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arXiv:1206.4787 · Physical Review Letters

Relaxation Dynamics of Disordered Spin Chains: Localization and the Existence of a Stationary State

Simone Ziraldo, Alessandro Silva, Giuseppe E. Santoro

Published Dec 11, 2012• 48 citations
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Abstract

We study the unitary relaxation dynamics of disordered spin chains following a sudden quench of the Hamiltonian. We give analytical arguments, corroborated by specific numerical examples, to show that the existence of a stationary state depends crucially on the spectral and localization properties of the final Hamiltonian, and not on the initial state. We test these ideas on integrable one-dimensional models of the Ising or XY class, but argue more generally on their validity for more complex (nonintegrable) models.

DOI
10.1103/physrevlett.109.247205

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We study the unitary relaxation dynamics of disordered spin chains following a sudden quench of the Hamiltonian. We give analytical arguments, corroborated by specific numerical examples, to show that the existence of a stationary state depends crucially on the spectral and localization properties of the final Hamiltonian, and not on the initial state. We test these ideas on integrable one-dimensional models of the Ising or XY class, but argue more generally on their validity for more complex (nonintegrable) models.
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In the present thesis we study the unitary dynamics and the thermalization properties of free-fermion-like Hamiltonians after a sudden quantum quench in presence of disorder. 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, one-body versus many-body) and on the localization properties of the final Hamiltonian. We then show an extension of the Wang-Landau algorithm which allows the computation of weighted distributions associated to quantum quenches, like the diagonal and the GGE ensemble expectation-value distributions. We present results on three one-dimensional models, the Anderson model, a disordered one-dimensional fermionic chain with long-range hopping, and the disordered Ising/XY spin chain.
Abstract We review the dynamics after quantum quenches in integrable quantum spin chains. We give a pedagogical introduction to relaxation in isolated quantum systems, and discuss the description of the steady state by (generalized) Gibbs ensembles. We then turn to general features in the time evolution of local observables after the quench, using a simple model of free fermions as an example. In the second part we present an overview of recent progress in describing quench dynamics in two key paradigms for quantum integrable models, the transverse field Ising chain and the anisotropic spin-1/2 Heisenberg chain.
We consider the time evolution following a quantum quench in spin-1/2 chains. It is well known that local conservation laws constrain the dynamics and, eventually, the stationary behavior of local observables. We show that some widely studied models, like the quantum XY model, possess extra families of local conservation laws in addition to the translation invariant ones. As a consequence, the additional charges must be included in the generalized Gibbs ensemble that describes the stationary properties. The effects go well beyond a simple redefinition of the stationary state. The time evolution of a non-translation invariant state under a (translation invariant) Hamiltonian with a perturbation that weakly breaks the hidden symmetries underlying the extra conservation laws exhibits pre-relaxation. In addition, in the limit of small perturbation, the time evolution following pre-relaxation can be described by means of a time-dependent generalized Gibbs ensemble.
We investigate the time evolution of the density of kinks in the spin-1/2 quantum Ising spin chain after a sudden quench in the transverse field strength, and find that it relaxes to a value which depends on the initial and the final values of the transverse field, with an oscillating power-law decay. We provide analytical estimates of the long-time behavior and of the asymptotic value reached after complete relaxation, and discuss the role of quantum criticality in the quench dynamics. We show that, for a dynamics at the critical point, the residual density of kinks after the quench can be described by equilibrium statistical mechanics at a finite temperature dictated by the energy of the state after the quench. On the other hand, outside of criticality it does not exhibit thermalization.
We provide numerical evidence that after a local quench in an isolated infinite quantum spin chain, the quantum state locally relaxes to the ground state of the post-quenched Hamiltonian, i.e. dissipates. This is a consequence of the unitary quantum dynamics. A mechanism similar to the eigenstate thermalization hypothesis is shown to be responsible for the dissipation observed. We also demonstrate that integrability obstructs dissipation. The numerical simulations are done directly in the thermodynamic limit with a time-evolution algorithm based on matrix product states. The area law of entanglement entropy is observed to hold after the local quench. As a result, the simulations can be performed for long times with small bond dimensions. Various local quenches on the Ising chain and the three-state Potts chain are studied.