Breakdown Of Ergodicity Research Articles

Tuning between Continuous Time Crystals and Many-Body Scars in Long-Range XYZ Spin Chains.

Persistent oscillatory dynamics in nonequilibrium many-body systems is a tantalizing manifestation of ergodicity breakdown that continues to attract much attention. Recent works have focused on two classes of such systems: discrete time crystals and quantum many-body scars (QMBS). While both systems host oscillatory dynamics, its origin is expected to be fundamentally different: discrete time crystal is a phase of matter which spontaneously breaks the Z_{2} symmetry of the external periodic drive, while QMBS span a subspace of nonthermalizing eigenstates forming an su(2) algebra representation. Here, we ask a basic question: is there a physical system that allows us to tune between these two dynamical phenomena? In contrast to much previous work, we investigate the possibility of a continuous time crystal (CTC) in undriven, energy-conserving systems exhibiting prethermalization. We introduce a long-range XYZ spin model and show that it encompasses both a CTC phase as well as QMBS. We map out the dynamical phase diagram using numerical simulations based on exact diagonalization and time-dependent variational principle in the thermodynamic limit. We identify a regime where QMBS and CTC order coexist, and we discuss experimental protocols that reveal their similarities as well as key differences.

Emergence of Hilbert Space Fragmentation in Ising Models with a Weak Transverse Field.

The transverse-field Ising model is one of the fundamental models in quantum many-body systems, yet a full understanding of its dynamics remains elusive in higher than one dimension. Here, we show for the first time the breakdown of ergodicity in d-dimensional Ising models with a weak transverse field in a prethermal regime. We demonstrate that novel Hilbert-space fragmentation occurs in the effective nonintegrable model with d≥2 as a consequence of only one emergent global conservation law of the domain wall number. Our results indicate nontrivial initial-state dependence for nonequilibrium dynamics of the Ising models with a weak transverse field.

Unsupervised detection of decoupled subspaces: Many-body scars and beyond

Highly excited eigenstates of quantum many-body systems are typically featureless thermal states. Some systems, however, possess a small number of special, low-entanglement eigenstates known as quantum scars. We introduce a quantum-inspired machine-learning platform based on a quantum variational autoencoder (QVAE) that detects families of scar states in spectra of many-body systems. Unlike a classical autoencoder, QVAE performs a parametrized unitary operation, allowing us to compress a single eigenstate into a smaller number of qubits. We demonstrate that the autoencoder trained on a scar state is able to detect the whole family of scar states sharing common features with the input state. We identify families of quantum many-body scars in the PXP model beyond the ${\mathbb{Z}}_{2}$ and ${\mathbb{Z}}_{3}$ families and find dynamically decoupled subspaces in the Hilbert space of disordered, interacting spin-ladder model. The possibility of an automatic detection of subspaces of scar states opens new pathways in studies of models with a weak breakdown of ergodicity and fragmented Hilbert spaces.

A generalization of the standard map and its statistical characterization

From the statistical mechanical point of view, area-preserving maps have great potential and importance. These maps exhibit chaotic and regular behavior separately or together in the available phase space as the control parameter changes. Several works on these maps, e.g., the standard map and the web map, have shown that ergodicity breakdown causes the statistical mechanical framework that describes the dynamics of the system to change. In this paper, for a novel generalization of the standard map, which we define by generalizing the periodic function used in its definition, we verify that a q-Gaussian with qsimeq 1.935 for the probability distribution of sum of the iterates of the system with initial conditions chosen from the nonergodic stability islands is robust. We also show that the probability distributions become more complicated and unexpected limiting behavior occurs for some parameter regimes.

Breakdown of Ergodicity and Self-Averaging in Polar Flocks with Quenched Disorder.

We show that spatial quenched disorder affects polar active matter in ways more complex and far reaching than heretofore believed. Using simulations of the 2D Vicsek model subjected to random couplings or a disordered scattering field, we find in particular that ergodicity is lost in the ordered phase, the nature of which we show to depend qualitatively on the type of quenched disorder: for random couplings, it remains long-range ordered, but qualitatively different from the pure (disorderless) case. For random scatterers, polar order varies with system size but we find strong non-self-averaging, with sample-to-sample fluctuations dominating asymptotically, which prevents us from elucidating the asymptotic status of order.

