Description
Title: Validity of linear response for gapless many-body quantum systems
Abstract: In this talk I will discuss the transport properties of interacting fermionic lattice models, at low temperature, exposed to external perturbations slowly varying in space and in time. For weak enough perturbations, the linear response of physical observables is typically computed using Kubo formula, obtained truncating the Duhamel expansion for the time-dependent dynamics at first order. A first nontrivial question is to explicitly compute Kubo formula, and to understand its dependence on the parameters of the model. A second, complementary question is to understand the rigorous validity of Kubo formula, namely to prove that all higher order corrections in the Duhamel expansion are subleading, uniformly in the size of the system. Both questions become particularly hard for gapless systems, which are relevant for describing metals. In this talk, I will present a rigorous derivation of Kubo formula for interacting one-dimensional Fermi systems, in the absence of a spectral gap. Our result provides an explicit expression for the linear response of the current and of the density operators, as well as rigorous quantitative estimates for all higher order corrections. The proof is based on the combination of a number of ingredients: the rewriting of the real-time Duhamel expansion in terms of Euclidean correlation functions; the renormalization group analysis of Euclidean multipoint correlation functions, which provides sharp estimates for such correlations on large scales; the use of conservation laws and of the associated Ward identities, to determine the explicit form of the leading term, and to prove a key cancellation for all higher order terms which reproduces a prediction of bosonization. Based on a joint work with Giuseppe Scola and with Harman Preet Singh.
