Divergence of Dynamical Conductivity at Certain Percolative Superconductor-Insulator Transitions

Random inductor-capacitor (LC) networks can 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 (4000 x 4000) square lattices, where \delta=p-p_c is the tuning parameter for the SIT. We confirm that the conductivity obeys a scaling form, so that the characteristic frequency scales as \Omega ~ |\delta|^{\nu z} with \nu z \approx 1.91, the superfluid stiffness scales as \Upsilon ~ |\delta|^t with t \approx 1.3, and the electric susceptibility scales as \chi_E ~ |\delta|^{-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 Im \sigma(\omega) at small \omega changes across the SIT. Most importantly, we find that right at the SIT Re \sigma(\omega) ~ \omega^{t/\nu z-1} ~ \omega^{-0.32}, so that the conductivity diverges in the DC limit, in contrast with most other classical and quantum models of SITs.