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arXiv:1205.0370 · New Journal of Physics

Many-body localization and thermalization in the full probability distribution function of observables

Elena Canovi, Davide Rossini, Rosario Fazio, Giuseppe E. Santoro, Alessandro Silva

Published Sep 21, 2012• 32 citations
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Abstract

We investigate the relation between thermalization following a quantum quench and many-body localization in quasi-particle space in terms of the long-time full distribution function of physical observables. In particular, expanding on our recent work (Canovi et al 2011 Phys. Rev. B 83 094431), we focus on the long-time behavior of an integrable XXZ chain subject to an integrability-breaking perturbation. After a characterization of the breaking of integrability and the associated localization/delocalization transition using the level spacing statistics and the properties of the eigenstates, we study the effect of integrability breaking on the asymptotic state after a quantum quench of the anisotropy parameter, looking at the behavior of the full probability distribution of the transverse and longitudinal magnetization of a subsystem. We compare the resulting distributions with those obtained in equilibrium at an effective temperature set by the initial energy. We find that, while the long-time distribution functions appear to always agree qualitatively with the equilibrium ones, quantitative agreement is obtained only when integrability is fully broken and the relevant eigenstates are diffusive in quasi-particle space.

DOI
10.1088/1367-2630/14/9/095020

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We investigate the relation between thermalization following a quantum quench and many-body localization in quasi-particle space in terms of the long-time full distribution function of physical observables. In particular, expanding on our recent work (Canovi et al 2011 Phys. Rev. B 83 094431), we focus on the long-time behavior of an integrable XXZ chain subject to an integrability-breaking perturbation. After a characterization of the breaking of integrability and the associated localization/delocalization transition using the level spacing statistics and the properties of the eigenstates, we study the effect of integrability breaking on the asymptotic state after a quantum quench of the anisotropy parameter, looking at the behavior of the full probability distribution of the transverse and longitudinal magnetization of a subsystem. We compare the resulting distributions with those obtained in equilibrium at an effective temperature set by the initial energy. We find that, while the long-time distribution functions appear to always agree qualitatively with the equilibrium ones, quantitative agreement is obtained only when integrability is fully broken and the relevant eigenstates are diffusive in quasi-particle space.
We conjecture that thermalization following a quantum quench in a strongly correlated quantum system is closely connected to many-body delocalization in the space of quasi-particles. This scenario is tested in the anisotropic Heisenberg spin chain with different types of integrability-breaking terms. We first quantify the deviations from integrability by analyzing the level spacing statistics and the inverse participation ratio of the system's eigenstates. We then focus on thermalization, by studying the dynamics after a sudden quench of the anisotropy parameter. Our numerical simulations clearly support the conjecture, as long as the integrability-breaking term acts homogeneously on the quasiparticle space, in such a way as to induce ergodicity over all the relevant Hilbert space.
We study the dynamics of the quantum Ising chain following a zero-temperature quench of the transverse field strength. Focusing on the behavior of two-point spin correlation functions, we show that the correlators of the order parameter display an effective asymptotic thermal behavior; i.e., they decay exponentially to zero with a phase coherence rate and a correlation length dictated by the equilibrium law with an effective temperature set by the energy of the initial state. On the contrary, the two-point correlation functions of the transverse magnetization or the density-of-kinks operator decay as a power law and do not exhibit thermal behavior. We argue that the different behavior is linked to the locality of the corresponding operator with respect to the quasiparticles of the model: nonlocal operators, such as the order parameter, behave thermally, while local ones do not. We study which features of the two-point correlators are a consequence of the integrability of the model by analyzing their robustness with respect to a sufficiently strong integrability-breaking term.
In this thesis two questions of equilibrium and non-equilibrium properties in many-body quantum mechanical systems are investigated. The first part is focused on probability distributions of quantum observables in many-body quantum systems. After an introduction to probability distributions, full counting statistics and the transverse field Ising model we give a brief overview over related experiments and then derive an expression for the probability distribution of the transverse field magnetization of a finite subsystem in any Gaussian state. We study the probability distribution in ground and thermal states as well as in a non-equilibrium setting after a quantum quench. We find an analytic expression for the time evolution after the quench and compare to numerics. The second part of the thesis is concerned with the stability of exact quasi-particle excitations of an integrable model after weak integrability breaking perturbations are introduced. For this we first discuss the stability of excitations in integrable systems and then give an introduction to the Heisenberg XXX-model in a magnetic field. After constructing exact excitations we calculate the decay rate in leading order perturbation theory using methods of integrability.
