Ergodicity Breaking Research Articles

Order and chaos in the SU(2) matrix model: Ergodicity and classical phases

We study the classical nonlinear dynamics of the SU(2) Yang-Mills matrix model introduced in [] as a low-energy approximation to two-color QCD. Restricting to the spin-0 sector of the model, we unearth an unexpected tetrahedral symmetry, which endows the dynamics with an extraordinarily rich structure. Among other things, we find that the spin-0 sector contains coexisting chaotic subsectors as well as nested chaotic “basins”, and displays alternation between regular and chaotic dynamics as energy is varied. The symmetries also grant us a considerable amount of analytic control which allows us to make several quantitative observations. We see that the classical spin-0 sector has a rich phase structure, arising from ergodicity breaking. Surprisingly, we find that many of these classical phases display numerous similarities to previously discovered phases of the spin-0 sector [], and we explore these similarities in a heuristic fashion. Published by the American Physical Society 2024

Krylov delocalization/localization across ergodicity breaking

Krylov delocalization/localization across ergodicity breaking

A functional renormalization group for signal detection and stochastic ergodicity breaking

Signal detection is one of the main challenges in data science. As often happens in data analysis, the signal in the data may be corrupted by noise. There is a wide range of techniques that aim to extract the relevant degrees of freedom from data. However, some problems remain difficult. This is notably the case for signal detection in almost continuous spectra when the signal-to-noise ratio is small enough. This paper follows a recent bibliographic line, which tackles this issue with field-theoretical methods. Previous analysis focused on equilibrium Boltzmann distributions for an effective field representing the degrees of freedom of data. It was possible to establish a relation between signal detection and Z2 -symmetry breaking. In this paper, we consider a stochastic field framework inspired by the so-called ‘model A’, and show that the ability to reach, or not reach, an equilibrium state is correlated with the shape of the dataset. In particular, by studying the renormalization group of the model, we show that the weak ergodicity prescription is always broken for signals that are small enough, when the data distribution is close to the Marchenko–Pastur law. This, in particular, enables the definition of a detection threshold in the regime where the signal-to-noise ratio is small enough.

Universal superdiffusion of random walks in media with embedded fractal networks of low diffusivity.

Diffusion in composite media with high contrasts between diffusion coefficients in fractal sets of inclusions and in their embedding matrices is modeled by lattice random walks (RWs) with probabilities p<1 of hops from fractal sites and 1 from matrix sites. Superdiffusion is predicted in time intervals that depend on p and with diffusion exponents that depend on the dimensions of matrix (E) and fractal (D_{F}) as ν=1/(2+D_{F}-E). This contrasts with the nonuniversal subdiffusion of RWs confined to fractal media. Simulations with four fractals show the anomaly at several time decades for p≲10^{-3} and the crossover to the asymptotic normal diffusion. These results show that superdiffusion can be observed in isotropic RWs with finite moments of hop length distributions and allow the estimation of the dimension of the inclusion set from the diffusion exponent. However, displacements within single trajectories have normal scaling, which shows transient ergodicity breaking.

Anomalous diffusion in external-force-affected deterministic systems.

This study investigates the impact of external forces on the movement of particles, specifically focusing on a type of box piecewise linear map that generates normal diffusion akin to Brownian motion. Through numerical methods, the research delves into the effects of two distinct external forces: linear forces linked to the particle's current position and periodic sinusoidal forces related to time. The results uncover anomalous dynamical behavior characterized by nonlinear growth in the ensemble-averaged mean-squared displacement (EAMSD), aging, and ergodicity breaking. Notably, the diffusion pattern of particles under linear external forces resembles an Ehrenfest double urn model, with its asymptotic EAMSD coinciding with the Langevin equationunder linear potential. Meanwhile, particle movement influenced by periodic sinusoidal forces corresponds to an inhomogeneous Markov chain, with its external force amplitude and diffusion coefficient function exhibiting a "multipeak" fractal structure. The study also provides insights into the formation of this structure through the turnstiles dynamics.