Disentangling roots of ergodicity breakdown by spectral analyses

The connection between ergodicity breaking for a forced system and its two-time velocity correlation function is demonstrated using the Khinchin theorem, where the particle dynamics is described by a generalized Langevin equation. Products of observables at a pair of time points are constructed that are correlated with their initial preparations. Three types of nonergodic behaviors are elucidated, specifically the velocity correlation function of the system either approaches a constant, oscillates with time around a non-vanishing constant or oscillates around zero-value at large times. The corresponding noise spectral densities behave as low-hindering and high-passing, band-passing as well as low-passing and high-hindering or simply band-hindering. When a harmonic potential is imposed on the system, the first situation can be transformed into ergodicity whereas the latter two types cannot. Furthermore, the application to the Debye Brownian oscillator as well as the verification of several famous models are discussed.

Breakdown of ergodicity in disordered U(1) lattice gauge theories

We show how U(1) lattice gauge theories display key signatures of ergodicity breaking in the presence of a random charge background. Contrary to the widely studied case of spin models, in the presence of Coulomb interactions, the spectral properties of such lattice gauge theories are very weakly affected by finite-volume effects. This allows to draw a sharp boundary for the ergodic regime, and thus the breakdown of quantum chaos for sufficiently strong gauge couplings, at the system sizes accessible via exact diagonalization. Our conclusions are independent on the value of a background topological angle, and are contrasted with a gauge theory with truncated Hilbert space, where instead we observe very strong finite-volume effects akin to those observed in spin chains.

Effect of Self-Oscillation on Escape Dynamics of Classical and Quantum Open Systems.

We study the effect of self-oscillation on the escape dynamics of classical and quantum open systems by employing the system-plus-environment-plus-interaction model. For a damped free particle (system) with memory kernel function expressed by Zwanzig (J. Stat. Phys. 9, 215 (1973)), which is originated from a harmonic oscillator bath (environment) of Debye type with cut-off frequency , ergodicity breakdown is found because the velocity autocorrelation function oscillates in cosine function for asymptotic time. The steady escape rate of such a self-oscillated system from a metastable potential exhibits nonmonotonic dependence on , which denotes that there is an optimal cut-off frequency makes it maximal. Comparing results in classical and quantum regimes, the steady escape rate of a quantum open system reduces to a classical one with decreasing gradually, and quantum fluctuation indeed enhances the steady escape rate. The effect of a finite number of uncoupled harmonic oscillators N on the escape dynamics of a classical open system is also discussed.

Generalization of the Kubo relation for confined motion and ergodicity breakdown.

A time-dependent generalized Kubo relation is derived by introducing the notion of a diffusion function for a particle confined in a harmonic potential. The relation reduces to the standard Kubo relation as a special case but holds for anomalous diffusion, nonergodic processes, and bounded motion. We analyze in detail the behaviors of the diffusion and memory functions and report a generalized Stokes-Einstein relation concerning anomalous diffusion. Furthermore, we demonstrate that when a high finite-frequency cutoff is imposed on the noise spectral density, a breakdown in ergodicity accompanied by the appearance of nonstationarity in the velocity autocorrelation function occurs in forced systems. This breakdown is taken as explicit evidence for either decay-spring-memory or recovering-force effects leading to nonexponential relaxation kinematics.

Self-assembled monolayer formation of pentamers-like molecules onto FCC(111) surfaces: the case of curcuminoids onto Au(111) surface

The adsorption of rigid straight electrically polarized pentamers over a FCC(111) surface is studied. The model was inspired by the deposition of 2-thiophene molecules over the Au(111) surface, which was previously characterized by experimental techniques and simulated under the frame of the density functional theory. We now obtain and report the charge distribution of the molecule which allows to propose a deposition model followed by Monte Carlo simulations over an ad-hoc lattice gas model. We show that for a certain value of the chemical potential there exists an isotropic-nematic phase transition which can explain the formation of a self-assembled monolayer like the one observed in the transmission electron microscopy images. An order parameter is defined to characterize the transition which presents a step-like behavior at a critical chemical potential value. The possible nature of the nematic transition in conjunction with an ergodicity breakdown is discussed as future work by means of statistical physics techniques.

