Critical Exponents of the Superfluid–Bose-Glass Transition in Three Dimensions

Recent experimental and numerical studies of the critical-temperature exponent $\varphi$ for the superfluid–Bose-glass universality in three-dimensional systems report strong violations of the key quantum critical relation, $\varphi =\nu z$, where $z$ and $\nu$ are the dynamic and correlation-length exponents, respectively; these studies question the conventional scaling laws for this quantum critical point. Using Monte Carlo simulations of the disordered Bose-Hubbard model, we demonstrate that previous work on the superfluid-to-normal-fluid transition-temperature dependence on the chemical potential (or the magnetic field, in spin systems), $T_c\propto (\mu -{\mu }_c{)}^{\varphi }$, was misinterpreting transient behavior on approach to the fluctuation region with the genuine critical law. When the model parameters are modified to have a broad quantum critical region, simulations of both quantum and classical models reveal that the $\varphi =\nu z$ law [with $\varphi =2.7(2)$, $z=3$, and $\nu =0.88(5)$] holds true, resolving the $\varphi$-exponent “crisis.”

Figure 1

Critical temperature of the hard-core Bose-Hubbard model as a function of chemical potential for disorder strength $\mathrm{\Delta }/t=16$ fitted to the $T_c=A(\mu -{\mu }_c{)}^{1.1}$ power law. The dashed line is to guide the eye.

Figure 2

Density at the thermal critical point of model 1 as a function of chemical potential for $\mathrm{\Delta }/t=16$. The dashed line is a linear fit.

Figure 3

Finite-size scaling plots for $⟨W^2⟩=\tilde{f}(L^{1/\nu }\delta )$ for system sizes $L=12$ (black), $L=14$ (red), $L=16$ (blue), $L=18$ (magenta), and $L=20$ (green) with fixed ratio $L_{\tau }=2L^3$. Data points are fitted with second-order polynomials. We do not observe corrections to scaling within our error bars.

Figure 4

Deducing $1/\nu$ from the linear fit of $\ln |\partial ⟨W^2⟩/\partial \mathrm{\Delta }|$ as a function of $\ln L$ using four points near the critical point, $\mathrm{\Delta }=8.8,9.0,9.2$, and 9.4. Error bars are based on the uncertainty of the fitting procedure, given the data points and their statistical error bars in Fig. 3.

Figure 5

Critical temperature of the $J$-current model as a function of disorder strength. Solid line is the power-law fit to the lowest transition temperatures assuming a known location of the quantum critical point. Dashed line is a power law originating from $\mathrm{\Delta }=8.83$.

Figure 6

Critical temperature dependence on disorder strength in the hard-core DBH at half-integer filling factor. The solid line is a fit of the last five points to the $A({\mathrm{\Delta }}_c-\mathrm{\Delta }{)}^{\varphi }$ law, with exponent $\varphi =2.7$ fixed at the value determined from simulations of the $J$-current model. From this fit we predict that the quantum critical point is located at ${\mathrm{\Delta }}_c\approx 24.67$. Error bars are shown, but they are smaller than the symbol size. Inset: Zoom in on the tail of the main plot.