Response Theory for Locally Gapped Systems

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 ${\Lambda }^{\text{gap}}\subset \Lambda$ , whenever the Liouvillian $-\text{i}[H_0,·]$ is almost invertible on local observables supported in ${\Lambda }^{\text{gap}}$ 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 $\epsilon V$ acts in a region which is further than $|log\epsilon |$ away from the non-gapped region $\Lambda \backslash \Lambda \text{gap}$ .