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 $ρ_0$ of a Hamiltonian $H_0$ is locally gapped in $Λ^{\mathrm{gap}} \subset Λ$, whenever the Liouvillian $- \mathrm{i} \, [H_0, \, \cdot \, ]$ is almost invertible on local observables supported in $Λ^{\mathrm{gap}}$ when tested in $ρ_0$. 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 $|\log ε|$ away from the non-gapped region $Λ\setminus Λ^{\mathrm{gap}}$.