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 use the finite-size, density-matrix-renormalization-group (FSDMRG) method to obtain the phase diagram of the one-dimensional
(d=1) extended Bose-Hubbard model for density $\ensuremath{\rho}=1$ in the $U\ensuremath{-}V$ plane, where
U and
V are, respectively, onsite and nearest-neighbor interactions. The phase diagram comprises three phases: superfluid (SF), Mott insulator (MI), and mass-density-wave (MDW). For small values of
U and
V, we get a reentrant SF-MI-SF phase transition. For intermediate values of interactions the SF phase is sandwiched between MI and MDW phases with continuous SF-MI and SF-MDW transitions. We show, by a detailed, finite-size scaling analysis, that the MI-SF transition is of Kosterlitz-Thouless (KT) type whereas the MDW-SF transition has both KT and two-dimensional Ising characters. For large values of
U and
V we get a direct, first-order, MI-MDW transition. The MI-SF, MDW-SF, and MI-MDW phase boundaries join at a bicritical point at $(U,V)=(8.5\ifmmode\pm\else\textpm\fi{}0.05,4.75\ifmmode\pm\else\textpm\fi{}0.05)$.
The validity of modified finite-size scaling above the upper critical dimension is demonstrated for the quantum phase transition whose dynamical critical exponent is z=2. We consider the N -component Bose-Hubbard model, which is exactly solvable and exhibits mean-field type critical phenomena in the large- N limit. The modified finite-size scaling holds exactly in that limit. However, the usual procedure, taking the large system-size limit with fixed temperature, does not lead to the expected (and correct) mean-field critical behavior because of the limited range of applicability of the finite-size scaling form. By quantum Monte Carlo simulation, it is shown that the modified finite-size scaling holds in the case of N=1.
We investigate the interaction-induced superfluid-to-Mott-insulator transition in the one-dimensional Bose-Hubbard model (BHM) for fillings
n=1,
n=2, and
n=3 by studying the single-particle gap, the fidelity susceptibility, and the amplitude of Bloch oscillations via density-matrix renormalization-group methods. We apply a generic scaling procedure for the gap, which allows us to determine the critical points with very high accuracy. We also study how the fidelity susceptibility behaves across the phase transition. Furthermore, we show that in the BHM, and in a system of spinless fermions, the amplitude of Bloch oscillations after a tilt of the lattice vanishes at the critical points. This indicates that Bloch oscillations can serve as a tool to detect the transition point in ongoing experiments with ultracold gases.
We study the quantum (zero-temperature) critical behaviors of confined particle systems described by the one-dimensional (1D) Bose-Hubbard model in the presence of a confining potential, at the Mott insulator to superfluid transitions, and within the gapless superfluid phase. Specifically, we consider the hard-core limit of the model, which allows us to study the effects of the confining potential by exact and very accurate numerical results. We analyze the quantum critical behaviors in the large trap-size limit within the framework of the trap-size scaling (TSS) theory, which introduces a new trap exponent $\ensuremath{\theta}$ to describe the dependence on the trap size. This study is relevant for experiments of confined quasi-1D cold atom systems in optical lattices. At the low-density Mott transition, TSS can be shown analytically within the spinless fermion representation of the hard-core limit. The trap-size dependence turns out to be more subtle in the other critical regions, when the corresponding homogeneous system has a nonzero filling
f, showing an infinite number of level crossings of the lowest states when increasing the trap size. At the
n=1 Mott transition, this gives rise to a modulated TSS: The TSS is still controlled by the trap-size exponent $\ensuremath{\theta}$, but it gets modulated by periodic functions of the trap size. Modulations of the asymptotic power-law behavior is also found in the gapless superfluid region, with additional multiscaling behaviors.