The study of symmetry restoration has recently emerged as a fruitful means to extract high-level information on the relaxation of quantum many-body systems. However, while the restoration of internal symmetries has been investigated intensively, that of spatial symmetries has hitherto only been considered in the context of random unitary circuits. Here, we present a complementary study of translation symmetry restoration in integrable systems. In particular, we consider a one-dimensional chain of spinless, noninteracting fermions quenched from a ν > 1 shift invariant state, and follow the local restoration of one-site shift invariance using the Frobenius distance Δ F A between the state on a subsystem and its symmetrized counterpart. Distinct from the case of random unitary circuits, where symmetry restoration occurs abruptly for times proportional to the subsystem size, we find that symmetry here is restored smoothly and over timescales of the order of the subsystem size squared. We also find that the so-called ‘‘quantum Mpemba effect’’ is readily observed. Most importantly, we show that—in contrast to the case of continuous internal symmetries—this discrete symmetry restoration is not qualitatively described by a quasiparticle picture for Δ F A , and therefore goes beyond the hydrodynamic description. Our results can be directly extended to higher dimensions.
The study of symmetry restoration has recently emerged as a fruitful means to extract high-level information on the relaxation of quantum many-body systems. However, while the restoration of internal symmetries has been investigated intensively, that of spatial symmetries has hitherto only been considered in the context of random unitary circuits. Here, we present a complementary study of translation symmetry restoration in integrable systems. In particular, we consider a one-dimensional chain of spinless, noninteracting fermions quenched from a ν > 1 shift invariant state, and follow the local restoration of one-site shift invariance using the Frobenius distance Δ F A between the state on a subsystem and its symmetrized counterpart. Distinct from the case of random unitary circuits, where symmetry restoration occurs abruptly for times proportional to the subsystem size, we find that symmetry here is restored smoothly and over timescales of the order of the subsystem size squared. We also find that the so-called ‘‘quantum Mpemba effect’’ is readily observed. Most importantly, we show that—in contrast to the case of continuous internal symmetries—this discrete symmetry restoration is not qualitatively described by a quasiparticle picture for Δ F A , and therefore goes beyond the hydrodynamic description. Our results can be directly extended to higher dimensions.
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.
In this thesis, we first present a review on the background and research related to our published works. We then present our first work, which investigates entanglement entropy free-fermionic systems of D-spatial dimensions using an effective quasiparticle description. In particular, we consider a quantum quench to the D-dimensional tight-binding model from an initial state invariant under translation by nu sites. We develop an efficient Monte Carlo scheme to implement the quasiparticle solution to arbitrary connected subsystems, and show that, in two dimensions, the shape and orientation of the subsystem with respect to the underlying lattice is able to effect qualitative behaviour of the entanglement growth. In our second work, we investigate the breaking and restoration of translation symmetry in the same quench setup restricted to the one-dimensional setting. We establish the presence of the so-called quantum Mpemba effect in this setting, which had hitherto been almost exclusively considered with respect to continuous U(1) symmetry breaking. We also apply a quasiparticle picture to our chosen measure of symmetry breaking and find that this picture does not agree with the solution obtained by exact diagonalisation. From this we conclude that the restoration of discrete symmetries in integrable systems goes beyond the leading hydrodynamic order. Finally, in this thesis we also present a brief review related to our most recent work, which considers the recently proposed hierarchical dual-unitary circuit in two dimensions. We then present the current iteration of this work, which defines the set of DU2 conditions on the two-dimensional gates that may be used to fully simplify two-point spatio-temporal correlation functions in terms of the repeat action of quantum channels. Furthermore, we identify directions of these correlations along the edge of the causal lightcone with no 1D counterpart that are generically non-vanishing with respect to these DU2 conditions. This is demonstrated for a particular family of gate with site dimension d=6.
