Quantum criticality in a dissipative (2+1)-dimensional $XY$ model of circulating currents in high-$T_c$ cuprates

We present large-scale Monte Carlo results for the dynamical critical exponent $z$ and the spatio-temporal two-point correlation function of a (2+1)-dimensional quantum $XY$ model with bond dissipation, proposed to describe a quantum critical point in high-$T_c$ cuprates near optimal doping. The phase variables of the model, originating with a parametrization of circulating currents within the CuO${}_2$ unit cells in cuprates, are compact, $\left\{{\theta }_{\mathbf{r},\tau }\right\}\in [-\pi ,\pi \rangle$. The dynamical critical exponent is found to be $z\approx 1$, and the spatio-temporal correlation functions are explicitly demonstrated to be isotropic in space-imaginary time. The model thus has a fluctuation spectrum where momentum and frequency enter on equal footing, rather than having the essentially momentum-independent marginal Fermi-liquid-like fluctuation spectrum previously reported for the same model.

Figure 1

(Color online) Finite-size analysis of the maximum $L_{\tau }^{*}$ of the Binder cumulant curves $g\left(L_{\tau }\right)$ as a function of spatial system size $L$. For the black data points, the dynamical critical exponents $z$ as given in Table are obtained from the slope of the fitting lines (dashed). The red (gray) points show similar results for site dissipation for comparison, where a fit of the three largest systems yields $z=1.84\left(3\right)$.

Figure 2

(Color online) Correlation functions at the critical point $K_c=0.28244$ for dissipation strength $\alpha =0.05$ and quantum coupling $K_{\tau }=0.6$. The system size $L=32$, $L_{\tau }=49\approx L_{\tau }^{*}$ corresponds to the rightmost data point of the midmost data series in Fig. 1. The correlation function is defined in the spatial direction as $g\left(x\right)=g\left(\right|\mathbf{r}-{\mathbf{r}}^{\prime }\left|\right)=C(\mathbf{r}-{\mathbf{r}}^{\prime },0)$ and in the temporal direction as $g\left(\tau \right)=C(\mathbf{0},\tau )$, with $C$ defined in Eq. (2). Also, $L_x\equiv L$. Error bars are smaller than the linewidth, and the dotted lines are guides to the eye.