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

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.

Figure 1

The time evolution of the current velocity $v\left(t\right)$ in the dynamics of the model of Eq. (7), where $L=40$ and $V/J=1.8$.

Figure 2

The time evolution of the overlaps $O_1\left(t\right)$ (blue solid line) and $O_{-1}\left(t\right)$ (black dashed line) in the dynamics of the model of Eq. (7), where $L=40$ and $V/J=1.8$.

Figure 3

The energy splitting versus the number of lattice sites $L$ for the 1D hardcore-Bose-Hubbard model of Eq. (7) at half filling. We take $V/J=1.8$ (black diamonds), $2.0$ (blue squares), and $2.2$ (red diamonds). The solid lines represent the best fits to the respective numerical data with the function of Eq. (6), where $(A,B)=(2.80,1.11)$, $(2.28,0.965)$, and $(1.84,0.823)$ for $V/J=1.8$, $2.0$, and $2.2$, respectively.

Figure 4

The exponent $B$ versus $V/J$ for the 1D hardcore-Bose-Hubbard model of Eq. (7) at half filling. $B$ is extracted by fitting the numerical data of $\Delta$ versus $L$ up to $L=100$. We use the fitting function of Eq. (6) with the logarithmic correction for red squares while we use Eq. (10) without the correction for the black circles. The solid lines are guides for the eyes. The dashed line represents $B=1$, which is the condition to determine the transition point.

Figure 5

The red circles represent the critical point ${[U/\left(\nu J\right)]}_c$ versus the filling factor. The solid line represents the best fit of the numerical data with the function of Eq. (2), where the numerical constants are obtained as $(a,b,c)=(2.16,0.97,2.13)$. The transverse axis is depicted in a logarithmic scale. Notice that the critical point exists only at integer fillings although the solid line is continuous. In the inset, the same data are presented together with the transition point calculated with the analytical formula of Eq. (12) (green dashed line).

Figure 6

The critical value ${[U/\left(D\nu J\right)]}_c$ versus the filling factor $\nu$ for 2D (blue squares) and 3D (black diamonds). The solid lines represent the best fit of the numerical data with the function of Eq. (2). The dashed lines represent the transition points calculated from the analytical formula of Eq. (12). The transverse axis is depicted in a logarithmic scale. The numerical data for 2D and 3D are taken from Ref. [17]. Notice that the critical point exists only at integer fillings although the solid line is continuous.