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 Schrödinger equation for the experimental setup recently done by Braun et.al. [Proc. Nat. Acad. Sci. 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" timet, there exists another important time, 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 τQ, 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 nearestneighbor 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. them, works in [27, 28] questioned the applicability of the KZ scaling theory to the QPT, whereas [29,30] concluded that the observed results were in good agreement with the KZ scaling law.In this paper, we focus on the two-dimensional (2D) Bose-Hubbard model (BHM) [33,34], which is a canonical model of the bosonic ultra-cold atomic gas systems in an optical lattice. In particular, we add nearestneighbor (NN) repulsions between atoms. Then, the resultant system is described by an extended Bose-Hubbard model (EBHM). As a result, a parameter region corresponding to the density wave (DW) appears in the ground-state phase diagram, in addition to the Mott insulator and SF. Near the half filling, there exists a firstorder phase transition between the SF and DW [35]. We shall study the quench dynamics of the EBHM on passing across the SF and DW phase boundary. There are only a few works for the dynamical properties of quantum systems at first-order phase transitions under a quench [36][37][38], and therefore detailed study on that problem is desired.This paper is organized as follows. In section 2, we introduce the EBHM and explain the Gutzwiller (GW) methods, which are used in the present work. In section 3, quench dynamics of the first-order phase transition from the DW to SF is studied. Behavior of SF and DW orders are investigated by solving the Schrödinger equation by means of time-dependent GW (tGW) methods. We focus on the order parameters, correlation length, vortex number, etc, in particular, scaling laws of these quantities with respect to the quench time τ Q . Contrary to the common expectation, we find that scaling laws hold for the correlation length and vortex density. In section 4, we give a possible explanation of the observed results from viewpoint of the SF-bubblenucleation process. W...

In this paper, we study a one-dimensional boson system in a superlattice potential. This system is experimentally feasible by using ultracold atomic gases, and attracts much attention these days. It is expected that the system has a topological phase called a topological Mott insulator (TMI). We show that in strongly-interacting cases, the competition between the superlattice potential and the on-site interaction leads to various TMIs with a non-vanishing integer Chern number. Compared to the hard-core case, the soft-core boson system exhibits rich phase diagrams including various non-trivial TMIs. By using the exact diagonalization, we obtain detailed bulk-global phase diagrams including the TMIs with high Chern numbers and also various non-topological phases. We also show that in adiabatic experimental setups, the strongly-interacting bosonic TMIs exhibit the topological particle transfer, i.e., the topological charge pumping phenomenon, similarly to weakly-interacting systems. The various TMIs are characterized by topological charge pumping as it is closely related to the Chern number, and therefore the Chern number is to be observed in feasible experiments.Most of the previous studies have focused on the existence of non-trivial topological phases. Some numerical studies by using the density-matrix renormalization group method (DMRG) confirmed the existence of the non-trivial topological phase called topological Mott insulator (TMI) [6,7,9]. TMI is classified by a topological number such as the Chern number and it has a gap in the bulk but a gapless excitation in the (spacial as well as phase diagram) boundaries. A quantum Monte-Carlo simulation was also carried out to detect the topological phase [10]. However, the existence was verified only in a limited parameter regime, and detailed global phase diagrams are still lacking. In particular, strongly-correlated bosonic topological states with high Chern number in the bulk have not been clarified yet in a global parameter regime, nor it is understood well how the competition between the superlattice potential and the on-site repulsion determines the ground state of the system. Also, there is one important question, i.e., how the topological phases in the obtained phase diagrams are related with the topological charge pumping as bulk topological properties.In this paper, we shall study the above problems in the strongly-interacting boson system by using the exact diagonalization [22,23], and show explicitly relation between the equilibrium topological phases and the topological charge pumping in the adiabatic process by following [24]. From the relation, the global phase diagram plays a role of a guide for detecting various topological charge pumping in the experiments.This paper is organized as follows. The target boson model in the 1D optical superlattice is explained in section 2. Section 3 studies the ground states of the system for the hard-core boson limit. We explain the single particle (SP) equation, which is related to the famous Harper equation in 2D ...

In the present paper, the Bose-Hubbard model (BHM) with the nearest-neighbor (NN) repulsions is studied from the view point of possible bosonic analogs of the fractional quantum Hall (FQH) state in the vicinity of the Mott insulator (MI). First, by means of the Gutzwiller approximation, we obtain the phase diagram of the BHM in a magnetic field. Then, we introduce an effective Hamiltonian describing excess particles on a MI and calculate the vortex density, momentum distribution and the energy gap. These calculations indicate that the vortex solid forms for small NN repulsions, but a homogeneous featureless 'Bose-metal' takes the place of it as the NN repulsion increases. We consider particular filling factors at which the bosonic FQH state is expected to form. Chern-Simons (CS) gauge theory to the excess particle is introduced, and a modified Gutzwiller wave function, which describes bosons with attached flux quanta, is introduced. The energy of the excess particles in the bosonic FQH state is calculated using that wave function, and it is compared with the energy of the vortex solid and Bose-metal. We found that the energy of the bosonic FQH state is lower than that of the Bose-metal and comparable with the vortex solid. Finally, we clarify the condition that the composite fermion appears by using CS theory on the lattice that we previously proposed for studying the electron FQH effect.

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.