Prethermalization for Deformed Wigner Matrices

TL;DR

This paper demonstrates that certain weakly perturbed Hamiltonians with Wigner matrices exhibit prethermalization, relaxing to a thermal state through an intermediate state with a lifetime proportional to the inverse square of the perturbation strength.

Contribution

It provides a rigorous proof of prethermalization in deformed Wigner matrices and derives a general relaxation formula for the dynamics.

Findings

Prethermalization occurs with a lifetime of order λ^{-2}.

The relaxation formula relates perturbed and unperturbed dynamics.

A two-resolvent law is established for deformed Wigner matrices.

Abstract

We prove that a class of weakly perturbed Hamiltonians of the form $H_\lambda = H_0 + \lambda W$, with $W$ being a Wigner matrix, exhibits prethermalization. That is, the time evolution generated by $H_\lambda$ relaxes to its ultimate thermal state via an intermediate prethermal state with a lifetime of order $\lambda^{-2}$. Moreover, we obtain a general relaxation formula, expressing the perturbed dynamics via the unperturbed dynamics and the ultimate thermal state. The proof relies on a two-resolvent law for the deformed Wigner matrix $H_\lambda$.