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arXiv:1110.4308 · Physical Review A

Superfluid-to-Mott-insulator transition in the one-dimensional Bose-Hubbard model for arbitrary integer filling factors

Ippei Danshita, Anatoli Polkovnikov

Published Dec 28, 2011• 40 citations

Abstract

We study the quantum phase transition between the superfluid and the Mott insulator in the one-dimensional (1D) Bose-Hubbard model. Using the time-evolving block-decimation method, we numerically calculate the tunneling splitting of two macroscopically distinct states with different winding numbers. From the scaling of the tunneling splitting with respect to the system size, we determine the critical point of the superfluid-to-Mott-insulator transition for arbitrary integer filling factors. We find that the critical values versus the filling factor in 1D, 2D, and 3D are well approximated by a simple analytical function. We also discuss the condition for determining the transition point from a perspective of the instanton method.

DOI: 10.1103/physreva.84.063637

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We study the quantum phase transition between the superfluid and the Mott insulator in the one-dimensional (1D) Bose-Hubbard model. Using the time-evolving block-decimation method, we numerically calculate the tunneling splitting of two macroscopically distinct states with different winding numbers. From the scaling of the tunneling splitting with respect to the system size, we determine the critical point of the superfluid-to-Mott-insulator transition for arbitrary integer filling factors. We find that the critical values versus the filling factor in 1D, 2D, and 3D are well approximated by a simple analytical function. We also discuss the condition for determining the transition point from a perspective of the instanton method.

We study the transition from a Mott insulator to a superfluid in both the two- and the three-dimensional Bose-Hubbard model at zero temperature, employing the method of the effective potential. Converting Kato's perturbation series into an algorithm capable of reaching high orders, we obtain accurate critical parameters for any integer filling factor. Our technique allows us to monitor both the approach to the mean-field limit by considering spatial dimensionalities

d>3

and to the quantum rotor limit of high filling, which refers to an array of Josephson junctions.

We study the ground state property of the one-dimensional Bose—Hubbard model using an imaginary time evolving body decimation algorithm. The single-particle density matrix is numerically calculated for a Mott insulating system and a superfluid system separately. By plotting the chemical potential versus the filling n = N/L for U/J = 20 and U/J = 0.1, we identify the Mott gap for U/J = 20 in filling n = 1. Lastly, we investigate the occupation number of the Bloch state with quasimomentum for a system deep in the Mott phase and in the superfluid phase respectively. The results indicate Bose condensation in the quasimomentum space.

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 report results of quantum Monte Carlo simulations of the Bose-Hubbard model in three dimensions. Critical parameters for the superfluid-to-Mott-insulator transition are determined with significantly higher accuracy than has been done in the past. In particular, the position of the critical point at filling factor

n=1

is found to be at

{(U∕t)}_{\mathrm{c}}=29.34(2)

, and the insulating gap $\ensuremath{\Delta}$ is measured with accuracy of a few percent of the hopping amplitude

t

. We obtain the effective mass of particle and hole excitations in the insulating state---with explicit demonstration of the emerging particle-hole symmetry and relativistic dispersion law at the transition tip---along with the sound velocity in the strongly correlated superfluid phase. These parameters are the necessary ingredients to perform analytic estimates of the low temperature $(T⪡\ensuremath{\Delta})$ thermodynamics in macroscopic samples. We present accurate thermodynamic curves, including these for specific heat and entropy, for typical insulating

(U∕t=40)

and superfluid

(t∕U=0.0385)

phases. Our data can serve as a basis for accurate experimental thermometry, and a guide for appropriate initial conditions if one attempts to use interacting bosons in quantum information processing.