We study the time evolution of a half-filled Hubbard layer coupled to a magnon bath after a quench of the Hubbard interaction. Qualitatively different regimes, regarding the asymptotic long-time dynamics, are identified and characterized within the mean-field approximation. In the absence of the bath, the dynamics of the closed system is similar to that of a quenched BCS condensate. Though the presence of the bath introduces an additional relaxation mechanism, our numerical results and analytical arguments show that equilibration with the bath is not necessarily attained within the approximations used. Instead, nonequilibrium states, similar to the ones observed in the closed system, can emerge at long times as a consequence of the competition between the intrinsic relaxation mechanism (Landau damping, for example), and the bath-induced dissipation.
The dynamics of interacting bosons in one dimension following the sudden switching on of a weak disordered potential is investigated. On time scales before quasiparticles scatter (prethermalized regime), the dephasing from random elastic forward scattering causes all correlations to decay exponentially fast, but the system remains far from thermal equilibrium. For longer times, the combined effect of disorder and interactions gives rise to inelastic scattering and to thermalization. A novel quantum kinetic equation accounting for both disorder and interactions is employed to study the dynamics…
We study the nonequilibrium dynamics of a one-dimensional topological Kondo insulator, modelled by a $p$-wave Anderson lattice model, following a quantum quench of the on-site interaction strength. Our goal is to examine how the quench influences the topological properties of the system, therefore our main focus is the time evolution of the string order parameter, entanglement spectrum and the topologically-protected edge states. We point out that postquench local observables can be well captured by a thermal ensemble up to a certain interaction strength. Our results demonstrate that the topol…
We study the relaxation of the Holstein model after a sudden switch-on of the interaction by means of the nonequilibrium dynamical mean field theory, with the self-consistent Migdal approximation as an impurity solver. We show that there exists a qualitative change in the thermalization dynamics as the interaction is varied in the weak-coupling regime. On the weaker interaction side of this crossover, the phonon oscillations are damped more rapidly than the electron thermalization timescale, as determined from the relaxation of the electron momentum distribution function. On the stronger inter…
Using numerically exact methods we examine the Fermi-Hubbard model on arbitrary cluster topology. We focus on the question which systems eventually equilibrate or even thermalize after an interaction quench when initially prepared in a state highly entangled between system and bath. We find that constants of motion in integrable clusters prevent equilibration to the thermal state. We discuss the size of fluctuations during equilibration and thermalization and the influence of integrability. The influence of real space topology and in particular of infinite-range graphs on equilibration and the…
We study the expectation values of observables and correlation functions at long times after a global quantum quench. Our focus is on metallic (`gapless') fermionic many-body models and small quenches. The system is prepared in an eigenstate of an initial Hamiltonian, and the time evolution is performed with a final Hamiltonian which differs from the initial one in the value of one global parameter. We first derive general relations between time-averaged expectation values of observables as well as correlation functions and those obtained in an eigenstate of the final Hamiltonian. Our results …
We study the thermalization, after sudden and slow quenches, of an interacting model having a quantum phase transition from a Sachdev-Ye-Kitaev (SYK) non-Fermi liquid (NFL) to a Fermi liquid (FL). The model has SYK fermions coupled to non-interacting lead fermions and can be realized in a graphene flake connected to external leads. After a sudden quench to the NFL, a thermal state is reached rapidly via collapse-revival oscillations of the quasiparticle residue of the lead fermions. In contrast, the quench to the FL, across the NFL-FL transition, leads to multiple prethermal regimes and much s…
In two spatial dimensions, topological order is robust for static deformations at zero temperature, while it is fragile at any finite temperature. How robust is topological order after a quantum quench? In this paper we show that topological order thermalizes under the unitary evolution after a quantum quench. If the quench preserves gauge symmetry, there is a residual topological entropy exactly like in the finite temperature case. We obtain this result by studying the time evolution of the topological 2-Rényi entropy in a fully analytical, exact way. These techniques can be then applied to s…
We study the dynamics and thermalization of strongly correlated fermions in finite one-dimensional lattices after a quantum quench. Our calculations are performed using exact diagonalization. We focus on one- and two-body observables such as the momentum distribution function [n(k)] and the density-density structure factor [N(k)], respectively, and study the effects of approaching an integrable point. We show that while the relaxation dynamics and thermalization of N(k) for fermions is very similar to the one of hardcore bosons, the behavior of n(k) is distinctively different. The latter obser…
We investigate the mechanisms of control and suppression of pre-thermalization in $N$-component alkaline earth gases. To this end, we compute the short-time dynamics of the instantaneous momentum distribution and the relative population for different initial conditions after an interaction quench, accounting for the 11 peffect of initial interactions. We find that switching on an interaction that breaks the SU$(N)$ symmetry of the initial Hamiltonian, thus allowing for the occurrence of spin-changing collisions, does not necessarily lead to a suppression of pre-thermalization. However, the sup…
Although most studies of strongly correlated systems away from equilibrium have focused on clean systems, it is well known that disorder may significantly modify observed properties in various nontrivial ways. The nonequilibrium interplay of interaction and disorder in these systems thus requires further investigation. In the present paper, we use the recently developed nonequilibrium DMFT+CPA embedding scheme, that combines both the dynamical mean field theory (DMFT) and the coherent potential approximation (CPA) nonequilibrium extensions, to characterize the relaxation and the thermalization…
