Finite-size scaling for quantum criticality above the upper critical dimension: Superfluid–Mott-insulator transition in three dimensions

The validity of modified finite-size scaling above the upper critical dimension is demonstrated for the quantum phase transition whose dynamical critical exponent is $z=2$. We consider the $N$-component Bose-Hubbard model, which is exactly solvable and exhibits mean-field type critical phenomena in the large-$N$ limit. The modified finite-size scaling holds exactly in that limit. However, the usual procedure, taking the large system-size limit with fixed temperature, does not lead to the expected (and correct) mean-field critical behavior because of the limited range of applicability of the finite-size scaling form. By quantum Monte Carlo simulation, it is shown that the modified finite-size scaling holds in the case of $N=1$.

Figure 1

(Color online) MFSS plots of single-component BH model where $\mu /U=0.1$ and $\beta t/L^2=0.375$. $\delta \equiv t/U-{\left(t/U\right)}_c$ with ${\left(t/U\right)}_c=0.088935$. (a) Compressibility, (b) susceptibility, and (c) superfluid density.

Figure 2

(Color online) Zero-temperature phase diagram of single-component BH model in three dimensions. Almost all the quantum critical points ($L=8$,12, and 16), which is estimated using the Mott gap, is from our previous paper [6]. FSS indicates the result of MFSS. Inset is an enlarged view of the region near the critical point estimated by MFSS in Fig. 1.

Figure 3

(Color online) Temperature dependence of $\chi$ at the quantum critical point estimated by MFSS $\left[\mu /U=0.1,{\left(t/U\right)}_c=0.088935\right]$. The solid line is $A{\left(T/t\right)}^{-3/2}$, where $A=0.89$. The data points are obtained for $L=24$, 32, and 48. There is no visible size dependence on this scale. The statistical error is smaller than the size of symbols.

Figure 4

(Color online) MFSS plots of susceptibility at $U/\left(t/Z\right)=1$. (a) The $x$ dependence of scaling function of susceptibility $P_{\chi }^{UZ/t}\left(x,y\right)$ and the solution of self-consistent Eq. (13) with $y\equiv \beta t/\left(Z{L}^2\right)=1$, (b) $y$ dependence of $P_{\chi }^{UZ/t}\left(x,y\right)$ and the solution of self-consistent Eq. (13) at quantum critical point $\delta =0$.