Weak ergodicity breaking transition in a randomly constrained model

Experiments in Rydberg atoms have recently found unusually slow decay from a small number of special initial states. We investigate the robustness of such long-lived states (LLS) by studying an ensemble of locally constrained random systems with tunable range μ. Upon varying μ, we find a transition between thermal and weakly nonergodic (supporting a finite number of LLS) phases. Furthermore, we demonstrate that the LLS observed in the experiments disappear upon the addition of small perturbations so that the transition reported here is distinct from known ones. We then show that the LLS dynamics explores only part of the accessible Hilbert space, thus corresponding to localization in Hilbert space. Published by the American Physical Society 2024

Nonmesonic Quantum Many-Body Scars in a 1D Lattice Gauge Theory.

We investigate the meson excitations (particle-antiparticle bound states) in quantum many-body scars of a 1D Z_{2} lattice gauge theory coupled to a dynamical spin-1/2 chain as a matter field. By introducing a string representation of the physical Hilbert space, we express a scar state |Ψ_{n,l}⟩ as a superposition of all string bases with an identical string number n and a total length l. For the small-l scar state |Ψ_{n,l}⟩, the gauge-invariant spin exchange correlation function of the matter field hosts an exponential decay as the distance increases, indicating the existence of stable mesons. However, for large l, the correlation function exhibits a power-law decay, signaling the emergence of nonmesonic excitations. Furthermore, we show that this mesonic-nonmesonic crossover can be detected by the quench dynamics, starting from two low-entangled initial states, respectively, which are experimentally feasible in quantum simulators. Our results expand the physics of quantum many-body scars in lattice gauge theories and reveal that the nonmesonic state can also manifest ergodicity breaking.

Arcsine Laws of Light.

We demonstrate that the time-integrated light intensity transmitted by a coherently driven resonator obeys Lévy's arcsine laws-a cornerstone of extreme value statistics. We show that convergence to the arcsine distribution is algebraic, universal, and independent of nonequilibrium behavior due to nonconservative forces or nonadiabatic driving. We furthermore verify, numerically, that the arcsine laws hold in the presence of frequency noise and in Kerr-nonlinear resonators supporting non-Gaussian states. The arcsine laws imply a weak ergodicity breaking which can be leveraged to enhance the precision of resonant optical sensors with zero energy cost, as shown in our companion manuscript [V. G. Ramesh et al., companion paper, Phys. Rev. Res. (2024).PPRHAI2643-1564]. Finally, we discuss perspectives for probing the possible breakdown of the arcsine laws in systems with memory.

Decay of long-lived oscillations after quantum quenches in gapped interacting quantum systems

The presence of long-lived oscillations in the expectation values of local observables after quantum quenches has recently attracted considerable attention in relation to weak ergodicity breaking. Here, we focus on an alternative mechanism that gives rise to such oscillations in a class of systems that support kinematically protected gapped excitations at zero temperature. An open question in this context is whether such oscillations will ultimately decay. We provide strong support for the decay hypothesis by considering spin models that can be mapped to systems of weakly interacting fermions, which in turn are amenable to an analysis by standard methods based on the Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) hierarchy. We find that there is a time scale beyond which the oscillations start to decay that grows as the strength of the quench is made small. Published by the American Physical Society 2024

Topologically stable ergodicity breaking from emergent higher-form symmetries in generalized quantum loop models

We present a set of generalized quantum loop models that provably exhibit topologically stable ergodicity breaking. These results hold for both periodic and open boundary conditions, and derive from a one-form symmetry (notably not being restricted to sectors of extremal one-form charge). We identify simple models in which this one-form symmetry can be emergent, giving rise to the aforementioned ergodicity breaking as an exponentially long-lived prethermal phenomenon. We unveil a web of dualities that connects these models, in certain limits, to models that have previously been discussed in the literature. We also identify nonlocal conserved quantities in such models, which correspond to patterns of system-spanning domain walls and are robust to the addition of arbitrary kk-local perturbations.

Diffusion of active Brownian particles under quenched disorder.

The motion of a single active particle in one dimension with quenched disorder under the external force is investigated. Within the tailored parameter range, anomalous diffusion that displays weak ergodicity breaking is observed, i.e., non-ergodic subdiffusion and non-ergodic superdiffusion. This non-ergodic anomalous diffusion is analyzed through the time-dependent probability distributions of the particle's velocities and positions. Its origin is attributed to the relative weights of the locked state (predominant in the subdiffusion state) and running state (predominant in the superdiffusion state). These results may contribute to understanding the dynamical behavior of self-propelled particles in nature and the extraordinary response of nonlinear dynamics to the externally biased force.

