Relaxation Dynamics of Disordered Spin Chains: Localization and the Existence of a Stationary State (original) (raw)
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Relaxation and thermalization after a quantum quench: Why localization is important
Physical Review B, 2013
We study the unitary dynamics and the thermalization properties of free-fermion-like Hamiltonians after a sudden quantum quench, extending the results of S. Ziraldo et al. [Phys. Rev. Lett. 109, 247205 (2012)]. With analytical and numerical arguments, we show that the existence of a stationary state and its description with a generalized Gibbs ensemble (GGE) depend crucially on the observable considered (local versus extensive) and on the localization properties of the final Hamiltonian. We present results on two one-dimensional (1D) models, the disordered 1D fermionic chain with long-range hopping and the disordered Ising/XY spin chain. We analytically prove that, while time averages of one-body operators are perfectly reproduced by GGE (even for finite-size systems, if time integrals are extended beyond revivals), time averages of many-body operators might show clear deviations from the GGE prediction when disorder-induced localization of the eigenstates is at play.
Effective Thermal Dynamics Following a Quantum Quench in a Spin Chain
Physical Review Letters, 2009
We study the nonequilibrium dynamics of the Quantum Ising Model following an abrupt quench of the transverse field. We focus on the onsite autocorrelation function of the order parameter, and extract the phase coherence time τ ϕ Q from its asymptotic behavior. We show that the initial state determines τ ϕ Q only through an effective temperature set by its energy and the final Hamiltonian. Moreover, we observe that the dependence of τ ϕ Q on the effective temperature fairly agrees with that obtained in thermal equilibrium as a function of the equilibrium temperature.
Relaxation, prethermalization, and diffusion in a noisy quantum Ising chain
We study the dynamics of thermalization resulting from a time-dependent noise in a Quantum Ising Chain subject to a sudden quench of the transverse magnetic field. For weak noise the dynamics shows a pre-thermalized state at intermediate time scales, eventually drifting towards an asymptotic infinite temperature steady state characterized by diffusive behavior. By computing analytically the density of kinks, as well as the transverse and longitudinal magnetic field correlators, we characterize these two regimes, their observability and their signatures in the various physical quantities. 98.80.Jk, 98.80.Cq, 98.80.Es The dynamics of relaxation towards thermal equilibrium has been one of the recurrent themes of theoretical physics in the past decades . The problem is of crucial importance in many contexts, ranging from condensed matter physics to cosmology: if we think of injecting suddenly, e.g. by an abrupt change of one of its parameters (a quantum quench), a finite amount of energy in an otherwise closed many-body system, under which conditions will the system reach a thermal steady state ? And how is the steady state going to be attained ? The first question has been thoroughly addressed in the literature [1-4]: on one hand it is natural to expect that scattering processes will in the long run lead to an ergodic, thermal redistribution of energy among the elementary degrees of freedom . An exception are however integrable systems, where multi-particle scattering processes are highly constrained as a result of conservation laws . As recently observed in experiments with quasi-1d Bose gases, thermalization in the usual sense will not occur [5] and the asymptotic state eventually attained by the system in the thermodynamic limit is expected to be described by an effective Generalized Gibbs ensemble (GGE) accounting for all conserved quantities .
Physical Review B, 2010
We study the dynamics of the quantum Ising chain following a zero-temperature quench of the transverse field strength. Focusing on the behavior of two-point spin correlation functions, we show that the correlators of the order parameter display an effective asymptotic thermal behavior, i.e., they decay exponentially to zero, with a phase coherence rate and a correlation length dictated by the equilibrium law with an effective temperature set by the energy of the initial state. On the contrary, the two-point correlation functions of the transverse magnetization or the density-of-kinks operator decay as a power-law and do not exhibit thermal behavior. We argue that the different behavior is linked to the locality of the corresponding operator with respect to the quasi-particles of the model: non-local operators, such as the order parameter, behave thermally, while local ones do not. We study which features of the two-point correlators are a consequence of the integrability of the model by analizing their robustness with respect to a sufficiently strong integrability-breaking term.