In his Comment [see preceding Comment, Phys. Rev. A 82, 037601 (2010)] on the paper by Roux [Phys. Rev. A 79, 021608(R) (2009)], Rigol argued that the energy distribution after a quench is not related to standard statistical ensembles and cannot explain thermalization. The latter is proposed to stem from what he calls the eigenstate thermalization hypothesis and which boils down to the fact that simple observables are expected to be smooth functions of the energy. In this Reply, we show that there is no contradiction or confusion between the observations and discussions of Roux and the expecte…
We study the non-equilibrium dynamics of a Luttinger liquid after a simultaneous quantum quench of the interaction and dissipative quench to the environment within the realm of the Lindblad equation. When the couplings to environment satisfy detailed balance, the system is destined to thermalize, which we follow using bosonization. The thermodynamic entropy of the system, which also encodes information about the entanglement with the environment, either exhibits a maximum and minimum or grows monotonically before reaching its thermal value in a universal fashion. The single-particle density ma…
We study thermalization in the holographic (1+1)-dimensional CFT after simultaneous generation of two high-energy excitations in the antipodal points on the circle. The holographic picture of such quantum quench is the creation of BTZ black hole from a collision of two massless particles. We perform holographic computation of entanglement entropy and mutual information in the boundary theory and analyze their evolution with time. We show that equilibration of the entanglement in the regions which contained one of the initial excitations is generally similar to that in other holographic quench …
We investigate quenches of holographic theories in a confined phase, where the energy injected is insufficient to reach the deconfined phase. In such quenches, thermalization is not associated with gravitational collapse and the formation of a black hole. Nevertheless, we attempt to characterize the late-time state of this scenario. We check a number of notions of thermalization that do not require horizon formation, and find no evidence for thermalization in our chosen parameters and initial states, even in the weakest sense. We find that the post-quench behaviour of both local and nonlocal o…
Synthetic horizons in models for quantum matter provide an alternative route to explore fundamental questions of modern gravitational theory. Here, we apply these concepts to the problem of emergence of thermal quantum states in the presence of a horizon, by studying ground-state thermalization due to instantaneous horizon creation in a gravitational setting and its condensed matter analogue. By a sudden quench to position-dependent hopping amplitudes in a one-dimensional lattice model, we establish the emergence of a thermal state accompanying the formation of a synthetic horizon. The resulti…
We describe the non-equilibrium dynamics of the Sachdev-Ye-Kitaev models of fermions with all-to-all interactions. These provide tractable models of the dynamics of quantum systems without quasiparticle excitations. The Kadanoff-Baym equations show that the final state is thermal, and their numerical analysis appears consistent with a thermalization rate proportional to the absolute temperature of the final state. We also obtain an exact analytic solution of the non-equilibrium dynamics in the large $q$ limit of a model with $q$ fermion interactions: in this limit, the thermalization of the fe…
We use time-dependent non-equilibrium dynamical mean-field theory with weak-coupling auxiliary-field continuous time quantum Monte Carlo as an impurity solver to study the thermalization behavior of the mass-imbalanced single-band Hubbard model after a quench of the Coulomb interaction from the non-interacting limit to a finite positive value. By contrast, when the Coulomb interaction in our model is increased under equilibrium conditions, the quasi-particle weight for spin-up and spin-down (the mass imbalance) electrons approach zero simultaneously, indicating the absence of an orbital select…
We derive in detail several universal features in the time evolution of entanglement entropy and other nonlocal observables in quenched holographic systems. The quenches are such that a spatially uniform density of energy is injected at an instant in time, exciting a strongly coupled CFT which eventually equilibrates. Such quench processes are described on the gravity side by the gravitational collapse of a thin shell that results in a black hole. Various nonlocal observables have a unified description in terms of the area of extremal surfaces of different dimensions. In the large distance lim…
Motivated by recent experiments in ultracold atomic gases that explore the nonequilibrium dynamics of interacting quantum many-body systems, we investigate the opposite limit of Landau's Fermi liquid paradigm: We study a Hubbard model with a sudden interaction quench, that is the interaction is switched on at time t=0. Using the flow equation method, we are able to study the real time dynamics for weak interaction U in a systematic expansion and find three clearly separated time regimes: i) An initial buildup of correlations where the quasiparticles are formed. ii) An intermediate quasi-steady…
We study the role of dipolar interactions in the standard protocol used to achieve dynamic nuclear polarization (DNP). In the so-called spin-temperature regime, where the interactions establish an effective thermodynamic behavior in the out-of-equilibrium stationary state, we provide numerical predictions for the level of hyperpolarization. We show that nuclear spins equilibrate to the effective spin-temperature established among the electron spins of radicals, as expected from the quantum theory of thermalization. Moreover, we present an analytical technique to estimate the spin temperature, …
In weakly perturbed systems that are close to integrability, thermalization can be delayed by the formation of prethermalization plateaus. We study the build-up of density-density correlations after a weak interaction quench in the Hubbard model in $d > 1$ dimensions using unitary perturbation theory. Starting from a pre-quench state at temperature $T$, we show that the prethermalization values of the post-quench correlations are equal to the equilibrium values of the interacting model at the same temperature $T$. This is explained by the local character of density-density correlations.