Classical discrete time crystals

The spontaneous breaking of time-translation symmetry in periodically driven quantum systems leads to a new phase of matter: discrete time crystals (DTC). This phase exhibits collective subharmonic oscillations that depend upon an interplay of non-equilibrium driving, many-body interactions, and the breakdown of ergodicity. However, subharmonic responses are also a well-known feature of classical dynamical systems ranging from predator-prey models to Faraday waves and AC-driven charge density waves. This raises the question of whether these classical phenomena display the same rigidity characteristic of a quantum DTC. In this work, we explore this question in the context of periodically driven Hamiltonian dynamics coupled to a finite-temperature bath, which provides both friction and, crucially, noise. Focusing on one-dimensional chains, where in equilibrium any transition would be forbidden at finite temperature, we provide evidence that the combination of noise and interactions drives a sharp, first-order dynamical phase transition between a discrete time-translation invariant phase and an activated classical discrete time crystal (CDTC) in which time-translation symmetry is broken out to exponentially-long time scales. Power-law correlations are present along a first-order line which terminates at a critical point. We analyze the transition by mapping it to the locked-to-sliding transition of a DC-driven charge density wave. Our work points to a classical limit for quantum time crystals, and raises several intriguing questions concerning the non-equilibrium universality class of the CDTC critical point.

Alternative characterization of the nematic transition in deposition of rods on two-dimensional lattices.

We revisit the problem of excluded volume deposition of rigid rods of length k unit cells over square lattices. Two new features are introduced: (a) two new short-distance complementary order parameters, called Π and Σ, are defined, calculated, and discussed to deal with the phases present as coverage increases; (b) the interpretation is now done beginning at the high-coverage ordered phase which allows us to interpret the low-coverage nematic phase as an ergodicity breakdown present only when k≥7. In addition the data analysis invokes both mutability (dynamical information theory method) and Shannon entropy (static distribution analysis) to further characterize the phases of the system. Moreover, mutability and Shannon entropy are compared, and we report the advantages and disadvantages they present for their use in this problem.

Mixed nonergodicity of a forced system and its non-stationary strength

We report on mixed ergodicity breakdown in a forced system described by a generalized Langevin equation. We use the density of states for noise proposed by Costa et al. (Europhys. Lett., 63 (2003) 173), i.e., with a long-wavelength limit for acoustic phonons but zero weight at low frequencies. The velocity autocorrelation function for a particle in a harmonic potential is shown to oscillate with time and hence, we introduce a parameter to measure the non-stationary strength of its nonergodicity. The coupling between the self-oscillation and forced oscillation is found to weaken nonstationarity when the potential frequency is located in the middle of the frequency band for the driving noise. The systems nonergodicity is also accompanied by a dependence of the total energy of the particle moving in a periodic potential on the initial preparation.

Entropic Approach to the Detection of Crucial Events.

In this paper, we establish a clear distinction between two processes yielding anomalous diffusion and noise. The first process is called Stationary Fractional Brownian Motion (SFBM) and is characterized by the use of stationary correlation functions. The second process rests on the action of crucial events generating ergodicity breakdown and aging effects. We refer to the latter as Aging Fractional Brownian Motion (AFBM). To settle the confusion between these different forms of Fractional Brownian Motion (FBM) we use an entropic approach properly updated to incorporate the recent advances of biology and psychology sciences on cognition. We show that although the joint action of crucial and non-crucial events may have the effect of making the crucial events virtually invisible, the entropic approach allows us to detect their action. The results of this paper lead us to the conclusion that the communication between the heart and the brain is accomplished by AFBM processes.

Interaction-Induced Characteristic Length in Strongly Many-Body Localized Systems**Supported by the National Natural Science Foundation of China under Grant Nos 11474356 and 91421304, and the Special Program for Applied Research on Super Computation of the NSFC-Guangdong Joint Fund (the second phase) under Grant No U1501501. R.Q.H. was supported by China Postdoctoral Science Foundation under Grant No 2015T80069. Computational resources were provided

A complete set of local integrals of motion (LIOM) is a key concept for describing many-body localization (MBL), which explains a variety of intriguing phenomena in MBL systems. For example, LIOM constrain the dynamics and result in ergodicity violation and breakdown of the eigenstate thermalization hypothesis. However, it is difficult to find a complete set of LIOM explicitly and accurately in practice, which impedes some quantitative structural characterizations of MBL systems. Here we propose an accurate numerical method for constructing LIOM, discover through the LIOM an interaction-induced characteristic length ξ+, and prove a ‘quasi-product-state’ structure of the eigenstates with that characteristic length ξ+ for MBL systems. More specifically, we find that there are two characteristic lengths in the LIOM. The first one is governed by disorder and is of Anderson-localization nature. The second one is induced by interaction but shows a discontinuity at zero interaction, showing a nonperturbative nature. We prove that the entanglement and correlation in any eigenstate extend not longer than twice the second length and thus the eigenstates of the system are the quasi-product states with such a localization length.

