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arXiv:1501.02005 · Physical Review B

Propagation of a single-hole defect in the one-dimensional Bose-Hubbard model

F. Andraschko, Jesko Sirker

Published Jun 18, 2015• 15 citations
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Abstract

We study nonequilibrium dynamics in the one-dimensional Bose-Hubbard model starting from an initial product state with one boson per site and a single-hole defect. We find that for parameters close to the quantum critical point, the hole splits into a core showing a very slow diffusive dynamics, and a fast mode which propagates ballistically. Using an effective fermionic model at large Hubbard interactions $U$, we show that the ballistic mode is a consequence of an interference between slow-holon and fast-doublon dynamics, which occurs once the hole defect starts propagating into the bosonic background at unit filling. Within this model, the signal velocity of the fast ballistic mode is given by the maximum slope of the dispersion of the doublon quasiparticle in good agreement with the numerical data. Furthermore, we contrast this global quench with the dynamics of a single-hole defect in the ground state of the Bose-Hubbard model and show that the dynamics in the latter case is very different even for large values of $U$. This can also be seen in the entanglement entropy of the time-evolved states which grows much more rapidly in time in the global quench case.

DOI
10.1103/physrevb.91.235132

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We study nonequilibrium dynamics in the one-dimensional Bose-Hubbard model starting from an initial product state with one boson per site and a single-hole defect. We find that for parameters close to the quantum critical point, the hole splits into a core showing a very slow diffusive dynamics, and a fast mode which propagates ballistically. Using an effective fermionic model at large Hubbard interactions UUU, we show that the ballistic mode is a consequence of an interference between slow-holon and fast-doublon dynamics, which occurs once the hole defect starts propagating into the bosonic background at unit filling. Within this model, the signal velocity of the fast ballistic mode is given by the maximum slope of the dispersion of the doublon quasiparticle in good agreement with the numerical data. Furthermore, we contrast this global quench with the dynamics of a single-hole defect in the ground state of the Bose-Hubbard model and show that the dynamics in the latter case is very different even for large values of UUU. This can also be seen in the entanglement entropy of the time-evolved states which grows much more rapidly in time in the global quench case.
We analyze the propagation of correlations after a sudden interaction change in a strongly interacting quantum system in contact with an environment. In particular, we consider an interaction quench in the Bose-Hubbard model, deep within the Mott-insulating phase, under the effect of dephasing. We observe that dissipation effectively speeds up the propagation of single-particle correlations while reducing their coherence. In contrast, for two-point density correlations, the initial ballistic propagation regime gives way to diffusion at intermediate times. Numerical simulations, based on a time-dependent matrix product state algorithm, are supplemented by a quantitatively accurate fermionic quasiparticle approach providing an intuitive description of the initial dynamics in terms of holon and doublon excitations.
We investigate the dynamics of correlation propagation in the one-dimensional Fermi-Hubbard model with SU(NNN) symmetry when the repulsive-interaction strength is quenched from a large value, at which the ground state is a Mott-insulator with 1/N1/N1/N filling, to an intermediate value. From approximate analytical insights based on a simple model that captures the essential physics of the doublon excitations, we show that entanglement in the initial state leads to collective enhancement of the propagation velocity vSU(N)v_{\text{SU}(N)}vSU(N)​ when N>2N>2N>2, becoming equal to the velocity of the Bose-Hubbard model in the large-NNN limit. These results are supported by numerical calculations of the density-density correlation in the quench dynamics for N=2,3,4,N=2,3,4,N=2,3,4, and 666.
We analyze the quench dynamics of a one-dimensional bosonic Mott insulator and focus on the time evolution of density correlations. For these we identify a pronounced propagation front, the velocity of which, once correctly extrapolated at large distances, can serve as a quantitative characteristic of the many-body Hamiltonian. In particular, the velocity allows the weakly interacting regime, which is qualitatively well described by free bosons, to be distinguished from the strongly interacting one, in which pairs of distinct quasiparticles dominate the dynamics. In order to describe the latter case analytically, we introduce a general approximation to solve the Bose-Hubbard Hamiltonian based on the Jordan-Wigner fermionization of auxiliary particles. This approach can also be used to determine the ground-state properties. As a complement to the fermionization approach, we derive explicitly the time-dependent many-body state in the noninteracting limit and compare our results to numerical simulations in the whole range of interactions of the Bose-Hubbard model.
