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arXiv:1910.07461

Localization in the Discrete Non-Linear Schr\"odinger Equation: mechanism of a First-Order Transition in the Microcanonical Ensemble

Giacomo Gradenigo, Stefano Iubini, Roberto Livi, Satya N. Majumdar

Published Oct 16, 2019• 2 citations
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Abstract

When constrained to high energies, the wave function of the non-linear Schrodinger equation on a lattice becomes localized. We demonstrate here that localization occours as a first-order transition in the microcanonical ensemble, in a region of phase space where the canonical and the microcanonical ensembles are not equivalent. Quite remarkably, the transition to the localized non-ergodic phase takes place in a region where the microcanonical entropy, $S_N$, grows sub-extensively with the system size $N$, namely $S_N \sim N^{1/3}$. This peculiar feature signals a shrinking of the phase space and exemplifies quite a typical scenario of broken ergodicity phases. In particular, the localized phase is characterized by a condensation phenomenon, where a finite fraction of the total energy is localized on a few lattice sites. The detailed study of $S_N(E)$ close to the transition value of the energy, $E_c$ , reveals that its first-order derivative, $\frac{\partial S_N(E)}{\partial E}|_{E=E_c}$, exhibits a jump, thus naturally accounting for a first-order phase transition from a thermalized phase to a localized one, which is properly assessed by studying the participation ratio of the site energy as an order parameter. Moreover, we show that $S_N$ predicts the existence of genuine negative temperatures in the localized phase. In a physical perspective this indicates that by adding energy to the system it condensates into increasingly ordered states.

