Quantitative studies of the critical regime near the superfluid-to-Mott-insulator transition

We investigate the critical behaviors of correlation length and critical exponents for strongly interacting bosons in a two-dimensional optical lattice via quantum Monte Carlo simulations. By comparing the full numerical results to those given by the effective theory, we quantitatively determine the critical regime where the universal scaling behaviors apply at a finite temperature, for both classical Berezinskii-Kosterlitz-Thouless transition and quantum phase transition from superfluid to Mott insulator. Our results represent the critical regime that should be observed in present experimental conditions.

Figure 1

The semilog plot of the correlation function $G_{i,j}=〈a_i^{†}{a}_j〉$ as a function of distance $r\equiv |i-j|$ where $i=0$ at $T/U=0.01$ to 0.1 from top to bottom are shown at $J/U=0.04$ and $\mu /U=0$. The periodic boundary condition with a $100\times 100$ square lattice is applied. The exponential decay at $T/U=0.05$ to 0.1 is shown here, while the algebraic decay at $T/U=0.01$ to 0.04 can be identified in a log-log plot.

Figure 2

The correlation length $\xi$ as a function of temperature, where $\xi$ is fitted as $G_{i,j}\propto exp(-r/\xi )$ at $T$ above $T_c$, $J/U=0.04$, and $\mu /U=0$ in a $100\times 100$ square lattice with periodic boundary condition. In the inset, the semilog plot of $\xi$ as a function of ${(T-T_c)}^{-1/2}$, the divergent law of $\xi$ as $T$ is decreasing, is shown to be $ln\left(\xi \right)\propto {(T-T_c)}^{-1/2}$, which agrees with the one in the BKT transition regime [39]. The $T_c$ is defined as the temperature that $〈W^2〉=4/\pi$.

Figure 3

The critical exponent $\eta$ as a function of T plotted at $\mu /U=0$ and $J/U=0.04$ (a) and 0.3 (b). The filled marks with solid curves are the slopes from fitting from the log-log plot of the correlation function as $G_{i,j}\propto r^{-{\eta }_{\mathrm{num}}}$. The blank marks with dotted curves are ${\eta }_{\mathrm{BKT}}$ calculated from superfluid density.

Figure 4

The log-log plot of $\xi$ to $|J-J_c|/U$ at $\mu /U=0.37$ and $T/U=0.001$ in a $128\times 128$ square lattice. The universality class of the 3D-XY model is characterized by the relation $\xi \propto |J-J_c{|}^{-\nu }$, where $\nu \approx 0.6715$ is the correlation length exponent as the solid line.

Figure 5

(a) The semilog plot of $\xi$ to ${(T-T_c)}^{-1/2}$ at $J/U=0.05974$ in a $128\times 128$ square lattice. The inset shows the locations of the parameters in a zero-temperature phase diagram. The BKT critical behavior of correlation length, $ln\left(\xi \right)\propto {(T-T_c)}^{-1/2}$, is linear in this plot. $\mu /U=0,0.1,0.2,0.3,0.33,0.35,0.37$, from bottom to top, respectively. (b) The log-log plot of $\xi$ to $T$ at the same parameters as (a). The quantum critical behavior of correlation length, $\xi \propto 1/T$, is linear in this plot and well followed as $\mu /U\to 0.37\sim {\mu }_c/U$.

Figure 6

(a) Temperature scales as a function of chemical potential, $\mu /U$, at $J=J_c=0.05974U$. Blue filled circles are the true phase transition boundary, $T_c$, while the brown diamonds indicate the upper bound for the BKT scaling regime, i.e., $T_{\xi }^{\text{BKT}}$. $ln\left(\xi \right)\propto {(T-T_c)}^{-1/2}$ up to 99% accuracy in the yellow shaded regime between the top two curves. Red squares indicate $T_{\xi }^{\text{QC}}$, above which (pink regime) is the universal scaling regime ($\xi \propto 1/T$) near the quantum critical point. (b) Temperature scales as a function of tunneling amplitude, $J/U$, at $\mu ={\mu }_c=0.37$. The left-hand side and right-hand side of $J_c=0.05974U$ are, respectively, the Mott insulator and superfluid phases at zero temperature. All other notations are the same as in (a). The regime near the quantum critical point is maximized in the insets, where we only show the $T_c$ and $T_{\xi }^{\text{QC}}$ for clearness. Note that the $T_c$ is shown slightly above zero at $(\mu ,J)=({\mu }_c,J_c)$ due to the numerical limitation. The overlap of the BKT regime and quantum critical regime near $J=J_c$ is simply due to the 1% tolerance of the numerical fitting results, without any physical implication.

Figure 7

The superfluid density, condensate density, and $〈a_0^{†}{a}_{L/2}〉$ as a function of temperature at $\mu /U=0$ and $J/U=0.04$ in a $100\times 100$ square lattice. The black squares are superfluid density obtained by measuring winding number, the red circles are the condensate density by taking $k=0$ in the Fourier transform of the correlation function, and blue triangles are $〈a_0^{†}{a}_{49}〉$.