Continuous symmetries lead to universal slow relaxation of correlation functions in quantum many-body systems. In this work, we study how local symmetry-breaking impurities affect the dynamics of these correlation functions using Brownian quantum circuits, which we expect to apply to generic nonintegrable systems with the same symmetries. While explicitly breaking the symmetry is generally expected to lead to eventual restoration of full ergodicity, we find that approximately conserved quantities that survive under such circumstances can still induce slow relaxation. This can be understood using a super-Hamiltonian formulation, where low-lying excitations determine the late-time dynamics and exact ground states correspond to conserved quantities. We show that in one dimension, symmetry-breaking impurities modify diffusive and subdiffusive behaviors associated with U ( 1 ) and dipole conservation at late times, e.g., by increasing power-law decay exponents of the decay of autocorrelation functions. This stems from the fact that for these symmetries, impurities are relevant in the renormalization-group sense, e.g., bulk impurities effectively disconnect the system, completely modifying both temporal and spatial correlations. On the other hand, for an impurity that disrupts strong Hilbert space fragmentation, the super-Hamiltonian only acquires an exponentially small gap, leading to prethermal plateaus in autocorrelation functions which extend for times that scale exponentially with the distance to the impurity. Overall, our approach systematically characterizes how symmetry-breaking impurities affect relaxation dynamics in symmetric systems.
Continuous symmetries lead to universal slow relaxation of correlation functions in quantum many-body systems. In this work, we study how local symmetry-breaking impurities affect the dynamics of these correlation functions using Brownian quantum circuits, which we expect to apply to generic nonintegrable systems with the same symmetries. While explicitly breaking the symmetry is generally expected to lead to eventual restoration of full ergodicity, we find that approximately conserved quantities that survive under such circumstances can still induce slow relaxation. This can be understood using a super-Hamiltonian formulation, where low-lying excitations determine the late-time dynamics and exact ground states correspond to conserved quantities. We show that in one dimension, symmetry-breaking impurities modify diffusive and subdiffusive behaviors associated with U ( 1 ) and dipole conservation at late times, e.g., by increasing power-law decay exponents of the decay of autocorrelation functions. This stems from the fact that for these symmetries, impurities are relevant in the renormalization-group sense, e.g., bulk impurities effectively disconnect the system, completely modifying both temporal and spatial correlations. On the other hand, for an impurity that disrupts strong Hilbert space fragmentation, the super-Hamiltonian only acquires an exponentially small gap, leading to prethermal plateaus in autocorrelation functions which extend for times that scale exponentially with the distance to the impurity. Overall, our approach systematically characterizes how symmetry-breaking impurities affect relaxation dynamics in symmetric systems.
Symmetry algebras of quantum many-body systems with locality can be understood using commutant algebras, which are defined as algebras of operators that commute with a given set of local operators. In this work, we show that these symmetry algebras can be expressed as frustration-free ground states of a local superoperator, which we refer to as a “super-Hamiltonian.” We demonstrate this for conventional symmetries such as Z 2 , U ( 1 ) , and S U ( 2 ) , where the symmetry algebras map to various kinds of ferromagnetic ground states, as well as for unconventional ones that lead to weak ergodicity-breaking phenomena of Hilbert-space fragmentation (HSF) and quantum many-body scars. In addition, we show that the low-energy excitations of this super-Hamiltonian can be understood as approximate symmetries, which in turn are related to slowly relaxing hydrodynamic modes in symmetric systems. This connection is made precise by relating the super-Hamiltonian to the superoperator that governs the operator relaxation in noisy symmetric Brownian circuits and this physical interpretation also provides a novel interpretation for Mazur bounds for autocorrelation functions. We find examples of gapped (gapless) super-Hamiltonians indicating the absence (presence) of slow modes, which happens in the presence of discrete (continuous) symmetries. In the gapless cases, we recover hydrodynamic modes such as diffusion, tracer diffusion, and asymptotic scars in the presence of U ( 1 ) symmetry, HSF, and a tower of quantum scars, respectively. In all, this demonstrates the power of the commutant-algebra framework in obtaining a comprehensive understanding of exact symmetries and associated approximate symmetries and hydrodynamic modes, and their dynamical consequences in systems with locality. Published by the American Physical Society 2024
The finite parts of a large, locally interacting many-body system prepared out of equilibrium eventually equilibrate. Characterizing the underlying mechanisms of this process and its timescales, however, is particularly hard as it requires one to decouple universal features from observable-specific ones. Recently, additional insight was gained by studying how certain symmetries of the dynamics that are broken by the initial state are restored at the level of the reduced state of a given subsystem. This provides a high-level, observable-independent probe. Until now, this idea has been applied to the restoration of internal symmetries, e.g., U(1) symmetries related to charge conservation. Here we show that the same logic can be applied to the restoration of space-time symmetries, and hence can be used to characterize the relaxation of fully generic systems. We illustrate this idea by considering the paradigmatic example of “generic” many-body dynamics, i.e., a local random unitary circuit, where our method leads to exact results. We show that the restoration of translation symmetry in these systems only happens on timescales that are proportional to the subsystem's volume. In fact, for large enough subsystems, the time of symmetry restoration becomes initial-state independent (as long as the latter breaks the symmetry at time zero) and coincides with the thermalization time. For intermediate subsystems, however, one can observe the “quantum Mpemba effect,” where the state of the system restores a symmetry faster if it is initially more asymmetric. We provide an exact characterization of this effect in a nonintegrable system.
