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 ρ ₀ ρ 0 of a Hamiltonian H₀ H 0 is locally gapped in ᵍᵃᵖ⊂ Λ gap ⊂ Λ , whenever the Liouvillian - i\, [H₀, \, · \, ] - i [ H 0 , · ] is almost invertible on local observables supported in ᵍᵃᵖ Λ gap when tested in ρ 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 ε V ε V acts in a region which is further than |log ε | | log ε | away from the non-gapped region gap Λ \ Λ gap .