We investigate the equilibrium and dynamical properties of the Bose-Hubbard model and the related particle-hole symmetric spin-1 model in the vicinity of the superfluid to Mott insulator quantum phase transition. We employ the following methods: exact-diagonalization, mean-field (Gutzwiller), cluster mean-field, and mean-field plus Gaussian fluctuations. In the first part of the paper we benchmark the four methods by analyzing the equilibrium problem and give numerical estimates for observables such as the density of double occupancies and their correlation function. In the second part, we study parametric ramps from the superfluid to the Mott insulator and map out the crossover from the regime of fast ramps, which is dominated by local physics, to the regime of slow ramps with a characteristic universal power law scaling, which is dominated by long wavelength excitations. We calculate values of several relevant physical observables, characteristic time scales, and an optimal protocol needed for observing universal scaling.
We numerically study the superfluid to Mott insulator transition for bosonic atoms in a one-dimensional lattice by exploiting a recently developed simulation method for strongly correlated systems. We demonstrate this method's accuracy and applicability to Bose-Hubbard model calculations by comparison with exact results for small systems. By utilizing the efficient scaling of this algorithm we then concentrate on systems of comparable size to those studied in experiments and in the presence of a magnetic trap. We investigate spatial correlations and fluctuations of the ground state as well as the nature and speed at which the superfluid component is built up when dynamically melting a Mott insulating state by ramping down the lattice potential. This is performed for slow ramping, where we find that the superfluid builds up on a time scale consistent with single-atom hopping and for rapid ramping where the buildup is much faster than can be explained by this simple mechanism. Our calculations are in remarkable agreement with the experimental results obtained by Greiner et al. [Nature (London) 415, 39 (2002)].
We investigate the temperature-dependent behavior emerging in the vicinity of the superfluid (SF) to Mott insulator (MI) transition of interacting bosons in a two-dimensional optical lattice, described by the Bose-Hubbard model. The equilibrium phase diagram at finite temperatures is computed by means of the cluster mean-field theory (CMF) where the effect of non-local correlations is analyzed systematically by finite-size scaling of the cluster size. The phase diagram exhibits a rich structure including a transition and a crossover of the SF and MI phases respectively to a normal fluid (NF) state at finite temperature. In order to characterize these phases, and the NF transition and crossover scales, we calculate, in addition to the condensate amplitude, the superfluid fraction, sound velocity and compressibility. The phase boundaries obtained by CMF with finite-size scaling agree quantitatively with quantum Monte Carlo (QMC) results as well as with experiments. The von Neumann entanglement entropy of a cluster exhibits critical enhancement near the SF-MI quantum critical point (QCP). We also discuss the behavior of the transition lines near this QCP at the particle-hole symmetric point located at the tip of a Mott lobe as well as away from particle-hole symmetry.
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.
We develop a projection operator formalism for studying both the zero-temperature equilibrium phase diagram and the nonequilibrium dynamics of the Bose-Hubbard model. Our work, which constitutes an extension of that of Trefzger and Sengupta [Phys. Rev. Lett. 106, 095702 (2011)], shows that the method provides an accurate description of the equilibrium zero-temperature phase diagram of the Bose-Hubbard model for several lattices in two and three dimensions. We show that the accuracy of this method increases with the coordination number
z0 of the lattice and reaches to within
0.5 of quantum Monte Carlo data for lattices with
z0=6. We compute the excitation spectra of the bosons using this method in the Mott and the superfluid phases and compare our results with mean-field theory. We also show that the same method may be used to analyze the nonequilibrium dynamics of the model both in the Mott phase and near the superfluid-insulator quantum critical point where the hopping amplitude
J and the on-site interaction
U satisfy ${z}_{0}J/U\ensuremath{\ll}1$. In particular, we study the nonequilibrium dynamics of the model both subsequent to a sudden quench of the hopping amplitude
J and during a ramp from
Ji to
Jf characterized by a ramp time $\ensuremath{\tau}$ and exponent $\ensuremath{\alpha}$: $J(t)={J}_{i}+({J}_{f}\ensuremath{-}{J}_{i}){(t/\ensuremath{\tau})}^{\ensuremath{\alpha}}$. We compute the wave function overlap
F, the residual energy
Q, the superfluid order parameter $\ensuremath{\Delta}(t)$, the equal-time order parameter correlation function
C(t), and the defect formation probability
P for the above-mentioned protocols and provide a comparison of our results to their mean-field counterparts. We find that
Q,
F, and
P do not exhibit the expected universal scaling. We explain this absence of universality and show that our results for linear ramps compare well with the recent experimental observations.
The Bose-Hubbard model describes the physics of a system of bosonic ultracold atoms in an optical lattice, in which a phase transition is present between a superfluid phase and a Mott insulator one. \nThe exact solution of this Hamiltonian is only feasible to find the ground-state of small systems, while other techniques (as mean-field schemes or quantum Monte Carlo) are necessary to study systems of larger size. \n \nAs a first application, we study the trapped model - relevant for the comparison with current experiments - through an inhomogeneous mean-field scheme. We describe some signatures of the phase crossover between superfluid and Mott insulator. In particular, the visibility of the quasimomentum distribution shows some kinks as a function of the lattice depth; we describe these features and we link them with the ones observed in other works in the literature. \n \nAs a second application, we use quantum Monte Carlo techniques to study the one-dimensional Bose-Hubbard model with long-range interactions and we focus on the appearance of the Haldane insulating phase, distinguishable from the Mott one through the presence of non-local hidden order. \nNon-local correlation functions are also used to describe the difference between the superfluid phase and the Mott insulator one.
Spectral properties of the Bose-Hubbard model and a recently proposed coupled-cavity model are studied by means of quantum Monte Carlo simulations in one dimension. Both models exhibit a quantum phase transition from a Mott insulator to a superfluid phase. The dynamic structure factor $S(k,\ensuremath{\omega})$ and the single-particle spectrum $A(k,\ensuremath{\omega})$ are calculated, focusing on the parameter region around the phase transition from the Mott insulator with density one to the superfluid phase, where correlations are important. The strongly interacting nature of the superfluid phase manifests itself in terms of additional gapped modes in the spectra. Comparison is made to recent analytical work on the Bose-Hubbard model. Despite some subtle differences due to the polaritonic particles in the cavity model, the gross features are found to be very similar to the Bose-Hubbard case. For the polariton model, emergent particle-hole symmetry near the Mott lobe tip is demonstrated and temperature and detuning effects are analyzed. A scaling analysis for the generic transition suggests mean-field exponents, in accordance with field-theory results.