Relaxation in a Completely Integrable Many-Body Quantum System: An Ab Initio Study of the Dynamics of the Highly Excited States of Lattice Hard-Core Bosons

TL;DR

The relaxation hypothesis is confirmed through an ab initio numerical investigation of the dynamics of hard-core bosons on a one-dimensional lattice, and a natural extension of the Gibbs ensemble to integrable systems results in a theory that is able to predict the mean values of physical observables after relaxation.

Abstract

In this Letter we pose the question of whether a many-body quantum system with a full set of conserved quantities can relax to an equilibrium state, and, if it can, what the properties of such state are. We confirm the relaxation hypothesis through a thorough ab initio numerical investigation of the dynamics of hard-core bosons on a one-dimensional lattice. Further, a natural extension of the Gibbs ensemble to integrable systems results in a theory that is able to predict the mean values of physical observables after relaxation. Finally, we show that our generalized equilibrium carries more memory of the initial conditions than the usual thermodynamic one. This effect may have many experimental consequences, some of which having already been observed in the recent experiment on the non-equilibrium dynamics of one-dimensional hard-core bosons in a harmonic potential [T. Kinoshita, T. Wenger, D. S. Weiss, Nature (London) 440, 900 (2006)].

Figures (2)

• Figure 1: (color online). Momentum distribution of $N=30$ hard-core bosons undergoing a free expansion from an initial zero-temperature hard-wall box of size $L_{\hbox{\small in.}} = 150$ to the final hard-wall box of size $L=600$. The initial box is situated in the middle of the final one. (a) Approach to equilibrium. (b) Equilibrium (quasi-)momentum distribution after relaxation in comparison with the predictions of the grand-canonical and of the fully constrained (\ref{['fully_constrained_rho']}) thermodynamical ensembles. The prediction of the fully constrained ensemble is virtually indistinct from the results of the dynamical simulation; see the inset for a measure of the accuracy. (An animation of the time evolution is posted on line movies.)
• Figure 2: (color online). Time evolution of the (quasi-)momentum distribution (a) and the (quasi-)momentum distribution after relaxation (b) of $N=30$ hard-core bosons undergoing a free expansion from an initial zero-temperature superlattice with period four of half-depth $A=8J$ and bound by a hard-wall box of size $L=600$, to the final flat-bottom box ($A=0$) of the same size. The discrepancy between the result of time propagation and the prediction of the fully constrained ensemble (\ref{['fully_constrained_rho']}) (also shown in (b)) is less than the width of the line. Momentum peaks remain well-resolved during the whole duration of propagation; $t_{fin.} = 3000 \hbar/J$ for the subfigure (b). (An animation for the time evolution can also be found in movies.)