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arXiv:1606.05650 · Physical review. B./Physical review. B

Partial breakdown of quantum thermalization in a Hubbard-like model

James R. Garrison, Ryan V. Mishmash, Matthew P. A. Fisher

Published Feb 17, 2017• 28 citations
arXivHTMLPDF

Abstract

In recent years, physicists have made great strides toward better understanding isolated many-body quantum systems. Already, two distinct phases are well known: systems that are ergodic and thermalize completely, and, by contrast, systems that exhibit many-body localization (MBL) due to a strong disorder potential, which prevents thermalization. It is tempting to wonder whether a phase of matter could exist between the extremes of full thermalization and MBL in a generic, i.e., nonintegrable, isolated many-body quantum system. Here, the authors provide evidence that such a nonergodic phase can be realized in a translationally invariant itinerant electron model, with both charge and spin degrees of freedom. Exact diagonalization calculations on finite system sizes reveal that there exists a ``spin band'' of nonthermal eigenstates. These states have entanglement entropy that scales as an overall ``volume law,'' but after a partial measurement of the spin on each site the scaling is modified to an ``area law,'' thus suggesting the states are not ergodic. These properties and others suggest that this model realizes a new nonthermal phase of matter, known as a quantum disentangled liquid (QDL). The putative existence of this phase has striking implications for the foundations of quantum statistical mechanics.

