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arXiv:1211.6731 · Physical Review B

Finite-size corrections versus relaxation after a sudden quench

Maurizio Fagotti

Published Apr 4, 2013• 69 citations
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Abstract

We consider the time evolution after sudden quenches of global parameters in translational invariant Hamiltonians and study the time average expectation values and entanglement entropies in finite chains. We show that in noninteracting models the time average of spin correlation functions is asymptotically equal to the infinite time limit in the infinite chain, which is known to be described by a generalized Gibbs ensemble. The equivalence breaks down considering nonlocal operators, and we establish that this can be traced back to the existence of conservation laws common to the Hamiltonian before and after the quench. We develop a method to compute the leading finite-size corrections. We find that the finite-size corrections are generally large in observables with large relaxation timescales.

DOI
10.1103/physrevb.87.165106

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We consider the time evolution after sudden quenches of global parameters in translational invariant Hamiltonians and study the time average expectation values and entanglement entropies in finite chains. We show that in noninteracting models the time average of spin correlation functions is asymptotically equal to the infinite time limit in the infinite chain, which is known to be described by a generalized Gibbs ensemble. The equivalence breaks down considering nonlocal operators, and we establish that this can be traced back to the existence of conservation laws common to the Hamiltonian before and after the quench. We develop a method to compute the leading finite-size corrections. We find that the finite-size corrections are generally large in observables with large relaxation timescales.
We consider the time evolution following a quantum quench in spin-1/2 chains. It is well known that local conservation laws constrain the dynamics and, eventually, the stationary behavior of local observables. We show that some widely studied models, like the quantum XY model, possess extra families of local conservation laws in addition to the translation invariant ones. As a consequence, the additional charges must be included in the generalized Gibbs ensemble that describes the stationary properties. The effects go well beyond a simple redefinition of the stationary state. The time evolution of a non-translation invariant state under a (translation invariant) Hamiltonian with a perturbation that weakly breaks the hidden symmetries underlying the extra conservation laws exhibits pre-relaxation. In addition, in the limit of small perturbation, the time evolution following pre-relaxation can be described by means of a time-dependent generalized Gibbs ensemble.
We consider the time evolution following a quantum quench in spin-1/2 chains. It is well known that local conservation laws constrain the dynamics and, eventually, the stationary behavior of local observables. We show that some widely studied models, such as the quantum XY model, possess extra families of local conservation laws in addition to the translation invariant ones. As a consequence, the additional charges must be included in the generalized Gibbs ensemble that describes the stationary properties. The effects go well beyond a simple redefinition of the stationary state. The time evolution of a non-translation-invariant state under a (translation-invariant) Hamiltonian with a perturbation that weakly breaks the hidden symmetries underlying the extra conservation laws exhibits pre-relaxation. In addition, in the limit of small perturbation, the time evolution following pre-relaxation can be described by means of a time-dependent generalized Gibbs ensemble.
We consider the reduced density matrix (RDM) ${\ensuremath{\rho}}_{A}(t)$ for a finite subsystem AAA after a global quantum quench in the infinite transverse-field Ising chain. It has been recently shown that the infinite time limit of ${\ensuremath{\rho}}_{A}(t)$ is described by the RDM ${\ensuremath{\rho}}_{\mathrm{GGE},A}$ of a generalized Gibbs ensemble. Here, we present some details on how to construct this ensemble in terms of local integrals of motion, and show its equivalence to the expression in terms of mode occupation numbers widely used in the literature. We then address the question of how ${\ensuremath{\rho}}_{A}(t)$ approaches ${\ensuremath{\rho}}_{\mathrm{GGE},A}$ as a function of time. To that end, we introduce a distance on the space of density matrices and show that it approaches zero as a universal power law ${t}^{\ensuremath{-}3/2}$ in time. As the RDM completely determines all local observables within AAA, this provides information on the relaxation of correlation functions of local operators. We then address the issue of how well a truncated generalized Gibbs ensemble with a finite number of local higher conservation laws describes a given subsystem at late times. We find that taking into account only local conservation laws with a range at most comparable to the subsystem size provides a good description. However, excluding even a single one of the most local conservation laws in general completely spoils this agreement.
We consider dynamic (non-equal-time) correlation functions of local observables after a quantum quench. We show that, in the absence of long-range interactions in the final Hamiltonian, the dynamics is determined by the same ensemble that describes static (equal-time) correlations. For many integrable models, static correlation functions of local observables after a quantum quench relax to stationary values, which are described by a generalized Gibbs ensemble. The same generalized Gibbs ensemble then determines dynamic correlation functions, and the basic form of the fluctuation dissipation theorem holds, although the absorption and emission spectra are not simply related as in the thermal case. For quenches in the transverse field Ising chain, we derive explicit expressions for the time evolution of dynamic order parameter correlators after a quench.
We consider global quenches in the quantum XY chain in a transverse field and study the nonequilibrium relaxation of the magnetization and the correlation function as well as the entanglement entropy in finite systems. For quenches in the ordered phase, the exact results are well described by a semiclassical theory (SCT) in terms of ballistically moving quasiparticles. In the thermodynamic limit the SCT is exact for the entanglement entropy and its modified version following the method of Calabrese, Essler and Fagotti: arXiv:1204.3911 is exact for the magnetization and the correlation function, too. The stationary correlation function is shown to be described by a generalized Gibbs ensemble.
We investigate the out-of-equilibrium dynamics of the one-dimensional quantum Ising model after a sudden quench of the transverse magnetic field. While for a translationally invariant system the statistical description of the asymptotic order parameter correlations after the quench can be performed in terms of the generalized Gibbs ensemble, we show that a breaking of translational invariance, e.g., by perturbing the boundary conditions, disrupts its validity. This effect, which of course vanishes in the thermodynamic limit, is shown to be very important in the presence of disorder.