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We introduce a notion of a \emph{local gap} for interacting many-body quantum lattice systems and prove the validity of response theory and Kubo's formula for localized perturbations in such settings. On a high level, our result shows that the usual spectral gap condition, concerning the system as a whole, is not a necessary condition for understanding local properties of the system. More precisely, we say that an equilibrium state $\rho_0$ of a Hamiltonian $H_0$ is locally gapped in $\Lambda^{\mathrm{gap}} \subset \Lambda$, whenever the Liouvillian $- \mathrm{i} \, [H_0, \, \cdot \, ]$ is almost invertible on local observables supported in $\Lambda^{\mathrm{gap}}$ when tested in $\rho_0$. To put this into context, we provide other alternative notions of a local gap and discuss their relations. The validity of response theory is based on the construction of \emph{non-equilibrium almost stationary states} (NEASSs). By controlling locality properties of the NEASS construction, we show that response theory holds to any order, whenever the perturbation \(\epsilon V\) acts in a region which is further than $|\log \epsilon|$ away from the non-gapped region $\Lambda \setminus \Lambda^{\mathrm{gap}}$.

We prove that the bulk magnetization is equal to the edge current in the thermodynamic limit for a large class of models of lattice fermions with finite-range interactions satisfying local indistinguishability of the Gibbs state, a condition known to hold for sufficiently high temperatures. Our result implies that edge currents in such systems are determined by bulk properties and are therefore stable against large perturbations near the boundaries. Moreover, the equality persists also after taking the derivative with respect to the chemical potential. We show that this form of bulk-edge correspondence is essentially a consequence of homogeneity in the bulk and locality of the Gibbs state. An important intermediate result is a new version of Bloch's theorem for two-dimensional systems, stating that persistent currents vanish in the bulk., Comment: 31 pages, 5 figures; v2: extended explanation of (21), corrected typos

In this paper we show that whenever a Gibbs state satisfies decay of correlations, then it is stable, in the sense that local perturbations influence the Gibbs state only locally, and it is local, namely it satisfies local indistinguishability. These implications hold true in any dimensions, only require locality of the Hamiltonian and rely on Lieb-Robinson bounds. Then, we explicitly apply our results to quantum spin systems in any dimension with short-range interactions at high enough temperature, where decay of correlations is known to hold. Furthermore, our results are applied to Gibbs states of finite one-dimensional spin chains with translation-invariant and exponentially decaying interactions, for which we also show that decay of correlations holds true above a threshold temperature that goes to zero in the limit of finite-range interactions. Our proofs are based on a detailed analysis of the locality properties of the quantum belief propagation for Gibbs states., Comment: 39 pages, 6 figures; v2: added section about SLT perturbations, updated and added references, fixed typos

We review recent results on adiabatic theory for ground states of extended gapped fermionic lattice systems under several different assumptions. More precisely, we present generalized super-adiabatic theorems for extended but finite as well as infinite systems, assuming either a uniform gap or a gap in the bulk above the unperturbed ground state. The goal of this note is to provide an overview of these adiabatic theorems and briefly outline the main ideas and techniques required in their proofs., Comment: 27 pages, 1 figure; v2: corrected typos and updated references; closely resembles the published version

Based on a result by Yarotsky (J. Stat. Phys. 118, 2005), we prove that localized but otherwise arbitrary perturbations of weakly interacting quantum spin systems with uniformly gapped on-site terms change the ground state of such a system only locally, even if they close the spectral gap. We call this a strong version of the local perturbations perturb locally (LPPL) principle which is known to hold for much more general gapped systems, but only for perturbations that do not close the spectral gap of the Hamiltonian. We also extend this strong LPPL-principle to Hamiltonians that have the appropriate structure of gapped on-site terms and weak interactions only locally in some region of space. While our results are technically corollaries to a theorem of Yarotsky, we expect that the paradigm of systems with a locally gapped ground state that is completely insensitive to the form of the Hamiltonian elsewhere extends to other situations and has important physical consequences., Comment: 9 pages, 2 figures; v1->v2: added references, v2->v3: updated references