We investigate signatures of non-ergodic behavior in the real-time evolution of a one-dimensional Bose-Hubbard model, where the initial state is a doubly occupied density-wave state. We show that the occupation dynamics at strong interactions is dominated by doublon-holon exchange which leads to a domain wall excitation and propagation. The latter manifests as a negated staggered pattern in the density-density correlations. While the single-particle and the pair correlation functions show highly localized correlations that decay rapidly away from the nearest neighbor. We show that the time scale of the domain-wall excitations depends on the inverse of the interaction strength and therefore dictates the slow relaxation dynamics. In the presence of a parabolic trap, the occupation dynamics at the edges become frozen and further suppresses the propagation of correlations. This suppression happens even for trap strengths weaker than the tunneling rate. We also show that the model can be mapped to an antiferromagnetic transverse-field Ising model in the limit of strong interactions and that the correlation-propagation velocity in the original model is well captured by the group velocity of the spin-wave excitation in the effective spin model.
We investigate the expansion of bosons and fermions in a homogeneous lattice after a sudden removal of the trapping potential using exact numerical methods. As a main result, we show that in one dimension, both bosonic and fermionic Mott insulators expand with the same velocity, irrespective of the interaction strength, provided the expansion starts from the ground state of the trapped gas. Furthermore, their density profiles become identical during the expansion; the asymptotic density dynamics is identical to that of initially localized, noninteracting particles, and the asymptotic velocity distribution is flat. The expansion velocity for initial correlated Mott insulating states is therefore independent of the interaction strength and particle statistics. Interestingly, this nonequilibrium dynamics is sensitive to the interaction driven quantum phase transition in the Bose-Hubbard model; while being constant in the Mott phase, the expansion velocity decreases in the superfluid phase and vanishes for large systems in the noninteracting limit. These results are compared to the setup of a recent experiment [Ronzheimer et al., Phys. Rev. Lett. 110, 205301 (2013)], where the trap opening was combined with an interaction quench from infinitely strong interactions to finite values. In the latter case, the interaction quench breaks the universal dynamics in the asymptotic regime and the expansion depends on the interaction strength. We carry out an analogous analysis for a two-component Fermi gas, with similar observations. In addition, we study the effect of breaking the integrability of hard-core bosons in different ways; while the fast ballistic expansion from the ground state of Mott insulators in one dimension remains unchanged for finite interactions, we observe strong deviations from this behavior on a two-leg ladder even in the hard-core case. This change in dynamics bares similarities with the dynamics in the dimensional crossover from one to two dimensions observed in the aformentioned experimental study.
We present a nonequilibrium Green's-functional approach to study the dynamics following a quench in weakly interacting Bose-Hubbard model (BHM). The technique is based on the self-consistent solution of a set of equations which represents a particular case of the most general set of Hedin's equations for the interacting single-particle Green's function. We use the ladder approximation as a skeleton diagram for the two-particle scattering amplitude useful, through the self-energy in the Dyson equation, for finding the interacting single-particle Green's function. This scheme is then implemented numerically by a parallelized code. We exploit this approach to study the correlation propagation after a quench in the interaction parameter, for one and two dimensions. In particular, we show how our approach is able to recover the crossover from the ballistic to the diffusive regime by increasing the boson-boson interaction. Finally we also discuss the role of a thermal initial state on the dynamics both for one- and two-dimensional BHMs, finding that, surprisingly, at high temperature a ballistic evolution is restored.