Critical Exponents of Dynamical Conductivity in 2D Percolative Superconductor-Insulator Transitions: Three Universality Classes

We simulate three types of random inductor-capacitor (LC) networks on 4000x4000 lattices. We calculate the dynamical conductivity using an equation-of-motion method in which timestep error is eliminated and windowing error is minimized [1]. We extract the critical exponent $a$ such that $\sigma(\omega) \propto \omega^{-a}$ at low frequencies. The results suggest that there are three different universality classes. The $L_{ij} C_i$ model, with capacitances from each site to ground, has $a=0.32$. The $L_{ij} C_{ij}$ model, with capacitances along bonds, has $a=0$. The $L_{ij} C_i C_{ij}$ model, with both types of capacitances, has $a=0.30$. This implies that classical percolative 2D superconductor-insulator transitions (SITs) generically have $\sigma(\omega) \rightarrow \infty$ as $\omega \rightarrow 0$. Therefore, experiments that give a constant conductivity as $\omega \rightarrow 0$ must be explained in terms of quantum effects. $~~~~~~~$ [1. Yen Lee Loh, Rajesh Dhakal, John F. Neis and Evan M. Moen, ``Divergence of dynamical conductivity at certain percolative superconductor-insulator transitions'', Journal of Physics: Condensed Matter 26, 50 (2014)]