Response theory for locally gapped systems

TL;DR

This paper introduces a local gap concept for quantum lattice systems, proving response theory and Kubo's formula applicability for localized perturbations without requiring a global spectral gap.

Contribution

It defines a local gap condition and demonstrates that response theory holds under this weaker assumption, expanding understanding of local properties in quantum many-body systems.

Findings

Response theory valid for local perturbations under local gap conditions

Construction of non-equilibrium almost stationary states (NEASSs)

Response theory holds to all orders with perturbations sufficiently far from the non-gapped region

Abstract

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 $\Lambda \setminus \Lambda^{\mathrm{gap}}$.