Probing the Possibilities of Ergodicity in the 1D Spin-1/2 XY Chain with Quench Dynamics
Scientific Reports
Ergodicity sits at the heart of the connection between statistical mechanics and dynamics of a physical system. By fixing the initial state of the system into the ground state of the Hamiltonian at zero temperature and tuning a control parameter, we consider the occurrence of the ergodicity with quench dynamics in the one-dimensional (1D) spin-1/2 XY model in a transverse magnetic field. The ground-state phase diagram consists of two ferromagnetic and paramagnetic phases. It is known the magnetization in this spin system is non-ergodic. We set up two different experiments as we call them single and double quenches and test the dynamics of the magnetization along the Z-axis and the spinspin correlation function along the X-axis which are the order parameters of the zero-temperature phases. Our exact results reveal that for single quenches at zero-temperature, the ergodicity depends on the initial state and the order parameter. In single quenches for a given order parameter, ergodicity will be observed with an ergodic-region for quenches from another phase, non-correspond to the phase of the order parameter, into itself. In addition, a quench from a ground-state phase point corresponding to the order parameter into or very close to the quantum critical point, h c = 1.0, discloses an ergodic behavior. Otherwise, for all other single quenches, the system behaves non-ergodic. Interestingly on the other setup, a double quench on a cyclic path, ergodicity is completely broken for starting from the phase corresponding to the order parameter. Otherwise, it depends on the first quenched point, and the quench time T when the model spent before a second quench in the way back which gives an ability to controlling the ergodicity in the system. Therefore, and contrary to expectations, in the mentioned model the ergodicity can be observed with probing quench dynamics at zero-temperature. Our results provide further insight into the zero-temperature dynamical behavior of quantum systems and their connections to the ergodicity phenomenon. One of the most controversial topics is how the statistical mechanics behavior could emerge in quantum-mechanical systems evolving under unitary dynamics 1-12. Historically, von Neumann was the first one that worked on the topic. Instead of physical state (or wave function) of the system, he focused on macroscopic observables and introduced the quantum ergodic theorem. The quantum ergodic theorem says every initial wave function from a microcanonical energy shell evolves so that for most times, in the long run, the joint probability distribution of commuting macroscopic observables obtained from the unitarily time-evolved wave function is close to the microcanonical distribution of commuting observables. Study of quantum ergodicity in spin systems has been of interest for a long time. In 1970, for the first time, Barouch and coworkers 13 studied the dynamics of the magnetization of the anisotropic spin-1/2 XY chain. In fact they used a single quench at finite temperature where their initial and final states were thermal states. In addition, they did not probe all quenches. By a quench from the paramagnetic phase into itself they showed that the equilibrium is not reached at the final evolutionary time and then the magnetization is a non-ergodic observable. This non-ergodic behavior was later confirmed for the entanglement between the nearest neighbor pair spins of the evolved states 14. In addition to the 1D XY model, the non-ergodicity has been also studied in quantum chaos 15 , 1D XXZ model to show ergodicity breaking that can create a many-body localization 16 and its extended 17 , 1D system of spinless and interacting fermions with a disordered potential 18 , the anisotropic Dicke model 19 , and in a small quantum system consisting of three superconducting qubits by measuring the evolution of the entanglement entropy 20 .
Prethermalization in a Nonintegrable Quantum Spin Chain after a Quench
Physical Review Letters, 2013
We study the dynamics of a quantum Ising chain after the sudden introduction of a non-integrable long-range interaction. Via an exact mapping onto a fully-connected lattice of hard-core bosons, we show that a pre-thermal state emerges and we investigate its features by focusing on a class of physically relevant observables. In order to gain insight into the eventual thermalization, we outline a diagrammatic approach which complements the study of the previous quasi-stationary state and provides the basis for a self-consistent solution of the kinetic equation. This analysis suggests that both the temporal decay towards the pre-thermal state and the crossover to the eventual thermal one may occur algebraically.
Applicability of the generalized Gibbs ensemble after a quench in the quantum Ising chain
We investigate the out-of-equilibrium dynamics of the one-dimensional quantum Ising model after a sudden quench of the transverse magnetic field. While for a translationally invariant system the statistical description of the asymptotic order parameter correlations after the quench can be performed in terms of the generalized Gibbs ensemble, we show that a breaking of translational invariance, e.g., by perturbing the boundary conditions, disrupts its validity. This effect, which of course vanishes in the thermodynamic limit, is shown to be very important in the presence of disorder.
Quantum quenches, thermalization, and many-body localization
Physical Review B, 2011
We discuss how thermalization following a quantum quench in a strongly correlated quantum system is intimately connected to many-body localization in the space of quasi-particles. We test our picture in the anisotropic Heisenberg spin chain with an integrability-breaking term. We first quantify the deviations from integrability by analyzing the level spacing statistics and the characteristics of the system eigenstates. We then focus on thermalization by studying the dynamics after a sudden quench of the anisotropy parameter. PACS numbers: 75.10.Jm, 72.15.Rn, 05.45.Mt The understanding of ergodicity and thermalization in quantum systems is one of the most intriguing problems in quantum physics. Starting with the 1929 paper of von Neumann [1], various attempts have been made towards the characterization of ergodic behavior in quantum systems , and the establishment of a link with the notion of quantum chaos . Theoretical interest in these issues resurfaced periodically until very recently, when an experimental study of the non-equilibrium dynamics of a quasi-one-dimensional condensate clearly demonstrated the lack of thermalization/ergodicity in a quantum many-body system . The attribution of this observation to quantum integrability generated a lot of interest on its connections with ergodicity and thermalization in strongly-correlated quantum systems .
Physical Review B, 2009
We consider a linear quench from the paramagnetic to ferromagnetic phase in the quantum Ising chain interacting with a static spin environment. Both decoherence from the environment and nonadiabaticity of the evolution near a critical point excite the system from the final ferromagnetic ground state. For weak decoherence and relatively fast quenches the excitation energy, proportional to the number of kinks in the final state, decays like an inverse square root of a quench time, but slow transitions or strong decoherence make it decay in a much slower logarithmic way. We also find that fidelity between the final ferromagnetic ground state and a final state after a quench decays exponentially with a size of a chain, with a decay rate proportional to average density of excited kinks, and a proportionality factor evolving from 1.3 for weak decoherence and fast quenches to approximately 1 for slow transitions or strong decoherence. Simultaneously, correlations between kinks randomly distributed along the chain evolve from a near-crystalline anti-bunching to a Poissonian distribution of kinks in a number of isolated Anderson localization centers randomly scattered along the chain.
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