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arXiv:1208.1121 · Physical Review E

Gaussian equilibration

Lorenzo Campos Venuti, Paolo Zanardi A

Published Jan 7, 2013• 38 citations
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Abstract

A finite quantum system evolving unitarily equilibrates in a probabilistic fashion. In the general many-body setting the time fluctuations of an observable A are typically exponentially small in the system size. We consider here quasifree Fermi systems where the Hamiltonian and observables are quadratic in the Fermi operators. We first prove a bound on the temporal fluctuations ΔA(2) and then map the equilibration dynamics to a generalized classical XY model in the infinite temperature limit. Using this insight, we conjecture that, in most cases, a central limit theorem can be formulated, leading to what we call Gaussian equilibration: observables display a Gaussian distribution with relative error ΔA/A[over ¯]=O(L(-1/2)), where L is the dimension of the single-particle space. The conjecture, corroborated by numerical evidence, is proven analytically under mild assumptions for the magnetization in the quantum XY model and for a class of observables in a tight-binding model. We also show that the variance is discontinuous at the transition between a quasifree model and a nonintegrable one.

DOI
10.1103/physreve.87.012106

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A finite quantum system evolving unitarily equilibrates in a probabilistic fashion. In the general many-body setting the time fluctuations of an observable A are typically exponentially small in the system size. We consider here quasifree Fermi systems where the Hamiltonian and observables are quadratic in the Fermi operators. We first prove a bound on the temporal fluctuations ΔA(2) and then map the equilibration dynamics to a generalized classical XY model in the infinite temperature limit. Using this insight, we conjecture that, in most cases, a central limit theorem can be formulated, leading to what we call Gaussian equilibration: observables display a Gaussian distribution with relative error ΔA/A[over ¯]=O(L(-1/2)), where L is the dimension of the single-particle space. The conjecture, corroborated by numerical evidence, is proven analytically under mild assumptions for the magnetization in the quantum XY model and for a class of observables in a tight-binding model. We also show that the variance is discontinuous at the transition between a quasifree model and a nonintegrable one.
We study the equilibration dynamics of closed finite quantum systems and address the question of the time needed for the system to equilibrate. In particular, we focus on the scaling of the equilibration time Teq{T}_{\mathrm{eq}}Teq​ with the system size LLL. For clean systems, we give general arguments predicting Teq=O(L0){T}_{\mathrm{eq}}=O({L}^{0})Teq​=O(L0) for clustering initial states, while for small quenches around a critical point we find ${T}_{\mathrm{eq}}=O({L}^{\ensuremath{\zeta}})$ where $\ensuremath{\zeta}$ is the dynamical critical exponent. We then analyze noisy systems where exponentially large time scales are known to exist. Specifically, we consider the tight-binding model with diagonal impurities and give numerical evidence that in this case ${T}_{\mathrm{eq}}\ensuremath{\sim}B{e}^{C{L}^{\ensuremath{\psi}}}$ where $B,C,\phantom{\rule{0.16em}{0ex}}\ensuremath{\psi}$ are observable-dependent constants. Finally, we consider another noisy system whose evolution dynamics is randomly sampled from a circular unitary ensemble. Here, we are able to prove analytically that Teq=O(1){T}_{\mathrm{eq}}=O(1)Teq​=O(1), thus showing that noise alone is not sufficient for slow equilibration dynamics.
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.
It has been proposed to investigate the equilibration properties of a small isolated quantum system by means of the matrix of asymptotic transition probabilities in some preferential basis. The trace TTT of this matrix measures the degree of equilibration of the system prepared in a typical state of the preferential basis. This quantity may vary between unity (ideal equilibration) and the dimension NNN of the Hilbert space (no equilibration at all). Here we analyze several examples of simple systems where the behavior of TTT can be investigated by analytical means. We first study the statistics of TTT when the Hamiltonian governing the dynamics is random and drawn from a distribution invariant under the group U(N)(N)(N) or O(N)(N)(N). We then investigate a quantum spin SSS in a tilted magnetic field making an arbitrary angle with the preferred quantization axis, as well as a tight-binding particle on a finite electrified chain. The last two cases provide examples of the interesting situation where varying a system parameter -- such as the tilt angle or the electric field -- through some scaling regime induces a continuous transition from good to bad equilibration properties.
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.
An isolated quantum system evolves reversibly, yet a small subsystem can lose nearly all information about its initial state. Conserved quantities may retain some of this information, depending on how strongly they overlap with local observables. Here we consider a subsystem in an arbitrary initial state coupled to a maximally mixed bath. We show that the worst-input mean-square equilibration error, maximised over window origins, is bounded from above and below by the mean square of a complete local autocorrelation, with the stationary part subtracted. The two mean-square quantities are linearly comparable. A correlation record over a finite interval therefore controls the equilibration error in every interval of the same length, and the result extends to nonnegative observation weights with nonnegative Fourier transform. The result allows arbitrary degeneracies of energies and gaps. We apply it to two static Hamiltonian families. For fixed spin subsystems in the one-particle sector of a bounded-range periodic clock, fixed-accuracy thermalisation is guaranteed on windows that grow only slightly faster than linearly with system size, despite exponentially small positive energies. A random commuting model with logarithmic interaction range retains an extensive set of conserved observables. Its expected mean-square error is bounded by a finite-size term and an inverse-time term. The inverse-time dependence is unavoidable because of the initial transient. These examples show how local relaxation can coexist with exact conservation laws and a finely spaced spectrum.
We prove that quantum many-body systems on a one-dimensional lattice locally relax to Gaussian states under non-equilibrium dynamics generated by a bosonic quadratic Hamiltonian. This is true for a large class of initial states—pure or mixed—which have to satisfy merely weak conditions concerning the decay of correlations. The considered setting is a proven instance of a situation where dynamically evolving closed quantum systems locally appear as if they had truly relaxed, to maximum entropy states for fixed second moments. This furthers the understanding of relaxation in suddenly quenched quantum many-body systems. The proof features a non-commutative central limit theorem for non-i.i.d. random variables, showing convergence to Gaussian characteristic functions, giving rise to trace-norm closeness. We briefly link our findings to the ideas of typicality and concentration of measure.