Understanding relaxation in isolated quantum many-body systems remains a central challenge. Recently, the quantum Mpemba effect (QME), a counterintuitive relaxation phenomenon, has attracted considerable attention and has been extensively studied in systems with global symmetries. Here, we study the QME in gauge theories with massive local gauge symmetries. In the lattice Schwinger model, we demonstrate that the gauge structure of the reduced density matrix of a subsystem is entirely determined by the initial state and remain unchanged during the time evolution. We then investigate whether gauge symmetry can be dynamically restored following a symmetric quench. Analytical and numerical results show that when the Maxwell term is zero, gauge symmetry restoration fails due to the emergence of a peculiar conservation law. However, for any finite Maxwell term, subsystem gauge symmetry is restored in the thermodynamic limit. Based on these results, we systematically construct a families of initial states exhibiting the QME. We further explore the QME in the quantum link model-a truncated lattice Schwinger model, which has been realized in experiments. Moreover, we propose an experimentally accessible order parameter that correctly captures the QME. Our work demonstrates the generality of the quantum Mpemba effect even in the local gauge symmetries, and are directly relevant to ongoing quantum simulation experiments of gauge theories.
Entanglement asymmetry, which serves as a diagnostic tool for symmetry breaking and a proxy for thermalization, has recently been proposed and studied in the context of symmetry restoration for quantum many-body systems undergoing a quench. In this Letter, we investigate symmetry restoration in various symmetric random quantum circuits, particularly focusing on the U(1) symmetry case. In contrast to nonsymmetric random circuits where the U(1) symmetry of a small subsystem can always be restored at late times, we reveal that symmetry restoration can fail in U(1)-symmetric circuits for certain weak symmetry-broken initial states in finite-size systems. In the early-time dynamics, we observe an intriguing quantum Mpemba effect implying that symmetry is restored faster when the initial state is more asymmetric. Furthermore, we also investigate the entanglement asymmetry dynamics for SU(2) and Z_{2} symmetric circuits and identify the presence and absence of the quantum Mpemba effect for the corresponding symmetries, respectively. A unified understanding of these results is provided through the lens of quantum thermalization with conserved charges.
The Mpemba effect, where a state prepared farther from equilibrium relaxes faster to equilibrium than one prepared closer, has a quantum counterpart where relaxation is resolved by conserved charge. However, the fate of the quantum Mpemba effect in systems with long-range interactions remains an open question. Here, we study the quantum Mpemba effect in long-ranged, U(1)-symmetric random unitary circuits. Using annealed Rényi-2 entanglement asymmetry computed via replica tensor networks and exact diagonalization, we track the symmetry restoration from three types of tilted product states: ferromagnetic, antiferromagnetic, and ferromagnetic with a central domain wall. The quantum Mpemba effect is present for tilted ferromagnetic states at all interaction ranges, but absent for tilted antiferromagnetic states, and occurs for the domain-wall state only in effectively short-ranged circuits, where the Mpemba time tM is found to scale with the subsystem size NA as tM∼NAz, with the dynamical exponent z=min(α−1,2). These results reveal how the quantum Mpemba effect is governed by the interplay between interaction range and initial-state charge bias in long-ranged chaotic systems.
The quantum Mpemba effect is the counterintuitive nonequilibrium phenomenon wherein the dynamic restoration of a broken symmetry occurs more rapidly when the initial state exhibits a higher degree of symmetry breaking. The effect has been recently discovered theoretically and observed experimentally in the framework of global quantum quenches, but so far it has only been investigated in one-dimensional systems. Here we focus on a two-dimensional free-fermion lattice employing the entanglement asymmetry as a measure of symmetry breaking. Our investigation begins with the ground-state analysis of a system featuring nearest-neighbor hoppings and superconducting pairings, the latter breaking explicitly the U(1) particle-number symmetry. We compute analytically the entanglement asymmetry of a periodic strip using dimensional reduction, an approach that allows us to adjust the extent of the transverse size, achieving a smooth crossover between one and two dimensions. Further applying the same method, we study the time evolution of the entanglement asymmetry after a quench to a Hamiltonian with only nearest-neighbor hoppings, preserving the particle-number symmetry which is restored in the stationary state. We find that the quantum Mpemba effect is strongly affected by the size of the system in the transverse dimension, with the potential to either enhance or spoil the phenomenon depending on the initial states. We establish the conditions for its occurrence based on the properties of the initial configurations, extending the criteria found in the one-dimensional case.