Entanglement asymmetry provides a quantitative measure of symmetry breaking in many-body quantum states. Focusing on inhomogeneous U(1) charges, such as dipole and multipole moments, we show that the typical asymmetry is bounded by a universal fraction of its maximal value. Multipole charges naturally arise in systems with Hilbert-space fragmentation, where the dynamics splits into exponentially many disconnected sectors. Using the commutant algebra formalism, we generalize entanglement asymmetry to account for fragmentation. While the asymmetry grows logarithmically for conventional symmetries, it can scale extensively in fragmented systems and distinguish classical from quantum fragmentation. We derive general upper bounds for the asymmetry and identify states that saturate them. To study the typical behavior of the asymmetry, we consider the ensemble of random matrix product states. By identifying the bond dimension with an effective time parameter, we qualitatively reproduce recent results on asymmetry dynamics in random quantum circuits, suggesting a universal behavior for the asymmetry of U(1) charges in local ergodic systems.
Conservation laws in a quantum many-body system play a direct role in its dynamic behavior. Understanding the effect of weakly breaking a conservation law due to coherent and incoherent errors is thus crucial, e.g., in the realization of reliable quantum simulators. In this work, we perform exact numerics and time-dependent perturbation theory to study the dynamics of \textit{symmetry violation} in quantum many-body systems with slight coherent (at strength λ) or incoherent (at strength γ) breaking of their local and global symmetries. We rigorously prove the symmetry violation to be a divergence measure in Hilbert space. Based on this, we show that symmetry breaking generically leads to a crossover in the divergence growth from diffusive behavior at onset times to ballistic or hyperballistic scaling at intermediate times, before diffusion dominates at long times. More precisely, we show that for local errors the leading coherent contribution to the symmetry violation cannot be of order lower than ∝λt2 while its leading-order incoherent counterpart is typically of order ∝γt. This remarkable interplay between unitary and incoherent gauge-breaking scalings is also observed at higher orders in projectors onto symmetry (super)sectors. Due to its occurrence at short times, the diffusive-to-ballistic crossover is expected to be readily accessible in modern ultracold-atom and NISQ-device experiments.
We study operator dynamics in many-body quantum systems, focusing on generic features of systems that are ergodic, spatially extended, and lack conserved densities. Quantum circuits of various types provide simple models for such systems. We focus on Floquet quantum circuits, comparing their behavior with what has been found previously for circuits that are random in time. Floquet circuits, which have discrete time-translation symmetry, represent an intermediate case between circuits that are random in time and lack any symmetry, and systems with a time-independent Hamiltonian and continuous time-translation invariance. By making this comparison, one of our aims is to identify signatures of time-translation symmetry in Floquet operator dynamics. To characterize behavior we examine a variety of quantities in solvable models and numerically: operator autocorrelation functions; the partial spectral form factor; the out-of-time-order correlator (OTOC); and the paths in operator space that make the dominant contributions to the ensemble-averaged autocorrelation functions. Our most striking result is that ensemble-averaged autocorrelation functions show behavior that is distinctively different in Floquet systems compared to systems in which successive time-steps are independent. Specifically, while average autocorrelation functions decay on a microscopic timescale for circuits that are random in time, in Floquet systems they have a late-time tail with a duration that grows parametrically with the size of the operator support. In the simplest models this tail is separated from the initial decay by a minimum, so that the average autocorrelation function has an intermediate-time peak. The existence of these tails provides a way to understand deviations of the spectral form factor from random matrix behavior at times shorter than the Thouless time. In contrast to this feature in autocorrelation functions, we find no new aspects to the behavior of OTOCs for Floquet models compared to random-in-time circuits. We show that this difference between averaged autocorrelation functions and OTOCs can be understood in terms of the paths in operator space that contribute to the two quantities: paths for the former retain a limited support at late times, while paths for the latter are dominated by operator spreading.
We study unitary quantum dynamics in noisy Brownian models with global continuous symmetries, such as U(1) and SU(2), focusing on Rényi entanglement entropies and hydrodynamic and non-hydrodynamic correlators. By mapping the averaged late-time dynamics to the low-energy physics of effective replica Hamiltonians, we find that the evolution is controlled by the quantum geometry of their ground-state manifolds, which is directly related to the geometry of k-commutants---the symmetry algebra of k replicas of the system. In interacting systems, these k-commutants are generically determined solely by the symmetries of the system, independent of microscopic details of the noisy evolution. This allows us to use the time-dependent variational principle (TDVP) to provide simple geometric explanations for the sub-ballistic Rényi entanglement growth and the anomalous decay of non-hydrodynamic correlators in interacting systems with continuous symmetries. We find this behavior to be intimately connected to singularities within the k-commutant manifolds, arising from frozen ``void'' states in the Hilbert space that exist due to continuous on-site symmetries. This also demystifies the important role of voids in the dynamics of these observables, previously identified in U(1) symmetric systems. We compare these behaviors in interacting systems with Abelian and non-Abelian continuous symmetries and in free-fermion systems, which differ in the geometry of their k-commutants. Ultimately, this work provides a general geometric framework for systematically studying observables in noisy systems with continuous symmetries, including Haar-random circuits.