \in \mathcal {D}_{f_1}$. From Corollary B.2, we have
Similarly, again applying Corollary B.2 with 
$\lambda = \lambda _2$ for 
$M \le \left \lfloor \frac {\lambda N}{2}\right \rfloor $, we have
On the other hand, by application of Proposition B.1 and Corollary B.2 in combination with Lemma C.2 from [Reference Henheik and Teufel15], for 
$M \ge \left \lfloor \frac {\lambda N}{2}\right \rfloor $, we have
Therefore, for 
$M \ge \left \lfloor \frac {\lambda N}{2}\right \rfloor $, we have
By collecting the estimates above and using the conditions on 
$f_1$ and 
$f_2$, we find that
where 
$h_{t,s}(N) \to 0$ as 
$N \to \infty $ uniformly for s and t in compacts.
Lemma B.5 Quasi-locality of conjugation with unitaries
Let 
$S \in \mathcal {L}_{I, \zeta , 0 }$ have a thermodynamic limit and let 
$\mathrm {e}^{\mathrm {i}\mathcal {L}_{S(t)}}$ denote the automorphism on 
$\mathcal {A}$ defined by
for 
$A\in \mathcal {A}_{\mathrm {loc}}$. Let 
$f_1,f_2 :[0,\infty ) \to (0,\infty )$ be bounded, nonincreasing functions with 
$\lim _{t\to \infty } f_i(t) = 0$, for 
$i = 1,2$, such that
$$ \begin{align*} \sup_{N \in \mathbb{N}} \frac{f_1(N-\lfloor \frac{N}{2} \rfloor)}{f_2(N)} < \infty,\quad\mbox{and}\quad \sup_{N \in \mathbb{N}} \frac{N^d \, \zeta(\left\lfloor \frac{N}{2} \right\rfloor)}{f_2(N)} < \infty. \end{align*} $$Then 
$\mathrm {e}^{\mathrm {i}\mathcal {L}_{S(t)}}: \mathcal {D}_{f_1} \to \mathcal {D}_{f_2}$ is a bounded operator and the sequence 
$\mathrm {e}^{\mathrm {i}\mathcal {L}^{\Lambda _N}_{S(t)}}: \mathcal {D}_{f_1} \to \mathcal {D}_{f_2}$ of operators is uniformly bounded, both uniformly in 
$t \in I$. More precisely, there exists a constant 
$C_{f_1,f_2}$ such that
for all 
$A \in \mathcal {D}_{f_1}$. Besides the indicated dependence on 
$f_1$ and 
$f_2$, the constant 
$C_{f_1,f_2}$ only depends on 
$\zeta $ and the norm 
$\Vert S \Vert _{I, \zeta , 0 }$.
Proof. As a conjugation with unitaries can be viewed as a time evolution at a frozen time, we have by application of Corollary B.2 and Lemma C.2 from [Reference Henheik and Teufel15] (analogous to the proof of Lemma 4.5 in [Reference Moon and Ogata22]) that
for 
$N,k \in \mathbb {N}$ with 
$k<N$, uniformly in 
$t \in I$. Using this bound with 
$k = N- \left \lfloor \frac {N}{2}\right \rfloor $ yields the claim. The boundedness of 
$\mathrm {e}^{\mathrm {i}\mathcal {L}^{\Lambda _N}_{S(t)}}$ can be proven in the same way.
B.2 Derivations
Proposition B.6 Commutator bounds and convergence of derivations
Let 
$H_0 \in \mathcal {L}_{I, \zeta , 0 }$ and 
$v \in \mathcal {V}_I$ both have a thermodynamic limit. Define the induced family of local derivations as
Then for all 
$k \in \mathbb {N}$ and for all 
$X, Y \subset \Lambda _k$, 
$A \in \mathcal {A}_X$ and 
$B \in \mathcal {A}_Y$ it holds that
If 
$H_0$ and v have a rapid thermodynamic limit with exponent 
$\gamma \in (0,1)$, then there exist 
$\lambda _1>0$, 
$\lambda _2 \in (0,1)$ and 
$C<\infty $ such that for all 
$l,k \in \mathbb {N}$ with 
$l \ge k$, 
$X \subset \Lambda \left (k \right )$ and 
$A \in \mathcal {A}_X$,
In every case above, the constant C only depends on 
$\zeta $, 
$\Vert \Phi _{H_0}\Vert _{I,\zeta , 0 }$ and 
$C_v$.
