Response theory for locally gapped systems

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 $\ddot{I}{}_0$ of a Hamiltonian $H_0$ is locally gapped in $\hat{I}{›}^{\mathrm{gap}}\subset \hat{I}›$, whenever the Liouvillian $-\mathrm{i}[H_0,\cdot ]$ is almost invertible on local observables supported in $\hat{I}{›}^{\mathrm{gap}}$ when tested in $\ddot{I}{}_0$. 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 \(εV\) acts in a region which is further than $|\log \hat{I}\mu |$ away from the non-gapped region $\hat{I}›\setminus \hat{I}{›}^{\mathrm{gap}}$.