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 $$\rho _0$$ ρ 0 of a Hamiltonian $$H_0$$ H 0 is locally gapped in $${\varLambda ^{\textrm{gap}}}\subset \varLambda $$ Λ gap ⊂ Λ , whenever the Liouvillian $$- \textrm{i}\, [H_0, \, \cdot \, ]$$ - i [ H 0 , · ] is almost invertible on local observables supported in $${\varLambda ^{\textrm{gap}}}$$ Λ gap when tested in $$\rho _0$$ 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 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 $$\varepsilon V$$ ε V acts in a region which is further than $$|\log \varepsilon |$$ | log ε | away from the non-gapped region $$\varLambda \backslash \varLambda \textrm{gap}$$ Λ \ Λ gap .