Ergodicity breaking from Rydberg clusters in a driven-dissipative many-body system.

It is challenging to probe ergodicity breaking trends of a quantum many-body system when dissipation inevitably damages quantum coherence originated from coherent coupling and dispersive two-body interactions. Rydberg atoms provide a test bed to detect emergent exotic many-body phases and nonergodic dynamics where the strong Rydberg atom interaction competes with and overtakes dissipative effects even at room temperature. Here, we report experimental evidence of a transition from ergodic toward ergodic breaking dynamics in driven-dissipative Rydberg atomic gases. The broken ergodicity is featured by the long-time phase oscillation, which is attributed to the formation of Rydberg excitation clusters in limit cycle phases. The broken symmetry in the limit cycle is a direct manifestation of many-body collective effects, which is verified experimentally by tuning atomic densities. The reported result reveals that Rydberg many-body systems are a promising candidate to probe ergodicity breaking dynamics, such as limit cycles, and enable the benchmark of nonequilibrium phase transition.

Transition to anomalous dynamics in a simple random map.

The famous doubling map (or dyadic transformation) is perhaps the simplest deterministic dynamical system exhibiting chaotic dynamics. It is a piecewise linear time-discrete map on the unit interval with a uniform slope larger than one, hence expanding, with a positive Lyapunov exponent and a uniform invariant density. If the slope is less than one, the map becomes contracting, the Lyapunov exponent is negative, and the density trivially collapses onto a fixed point. Sampling from these two different types of maps at each time step by randomly selecting the expanding one with probability p, and the contracting one with probability 1-p, gives a prototype of a random dynamical system. Here, we calculate the invariant density of this simple random map, as well as its position autocorrelation function, analytically and numerically under variation of p. We find that the map exhibits a non-trivial transition from fully chaotic to completely regular dynamics by generating a long-time anomalous dynamics at a critical sampling probability pc, defined by a zero Lyapunov exponent. This anomalous dynamics is characterized by an infinite invariant density, weak ergodicity breaking, and power-law correlation decay.

Nonthermal entanglement dynamics in a dipole-facilitated glassy model with disconnected subspaces

We construct a dipole-facilitated kinetic constraint to partition the Hilbert space into three disconnected subspaces, two of which are nonthermal and the other acts as an intrinsic thermal bath. The resulting glassy system freely oscillates in nonthermal subspaces, making the quantum entanglement perform like a substantial qubit. The spatially spreading entanglement, quantified by concurrence, fidelity, and 2-Rényi entropy, is found to be spontaneously recovered, which is absent in other reference models. Under low-frequency random flip noise, this reversible hydrodynamics of entanglement holds high fidelity and volume law, while at high frequency thermalization unusually occurs leading to a strange phase transition. Our work offers an elaborate space structure for realizing ergodicity breaking and controllable entanglement dynamics. Published by the American Physical Society 2024

Disorder-tunable entanglement at infinite temperature.

Emerging quantum technologies hold the promise of unravelling difficult problems ranging from condensed matter to high-energy physics while, at the same time, motivating the search for unprecedented phenomena in their setting. Here, we use a custom-built superconducting qubit ladder to realize non-thermalizing states with rich entanglement structures in the middle of the energy spectrum. Despite effectively forming an "infinite" temperature ensemble, these states robustly encode quantum information far from equilibrium, as we demonstrate by measuring the fidelity and entanglement entropy in the quench dynamics of the ladder. Our approach harnesses the recently proposed type of non-ergodic behavior known as "rainbow scar," which allows us to obtain analytically exact eigenfunctions whose ergodicity-breaking properties can be conveniently controlled by randomizing the couplings of the model without affecting their energy. The on-demand tunability of quantum correlations via disorder allows for in situ control over ergodicity breaking, and it provides a knob for designing exotic many-body states that defy thermalization.