We study the expectation values of observables and correlation functions at long times after a global quantum quench. Our focus is on metallic (`gapless') fermionic many-body models and small quenches. The system is prepared in an eigenstate of an initial Hamiltonian, and the time evolution is performed with a final Hamiltonian which differs from the initial one in the value of one global parameter. We first derive general relations between time-averaged expectation values of observables as well as correlation functions and those obtained in an eigenstate of the final Hamiltonian. Our results are valid to linear and quadratic order in the quench parameter g and generalize prior insights in several essential ways. This allows us to develop a phenomenology for the thermalization of local quantities up to a given order in g. Our phenomenology is put to a test in several case studies of one-dimensional models representative of four distinct classes of Hamiltonians: quadratic ones, effectively quadratic ones, those characterized by an extensive set of (quasi-) local integrals of motion, and those for which no such set is known (and believed to be nonexistent). We show that for each of these models, all observables and correlation functions thermalize to linear order in g. The more local a given quantity, the longer the linear behavior prevails when increasing g. Typical local correlation functions and observables for which the term O(g) vanishes thermalize even to order g^2. Our results show that lowest order thermalization of local observables is an ubiquitous phenomenon even in models with extensive sets of integrals of motion.
The eigenstate thermalization hypothesis provides one picture of thermalization in a quantum system by looking at individual eigenstates. However, it is also important to consider how local observables reach equilibrium values dynamically. Quench protocol is one of the settings to study such questions. A recent numerical study [Ba\~nuls, Cirac, and Hastings, Phys. Rev. Lett. 106, 050405 (2007)] of a nonintegrable quantum Ising model with longitudinal field under such a quench setting found different behaviors for different initial quantum states. One particular case called the ``weak-thermalization'' regime showed apparently persistent oscillations of some observables. Here we provide an explanation of such oscillations. We note that the corresponding initial state has low energy density relative to the ground state of the model. We then use perturbation theory near the ground state and identify the oscillation frequency as essentially a quasiparticle gap. With this quasiparticle picture, we can then address the long-time behavior of the oscillations. Upon making additional approximations which intuitively should only make thermalization weaker, we argue that the oscillations nevertheless decay in the long-time limit. As part of our arguments, we also consider a quench from a BEC to a hard-core boson model in one dimension. We find that the expectation value of a single-boson creation operator oscillates but decays exponentially in time, while a pair-boson creation operator has oscillations with a ${t}^{\ensuremath{-}3/2}$ decay in time. We also study dependence of the decay time on the density of bosons in the low-density regime and use this to estimate decay time for oscillations in the original spin model.
We study the expectation values of observables and correlation functions at long times after a global quantum quench. Our focus is on metallic (``gapless'') fermionic many-body models and small quenches. The system is prepared in an eigenstate of an initial Hamiltonian, and the time evolution is performed with a final Hamiltonian which differs from the initial one in the value of one global parameter. We first derive general relations between time-averaged expectation values of observables as well as correlation functions and those obtained in an eigenstate of the final Hamiltonian. Our results are valid to linear and quadratic order in the quench parameter ggg and generalize prior insights in several essential ways. This allows us to develop a phenomenology for the thermalization of local quantities up to a given order in ggg. Our phenomenology is put to a test in several case studies of one-dimensional models representative of four distinct classes of Hamiltonians: quadratic ones, effectively quadratic ones, those characterized by an extensive set of (quasi-) local integrals of motion, and those for which no such set is known (and believed to be nonexistent). We show that for each of these models, all observables and correlation functions thermalize to linear order in ggg. The more local a given quantity, the longer the linear behavior prevails when increasing ggg. Typical local correlation functions and observables for which the term O(g)\mathcal{O}(g)O(g) vanishes thermalize even to order g2{g}^{2}g2. Our results show that lowest-order thermalization of local observables is an ubiquitous phenomenon even in models with extensive sets of integrals of motion.