Relaxation of phonons in the Lieb-Liniger gas by dynamical refermionization

TL;DR

This paper studies how phonons in the Lieb-Liniger gas relax from an out-of-equilibrium Gaussian state to a non-thermal stationary state, using a mapping to a free Fermi gas and bosonization techniques.

Contribution

It introduces a method to characterize the stationary state of the Lieb-Liniger gas after relaxation by mapping to a non-interacting Fermi gas and applying bosonization.

Findings

The stationary phonon distribution can be computed analytically.

The relaxation dynamics depend on the initial phonon state.

Results agree with exact solutions in the hard-core limit.

Abstract

We investigate the Lieb-Liniger gas initially prepared in an out-of-equilibrium state that is Gaussian in terms of the phonons. Because the phonons are not exact eigenstates of the Hamiltonian, the gas relaxes to a stationary state at very long times. Thanks to integrability, that stationary state needs not be a thermal state. We characterize the stationary state of the gas after relaxation and compute its phonon population distribution. Technically, this follows from the mapping between the exact eigenstates of the Lieb-Liniger Hamiltonian and those of a non-interacting Fermi gas -- a mapping provided by the Bethe equations -- , as well as on bosonization formulas valid in the low-energy sector of the Hilbert space. We apply our results to the case where the initial state is an excited coherent state for a single phonon mode, and we compare them to exact results obtained in the hard-core limit.