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arXiv:0803.2947 · Journal of Statistical Mechanics Theory and Experiment

Spreading of correlations and entanglement after a quench in the one-dimensional Bose–Hubbard model

Andreas Martin Läuchli, Corinna Kollath

Published May 30, 2008• 323 citations
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Abstract

We investigate the spreading of information in a one-dimensional Bose–Hubbard system after a sudden parameter change. In particular, we study the time evolution of correlations and entanglement following a quench. The investigated quantities show a light-cone-like evolution, i.e. the spreading with a finite velocity. We discuss the relation of this velocity to other characteristic velocities of the system, like the sound velocity. The entanglement is investigated using two different measures, the von Neumann entropy and mutual information. Whereas the von Neumann entropy grows rapidly with time the mutual information between two small subsystems can decrease as well after an initial increase. Additionally we show that the static von Neumann entropy characterizes the location of the quantum phase transition.

DOI
10.1088/1742-5468/2008/05/p05018

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We investigate the spreading of information in a one-dimensional Bose–Hubbard system after a sudden parameter change. In particular, we study the time evolution of correlations and entanglement following a quench. The investigated quantities show a light-cone-like evolution, i.e. the spreading with a finite velocity. We discuss the relation of this velocity to other characteristic velocities of the system, like the sound velocity. The entanglement is investigated using two different measures, the von Neumann entropy and mutual information. Whereas the von Neumann entropy grows rapidly with time the mutual information between two small subsystems can decrease as well after an initial increase. Additionally we show that the static von Neumann entropy characterizes the location of the quantum phase transition.
We investigate the spreading of information in a Bose-Hubbard system after a sudden parameter change. In particular, we study the time-evolution of correlations and entanglement following a quench. The investigated quantities show a light-cone like evolution, i.e. the spreading with a finite velocity. We discuss the relation of this veloctiy to other characteristic velocities of the system, like the sound velocity. The entanglement is investigated using two different measures, the von-Neuman entropy and mutual information. Whereas the von-Neumann entropy grows rapidly with time the mutual information can as well decrease after an initial increase. Additionally we show that the static von Neuman entropy characterises the location of the quantum phase transition.
We study the spreading of density-density correlations in Bose-Hubbard models after a quench of the interaction strength, using time-dependent variational Monte Carlo simulations. It gives access to unprecedented long propagation times and to dimensions higher than one. In both one and two dimensions, we find ballistic light-cone spreading of correlations and extract accurate values of the light-cone velocity in the superfluid regime. We show that the spreading of correlations is generally supersonic, with a light-cone propagating faster than sound modes but slower than the maximum group velocity of density excitations, except at the Mott transition, where all the characteristic velocities are equal. Further, we show that in two dimensions the correlation spreading is highly anisotropic and presents nontrivial interference effects.
We study the spreading of correlations in space and time after a quantum quench in the Bose-Hubbard model. We derive equations of motion for the single-particle Green's function within the contour-time formalism, allowing us to study dynamics in the strong-coupling regime. We discuss the numerical solutions of these equations and calculate the single-particle density matrix for quenches in the Mott phase. We demonstrate light-cone-like spreading of correlations in the Mott phase in one, two, and three dimensions and calculate the propagation velocities in each dimension.
Lieb-Robinson and related bounds set an upper limit on the speed at which information propagates in nonrelativistic quantum systems. Experimentally, light-cone-like spreading has been observed for correlations in the Bose-Hubbard model (BHM) after a quantum quench. Using a two-particle irreducible (2PI) strong-coupling approach to out-of-equilibrium dynamics in the BHM we calculate both the group and phase velocities for the spreading of single-particle correlations in one, two, and three dimensions as a function of interaction strength. Our results are in quantitative agreement with measurements of the speed of spreading of single-particle correlations in both the one- and two-dimensional BHM realized with ultracold atoms. They also are consistent with the claim that the phase velocity rather than the group velocity was observed in recent experiments in two dimensions. We demonstrate that there can be large differences between the phase and group velocities for the spreading of correlations and explore how the anisotropy in the velocity varies across the phase diagram of the BHM. Our results establish the 2PI strong-coupling approach as a powerful tool to study out-of-equilibrium dynamics in the BHM in dimensions greater than one.
We study correlations in space and time after a quantum quench in the Bose Hubbard model. We obtain a two particle irreducible effective action within the Schwinger-Keldysh formalism that allows us to study dynamics in both the superfluid and Mott insulating phases. We derive equations of motion for both the superfluid order parameter and two-point correlation functions. We discuss the numerical solutions of these equations and calculate the quasimomentum distribution for quenches in the Mott phase. We demonstrate light-cone like spreading of correlations in the Mott phase in both one and two dimensions and calculate the propagation velocity in each dimension.
"We studied the one-dimensional Bose-Hubbard model, by means of the Density Matrix Renormalization Group (DMRG). The U parameter, which contains the Coulombian repulsion information, is fixed to U = 1 and it is performed a swept over the hopping parameter t. The entanglement is investigated by implementing the one-site, two-site and block von Neumann entropy looking for which of these measures give account of the quantum phase transition from Mott insulator to superfluid. The one-site and two-site entropy do not give information of the phase transition of the system. In contrast, the block entropy shows, in the Mott insulator phase a behavior that saturates rapidly to a finite value and in the superfluid phase shows a logarithmic behavior that agrees with the mean field theory. From the derivative of the block entropy average as a function of t we found the phase transition takes place at tc =0.15."