Divergence of Dynamical Conductivity at Certain Percolative Superconductor-Insulator Transitions

Coarse-grained superconductor-insulator composites can be modeled as random inductor-capacitor (LC) networks, which exhibit percolative superconductor-insulator transitions (SITs). We use a simple and efficient algorithm to compute the dynamical conductivity $\sigma(\omega,p)$ of one type of LC network on large ($6144 \times 6144$) square lattices, where $\delta=p-p_c$ is the tuning parameter for the SIT [1]. We confirm that the conductivity obeys a scaling form near criticality, so that the characteristic frequency scales as $\Omega \propto \left|\delta\right|^{\nu z}$ with $\nu z \approx 1.91$, the superfluid stiffness scales as $\Upsilon \propto \left| \delta \right|^t$ with $t \approx 1.3$, and the electric susceptibility scales as $\chi_E \propto \left| \delta \right|^{-s}$ with $s = 2\nu z - t \approx 2.52$. In the insulating state, the low-frequency dissipative conductivity is exponentially small, whereas in the superconductor, it is linear in frequency. The sign of m $\sigma(\omega)$ at small $\omega$ changes across the SIT. Most importantly, right at the SIT, Re $\sigma(\omega) \propto \omega^{t/\nu z-1} \propto \omega^{-0.32}$, so that the quasi-dc conductivity $\sigma^*$ is infinite, in contrast with most other classical and quantum models of SITs.