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Modern Physics Letters B•Feb 20, 2019•15 citations
When constrained to high energies, the wave function of the non-linear Schrodinger equation on a lattice becomes localized. We demonstrate here that localization occours as a first-order transition in the microcanonical ensemble, in a region of phase space where the canonical and the microcanonical ensembles are not equivalent. Quite remarkably, the transition to the localized non-ergodic phase takes place in a region where the microcanonical entropy, SNS_NSN​, grows sub-extensively with the system size NNN, namely SN∼N1/3S_N \sim N^{1/3}SN​∼N1/3. This peculiar feature signals a shrinking of the phase space and exemplifies quite a typical scenario of broken ergodicity phases. In particular, the localized phase is characterized by a condensation phenomenon, where a finite fraction of the total energy is localized on a few lattice sites. The detailed study of SN(E)S_N(E)SN​(E) close to the transition value of the energy, EcE_cEc​ , reveals that its first-order derivative, ∂SN(E)∂E∣E=Ec\frac{\partial S_N(E)}{\partial E}|_{E=E_c}∂E∂SN​(E)​∣E=Ec​​, exhibits a jump, thus naturally accounting for a first-order phase transition from a thermalized phase to a localized one, which is properly assessed by studying the participation ratio of the site energy as an order parameter. Moreover, we show that SNS_NSN​ predicts the existence of genuine negative temperatures in the localized phase. In a physical perspective this indicates that by adding energy to the system it condensates into increasingly ordered states.
Abstract We perform a detailed numerical study of the very peculiar thermodynamic properties of the localized high-energy phase of the discrete non-linear Schrödinger equation (DNLSE). A numerical sampling of the microcanonical ensemble done by means of Hamiltonian dynamics reveals a new and subtle relation between the presence of the localized phase and a property of the system that we have called classical entanglement . Our main finding is that a quantity defined for our classical system in perfect analogy with the entanglement entropy of quantum ones, and that we have therefore called S ent , grows with the system size N in the localized phase as S ent ( N ) ∼ log ⁡ ( N ) , therefore revealing the presence of subtle non-local correlations between any finite portion of the system and the rest of it. This manifestation of classical entanglement beautifully captures the lack of system separability in the DNLSE localized phase, revealing how statistical correlations specific to the microcanonical ensemble and non-reproducible in the canonical one, may concur to determine a property totally analogous to the one produced by non-local quantum correlations.
The interplay between quenched disorder and interaction effects opens the possibility in a closed quantum many-body system of a phase transition at finite energy density between an ergodic phase, which is governed by the laws of statistical physics, and a localized one, in which the degrees of freedom are frozen and ergodicity breaks down. The possible existence of a quantum phase transition at finite energy density is strongly questioning our understanding of the fundamental laws of nature and has generated an active field of research called many-body localization. This thesis consists of three parts and is dedicated to the understanding and characterization of the phenomenon of many-body localization, approaching it from complementary facets. In particular, borrowing methods and tools from different fields, we analyze timely problems. The first part of the thesis is devoted to detecting the many-body localization transition and to characterize both the ergodic and the localized phase it separates. Here we provide a characterization from two different perspectives: the first one is based on the study of local entanglement properties. In the second one, using tools from quantum-chaos theory, we attempt to answer the question of understanding time-irreversibility, and thus probing the breaking of ergodicity. We analyze experimentally viable observables. Moreover, we propose two different quantities to distinguish an Anderson insulating phase from a many-body localized one, which is one of the issues in experiments. The second part focuses on understanding the existence of a putative subdiffusive multifractal phase. Analyzing the quantum dynamics of the system in this region of the phase diagram, we point out the importance of finite-size effects, questioning the existence of this multifractal phase. We speculate with a possible scenario in which the diffusivity and thus ergodicity could be restored in the thermodynamic limit. Furthermore, we find that the propagation is highly non-Gaussian, which could have an important effect on understanding the critical point of the according transition. We tackle this problem also from a different angle. A possible toy-model to understand many-body localization entails the Anderson model on a random-regular graph. Also in the latter model the possible existence of an intermediate multifractal phase has been conjectured. There, studying the survival return probability of a particle with time, we give a new characterization of multifractal phases and give indication of the possible existence of this phase. Nevertheless, we also outline possible caveats. In the last part of this thesis we study the interplay between symmetry and correlated disorder in a non-interacting fermionic system. We show another possible mechanism for breaking localization. In particular, we focus on studying information and particle transport, emphasizing how the two types of propagation can be different.
We consider the phenomenon of condensation of a globally conserved quantity $H={\ensuremath{\sum}}_{i=1}^{N}{\ensuremath{\epsilon}}_{i}$ distributed on NNN sites, occurring when the density h=H/Nh=H/Nh=H/N exceeds a critical density hc{h}_{c}hc​. We numerically study the dependence of the participation ratio ${Y}_{2}=\ensuremath{\langle}{\ensuremath{\epsilon}}_{i}^{2}\ensuremath{\rangle}/(N{h}^{2})$ on the size NNN of the system and on the control parameter $\ensuremath{\delta}=(h\ensuremath{-}{h}_{c})$, for various models: (i) a model with two conservation laws, derived from the discrete nonlinear Schr\"odinger equation; (ii) the continuous version of the zero-range process class, for different forms of the function $f(\ensuremath{\epsilon})$ defining the factorized steady state. Our results show that various localization scenarios may appear for finite NNN and close to the transition point. These scenarios are characterized by the presence or the absence of a minimum of Y2{Y}_{2}Y2​ when plotted against NNN and by an exponent $\ensuremath{\gamma}\ensuremath{\ge}2$ defined through the relation ${N}^{*}\ensuremath{\simeq}{\ensuremath{\delta}}^{\ensuremath{-}\ensuremath{\gamma}}$, where N∗{N}^{*}N∗ separates the delocalized region ($N\ensuremath{\ll}{N}^{*}$, Y2{Y}_{2}Y2​ vanishes with increasing NNN) from the localized region ($N\ensuremath{\gg}{N}^{*}$, Y2{Y}_{2}Y2​ is approximately constant). We finally compare our results with the structure of the condensate obtained through the single-site marginal distribution.
The basic quantity for the description of the statistical properties of physical systems is the density of states or, equivalently, the microcanonical entropy. Macroscopic quantities of a system in equilibrium can be computed directly from the entropy. Response functions such as the susceptibility are related, for example, to the curvature of the entropy surface. Interestingly, physical quantities in the microcanonical ensemble show characteristic properties of phase transitions in finite systems. In this article we investigate these characteristics for finite Ising systems. The singularities in microcanonical quantities, which signal a continuous phase transition in infinite systems, are characterized by classical critical exponents. Estimates of the non-classical exponents which emerge only in the thermodynamic limit can nevertheless be obtained by analysing effective exponents or by applying a microcanonical finite-size scaling theory. This is explicitly demonstrated for two- and three-dimensional Ising systems.
In the presence of strong enough disorder one-dimensional systems of interacting spinless fermions at nonzero filling factor are known to be in a many-body localized phase. When represented in `Fock space,' the Hamiltonian of such a system looks like that of a single `particle' hopping on a Fock lattice in the presence of a random disordered potential. The coordination number of the Fock lattice increases linearly with the system size LLL in one dimension. Thus in the thermodynamic limit $L\ensuremath{\rightarrow}\ensuremath{\infty}$, the disordered interacting problem in one dimension maps on to an Anderson model with infinite coordination number. Despite this, this system displays localization which appears counterintuitive. A close observation of the on-site disorder potentials on the Fock lattice reveals a large degree of correlation among them as they are derived from an exponentially smaller number of on-site disorder potentials in real space. This indicates that the correlations between the on-site disorder potentials on a Fock lattice has a strong effect on the localization properties of the corresponding many-body system. This intuition is also consistent with studies of quantum random energy model where the typical mid-spectrum states are ergodic and the on-site potentials in Fock space are completely uncorrelated. In this work we perform a systematic quantitative exploration of the nature of correlations of the Fock space potential required for localization. We study different functional variations of the disorder correlation in Fock lattice by analyzing the eigenspectrum obtained through exact diagonalization. Without changing the typical strength of the on-site disorder potential in Fock lattice we show that changing the correlation strength can induce thermalization or localization in systems. From among the various forms of correlations we study, we find that only the linear variation of correlations with Hamming distance in Fock space is able to drive a thermal-MBL phase transition where the transition is driven by the correlation strength. Systems with the other forms of correlations we study are found to be ergodic.
A long-range interacting Fermi chain placed in the uniform and the staggered magnetic field is studied via the micro-canonical approach. The relation between the entropy and the energy of the system is obtained by counting the number of microscopic states. We find that this system is non-ergodic and can exhibit first-order phase transition, second-order phase transition, or both. The microcanonical ensemble predicts negative specific heat regions and temperature jumps. Moreover, the global phase diagram of the system is constructed.