Proof. Involving Lemma 2.1, the first and second estimates are clear from Example 5.4 in [Reference Nachtergaele, Sims and Young24]. The first alternative of the third estimate is proven analogous to Theorem 3.8 (ii) in [Reference Nachtergaele, Sims and Young24] and choosing 
$\Lambda _M = \Lambda _{\lceil k - \lambda _1 k^{\gamma }\rceil }$ with 
$\lambda _1$ from Definition 3.1 (respectively the alternative characterisation from Lemma C.1). The second alternative can easily be concluded from the first. This latter form is used in the proof of the lemmata below.
As a consequence, 
$(\mathcal {L}_t^{\Lambda _k}(A))_{k \in \mathbb {N}}$ is a Cauchy sequence for all 
$A \in \mathcal {A}_{\mathrm {loc}}$ uniformly for 
$t \in I$. The limiting derivation 
$(\mathcal {L}_t, D(\mathcal {L}_t))$ is closable (see Proposition 3.2.22 and Proposition 3.1.15 in [Reference Bratteli and Robinson8]) and its closure (denoted by the same symbol) is called the bulk derivation generated by 
$H = H_0+V_v$ at 
$t \in I$.
Since the finite-volume metrics 
$d^{\Lambda _k}(\cdot , \cdot )$ are compatible in the bulk, we obtain the corresponding statements also for the bulk derivation.
Corollary B.7 Infinite-volume derivation
Under the conditions of Proposition B.6, there exists and 
$C<\infty $ such that for all 
$X,Y \in \mathcal {P}_0(\mathbb {Z}^d)$, 
$A \in \mathcal {A}_X$ and 
$B \in \mathcal {A}_Y$,
If 
$H_0$ and v have a rapid thermodynamic limit, there exist 
$\lambda _1>0$, 
$\lambda _2 \in (0,1)$ such that
In every case above, the constant C depends only on 
$\zeta $, 
$\Vert \Phi _{H_0}\Vert _{I,\zeta , 0}$ and 
$C_v$.
The following two lemmata generalise Lemma 4.12 and Lemma 4.13 from [Reference Moon and Ogata22]. Here we use that whenever 
$\Psi _{H_0} \in \mathcal {B}^\circ _{I,\exp (-a\cdot ),0}$ satisfies (I2),
$$ \begin{align} \lim_{\delta \to 0} b(\delta) := \lim_{\delta \to 0} \sup_{\substack{t,t_0 \in I \\ 0< \vert t-t_0\vert < \delta}} \left\Vert \frac{\Psi_{H_0}(t)-\Psi_{H_0}(t_0)}{t-t_0} -\dot{\Psi}_{H_0}(t_0)\right\Vert{}^\circ_{\exp(-a\cdot), 0 }=0. \end{align} $$Lemma B.8 Quasi-locality of derivations I
Let 
$H_0 \in \mathcal {L}_{I,\zeta , 0 }$ and 
$v \in \mathcal {V}_I$ both have a thermodynamic limit, set 
$H=H_0+V_v$ and let 
$\mathcal {L}_s: D(\mathcal {L}_s) \to \mathcal {A}$ be the bulk derivation generated by H at 
$s \in I$. Let 
$f_1,f_2: [0,\infty ) \to (0,\infty )$ be bounded, nonincreasing functions with 
$\lim \limits _{t \to \infty }f_i(t) =0$, for 
$i= 1,2$, such that
$$ \begin{align*} \sum_{k=1}^{\infty} (k+1)^{d+1} f_1(k) < \infty \end{align*} $$and
$$ \begin{align*} \lim\limits_{N \to \infty} \frac{\sum_{k=\lfloor \frac{N}{2}\rfloor}^{\infty}(k+1)^{d+1} f_1(k)}{f_2(N)} = \lim\limits_{N \to \infty} \frac{N^{d+1} \zeta(N-\lfloor\frac{N}{2} \rfloor)}{f_2(N)} =0. \end{align*} $$Then 
$\mathcal {D}_{f_1} \subset D(\mathcal {L}_s)$ and 
$\mathcal {L}_t: \mathcal {D}_{f_1} \to \mathcal {D}_{f_2}$ is a bounded operator and the sequence 
$\mathcal {L}^{\Lambda _N}_t: \mathcal {D}_{f_1} \to \mathcal {D}_{f_2}$ of operators is uniformly bounded, both uniformly in 
$t \in I$. More precisely, there is a constant 
$C_{f_1,f_2}> 0 $ such that
for all 
$A \in \mathcal {D}_{f_1}$. Besides the indicated dependence on 
$f_1$ and 
$f_2$, the constant 
$C_{f_1,f_2}$ only depends on 
$\zeta $, 
${\Vert \Phi _{H_0} \Vert _{I,\zeta , 0 }}$ and 
$C_v$.
