Quantum Critical Dynamics Simulation of Dirty Boson Systems

Recently, the scaling result $z=d$ for the dynamic critical exponent at the Bose glass to superfluid quantum phase transition has been questioned both on theoretical and numerical grounds. This motivates a careful evaluation of the critical exponents in order to determine the actual value of $z$. We study a model of quantum bosons at $T=0$ with disorder in 2D using highly effective worm Monte Carlo simulations. Our data analysis is based on a finite-size scaling approach to determine the scaling of the quantum correlation time from simulation data for boson world lines. The resulting critical exponents are $z=1.8\pm 0.05$, $\nu =1.15\pm 0.03$, and $\eta =-0.3\pm 0.1$, hence suggesting that $z=2$ is not satisfied.

Figure 1

Selection of Monte Carlo results for the winding number fluctuation divided by $L_{\tau }^2$ as a function of $L_{\tau }$. Solid curves are polynomial fits to the data curves, from which the locations $L_{\tau }^{*}$ and sizes $(W^2/L_{\tau }^2{)}^{*}$ of the maxima can be determined. Inset: Equilibration test for $L=40$, $L_{\tau }=240$, $K=0.2477$.

Figure 2

Maximum values ${\Phi }^{*}=(W^2/{\alpha }_{\tau }^2{)}^{*}$ vs system size $L$ for different couplings $K$. For $z=1.78$ the data obeys ${\Phi }^{*}=\mathrm{const}$ at $K=K_c$ for $L\ge 16$, indicated by the horizontal dashed line, which estimates $K_c=0.2477$. The lower dashed line corresponds to $z=2$, which approximately describes the data for small sizes $L<16$ at $K=0.246$.

Figure 3

(a) Intersection plot for the scaled winding number function maxima ${\Phi }^{*}$, showing an intersection at $K_c=0.2477$. (b) FSS data collapse of the data in (a) obtained for $\nu =1.15$, $L\ge 16$. (c) Intersection plot for the scaled susceptibility $\chi /L^{2-\eta }$ evaluated at ${\alpha }_{\tau }=0.35$. The data curves for large sizes roughly intersect at $K=0.2477$ for $\eta =-0.29$, but with much larger corrections visible for small system sizes than for ${\Phi }^{*}$. (d) FSS data collapse of $\chi /L^{2-\eta }$ for $\nu =1.15$, $L\ge 16$.

Figure 4

FSS data collapses of MC data for the scaled winding number fluctuation $\Phi =W^2/{\alpha }_{\tau }^2$ and susceptibility $\chi /L^{2-\eta }$ as functions of ${\alpha }_{\tau }=L_{\tau }/L^z$ for $K_c=0.2477$, $z=1.8$, $\eta =-0.29$. Solid curves are polynomial fits to the data. Inset: Dependence of $z$ on the range of system sizes used in the estimate. $L_{\min }$ indicates the smallest in a sequence of size quadruplets in $L=8$, 10, 12, 16, 20, 30, 40, 60 used to estimate $z$, except for $L_{\min }=30$ which indicates sizes 30, 40, 60.