Matrix-valued Boltzmann Equation for the Hubbard Chain

TL;DR

This paper analyzes the matrix-valued Boltzmann equation for the Hubbard chain, revealing integrability, conservation laws, and stationary states, with numerical results showing rapid convergence to equilibrium.

Contribution

It provides a detailed analytical and numerical study of the matrix-valued Boltzmann equation for the Hubbard chain, including characterization of stationary states and convergence behavior.

Findings

H-theorem holds for the system

Existence of non-thermal stationary states

Rapid exponential convergence to stationarity

Abstract

We study, both analytically and numerically, the Boltzmann transport equation for the Hubbard chain with nearest neighbor hopping and spatially homogeneous initial condition. The time-dependent Wigner function is matrix-valued because of spin. The H-theorem holds. The nearest neighbor chain is integrable which, on the kinetic level, is reflected by infinitely many additional conservation laws and linked to the fact that there are also non-thermal stationary states. We characterize all stationary solutions. Numerically, we observe an exponentially fast convergence to stationarity and investigate the convergence rate in dependence on the initial conditions.