Quantum criticality in a dissipative (2+1)-dimensional $XY$ model of circulating currents in high-$T_{\rm c}$ 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, $\{ \theta_{vvr,\tau} \}$ $\in [-\pi,\pi \rangle$. 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.