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arXiv:1711.08882 · Physical Review A

Dynamics of a quantum phase transition in the Bose-Hubbard model: Kibble-Zurek mechanism and beyond

Keita Shimizu, Yoshihito Kuno, Takahiro Hirano, Ikuo Ichinose

Published Mar 28, 2018• 29 citations
arXivHTMLPDF

Abstract

In this paper, we study the dynamics of the Bose-Hubbard model by using time-dependent Gutzwiller methods. In particular, we vary the parameters in the Hamiltonian as a function of time, and investigate the temporal behavior of the system from the Mott insulator to the superfluid (SF) crossing a second-order phase transition. We first solve a time-dependent Schr\"odinger equation for the experimental setup recently done by Braun et al. [Proc. Natl. Acad. Sci. USA 112, 3641 (2015)] and show that the numerical and experimental results are in fairly good agreement. However, these results disagree with the Kibble-Zurek scaling. From our numerical study, we reveal a possible source of the discrepancy. Next, we calculate the critical exponents of the correlation length and vortex density in addition to the SF order parameter for a Kibble-Zurek protocol. We show that beside the ``freeze'' time $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{t}$, there exists another important time, ${t}_{\mathrm{eq}}$, at which an oscillating behavior of the SF amplitude starts. From calculations of the exponents of the correlation length and vortex density with respect to a quench time ${\ensuremath{\tau}}_{\mathrm{Q}}$, we obtain a physical picture of a coarsening process. Finally, we study how the system evolves after the quench. We give a global picture of dynamics of the Bose-Hubbard model.

DOI
10.1103/physreva.97.033626

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In this paper, we study the dynamics of the Bose-Hubbard model by using time-dependent Gutzwiller methods. In particular, we vary the parameters in the Hamiltonian as a function of time, and investigate the temporal behavior of the system from the Mott insulator to the superfluid (SF) crossing a second-order phase transition. We first solve a time-dependent Schr"odinger equation for the experimental setup recently done by Braun et al. [Proc. Natl. Acad. Sci. USA 112, 3641 (2015)] and show that the numerical and experimental results are in fairly good agreement. However, these results disagree with the Kibble-Zurek scaling. From our numerical study, we reveal a possible source of the discrepancy. Next, we calculate the critical exponents of the correlation length and vortex density in addition to the SF order parameter for a Kibble-Zurek protocol. We show that beside the ``freeze'' time t\ifmmode^\elseˆ\fi\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{t}t\ifmmode^\elseˆ\fi​, there exists another important time, teq{t}_{\mathrm{eq}}teq​, at which an oscillating behavior of the SF amplitude starts. From calculations of the exponents of the correlation length and vortex density with respect to a quench time \ensuremathτQ{\ensuremath{\tau}}_{\mathrm{Q}}\ensuremathτQ​, we obtain a physical picture of a coarsening process. Finally, we study how the system evolves after the quench. We give a global picture of dynamics of the Bose-Hubbard model.
In this paper, we study the nonequilibrium dynamics of the Bose–Hubbard model with the nearest-neighbor repulsion by using time-dependent Gutzwiller (GW) methods. In particular, we vary the hopping parameters in the Hamiltonian as a function of time, and investigate the dynamics of the system from the density wave (DW) to the superfluid (SF) crossing a first-order phase transition and vice versa. From the DW to SF, we find scaling laws for the correlation length and vortex density with respect to the quench time. This is a reminiscence of the Kibble–Zurek scaling for continuous phase transitions and contradicts the common expectation. We give a possible explanation for this observation. On the other hand from SF to DW, the system evolution depends on the initial SF state. When the initial state is the ground state obtained by the static GW methods, a coexisting state of the SF and DW domains forms after passing through the critical point. Coherence of the SF order parameter is lost as the system evolves. This is a phenomenon similar to the glass transition in classical systems. When the state starts from the SF with small local phase fluctuations, the system obtains a large size DW domain structure with thin domain walls.
The dynamics of quantum phase transitions pose one of the most challenging problems in modern many-body physics. Here, we study a prototypical example in a clean and well-controlled ultracold atom setup by observing the emergence of coherence when crossing the Mott insulator to superfluid quantum phase transition. In the 1D Bose-Hubbard model, we find perfect agreement between experimental observations and numerical simulations for the resulting coherence length. We, thereby, perform a largely certified analog quantum simulation of this strongly correlated system reaching beyond the regime of free quasiparticles. Experimentally, we additionally explore the emergence of coherence in higher dimensions, where no classical simulations are available, as well as for negative temperatures. For intermediate quench velocities, we observe a power-law behavior of the coherence length, reminiscent of the Kibble-Zurek mechanism. However, we find nonuniversal exponents that cannot be captured by this mechanism or any other known model.
In this paper, we study the dynamics of the Bose-Hubbard model with the nearest-neighbor repulsion by using time-dependent Gutzwiller methods. Near the unit filling, the phase diagram of the model contains density wave (DW), supersolid (SS), and superfluid (SF). The three phases are separated by two second-order phase transitions. We study ``slow-quench'' dynamics by varying the hopping parameter in the Hamiltonian as a function of time. In the phase transitions from the DW to SS and from the DW to SF, we focus on how the SF order forms and study scaling laws of the SF correlation length, vortex density, etc. The results are compared with the Kibble-Zurek scaling. On the other hand from the SF to DW, we study how the DW order evolves with generation of the domain walls and vortices. Measurement of first-order SF coherence reveals interesting behavior in the DW regime.
The ground state of the one-dimensional Bose-Hubbard model at unit filling undergoes the Mott-superfluid quantum phase transition. It belongs to the Kosterlitz-Thouless universality class with an exponential divergence of the correlation length in place of the usual power law. We present numerical simulations of a linear quench both from the Mott insulator to superfluid and back. The results satisfy the scaling hypothesis that follows from the Kibble-Zurek mechanism (KZM). In the superfluid-to-Mott quenches there is no significant excitation in the superfluid phase despite its gaplessness. Since all critical superfluid ground states are qualitatively similar, the excitation begins to build up only after crossing the critical point when the ground state begins to change fundamentally. The last process falls into the KZM framework.
The Bose-Hubbard model is the simplest model of interacting bosons on a lattice. It has recently been the focus of much attention due to the realization of this model with cold atoms in an optical lattice. The ability to tune parameters in the Hamiltonian as a function of time in cold atom systems has opened up the possibility of studying out-of-equilibrium dynamics, including crossing the quantum critical region of the model in a controlled way. In this paper, I give a brief introduction to the Bose Hubbard model, and its experimental realization and then give an account of theoretical and experimental efforts to understand out-of-equilibrium dynamics in this model, focusing on quantum quenches, both instantaneous and of finite duration. I discuss slow dynamics that have been observed theoretically and experimentally for some quenches from the superfluid phase to the Mott insulating phase and the picture of two timescales, one for fast local equilibration and another for slow global equilibration, that appears to characterize this situation. I also discuss the theoretical and experimental observation of the Lieb-Robinson bounds for a variety of quenches and the Kibble-Zurek mechanism in quenches from the Mott insulator to superfluid. I conclude with a discussion of open questions and future directions.
We study the dynamics of phase transitions in the one-dimensional Bose-Hubbard model. To drive the system from a Mott insulator to a superfluid phase, we change the tunneling frequency at a finite rate. We investigate the buildup of correlations during fast and slow transitions using variational wave functions, dynamical Bogoliubov theory, Kibble-Zurek mechanism, and numerical simulations. We show that time-dependent correlations satisfy characteristic scaling relations that can be measured in optical lattices filled with cold atoms.