Universal conductivity at a 2d superconductor-insulator transition: the effects of quenched disorder and Coulomb interaction

We calculate the zero-temperature universal electrical conductivity at a superconductor-insulator transition in two spatial dimensions. We focus on transitions in the universality class of the dirty 3d XY model. We use a dual model consisting of a single Dirac fermion at zero density coupled to a Chern-Simons gauge field in the presence of a quenched random mass, with or without an unscreened Coulomb interaction. Our calculation is performed in a $1\mathrm{/}{N}_f$ expansion, where $N_f$ is the number of Dirac fermions. At zeroth order, the model exhibits particle-vortex self-dual electrical transport with ${\sigma }_{xx}\lesssim (2e{)}^2\mathrm{/}h$ and small, but finite ${\sigma }_{xy}$. Corrections of $\mathcal{O}(1\mathrm{/}{N}_f)$ due to fluctuations in the Chern-Simons gauge field and disorder produce violations of self-duality. We find these violations to be milder when the Coulomb interaction is present.