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arXiv:0912.3357 · Physical Review A

Universality in the equilibration of quantum systems after a small quench

Lorenzo Campos Venuti, Paolo Zanardi A

Published Mar 11, 2010• 37 citations
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Abstract

A sudden change in the Hamiltonian parameter drives a quantum system out of equilibrium. For a finite-size system, expectations of observables start fluctuating in time without converging to a precise limit. A new equilibrium state emerges only in the probabilistic sense, when the probability distribution for the observable expectations over long times concentrates around their mean value. In this paper we study the full statistic of generic observables after a small quench. When the quench is performed around a regular (i.e., noncritical) point of the phase diagram, generic observables are expected to be characterized by Gaussian distribution functions (``good equilibration''). Instead, when quenching around a critical point a new, universal, double-peaked distribution function emerges for relevant perturbations. Our analytic predictions are numerically checked for a nonintegrable extension of the quantum Ising model.

DOI
10.1103/physreva.81.032113

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A sudden change in the Hamiltonian parameter drives a quantum system out of equilibrium. For a finite-size system, expectations of observables start fluctuating in time without converging to a precise limit. A new equilibrium state emerges only in the probabilistic sense, when the probability distribution for the observable expectations over long times concentrates around their mean value. In this paper we study the full statistic of generic observables after a small quench. When the quench is performed around a regular (i.e., noncritical) point of the phase diagram, generic observables are expected to be characterized by Gaussian distribution functions (``good equilibration''). Instead, when quenching around a critical point a new, universal, double-peaked distribution function emerges for relevant perturbations. Our analytic predictions are numerically checked for a nonintegrable extension of the quantum Ising model.
Out-of-equilibrium quantum systems display complex temporal patterns. Such time fluctuations are generically exponentially small in the system volume and therefore can be safely ignored in most of the cases. However, if one consider small quench experiments, time fluctuations can be greatly enhanced. We show that time fluctuations may become stronger than other forms of equilibrium quantum fluctuations if the quench is performed close to a critical point. For sufficiently relevant operators the full distribution function of dynamically evolving observable expectation values becomes a universal function uniquely characterized by the critical exponents and the boundary conditions. At regular points of the phase diagram and for nonsufficiently relevant operators the distribution becomes Gaussian. Our predictions are confirmed by an explicit calculation on the quantum Ising model.
Closed quantum systems evolve unitarily and therefore cannot converge in a strong sense to an equilibrium state starting out from a generic pure state. Nevertheless for large system size one observes temporal typicality. Namely, for the overwhelming majority of the time instants, the statistics of observables is practically indistinguishable from an effective equilibrium one. In this paper we consider the Loschmidt echo (LE) to study this sort of unitary equilibration after a quench. We draw several conclusions on general grounds and on the basis of an exactly solvable example of a quasifree system. In particular we focus on the whole probability distribution of observing a given value of the LE after waiting a long time. Depending on the interplay between the initial state and the quench Hamiltonian, we find different regimes reflecting different equilibration dynamics. When the perturbation is small and the system is away from criticality the probability distribution is Gaussian. However close to criticality the distribution function approaches a double-peaked, universal form.
The nonequilibrium dynamics of a system that is located in the vicinity of a quantum critical point is affected by the critical slowing down of order-parameter correlations with the potential for novel out-of-equilibrium universality. After a quantum quench, i.e., a sudden change of a parameter in the Hamiltonian, such a system is expected to almost instantly fall out of equilibrium and undergo aging dynamics, i.e., dynamics that depends on the time passed since the quench. Investigating the quantum dynamics of an NNN-component ${\ensuremath{\varphi}}^{4}$ model coupled to an external bath, we determine this universal aging and demonstrate that the system undergoes a coarsening, governed by a critical exponent that is unrelated to the equilibrium exponents of the system. We analyze this behavior in the large-NNN limit, which is complementary to our earlier renormalization-group analysis, allowing in particular the direct investigation of the order-parameter dynamics in the symmetry-broken phase and at the upper critical dimension. By connecting the long-time limit of fluctuations and response, we introduce a distribution function that shows that the system remains nonthermal and exhibits quantum coherence even on long time scales.
The nonequilibrium dynamics of an isolated quantum system after a sudden quench to a dynamical critical point is expected to be characterized by scaling and universal exponents due to the absence of time scales. We explore these features for a quench of the parameters of a Hamiltonian with O(N) symmetry, starting from a ground state in the disordered phase. In the limit of infinite N, the exponents and scaling forms of the relevant two-time correlation functions can be calculated exactly. Our analytical predictions are confirmed by the numerical solution of the corresponding equations. Moreover, we find that the same scaling functions, yet with different exponents, also describe the coarsening dynamics for quenches below the dynamical critical point.
In this work we investigate the equilibration dynamics after a sudden Hamiltonian quench of a quantum spin system initially prepared in a thermal state. To characterize the equilibration we evaluate the Loschmidt echo, a global measure for the degree of distinguishability between the initial and time-evolved quenched states. We present general results valid for small quenches and detailed analysis of the quantum XYXYXY chain. The result is that quantum criticality manifests, even at small but finite temperatures, in a universal double-peaked form of the echo statistics and poor equilibration for sufficiently relevant perturbations. In addition, for this model we find a tight lower bound on the Loschmidt echo in terms of the purity of the initial state and the more easily evaluated Hilbert-Schmidt inner product between initial and time-evolved quenched states. This bound allows us to relate the time-averaged Loschmidt echo with the purity of the time-averaged state, a quantity that has been shown to provide an upper bound on the variance of observables.
The equilibration dynamics of a closed quantum system is encoded in the long-time distribution function of generic observables. In this Letter we consider the Loschmidt echo generalized to finite temperature, and show that we can obtain an exact expression for its long-time distribution for a closed system described by a quantum XY chain following a sudden quench. In the thermodynamic limit the logarithm of the Loschmidt echo becomes normally distributed, whereas for small quenches in the opposite, quasicritical regime, the distribution function acquires a universal double-peaked form indicating poor equilibration. These findings, obtained by a central limit theorem-type result, extend to completely general models in the small-quench regime.