We investigate the implications of integrability for the existence of quantum disentangled liquid (QDL) states in the half-filled one-dimensional Hubbard model. We argue that there exist finite energy-density eigenstates that exhibit QDL behaviour in the sense of Grover & Fisher (2014 J. Stat. Mech. 2014 , P10010. ( doi:10.1088/1742-5468/2014/10/P10010 )). These states are atypical in the sense that their entropy density is smaller than that of thermal states at the same energy density. Furthermore, we show that thermal states in a particular temperature window exhibit a weaker form of the QDL property, in agreement with recent results obtained by strong-coupling expansion methods in Veness et al. (2016 ( http://arxiv.org/abs/1611.02075 )). This article is part of the themed issue ‘Breakdown of ergodicity in quantum systems: from solids to synthetic matter’.
We investigate the implications of integrability for the existence of quantum disentangled liquid (QDL) states in the half-filled one-dimensional Hubbard model. We argue that there exist finite energy-density eigenstates that exhibit QDL behaviour in the sense of Grover & Fisher (2014 J. Stat. Mech. 2014 , P10010. ( doi:10.1088/1742-5468/2014/10/P10010 )). These states are atypical in the sense that their entropy density is smaller than that of thermal states at the same energy density. Furthermore, we show that thermal states in a particular temperature window exhibit a weaker form of the QDL property, in agreement with recent results obtained by strong-coupling expansion methods in Veness et al. (2016 ( http://arxiv.org/abs/1611.02075 )). This article is part of the themed issue ‘Breakdown of ergodicity in quantum systems: from solids to synthetic matter’.
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.
This thesis focuses on the one-dimensional Hubbard model, considering both dynamical response functions and also particular entanglement measures of highly excited eigenstates. In Chapter 2 we discuss how to describe the optical conductivity at finite energy scales. In order to calculate the dynamical response of this interacting system, it is necessary to go beyond the conventional Luttinger liquid picture. Using mobile impurity models and integrability we make quantitative predictions which can be compared with numerics. In Chapter 3, we investigate the existence of the so-called "quantum disentangled liquid" (QDL) in the same system. The diagnostic for the QDL involves understanding the bipartite entanglement entropy of highly excited eigenstates. Here we use results from integrability and strong-coupling expansions to argue that such a state of matter is realised in the Hubbard chain in a specific sense.
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.
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.
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.
In the one-dimensional low-density Jaynes-Cummings Hubbard (JCH) model, we find that when the hopping strength is much smaller than the coupling strength, the average restricted energy gap ratio exhibits an abnormal statistical behavior that is neither a Poisson nor a Gaussian orthogonal ensemble. But the average half-chain entanglement entropy exhibits ergodicity, and the eigenstate thermalization hypothesis (ETH) is valid for the observable. These results are quite different from those of the standard JCH model. In addition, when the hopping and the coupling strengths are of the same order, quantum chaos still appears in the low-density JCH model, which is in contrast to the integrability of the one-dimensional hard-core bosons. Finally, the dipole-dipole interaction breaks the particle-hole symmetry and leads the abnormal statistical properties to be closer to those of the integrable system at the weak hopping strength limit, but the quantum chaos properties cannot be affected when the hopping strength is of the same order as the coupling strength. Our results demonstrate the counterintuitive behavior in the low-density JCH model and explain the physics behind them from the perspective of the energy spectrum.