2024/10/14 by Henheik, Joscha, Wessel, Tom
#81Q15 #81Q99 #81V70 #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Physics (quant-ph)
paper · doi:10.48550/arxiv.2410.10809
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, 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 ρ0 of a Hamiltonian H0 is locally gapped in Λgap ⊂ Λ, whenever the Liouvillian - i [H0, ⋅ ] is almost invertible on local observables supported in Λgap when tested in ρ0. 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 \(εV\) acts in a region which is further than |log ε| away from the non-gapped region Λ∖ Λgap.