In particular, let 
$H_0$ be given by an infinite-volume interaction 
$\Psi _{H_0} \in \mathcal {B}_{I, \exp (-a \cdot ), 0}^\circ $ that satisfies (I2). Then, with 
$b(\delta )$ defined in (B.3), it follows that
and
Proof. The proof of this lemma is analogous to the one of Lemma 4.12 in [Reference Moon and Ogata22]. By application of Corollary B.7, for any 
$A \in \mathcal {D}_{f_1}$ and 
$N,M \in \mathbb {N}$ with 
$N>M$, we have
This implies that
with 
$A \in \mathcal {D}_{f_1}$ is a Cauchy sequence in 
$\mathcal {A}$; hence, there exists a limit. Moreover,
converges to A in 
$\Vert \cdot \Vert $. Since the derivation is closed, 
$A \in \mathcal {D}_{f_1}$ belongs to the domain 
$D(\mathcal {L}_t)$ of 
$\mathcal {L}_t$ and
Hence, 
$\mathcal {D}_{f_1} \subset D(\mathcal {L}_t)$. Similar to the estimate in (B.4), one obtains
for any 
$A \in \mathcal {D}_f$ by considering a limit. Now, we are left to estimate
In the last line we used Corollary B.7 together with Lemma C.2 from [Reference Henheik and Teufel15]. Hence, we have proven the claim. The boundedness of the sequence 
$\mathcal {L}_t^{\Lambda _N}$ can be proven in the same way.
Lemma B.9 Quasi-locality of derivations II
Let 
$H_0 \in \mathcal {L}_{I,\zeta , 0 }$ and 
$v \in \mathcal {V}_I$ both have a rapid thermodynamic limit with exponent 
$\gamma \in (0,1)$, set 
$H=H_0+V_v$ and let 
$\mathcal {L}_t: D(\mathcal {L}_t) \to \mathcal {A}$ be the derivation generated by H at 
$t \in I$. Let 
$f_1,f_2: [0,\infty ) \to (0,\infty )$ be bounded, nonincreasing functions with 
$\lim \limits _{s\to \infty }f_i(s) =0$, for 
$i=1,2$, such that
$$ \begin{align*} \sum_{k=1}^{\infty} (k+1)^{d+1} \sqrt{f_1}(k) < \infty \end{align*} $$and
$$ \begin{align*} \lim\limits_{N \to \infty} \frac{\sum_{k=\lfloor \frac{N}{2}\rfloor}^{\infty}(k+1)^{d+1} \sqrt{f_1}(k)}{f_2^2(N)} = \lim\limits_{N \to \infty} \frac{N^{d+1} \zeta_{\gamma}(N - \lfloor \frac{N}{2} \rfloor)}{f_2^2(N)} =0. \end{align*} $$Then 
$\mathcal {D}_{f_1} \subset D(\mathcal {L}_t)$, 
$\mathcal {L}_{t} : \mathcal {D}_{f_1} \to \mathcal {D}_{f_2}$ and 
$\mathcal {L}^{\Lambda _N}_{t} : \mathcal {D}_{f_1} \to \mathcal {D}_{f_2}$ are bounded operators and 
$\mathcal {L}_{t}^{\Lambda _N} \xrightarrow {N \to \infty } \mathcal {L}_{t}$ in operator norm uniformly for all 
$t \in I$. In particular, for each 
$A \in \mathcal {D}_{f_1}$, 
$t \mapsto \mathcal {L}_t(A)$ is continuous with respect to the norm 
$\Vert \cdot \Vert _{f_2} $.
