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arXiv:cond-mat/0412475 · Annals of Physics

Generalized Gibbs ensembles for time-dependent processes

Ph. Chomaz, Francesca Gulminelli, Olivier Juillet

Published Jul 18, 2005• 16 citations
arXivHTMLPDF
DOI
10.1016/j.aop.2005.05.005

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Generalized Gibbs ensembles have been used as powerful tools to describe the steady state of integrable many-particle quantum systems after a sudden change of the Hamiltonian. Here, we demonstrate numerically that they can be used for a much broader class of problems. We consider integrable systems in the presence of weak perturbations which break both integrability and drive the system to a state far from equilibrium. Under these conditions, we show that the steady state and the time evolution on long timescales can be accurately described by a (truncated) generalized Gibbs ensemble with time-dependent Lagrange parameters, determined from simple rate equations. We compare the numerically exact time evolutions of density matrices for small systems with a theory based on block-diagonal density matrices (diagonal ensemble) and a time-dependent generalized Gibbs ensemble containing only a small number of approximately conserved quantities, using the one-dimensional Heisenberg model with perturbations described by Lindblad operators as an example.
In this work we investigate the nonequilibrium dynamics of closed quantum systems. In particular we focus on the stationary properties of integrable systems. Here we show how a generalized Gibbs ensemble (GGE) can be constructed as the best approximation to the time-dependent density matrix. Our procedure allows for a systematic construction of the GGE by a constrained minimization of the distance between the latter and the true state. Moreover, we show that the entropy of the GGE is a direct measure for the quality of the approximation. We apply our method to a quenched hard core Bose gas. Whereas a correlated GGE properly describes all stationary nonlocal correlations, a simple harmonic GGE is sufficient to completely describe reduced local states.
In recent years, statistical physics tools have been used to study the dynamics of macroscopic quantum and classical integrable models. These studies have shown that, in the thermodynamic limit, long term time-averages of local observables equal statistical averages calculated with a Generalized Gibbs Ensemble (GGE). In the latter, the weight in phase space involves all constants of motion multiplied by Lagrange multipliers which are fixed by their initial values. We argue that the GGE can be interpreted as a proper Boltzmann measure of a different (non-standard) mechanical model. This measure can therefore be sampled with stochastic methods implemented, for example, as a Langevin process. We tailor the deterministic force in the Langevin equations so that at long times the stochastic trajectories sample the desired measure. We use the integrable Neumann Model -- the model of a classical particle constrained to move on a sphere under anisotropic harmonic potentials -- as a benchmark. We test the method in a setup where the Lagrange multipliers are fixed by the GGE governing the asymptotic dynamics of the isolated Neumann model following a quench. This problem has different phases depending on the initial conditions and quench protocol and, in all cases, with the stochastic evolution of an adequately tuned Langevin process we reproduce the asymptotic values of relevant observables.