Matrix-valued Boltzmann equation for the non-integrable Hubbard chain

TL;DR

This paper derives a matrix-valued Boltzmann equation for a non-integrable Hubbard chain, showing how added long-range hopping affects the relaxation dynamics and convergence to thermal equilibrium.

Contribution

It introduces a Boltzmann kinetic framework for the non-integrable Hubbard chain, analyzing how longer-range hopping influences stationary states and relaxation behavior.

Findings

Degeneracy of stationary states is lifted with longer-range hopping.

Convergence to thermal equilibrium is exponentially fast.

Small next-nearest neighbor hopping causes rapid relaxation to quasi-stationary states before equilibrium.

Abstract

The standard Fermi-Hubbard chain becomes non-integrable by adding to the nearest neighbor hopping additional longer range hopping amplitudes. We assume that the quartic interaction is weak and investigate numerically the dynamics of the chain on the level of the Boltzmann type kinetic equation. Only the spatially homogeneous case is considered. We observe that the huge degeneracy of stationary states in case of nearest neighbor hopping is lost and the convergence to the thermal Fermi-Dirac distribution is restored. The convergence to equilibrium is exponentially fast. However for small n.n.n. hopping amplitudes one has a rapid relaxation towards the manifold of quasi-stationary states and slow relaxation to the final equilibrium state.