Proof. The proof of this lemma is analogous to the one of Lemma 4.13 in [Reference Moon and Ogata22]. From Corollary B.7, combined with Lemma B.8, we have for 
$A \in \mathcal {D}_{f_1}$ that
which vanishes as 
$N \to \infty $. Therefore,
$$ \begin{align*} \lim\limits_{N \to \infty} \sup_{t \in I} \left\Vert \mathcal{L}_t - \mathcal{L}_t^{\Lambda_N} \right\Vert{}_{\mathcal{L}(\mathcal{D}_{f_1},\mathcal{A} )} = 0 . \end{align*} $$Furthermore, for 
$A \in \mathcal {D}_{f_1}$, we have by application of Lemma B.8 that
which implies the claim.
B.3 Inverse Liouvillian
Finally, we show that also 
$\mathcal {I}_t$ defined in Appendix A preserves quasi-locality of observables. Similar statements hold for 
$\mathcal {J}_t$. The following lemma generalises Lemma 4.6 from [Reference Moon and Ogata22].
Lemma B.10 Quasi-locality of the inverse Liouvillian
Let 
$H_0 \in \mathcal {L}_{I, \exp (-a \cdot ),0 }$ have a thermodynamic limit. For 
$f_1, f_2$ as in Lemma B.3, 
$\mathcal {I}_t : \mathcal {D}_{f_1} \to \mathcal {D}_{f_2}$ is a bounded operator and the sequence 
$\mathcal {I}^{\Lambda _N}_t : \mathcal {D}_{f_1} \to \mathcal {D}_{f_2}$ is uniformly bounded, both uniformly in 
$t \in I$. More precisely, there exists a positive constant 
$C_{f_1,f_2}$ such that for all 
$ A \in \mathcal {D}_{f_1} $,
Besides the indicated dependence on 
$f_1$ and 
$f_2$, the constant 
$C_{f_1,f_2}$ only depends on a and 
${\Vert \Phi _0 \Vert _{I,\exp (-a \, \cdot ), 0 }}$.
If 
$H_0$ has a rapid thermodynamic limit with exponent 
$\gamma \in (0,1)$ and if the functions 
$f_1$and 
$f_2$ satisfy, in addition, the requirements of Lemma B.4, then 
${\mathcal {I}_t^{\Lambda _N} \xrightarrow {N \to \infty } \mathcal {I}_t}$ in operator norm uniformly in 
$t \in I$.
The above statements also hold for the derivatives 
$\frac {\mathrm {d}^k}{\mathrm {d} t^k} \mathcal {I}_t$ (respectively 
$\frac {\mathrm {d}^k}{\mathrm {d} t^k} \mathcal {I}^{\Lambda _N}_t$) after suitably replacing the conditions on 
$f_1$ and 
$f_2$ from Lemma B.3 and Lemma B.4.
Proof. The first part of this lemma is a simple consequence of Lemma B.3 using the properties of the function 
$b_{f_1,f_2}$. The second part follows from Lemma B.4 and Lemma B.3 in combination with the dominated convergence theorem.
The statements on the derivatives follow with the aid of Lemma B.3, Lemma B.4, Lemma B.8 and Lemma B.9 by noticing that the first derivative is given by
and the higher derivatives have a similar form. More precisely, for this first derivative, sufficient conditions on 
$f_1$ and 
$f_2$ for the boundedness of 
$\frac {\mathrm {d}}{\mathrm {d}t} \mathcal {I}_t : \mathcal {D}_{f_1} \to \mathcal {D}_{f_2}$ are given by
$$ \begin{align*} \int_{0}^\infty \mathrm{d}s \ w_g(s) \frac{ (2 s)^2 }{f_2(4v_a s) \cdot \sqrt{f_1}(4v_as)} < \infty, \end{align*} $$
$$ \begin{align*} \lim\limits_{N \to \infty} \frac{\sum_{k=\lfloor \frac{N}{2}\rfloor}^{\infty}(k+1)^{d+1} \sqrt{f_1}(k)}{f_2^2(N)} = \lim\limits_{N \to \infty} \frac{N^{d+1} {\mathrm{e}}^{-a\frac{N}{4}}}{f_2^2(N)} =0. \end{align*} $$This follows by application of Lemma B.3 with 
$(f_1,\sqrt {f_1})$, Lemma B.8 with 
$(\sqrt {f_1}, f_2^2)$ and again Lemma B.3 with 
$(f_2^2,f_2)$. For higher derivatives, the powers of 
$f_1$ in the conditions above get (polynomially) decreased, whereas the powers of 
$f_2$ and 
$ t $ get (polynomially) increased. Using Lemma B.9, one can easily establish sufficient conditions on 
$f_1$ and 
$f_2$ for the convergence 
${\frac {\mathrm {d}^k}{\mathrm {d}t^k} \mathcal {I}_t^{\Lambda _N} \xrightarrow {N \to \infty } \frac {\mathrm {d}^k}{\mathrm {d}t^k} \mathcal {I}_t}$ in operator norm involving even higher (respectively lower) powers of 
$f_2$, s and 
$f_1$.
