Critical Properties of the Superfluid—Bose-Glass Transition in Two Dimensions

We investigate the superfluid (SF) to Bose-glass (BG) quantum phase transition using extensive quantum Monte Carlo simulations of two-dimensional hard-core bosons in a random box potential. $T=0$ critical properties are studied by thorough finite-size scaling of condensate and SF densities, both vanishing at the same critical disorder $W_c=4.80(5)$. Our results give the following estimates for the critical exponents: $z=1.85(15)$, $\nu =1.20(12)$, $\eta =-0.40(15)$. Furthermore, the probability distribution of the SF response $P(\ln {\rho }_{\mathrm{SF}})$ displays striking differences across the transition: while it narrows with increasing system sizes $L$ in the SF phase, it broadens in the BG regime, indicating an absence of self-averaging, and at the critical point $P(\ln {\rho }_{\mathrm{SF}}+z\ln L)$ is scale invariant. Finally, high-precision measurements of the local density rule out a percolation picture for the SF-BG transition.

Figure 1

Scaling analysis of the SF ${\rho }_{\mathrm{SF}}$ (top) and BEC ${\rho }_0$ (bottom) densities. Solid lines show best fits to the universal scaling functions Eqs. (2) and (3) for the full data set with $z\simeq 1.85$, $W_c^{\mathrm{SF}}\simeq 4.8$, $W_c^0\simeq 4.79$, $z+\eta \simeq 1.42$, ${\nu }_{\mathrm{SF}}\simeq 1.1$, ${\nu }_0\simeq 1.2$, and $G_{\mathrm{SF}|0}$ 3rd order polynomials. The distance from the critical point $W_c=4.80(5)$ (grey area), when rescaled by $L^{-1/\nu }$ with $\nu =1.2$, yields a perfect data collapse (insets).

Figure 2

Histogram of QMC estimates for $\ln {\rho }_{\mathrm{SF}}$ performed over ${\mathcal{N}}_s\sim {10}^4$ disordered samples for each size $L$. (a) In the SF regime $W=4$, distributions get narrower with increasing $L$ whereas in the BG phase (d) for $W=4.95$ they broaden. At criticality $W_c=4.8$ [(b),(c)] the broadening stops above $L=20$ and $P(\ln {\rho }_{\mathrm{SF}}+z\ln L)$ displays a good collapse using $z=1.85$. The insets show the ratio $\mathcal{R}={\rho }_{\mathrm{SF}}^{\text{avg}}/{\rho }_{\mathrm{SF}}^{\text{typ}}$ vs system size $L$.

Figure 3

Typical SF density (a) plotted as ${\rho }_{\mathrm{SF}}^{\text{typ}}\times L^z$ vs $W$ where the crossing at $W_c=4.8$ has a visible drift, captured by ${\mathcal{G}}_{\mathrm{SF}}^{\text{typ}}[L^{1/\nu }(W-W_c)]+c{L}^{-y}$ with fixed $\nu =1.2$ and $z=1.85$, and an estimated irrelevant exponent $y=0.97(4)$. In panel (b), ${\rho }_{\mathrm{SF}}^{\text{typ}}/{\rho }_{\mathrm{SF}}^{\text{typ},W_c}$ plotted against $L|W-W_c{|}^{\nu }$ exhibits an almost perfect collapse of the data for $4\le W\le 5.5$ and $12\le L\le 32$ with no additional parameters.

Figure 4

Corrected standard deviation of the logarithm of the SF response ${\tilde{\sigma }}_{\ln {\rho }_{\mathrm{SF}}}={\sigma }_{\ln {\rho }_{\mathrm{SF}}}-{\sigma }^c$ vs system size in units of the typical length scale $\xi =|W-W_c{|}^{-\nu }$. In the SF phase $W<4.8$, $\tilde{\sigma }$ tends to $-{\sigma }^c$ as $1/L$ (dashed line), whereas in the BG regime $W>4.8$, $\tilde{\sigma }$ grows as $L^{\omega }$ (full black line) with $\omega =0.5(2)$. Inset: zoom on the BG regime.

Figure 5

Local particle (hole) densities ${\rho }_i$ ($1-{\rho }_i$) plotted vs local chemical potentials $|{\mu }_i|$. Small blue (orange) points show QMC results of ${\mathcal{N}}_s=150$ samples of size $16\times 16$ measured at $\beta t=1024$ with $N_{\mathrm{MC}}={10}^5$ steps. The red data points show averages over windows of the chemical potential of size $\overline{\mu }-0.1\le {\mu }_i\le \overline{\mu }+0.1$ and the clean ($W=0$) result is shown by the dashed line, yielding exactly zero for $|\mu |>2$ for $\mathrm{Min}({\rho }_i,1-{\rho }_i)$ [69]. The inset quantifies the incompressible fraction, i.e., the fraction of sites with ${\rho }_i=0$ or ${\rho }_i=1$ as a function of MC steps showing that in the exact limit of infinite Markov chains the incompressible fraction tends to zero.