Scaling of the gap, fidelity susceptibility, and Bloch oscillations across the superfluid-to-Mott-insulator transition in the one-dimensional Bose-Hubbard model

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.

Figure 1

(a) Contour plot of the sum of squared residuals $S(b,U_c)$ for $n=1$. The white arrow signals the location of the minimum value of $S$. The white lines are equally spaced contour lines where $S$ is constant. (b) Best collapse of the data for $E_g^{*}\left(L\right)$ vs $x_L$ corresponding to $U_c=3.279$, $b=5.2$, and $C\to \infty$. The inset shows the rescaled gap vs $U$. A similar analysis for $n=2$ [$n=3$] is presented in panels (c) and (d) [(e) and (f)]. $U$ and $E_g^{*}$ are presented in units of $J$, whereas $b$ and $S$ are shown in units of $J^{1/2}$ and $J^2$, respectively.

Figure 2

Fidelity susceptibility for different system sizes at integer filling $n=1$ (red symbols and dashed lines), $n=2$ (green symbols and solid lines), and $n=3$ (blue symbols and dot-dashed lines). The plot shows data for $L=40$ (square), $L=80$ (circle), and $L=120$ (triangle), the lines are spline interpolations and serve as a guide to the eye. The thick solid black lines (diamonds) are the result of a finite-size extrapolation using a quadratic fit. The vertical dotted lines indicate the position of the quantum critical points obtained using the scaling analysis of the gap described in the text. $U$ and $\chi$ are presented in units of $J$ and $J^{-2}$, respectively.

Figure 3

COM for (a) spinless fermions ($L=20$, $\Omega =1$) after tilting the lattice for the indicated values of $V=0,...,3$. (b) The BHM ($L=14$, $\Omega =1$) after tilting the lattice for the indicated values of $U=1,...,5$. The red stars and blue circles denote maxima and minima of the oscillations, respectively. The center of mass is presented in units of the lattice spacing $a$, while time $t$ is measured in units of $\hslash /J$.

Figure 4

Amplitude of the first oscillation of the COM after tilting the lattice. (a) Spinless fermions at half filling ($L=20$ and $L=40$) and $\Omega =1$. (b) BHM at filling $n=1$ ($L=14$ and $L=20$) and $\Omega =1$. The inset shows results for $L=14$ at filling $n=2$. (c) The logarithm of the amplitude square of the Fourier transform of the COM oscillations for spinless fermions at half filling with $L=20$ and $\Omega =1$. (d) Same as in (c) but for the BHM at $n=1$ with $L=14$ and $\Omega =1$. The amplitudes of COM oscillations are measured in units of $a$, $V$, and $U$ in units of $J$, and $\omega$ in units of $J/\hslash$