B.4 Sequences of suitable weight functions
Lemma B.11. Let 
$v_a>0$ (Lieb–Robinson velocity), 
$\zeta \in \mathcal {S}$, 
$R\in (1,\infty )$ and define the set 
$\mathcal {S}_\zeta ^R$ of ‘admissible’ functions as
$$ \begin{align*} \mathcal{S}_\zeta^R:=\left\{ f \in \mathcal{S}\,\Big|\, \int_{0}^\infty \hspace{-1mm} \mathrm{d}t \; w_g(s) \left(\frac{ s }{f(4v_as) }\right)^R < \infty, \ \lim\limits_{N \to \infty} \frac{N^{d+1} \,\zeta(N)}{ f(N) ^R} =0 \right\} . \end{align*} $$Then there exists an infinite sequence 
$\left (f_j\right )_{j \in \mathbb {N}} \subset \mathcal {S}_\zeta ^{R}$ satisfying
$$ \begin{align*} \lim\limits_{N \to \infty} \frac{\sum_{k=\lfloor \frac{N}{2}\rfloor}^{\infty}(k+1)^{d+1} \, f_j(k) ^\alpha}{ f_{j+1}(N)^\beta} = 0 \quad \forall \alpha, \beta \in \left[\tfrac{1}{R},R\right] \quad \mathrm{and} \quad \forall j \in \mathbb{N}. \end{align*} $$Proof. According to Lemma A.1 from [Reference Monaco and Teufel20], 
$\mathcal {S}_\zeta ^R $ is nonempty. So, pick any 
$f \in \mathcal {S}_\zeta ^R$ and set
$$ \begin{align*} f_j(s) := f(s) ^{\frac{1}{ (5R^2)^j } } \end{align*} $$for any 
$j \in \mathbb {N}$. Obviously, 
$f_j \in \mathcal {S}_\zeta ^R$ and we estimate
$$ \begin{align*} \frac{\sum_{k=\lfloor \frac{N}{2}\rfloor}^{\infty}(k+1)^{d+1} f_j(k) ^\alpha}{ f_{j+1}(N) ^\beta} & \le \left( \sum_{k=\lfloor \frac{N}{2}\rfloor}^{\infty}(k+1)^{d+1} f_j(k) ^{\frac{1}{2R}} \right) \frac{ f_{j}\left(\lfloor N/2 \rfloor \right) ^{\frac{1}{2R}}}{ f_{j+1}(N) ^R} \\ &\le C_j \, \frac{ f_{j}\left(N/2 \right) ^{\frac{1}{2R}}}{ f_{j+1}(N/2) ^{2R}}\; =\; C_j \, f(N/2)^{\frac{1}{(5R^2)^j} { \frac{1}{10R}}}\xrightarrow{N \to \infty} 0 \end{align*} $$uniformly in 
$\alpha , \beta \in \left [\tfrac {1}{R},R\right ]$ and for any 
$j \in \mathbb {N}$. In the second inequality, we used that 
$f_{j+1}$ is logarithmically superadditive and 
$ f_j(1/2) ^{\frac {1}{2R}}> 0$.
C Operations preserving the rapid thermodynamic limit
In this appendix, we verify that operator families that are obtained by taking commutators and inverse Liouvillians (see Appendix A) of operator families and Lipschitz potentials having a rapid thermodynamic limit again have a rapid thermodynamic limit with the same exponent. The following statements are slight modifications of technical lemmata from [Reference Henheik and Teufel15].
