Adiabatic Evolution of Low-Temperature Many-Body Systems

TL;DR

This paper develops a rigorous framework for analyzing the adiabatic evolution of low-temperature many-body fermionic systems, providing convergent expansions and proving the validity of linear response theory.

Contribution

It introduces a convergent expansion method for the evolution of local observables in fermionic lattice models under slow perturbations at low temperatures, with uniform convergence results.

Findings

Proves closeness of evolved states to instantaneous Gibbs states at low temperatures.

Establishes the validity of linear response in the studied systems.

Develops a rigorous Wick rotation technique for real-time dynamics.

Abstract

We consider finite-range, many-body fermionic lattice models and we study the evolution of their thermal equilibrium state after introducing a weak and slowly varying time-dependent perturbation. Under suitable assumptions on the external driving, we derive a representation for the average of the evolution of local observables via a convergent expansion in the perturbation, for small enough temperatures. Convergence holds for a range of parameters that is uniform in the size of the system. Under a spectral gap assumption on the unperturbed Hamiltonian, convergence is also uniform in temperature. As an application, our expansion allows us to prove closeness of the time-evolved state to the instantaneous Gibbs state of the perturbed system, in the sense of expectation of local observables, at zero and at small temperatures. As a corollary, we also establish the validity of linear response. Our strategy is based on a rigorous version of the Wick rotation, which allows us to represent the Duhamel expansion for the real-time dynamics in terms of Euclidean correlation functions, for which precise decay estimates are proved using fermionic cluster expansion.