Response Theory for Locally Gapped Systems

Abstract

We introduce a notion of a 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, which concerns 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 \({\varLambda ^{\textrm{gap}}}\subset \varLambda \), whenever the Liouvillian \(- \textrm{i}\, [H_0, \, \cdot \, ]\) is almost invertible on local observables supported in \({\varLambda ^{\textrm{gap}}}\) 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 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 \(\varepsilon V\) acts in a region which is further than \(|\log \varepsilon |\) away from the non-gapped region \(\varLambda \backslash \varLambda \textrm{gap}\).