Lemma C.1 Alternative characterisation of having a rapid thermodynamic limit
The time-dependent interaction 
$\Phi \in \mathcal {B}_{I,\zeta ,n}$ has a rapid thermodynamic limit if and only if it satisfies the following Cauchy property:
$$ \begin{align*} \forall i \in \mathbb{N}_0 \ \ &\exists \lambda, C>0 \ \ \forall M\in \mathbb{N} \ \ \forall \, k,l \, \ge \, M + \lambda M^{\gamma} : \\ &\sup_{t \in I}\left\Vert \frac{\mathrm{d}^i}{\mathrm{d} t^i}\left( \Phi^{\Lambda_l}- \Phi^{ \Lambda_k}\right) (t) \, \right\Vert{}_{\zeta^{(i)}, n,\Lambda_M} \le C \, \zeta^{(i)}(M^{\gamma}) =: C \zeta^{(i)}_{\gamma}(M). \nonumber \end{align*} $$Proof. The proof is completely analogous to the one of Lemma D.1 in [Reference Henheik and Teufel15].
Lemma C.2 Commutator of SLT operators
Let 
$A, B \in \mathcal {L}_{I,\zeta , n+d }$ have a rapid thermodynamic limit with exponent 
$\gamma \in (0,1)$. Then the commutator 
$[A,B] \in \mathcal {L}_{I,\zeta , n }$ also has a rapid thermodynamic limit with exponent 
$\gamma $. In particular, if 
$A,B \in \mathcal {L}_{I,\mathcal {S}, \infty }$ both have a thermodynamic limit with exponent 
$\gamma $, then 
$[A,B] \in \mathcal {L}_{I,\mathcal {S}, \infty }$ also has a thermodynamic limit with exponent 
$\gamma $.
Proof. Using Lemma C.1, the proof is completely analogous to the one of Lemma D.2 in [Reference Henheik and Teufel15].
Lemma C.3 Commutator with Lipschitz potential
Let 
$A \in \mathcal {L}_{I,\zeta , n+d+1 }$ and 
$v \in \mathcal {V}_I$ both have a rapid thermodynamic limit with exponent 
$\gamma \in (0,1)$. Then the commutator 
$[A,V_v] \in \mathcal {L}_{I,\zeta , n}$ also has a rapid thermodynamic limit with exponent 
$\gamma $. In particular, if 
$A \in \mathcal {L}_{I,\mathcal {S}, \infty }$ has a rapid thermodynamic limit with exponent 
$\gamma $, then 
$[A,V_v] \in \mathcal {L}_{I,\mathcal {S}, \infty }$ also has a rapid thermodynamic limit with exponent 
$\gamma $.
Proof. Using Lemma C.1, the proof is completely analogous to the one of Lemma D.3 in [Reference Henheik and Teufel15].
Lemma C.4 Inverse Liouvillian
Let 
$H \in \mathcal {L}_{I,\exp (-a \, \cdot ), \infty }$ and B either an SLT operator in 
$\mathcal {L}_{I,\mathcal {S}, \infty }$ or a Lipschitz potential. Assume that H and 
$[H,B] \in \mathcal {L}_{I,\mathcal {S}, \infty }$ both have a rapid thermodynamic limit with exponent 
$\gamma \in (0,1)$. Then 
$\mathcal {I}_{H}(B) \in \mathcal {L}_{I,\mathcal {S}, \infty }$ also has a rapid thermodynamic limit with exponent 
$\gamma $.
Proof. Using Lemma C.1, the proof is analogous to the one of Lemma D.4 in [Reference Henheik and Teufel15]. But there is one crucial point to take care of: One arrives at
where we choose 
$M' = M + M^{\gamma }$ (i.e., 
$\lambda \to \lambda + 1$ in Definition 3.1, respectively Lemma C.1) and 
$T = \frac {aM^{\gamma }}{8 \Vert \Phi _H\Vert }$, such that the term in the square brackets decays faster than any polynomial as 
$M \to \infty $. So, after a possible adjustment of 
$\zeta \in \mathcal {S}$, which is legal by Lemma A.1 in [Reference Monaco and Teufel20], the proof comes to an end in the same manner as in Lemma D.4 in [Reference Henheik and Teufel15].
Conflict of interest
None.
Funding statement
J.H. acknowledges partial financial support by the ERC Advanced Grant ‘RMTBeyond’ No. 101020331. Support for publication costs from the Deutsche Forschungsgemeinschaft and the Open Access Publishing Fund of the University of Tübingen is gratefully acknowledged.
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