Thermalization play a central role in out-of-equilibrium physics of ultracold atoms or electronic transport phenomena. On the other hand, entanglement concepts have proven to be extremely useful to investigate quantum phases of matter. Here, it is argued that **bipartite** entanglement measures provide key information on out-of-equilibrium states and might therefore offer stringent thermalization criteria. This is illustrated by considering a global quench in an (extended) XXZ spin-1/2 chain across its (zero-temperature) quantum critical point. A non-local **bipartition** of the chain **preser…
After a sudden disruption, weakly interacting quantum systems first relax to a prethermalized state that can be described by perturbation theory and a generalized Gibbs ensemble. Using these properties of the prethermalized state we perturbatively derive a kinetic equation which becomes a quantum Boltzmann equation in the scaling limit of vanishing interaction. Applying this to interaction quenches in the fermionic Hubbard model we find that the momentum distribution relaxes to the thermal prediction of statistical mechanics. For not too large interaction, this two-stage scenario provides a qu…
Via the hierarchy of correlations, we study the strongly interacting Fermi-Hubbard model in the Mott insulator state and couple it to a Markovian environment which constantly monitors the particle numbers $\hat n_μ^\uparrow$ and $\hat n_μ^\downarrow$ for each lattice site $μ$. As expected, the environment induces an imaginary part $γ$ (i.e., decay rate) of the quasi-particle frequencies $ω_{\mathbf{k}}\toω_{\mathbf{k}}-iγ$ and tends to diminish the correlations between lattice sites. Surprisingly, the environment does also steer the state of the system on intermediate time scales $\mathcal{O}(…
We employ the Quench Action Method (QAM) for a recently considered geometrical quantum quench: two free fermionic chains initially at different temperatures are joined together in the middle and let evolve unitarily with a translation invariant Hamiltonian. We show that two different stationary regimes are reached at long times, depending on the interplay between the observation time scale T and the total length L of the system. We show the emergence of a non-equilibrium steady state (NESS) supporting an energy current for observation time T much smaller than the system size L. We then identif…
We review exact approaches and recent results related to the relaxation dynamics and description after relaxation of various one-dimensional lattice systems of hard-core bosons after a sudden quench. We first analyze the integrable case, where the combination of analytical insights and computational techniques enable one to study large system sizes. Thermalization does not occur in this regime. However, after relaxation, observables can be described by a generalization of the Gibbs ensemble. We then utilize full exact diagonalization to study what happens as integrability is broken. We show th…
We consider an open system near a quantum critical point that is suddenly moved towards the critical point. The bath-dominated diffusive non-equilibrium dynamics after the quench is shown to follow scaling behavior, governed by a critical exponent that emerges in addition to the known equilibrium critical exponents. We determine this exponent and show that it describes universal prethermalized coarsening dynamics of the order parameter in an intermediate time regime. Implications of this quantum critical prethermalization are a powerlaw rise of order and correlations after an initial collapse …
We show that the non-equilibrium time-evolution after interaction quenches in the one dimensional, integrable Hubbard model exhibits a dynamical transition in the half-filled case. This transition ceases to exist upon doping. Our study is based on systematically extended equations of motion. Thus it is controlled for small and moderate times; no relaxation effects are neglected. Remarkable similarities to the quench dynamics in the infinite dimensional Hubbard model are found suggesting dynamical transitions to be a general feature of quenches in such models.
