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arXiv:1409.6558 · Physical Review B

Quantum quench in long-range field theories

M. A. Rajabpour, Spyros Sotiriadis

Published Jan 21, 2015• 44 citations
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Abstract

We study the equilibration properties of observables in long-range field theories after the mass quench in $d=1$, 2, and 3 dimensions. We classify the regimes where we expect equilibration or its absence in different dimensions. In addition, we study the effect of the initial state in the equilibration properties of our system. In the case of free field theories, we show that whenever we have equilibration, the long-time limit of correlations can be described by the generalized Gibbs ensemble. We prove that all integrals of motion in our system are nonlocal.

DOI
10.1103/physrevb.91.045131

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We study the equilibration properties of observables in long-range field theories after the mass quench in d=1d=1d=1, 2, and 3 dimensions. We classify the regimes where we expect equilibration or its absence in different dimensions. In addition, we study the effect of the initial state in the equilibration properties of our system. In the case of free field theories, we show that whenever we have equilibration, the long-time limit of correlations can be described by the generalized Gibbs ensemble. We prove that all integrals of motion in our system are nonlocal.
One of the fundamental principles of statistical physics is that only partial information about a system's state is required for its macroscopic description. This is not only true for thermal ensembles, but also for the unconventional ensemble, known as Generalized Gibbs Ensemble (GGE), that is expected to describe the relaxation of integrable systems after a quantum quench. By analytically studying the quench dynamics in a prototypical one-dimensional critical model, the massless free bosonic field theory, we find evidence of a novel type of equilibration characterized by the preservation of an enormous amount of memory of the initial state that is accessible by local measurements. In particular, we show that the equilibration retains memory of non-Gaussian initial correlations, in contrast to the case of massive free evolution which erases all such memory. The GGE in its standard form, being a Gaussian ensemble, fails to predict correctly the equilibrium values of local observables, unless the initial state is Gaussian itself. Our findings show that the equilibration of a broad class of quenches whose evolution is described by Luttinger liquid theory with an initial state that is non-Gaussian in terms of the bosonic field, is not correctly captured by the corresponding bosonic GGE, raising doubts about the validity of the latter in general one-dimensional gapless integrable systems such as the Lieb-Liniger model. We also propose that the same experiment by which the GGE was recently observed [Langen et al., Science 348 (2015) 207-211] can also be used to observe its failure, simply by starting from a non-Gaussian initial state.
Abstract Ground states ofinteracting QFTs are non-Gaussian states, i.e. their connected n -point correlation functions do not vanish for , in contrast to the free QFT case. We show that, when the ground state of an interacting QFT evolves under a free massive QFT for a long time (a scenario that can be realised by a quantum quench), the connected correlation functions decay and all local physical observables equilibrate to values that are given by a Gaussian density matrix that retains memory only of the two-point initial correlation function. The argument hinges upon the fundamental physical principle of cluster decomposition, which is valid for the ground state of a general QFT. An analogous result was already known to be valid in the case of d = 1 spatial dimensions, where it is a special case of the so-called generalised Gibbs ensemble (GGE) hypothesis, and we now generalise it to higher dimensions. Moreover, in the case of massless free evolution, despite the fact that the evolution may lead not to equilibration but instead to unbounded increase of correlations with time, the GGE gives correctly the leading-order asymptotic behaviour of correlation functions in the thermodynamic and large time limit. The demonstration is performed in the context of a bosonic relativistic QFT, but the arguments apply more generally.
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 study the non-equilibrium time evolution of an integrable field theory in 1+1 dimensions after sudden variation of a global parameter of the Hamiltonian. For a class of quenches defined in the text, we compute the long time limit of the one point function of a local operator as a series of form factors. Even if some subtleties force us to handle this result with care, there is strong evidence that for long times the expectation value of any local operator can be described by a generalized Gibbs ensemble with a different effective temperature for each eigenmode.
In this PhD thesis we investigate some properties of one-dimensional quantum systems, focusing on two important aspects of integrable models: Their entanglement properties at equilibrium and their dynamical correlators after a quantum quench. The first part of the thesis will therefore be devoted to the study of the entanglement entropy in one-dimensional integrable systems, with a special focus on the XYZ spin-1/2 chain, which, in addition to being integrable, is also an interacting theory with non perturbative solutions. We derive its bipartite Renyi entropies in the thermodynamic limit and its behaviour in different phases and for different values of the mass-gap is analysed, both analytically and numerically. In particular it is worth mentioning that the numerical analysis of the entropies of the XYZ model presented in this dissertation has never been published in literature. In the second part of the thesis we study the dynamics of correlators after a quantum quench, preparing the system in a squeezed coherent initial state. The emphasis will be on the Transverse Field Ising Chain and the O(3) non-linear sigma model, the latter studied by means of the semi-classical theory, the former by a form-factor approach. Moreover in the last chapter we outline a general result about the dynamics of correlation functions of local observables after a quantum quench. In particular we show that if there are not long-range interactions in the final Hamiltonian, then the dynamics of the model at long times (non equal- time correlations) is described by the same statistical ensemble that describes its statical properties (equal-time correlations). For the Transverse Field Ising Chain this result means that its dynamics is described by a Generalized Gibbs Ensemble.