DOI
10.1103/physrevb.95.054204

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In recent years, physicists have made great strides toward better understanding isolated many-body quantum systems. Already, two distinct phases are well known: systems that are ergodic and thermalize completely, and, by contrast, systems that exhibit many-body localization (MBL) due to a strong disorder potential, which prevents thermalization. It is tempting to wonder whether a phase of matter could exist between the extremes of full thermalization and MBL in a generic, i.e., nonintegrable, isolated many-body quantum system. Here, the authors provide evidence that such a nonergodic phase can be realized in a translationally invariant itinerant electron model, with both charge and spin degrees of freedom. Exact diagonalization calculations on finite system sizes reveal that there exists a spin band'' of nonthermal eigenstates. These states have entanglement entropy that scales as an overall volume law,'' but after a partial measurement of the spin on each site the scaling is modified to an ``area law,'' thus suggesting the states are not ergodic. These properties and others suggest that this model realizes a new nonthermal phase of matter, known as a quantum disentangled liquid (QDL). The putative existence of this phase has striking implications for the foundations of quantum statistical mechanics.
A very fundamental problem in quantum statistical mechanics involves whether—and how—an isolated quantum system will reach thermal equilibrium after waiting a long time. In quantum systems that do thermalize, the long-time expectation value of any “reasonable” operator will match its predicted value in the canonical ensemble. The Eigenstate Thermalization Hypothesis (ETH) posits that this thermalization occurs at the level of each individual energy eigenstate; in fact, any single eigenstate in a microcanonical energy window will predict the expectation values of such operators exactly. In the first part of this dissertation, we identify, for a generic model system, precisely which operators satisfy ETH, as well as limits to the information contained in a single eigenstate. Remarkably, our results strongly suggest that a single eigenstate can contain information about energy densities—and therefore temperatures—far away from the energy density of the eigenstate.Additionally, we study the possible breakdown of quantum thermalization in a model of itinerant electrons on a one-dimensional chain, with both spin and charge degrees of freedom. This model exhibits peculiar properties in the entanglement entropy, the apparent scaling of which is modified from a “volume law” to an “area law” after performing a partial, site-wise measurement on the system. These properties and others suggest that this model realizes a new, non-thermal phase of matter, known as a Quantum Disentangled Liquid (QDL). The putative existence of this phase has striking implications for the foundations of quantum statistical mechanics.
We propose and explore a new finite temperature phase of translationally invariant multi-component liquids which we call a 'Quantum Disentangled Liquid' (QDL) phase. We contemplate the possibility that in fluids consisting of two (or more) species of indistinguishable quantum particles with a large mass ratio, the light particles might 'localize' on the heavy particles. We give a precise, formal definition of this QDL phase in terms of the finite energy density many-particle wavefunctions. While the heavy particles are fully thermalized, for a typical fixed configuration of the heavy particles, the entanglement entropy of the light particles satisfies an area law; this implies that the light particles have not thermalized. Equivalently, but more intuitively, if the positions of all the heavy particles are measured , the projected wavefunction for the unmeasured light particles has as an area law entanglement entropy. Thus, in a QDL phase, thermal equilibration is incomplete, and the canonical assumptions of statistical mechanics are not fully operative. The definition of the QDL phase for heavy/light particles can be readily generalized to other cases with two (or more) conserved currents, such as spin/charge in a system of spin-1/2 fermions (as in a Hubbard model). Indeed, we argue that the finite energy-density eigenstates of the t – J model will generically be in such a spin/charge QDL, although the fate of the QDL in the large U Hubbard model is uncertain. We explore the possibility of QDL in water, with the light proton degrees of freedom becoming 'localized' on the oxygen ions. While we do not presently know whether a local, generic Hamiltonian can have eigenstates of the QDL form, if not, then the non-thermal behavior discussed here will exist as an interesting crossover phenomena at a time scale that diverges as the ratio of the mass of the heavy to the light particles also diverges.
The transition between ergodic and many-body localization (MBL) phases lies at the heart of understanding quantum thermalization of many-body systems. Here, we predict a many-body critical (MBC) phase with finite-size scaling analysis in the one-dimensional extended Aubry-André-Harper-Hubbard model, which is different from both the ergodic phase and MBL phase, implying that the quantum system hosts three different fundamental phases in the thermodynamic limit. The level statistics in the MBC phase are well characterized by the so-called critical statistics, and the wave functions exhibit deep multifractal behavior only in the critical region. We further study the half-chain entanglement entropy and thermalization properties and show that the former, in the MBC phase, manifest a volume law scaling, while the many-body states violate the eigenstate thermalization hypothesis. The results are confirmed by the state-of-the-art numerical calculations with system size up to L=22. This work unveils a novel many-body phase which is extended but nonthermal.
Our expectation of thermalization is one borne of experience, and it is interesting to understand how conventional statistical mechanics may fail. The so-called quantum disentangled liquid (QDL) is a paradigm outside of the (Generalized) Gibbs description, but also without spatial disorder, precluding many-body localization. In this paper, the authors provide evidence that this QDL behavior exists in the Hubbard model. The QDL diagnostic involves bipartite entanglement entropies after a partial measurement. Both integrability in 1D and strong-coupling techniques are used to argue that the QDL diagnostic is indeed satisfied.
Many-body localization (MBL) enables new nonergodic dynamical phases of matter that avoid the fate of thermalization. Do other ergodicity-breaking phases beyond the MBL paradigm exist? Here, the authors introduce a method for constructing and probing the stability of candidate nonergodic phases. As an example, they consider disordered spin systems, in which a non-Abelian SU(2) symmetry precludes the conventional MBL phase that is characterized by quasilocal integrals of motion and area-law entanglement entropy of eigenstates. Instead, the authors construct a candidate nonergodic phase characterized by eigenstates having logarithmic scaling of entanglement entropy. They find that the phase is only marginally unstable and, therefore, nonthermal yet non-MBL dynamics can be observed in such disordered systems.
Many-body localized (MBL) systems lie outside the framework of statistical mechanics, as they fail to equilibrate under their own quantum dynamics. Even basic features of MBL systems, such as their stability to thermal inclusions and the nature of the dynamical transition to thermalizing behaviour, remain poorly understood. We study a simple central spin model to address these questions: a two-level system interacting with strength J with N ≫1 localized bits subject to random fields. On increasing J , the system transitions from an MBL to a delocalized phase on the vanishing scale J c ( N )∼1/ N , up to logarithmic corrections. In the transition region, the single-site eigenstate entanglement entropies exhibit bimodal distributions, so that localized bits are either ‘on’ (strongly entangled) or ‘off’ (weakly entangled) in eigenstates. The clusters of ‘on’ bits vary significantly between eigenstates of the same sample, which provides evidence for a heterogeneous discontinuous transition out of the localized phase in single-site observables. We obtain these results by perturbative mapping to bond percolation on the hypercube at small J and by numerical exact diagonalization of the full many-body system. Our results support the arguments that the MBL phase is unstable in systems with short-range interactions and quenched randomness in dimensions d that are high but finite. This article is part of the themed issue ‘Breakdown of ergodicity in quantum systems: from solids to synthetic matter’.