We study thermalization in a one-dimensional quantum system consisting of a noninteracting fermionic chain with each site of the chain coupled to an additional bath site. Using a density matrix renormalization group algorithm we investigate the time evolution of observables in the chain after a quantum quench. For low densities we show that the intermediate time dynamics can be quantitatively described by a system of coupled equations of motion. For higher densities our numerical results show a prethermalization for local observables at intermediate times and a full thermalization to the grand…
We study the coherent non-equilibrium dynamics of interacting two-dimensional systems after a quench from a trivial to a topological Chern insulator phase. While the many-body wavefunction is constrained to remain topologically trivial under local unitary evolution, we find that the Hall response of the system can dynamically approach a thermal value of the post-quench Hamiltonian, even though the efficiency of this thermalization process is shown to strongly depend on the microscopic form of the interactions. Quite remarkably, the effective temperature of the steady state Hall response can be…
By combining the Baeriswyl wavefunction with equilibrium and time-dependent variational principles, we develop a non-equilibrium formalism to study quantum quenches for two dimensional spinless fermions with nearest-neighbour hopping and repulsion. The variational ground state energy and the short time dynamics agree convincingly with the results of numerically exact simulations. We find that depending on the initial and final interaction strength, the quenched system either exhibits undamped oscillations or relaxes to a time independent steady state. The time averaged expectation value of the…
We study the dynamics of thermalization following a quantum quench using tensor-network methods. Contrary to the common belief that the rapid growth of entanglement and the resulting exponential growth of the bond dimension restricts simulations to short times, we demonstrate that the long time limit of local observables can be well captured using the time-dependent variational principle. This allows to extract transport coefficients such as the energy diffusion constant from simulations with rather small bond dimensions. We further study the characteristic of the chaotic wave that precedes th…
Entanglement is usually quantified by von Neumann entropy, but its properties are much more complex than what can be expressed with a single number. We show that the three distinct dynamical phases known as thermalization, Anderson localization, and many-body localization are marked by different patterns of the spectrum of the reduced density matrix for a state evolved after a quantum quench. While the entanglement spectrum displays Poisson statistics for the case of Anderson localization, it displays universal Wigner-Dyson statistics for both the cases of many-body localization and thermaliza…
We discuss recent results in the study of the evolution of strongly coupled field theories in the presence of time dependent couplings using the holographic correspondence. The aim is to understand (i) thermalization and (ii) universal behavior when the coupling crosses a critical point. Our emphasis is on situations where a subset of bulk fields can be treated in a probe approximation. We consider two different setups. In the first, defect conformal field theories are described by probe branes in AdS space-times, and an initial vacuum state evolves due to a time dependent coupling in the prob…
We present holographic descriptions of thermalization in conformal field theories using probe D-branes in AdS X S space-times. We find that the induced metrics on Dp-brane worldvolumes which are rotating in an internal sphere direction have horizons with characteristic Hawking temperatures even if there is no black hole in the bulk AdS. The AdS/CFT correspondence applied to such systems indeed reveals thermal properties such as Brownian motions and AC conductivities in the dual conformal field theories. We also use this framework to holographically analyze time-dependent systems undergoing a q…
In this work, we study the time evolution of the entanglement hamiltonian during the process of thermalization in a (1+1)-dimensional conformal field theory (CFT) after a quantum quench from a special class of initial states. In particular, we focus on a subsystem which is a finite interval at the end of a semi-infinite line. Based on conformal mappings, the exact forms of both entanglement hamiltonian and entanglement spectrum of the subsystem can be obtained. Aside from various interesting features, it is found that in the infinite time limit the entanglement hamiltonian and entanglement spe…
The Lieb-Robinson bound implies that the unitary time evolution of an operator can be restricted to an effective light cone for any Hamiltonian with short-range interactions. Here we present a very efficient renormalization group algorithm based on this light cone structure to study the time evolution of prepared initial states in the thermodynamic limit in one-dimensional quantum systems. The algorithm does not require translational invariance and allows for an easy implementation of local conservation laws. We use the algorithm to investigate the relaxation dynamics of double occupancies in …
We consider the time evolution of entanglement entropy after a global quench in a strongly coupled holographic system, whose subsequent equilibration is described in the gravity dual by the gravitational collapse of a thin shell of matter resulting in a black hole. In the limit of large regions of entanglement, the evolution of entanglement entropy is controlled by the geometry around and inside the event horizon of the black hole, resulting in regimes of pre-local- equilibration quadratic growth (in time), post-local-equilibration linear growth, a late-time regime in which the evolution does …
