We use quantum quenches to study the dynamics and thermalization of hardcore bosons in finite one-dimensional lattices. We perform exact diagonalizations and find that, far away from integrability, few-body observables thermalize. We then study the breakdown of thermalization as one approaches an integrable point. This is found to be a smooth process in which the predictions of standard statistical mechanics continuously worsen as the system moves toward integrability. We establish a direct connection between the presence or absence of thermalization and the validity or failure of the eigenstate thermalization hypothesis, respectively.
Deep Dive into Breakdown of thermalization in finite one-dimensional systems.
We use quantum quenches to study the dynamics and thermalization of hardcore bosons in finite one-dimensional lattices. We perform exact diagonalizations and find that, far away from integrability, few-body observables thermalize. We then study the breakdown of thermalization as one approaches an integrable point. This is found to be a smooth process in which the predictions of standard statistical mechanics continuously worsen as the system moves toward integrability. We establish a direct connection between the presence or absence of thermalization and the validity or failure of the eigenstate thermalization hypothesis, respectively.
Little more than fifty years ago, Fermi, Pasta, and Ulam (FPU) [1] set up a numerical experiment to prove the ergodic hypothesis for a one-dimensional (1D) lattice of harmonic oscillators once nonlinear couplings were added. Much to their surprise, the system exhibited long-time periodic dynamics with no signals of ergodicity. This behavior could not be explained in terms of Poincaré recurrences and motivated intense research [2], which ultimately gave rise to the modern chaos theory. It led to the discovery of solitons (stable solitary waves) in nonlinear systems and to the understanding of thermalization in terms of dynamical chaos [2]. In the latter scenario, there is a threshold below which the interactions breaking integrability are ineffective in producing chaotic behavior and the system cannot be described by standard statistical mechanics [3]. The FPU numerical calculations happened to be below that threshold [4].
More recently, experiments with ultracold gases in 1D geometries have challenged our understanding of the quantum domain [5]. After bringing a nearly isolated system out of equilibrium, no signals of relaxation to the expected thermal equilibrium distribution were observed. Some insight can be gained in the framework of integrable quantum systems [6], but then it remains the question of why thermalization did not occur even when the system was supposed to be away from integrability. In the latter regime, thermalization is expected to occur [7,8]. This new experimental result [5] has opened many questions such as: Will thermalization occur if one waits longer? Is there a threshold after which thermalization will occur? In this work we address some of these questions using numerical experiments.
In the limit in which the quantum system is integrable, it has been shown numerically [6] that observables such as the ones measured experimentally relax to an equilibrium distribution different from the thermal one. That a novel distribution is generated is the result of the conserved quantities that render the system integrable, and can be characterized by a generalization of the Gibbs ensemble (GGE) [6]. Several works since then have addressed the relevance and limitations of the GGE to various integrable systems and classes of observables [9]. Much less is known away from integrability where fewer analytical tools are available and numerical computations become more demanding. Early works in 1D have provided mixed re-sults; thermalization was observed in some regimes and not in others [10]. In two dimensions, thermalization was unambiguously shown to occur [8] and could be understood on the basis of the eigenstate thermalization hypothesis [7]. Recent works have also pointed out a possible intermediate quasisteady regime that could occur before thermalization in a class of fermionic systems [11]. Here we study how breaking integrability affects the thermalization of correlated bosons in a 1D lattice after a quantum quench.
We consider impenetrable bosons in a periodic 1D lattice with nearest-neighbor hopping t and repulsive interaction V , and next-nearest-neighbor hopping t ′ and repulsive interaction V ′ . The Hamiltonian reads [12]
When t ′ = V ′ = 0 this model is integrable. In order to understand how the proximity to the integrable point affects equilibration, we prepare an initial state that is an eigenstate of a system with t = t ini , V = V ini , t ′ , V ′ and then quench the nearest-neighbor parameters to t = t f in , V = V f in without changing t ′ , V ′ , i.e., we only change t ini , V ini → t f in , V f in . The same quench is then repeated for different values of t ′ , V ′ as one approaches t ′ = V ′ = 0. We have performed the exact time evolution of up to eight impenetrable bosons in lattices with up to 24 sites. Taking advantage of translational invariance, this required the full diagonalization of blocks in the Hamiltonian that contained up to 30,667 states. Does integrability, or its absence, affect the relaxation dynamics of experimentally relevant observables? To answer that question, we examine two of those observables: the momentum distribution function n(k) and the structure factor for the density-density correlations N (k) [12]. Since the initial state wavefunction can be expanded in the eigenstate basis of the final Hamiltonian H as |ψ ini = α C α |Ψ α , one finds that, if the spectrum is nondegenerate and incommensurate, the infinite time average of an observable Ô can be written as
where O αα are the matrix elements of Ô in the basis of the final Hamiltonian. This exact result can be thought as the prediction of a “diagonal ensemble”, where |C α | 2 is the weight of each state of this ensemble [8]. We then study the normalized area between our observables, during the time evolution, and their infinite time average, i.e., at each time we compute
and similarly for δN k . If, up to small fluctuations, n(k) and N (k) relax to a constant distribution, it must be the one
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