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arXiv:1103.1539 · Physical Review A

Quantum quench dynamics of the Bose-Hubbard model at finite temperatures

J. M. Zhang, Chao Shen, W. M. Liu

Published Jun 17, 2011• 17 citations
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Abstract

We study quench dynamics of the Bose-Hubbard model by exact diagonalization. Initially, the system is at thermal equilibrium and of a finite temperature. The system is then quenched by changing the on-site interaction strength $U$ suddenly. Both the single-quench and double-quench scenarios are considered. In the former case, the time-averaged density matrix and the real-time evolution are investigated. It is found that though the system thermalizes only in a very narrow range of the quenched value of $U$, it does equilibrate or relax well into a much larger range. Most importantly, it is proven that this is guaranteed for some typical observables in the thermodynamic limit. In order to test whether it is possible to distinguish the unitarily evolving density matrix from the time-averaged (thus time-independent), fully decohered density matrix, a second quench is considered. It turns out that the answer is affirmative or negative depending on whether the intermediate value of $U$ is zero or not.

DOI
10.1103/physreva.83.063622

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We study quench dynamics of the Bose-Hubbard model by exact diagonalization. Initially, the system is at thermal equilibrium and of a finite temperature. The system is then quenched by changing the on-site interaction strength UUU suddenly. Both the single-quench and double-quench scenarios are considered. In the former case, the time-averaged density matrix and the real-time evolution are investigated. It is found that though the system thermalizes only in a very narrow range of the quenched value of UUU, it does equilibrate or relax well into a much larger range. Most importantly, it is proven that this is guaranteed for some typical observables in the thermodynamic limit. In order to test whether it is possible to distinguish the unitarily evolving density matrix from the time-averaged (thus time-independent), fully decohered density matrix, a second quench is considered. It turns out that the answer is affirmative or negative depending on whether the intermediate value of UUU is zero or not.
Motivated by recent experiments, we study the relaxation dynamics and thermalization in the one-dimensional Bose-Hubbard model induced by a global interaction quench. Specifically, we start from an initial state that has exactly one boson per site and is the ground state of a system with infinitely strong repulsive interactions at unit filling. Using exact diagonalization and the density-matrix renormalization-group method, we compute the time dependence of such observables as the multiple occupancy and the momentum distribution function. Typically, the relaxation to stationary values occurs over just a few tunneling times. The stationary values are identical to the so-called diagonal ensemble on the system sizes accessible to our numerical methods and we further observe that the microcanonical ensemble describes the time averages of many observables reasonably well for small and intermediate interaction strength. The expectation values of observables in the canonical ensemble agree quantitatively with the time averages obtained from the quench at small interaction strengths, and qualitatively provide a good description even in parameter regimes where the microcanonical ensemble is not applicable due to finite-size effects. We discuss our numerical results in the framework of the eigenstate thermalization hypothesis. Moreover, we also observe that the diagonal and the canonical ensembles are practically identical for our initial conditions already on the level of their respective energy distributions for small interaction strengths. Finally, we discuss implications of our results for the interpretation of a recent sudden expansion experiment [Phys. Rev. Lett. 110, 205301 (2013)], in which the same interaction quench was realized.
We study thermalization of a two-component Bose-Hubbard model by exact diagonalization. Initially, the two components do not interact and are both at equilibrium but with different temperatures. As the onsite intercomponent interaction is turned on, perfect thermalization occurs. Remarkably, not merely those simple ``realistic'' physical observables thermalize but even the density matrix of the whole system---the time-averaged density matrix of the system can be well approximated by that of a canonical ensemble. We also discuss how symmetry of the quenched Hamiltonian affects the level spacing distribution and thermalization of the system.
We study the Bose-Hubbard and Fermi-Hubbard models in the (formal) limit of large coordination numbers $Z\ensuremath{\gg}1$. Via an expansion into powers of 1/Z1/Z1/Z, we establish a hierarchy of correlations which facilitates an approximate analytical derivation of the time evolution of the reduced density matrices for one and two sites, etc. With this method, we study the quantum dynamics (starting in the ground state) after a quantum quench, i.e., after suddenly switching the tunneling rate JJJ from zero to a finite value, which is still in the Mott regime. We find that the reduced density matrices approach a (quasi)equilibrium state after some time. For one lattice site, this state can be described by a thermal state (within the accuracy of our approximation). However, the (quasi)equilibrium state of the reduced density matrices for two sites including the correlations can not be described by a thermal state. Thus, real thermalization (if it occurs) should take a much longer time. This behavior has already been observed in other scenarios and is sometimes called ``prethermalization''. Finally, we compare our results to numerical simulations for finite lattices in one and two dimensions and find qualitative agreement.
We study the Bose and Fermi Hubbard model in the (formal) limit of large coordination numbers Z≫1Z\gg1Z≫1. Via an expansion into powers of 1/Z1/Z1/Z, we establish a hierarchy of correlations which facilitates an approximate analytical derivation of the time-evolution of the reduced density matrices for one and two sites etc. With this method, we study the quantum dynamics (starting in the ground state) after a quantum quench, i.e., after suddenly switching the tunneling rate JJJ from zero to a finite value, which is still in the Mott regime. We find that the reduced density matrices approach a (quasi) equilibrium state after some time. For one lattice site, this state can be described by a thermal state (within the accuracy of our approximation). However, the (quasi) equilibrium state of the reduced density matrices for two sites including the correlations cannot be described by a thermal state. Thus, real thermalization (if it occurs) should take much longer time. This behavior has already been observed in other scenarios and is sometimes called ``pre-thermalization.'' Finally, we compare our results to numerical simulations for finite lattices in one and two dimensions and find qualitative agreement.
When a quantum many-body system undergoes a quench, the time-averaged density matrix $\overline{\ensuremath{\rho}}$ governs the time-averaged expectation value of any observable. It is therefore the key object to look at when comparing results with equilibrium predictions. We show that the weights of $\overline{\ensuremath{\rho}}$ can be efficiently computed with Lanczos diagonalization for relatively large Hilbert spaces. As an application, we investigate the crossover from perturbative to nonperturbative quenches in the nonintegrable Bose-Hubbard model: in finite systems, an approximate Boltzmann distribution is observed for small quenches, while for larger ones the distributions do not follow standard equilibrium predictions. Studying thermodynamical features, such as the energy fluctuations and the entropy, show that $\overline{\ensuremath{\rho}}$ bears a memory of the initial state.
We study the nonequilibrium dynamics of the Luttinger model after a quantum quench, when the initial state is a finite temperature thermal equilibrium state. The diagonal elements of the density matrix in the steady state show thermal features for high temperature initial states only, otherwise they retain highly nonthermal character. The time evolution of Uhlmann fidelity, which measures the distance between the time evolved and initial states, is evaluated for arbitrary initial temperatures and quench protocols. In the long time limit, the overlap between the time evolved and initial system decreases exponentially with the temperature with a universal prefactor. Within perturbation theory, the statistics of final total energy and work are numerically evaluated in the case of a sudden quench, which yield identical distributions at zero temperature. In both statistics, temperature effects are more significant in small systems. The Dirac-$\ensuremath{\delta}$ peak at the adiabatic ground state energy remains present in the probability distribution of the total energy, but disappears from the work distribution at nonzero initial temperatures.