Landscapes of random diffusivity processes in harmonic potential

A huge number of experimental evidences support that Brownian yet non-Gaussian diffusion prevalently exists in biological and soft matter systems. To interpret the physical mechanism of the Brownian yet non-Gaussian diffusion, the superstatistical approach with random diffusivity and further general random diffusivity processes attract extensive attention. This paper focuses on the anomalous diffusion and nonergodic property of the random diffusivity processes confined in a harmonic potential. Based on the theoretical analyses in the framework of Langevin equation, we find the inherent disagreement between ensemble-averaged mean-squared displacement and time-averaged mean-squared displacement for any form of the random diffusivity D(t), which implies the ergodicity breaking of the confined dynamics. Further, the asymptotic distribution of the time-averaged mean-squared displacement is strictly derived and shows the explicit dependence on the time average of the diffusivity D(t), which means the randomness of the time-averaged mean-squared displacement is completely contributed by the random diffusivity D(t). These results hold true for a general diffusivity D(t), which has been verified through the simulations of three specific kinds of diffusivities D(t).

Viscoelastic active diffusion governed by nonequilibrium fractional Langevin equations: Underdamped dynamics and ergodicity breaking

In this work, we investigate the active dynamics and ergodicity breaking of a nonequilibrium fractional Langevin equation (FLE) with a power-law memory kernel of the form K(t)∼t−(2−2H), where 1/2<H<1 represents the Hurst exponent. The system is subjected to two distinct noises: a thermal noise satisfying the fluctuation–dissipation theorem and an active noise characterized by an active Ornstein–Uhlenbeck process with a propulsion memory time τA. We provide analytic solutions for the underdamped active fractional Langevin equation, performing both analytical and computational investigations of dynamic observables such as velocity autocorrelation, the two-time position correlation, ensemble- and time-averaged mean-squared displacements (MSDs), and ergodicity-breaking parameters. Our results reveal that the interplay between the active noise and long-time viscoelastic memory effect leads to unusual and complex nonequilibrium dynamics in the active FLE systems. Furthermore, the active FLE displays a new type of discrepancy between ensemble- and time-averaged observables. The active component of the system exhibits ultraweak ergodicity breaking where both ensemble- and time-averaged MSDs have the same functional form with unequal amplitudes. However, the combined dynamics of the active and thermal components of the active FLE system are eventually ergodic in the infinite-time limit. Intriguingly, the system has a long-standing ergodicity-breaking state before recovering the ergodicity. This apparent ergodicity-breaking state becomes exceptionally long-lived as H→1, making it difficult to observe ergodicity within practical measurement times. Our findings provide insight into related problems, such as the transport dynamics for self-propelled particles in crowded or polymeric media.

Probing chaos in the spherical p-spin glass model

We study the dynamics of a quantum p-spin glass model starting from initial states defined in microcanonical shells, in a classical regime. We compute different chaos estimators, such as the Lyapunov exponent and the Kolmogorov-Sinai entropy, and find a marked maximum as a function of the energy of the initial state. By studying the relaxation dynamics and the properties of the energy landscape we show that the maximal chaos emerges in correspondence with the fastest spin relaxation and the maximum complexity, thus suggesting a qualitative picture where chaos emerges as the trajectories are scattered over the exponentially many saddles of the underlying landscape. We also observe hints of ergodicity breaking at low energies, indicated by the correlation function and a maximum of the fidelity susceptibility.

Particle dynamics and ergodicity breaking in twisted-bilayer optical lattices

Particle dynamics and ergodicity breaking in twisted-bilayer optical lattices

Screening the Coulomb interaction leads to a prethermal regime in two-dimensional bad conductors

The absence of thermalization in certain isolated many-body systems is of great fundamental interest. Many-body localization (MBL) is a widely studied mechanism for thermalization to fail in strongly disordered quantum systems, but it is still not understood precisely how the range of interactions affects the dynamical behavior and the existence of MBL, especially in dimensions D > 1. By investigating nonequilibrium dynamics in strongly disordered D = 2 electron systems with power-law interactions ∝ 1/rα and poor coupling to a thermal bath, here we observe MBL-like, prethermal dynamics for α = 3. In contrast, for α = 1, the system thermalizes, although the dynamics is glassy. Our results provide important insights for theory, especially since we obtained them on systems that are much closer to the thermodynamic limit than synthetic quantum systems employed in previous studies of MBL. Thus, our work is a key step towards further studies of ergodicity breaking and quantum entanglement in real materials.