Validity of Kubo formula for 1d and quasi-1d lattice fermionic systems

We study the dynamical density and current responses of a class of non-interacting, one-dimensional gapless lattice Fermi systems subjected to density-type perturbations slowly varying in space and time, in the infinite-volume and zero-temperature limit. In addition, we analyse the edge density and current responses for quantum Hall lattice systems defined on a cylindrical geometry.

We establish the validity of linear response for these observables, thereby providing a justification of Kubo’s formula. The latter is computed explicitly in terms of the spectral properties at the Fermi points and the profile of the perturbation.

The proof of linear response validity on the relevant time scales is based on a detailed analysis of the imaginary-time Duhamel expansion. A crucial ingredient is a cancellation mechanism in the scaling limit of higher-order correlation functions that describe the nonlinear part of the response.

Joint work with Marcello Porta.