Electrode reactions in slowly relaxing media.

Standard models of reaction kinetics in condensed materials rely on the Boltzmann-Gibbs distribution for the population of reactants at the top of the free energy barrier separating them from the products. While energy dissipation and quantum effects at the barrier top can potentially affect the transmission coefficient entering the rate pre-exponential factor, much stronger dynamical effects on the reaction barrier are caused by the breakdown of ergodicity for populating the reaction barrier (violation of the Boltzmann-Gibbs statistics). When the spectrum of medium modes coupled to the reaction coordinate includes fluctuations slower than the reaction rate, such nuclear motions dynamically freeze on the reaction time scale and do not contribute to the activation barrier. Here we consider the consequences of this scenario for electrode reactions in slowly relaxing media. Changing the electrode overpotential speeds the electrode electron transfer up, potentially cutting through the spectrum of nuclear modes coupled to the reaction coordinate. The reorganization energy of electrochemical electron transfer becomes a function of the electrode overpotential, switching between the thermodynamic value at low rates to the nonergodic limit at higher rates. The sharpness of this transition depends on the relaxation spectrum of the medium. The reorganization energy experiences a sudden drop with increasing overpotential for a medium with a Debye relaxation but becomes a much shallower function of the overpotential for media with stretched exponential dynamics. The latter scenario characterizes the electron transfer in ionic liquids. The analysis of electrode reactions in room-temperature ionic liquids shows that the magnitude of the free energy of nuclear solvation is significantly below its thermodynamic limit. This result applies to reaction times faster than microseconds and is currently limited by the available dielectric relaxation data.

Atypical energy eigenstates in the Hubbard chain and quantum disentangled liquids.

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'.

A perspective on quantum integrability in many-body-localized and Yang-Baxter systems.

Two of the most active areas in quantum many-particle dynamics involve systems with an unusually large number of conservation laws. Many-body-localized systems generalize ideas of Anderson localization by disorder to interacting systems. While localization still exists with interactions and inhibits thermalization, the interactions between conserved quantities lead to some dramatic differences from the Anderson case. Quantum integrable models such as the XXZ spin chain or Bose gas with delta-function interactions also have infinite sets of conservation laws, again leading to modifications of conventional thermalization. A practical way to treat the hydrodynamic evolution from local equilibrium to global equilibrium in such models is discussed. This paper expands upon a presentation at a discussion meeting of the Royal Society on 7 February 2017. The work described was carried out with a number of collaborators, including Jens Bardarson, Vir Bulchandani, Roni Ilan, Christoph Karrasch, Siddharth Parameswaran, Frank Pollmann and Romain Vasseur.This article is part of the themed issue 'Breakdown of ergodicity in quantum systems: from solids to synthetic matter'.

Many-body localization: stability and instability.

Rare regions with weak disorder (Griffiths regions) have the potential to spoil localization. We describe a non-perturbative construction of local integrals of motion (LIOMs) for a weakly interacting spin chain in one dimension, under a physically reasonable assumption on the statistics of eigenvalues. We discuss ideas about the situation in higher dimensions, where one can no longer ensure that interactions involving the Griffiths regions are much smaller than the typical energy-level spacing for such regions. We argue that ergodicity is restored in dimension d>1, although equilibration should be extremely slow, similar to the dynamics of glasses.This article is part of the themed issue 'Breakdown of ergodicity in quantum systems: from solids to synthetic matter'.

Non-thermalization in trapped atomic ion spin chains.

Linear arrays of trapped and laser-cooled atomic ions are a versatile platform for studying strongly interacting many-body quantum systems. Effective spins are encoded in long-lived electronic levels of each ion and made to interact through laser-mediated optical dipole forces. The advantages of experiments with cold trapped ions, including high spatio-temporal resolution, decoupling from the external environment and control over the system Hamiltonian, are used to measure quantum effects not always accessible in natural condensed matter samples. In this review, we highlight recent work using trapped ions to explore a variety of non-ergodic phenomena in long-range interacting spin models, effects that are heralded by the memory of out-of-equilibrium initial conditions. We observe long-lived memory in static magnetizations for quenched many-body localization and prethermalization, while memory is preserved in the periodic oscillations of a driven discrete time crystal state.This article is part of the themed issue 'Breakdown of ergodicity in quantum systems: from solids to synthetic matter'.