Universal Conductivity in a Two-Dimensional Superfluid-to-Insulator Quantum Critical System

We compute the universal conductivity of the ($2+1$)-dimensional $XY$ universality class, which is realized for a superfluid-to-Mott insulator quantum phase transition at constant density. Based on large-scale Monte Carlo simulations of the classical ($2+1$)-dimensional $J$-current model and the two-dimensional Bose-Hubbard model, we can precisely determine the conductivity on the quantum critical plateau, $\sigma (\infty )=0.359(4){\sigma }_Q$ with ${\sigma }_Q$ the conductivity quantum. The universal conductivity curve is the standard example with the lowest number of components where the bottoms-up AdS/CFT correspondence from string theory can be tested and made to use [R. C. Myers, S. Sachdev, and A. Singh, Phys. Rev. D 83, 066017 (2011)]. For the first time, the shape of the $\sigma (i{\omega }_n)-\sigma (\infty )$ function in the Matsubara representation is accurate enough for a conclusive comparison and establishes the particlelike nature of charge transport. We find that the holographic gauge-gravity duality theory for transport properties can be made compatible with the data if temperature of the horizon of the black brane is different from the temperature of the conformal field theory. The requirements for measuring the universal conductivity in a cold gas experiment are also determined by our calculation.

Figure 1

Statistics of winding numbers squared for different system sizes $L=L_{\tau }$. Fitting the data using $a+b(K-K_c)L^{1/\nu }+c{L}^{-\omega }$ with $\nu =0.6717(3)$ and $a=0.5160(6)$ [19] leads to the critical point $K_c=0.3330670(2)$ of the $J$-current model [Eq. (2)].

Figure 2

Thermodynamic limit data for the conductivity in Matsubara representation for increasing values of $L_{\tau }$, bottom to top. The universal result is obtained by extrapolating the data to the $L_{\tau }\to \infty$ limit (black filled circles).

Figure 3

Comparison between the numerical result for the universal conductivity in Matsubara space and holographic conductivity evaluated at complex frequencies. We consider only the largest possible value of the holographic parameter $\gamma =1/12$ because it offers the best fit. The holographic curve can be made to fit the data much better if the temperature is scaled by a factor of 2.5. We also show the fit based on the 3rd order polynomial described in the text. The inset presents the possible universal real frequency conductivity curves obtained by analytical continuation of fitting functions.

Figure 4

Universal result for the conductivity in Matsubara representation from quantum Monte Carlo simulations for the Bose-Hubbard model. Temperature decreases from the bottom to the top, and the extrapolation to zero temperature is shown by black squares.

Figure 5

Real frequency conductivity of the Bose-Hubbard system at $T/t=0.2$ and $L=100$. Both MaxEnt and consistent constraints analytical continuation methods [23] are tested.