  At zero temperature, a two dimensional lattice of Majorana zero modes on
mesoscopic superconducting islands has a topologically ordered toric code
phase. Recently, a Landau field theory has been proposed for the system that
captures its different phases and the associated phase-transitions. It was
shown that with the increase of Josephson tunneling between the islands, a
continuous symmetry-breaking 3D-XY transition gets transformed into a discrete
symmetry-breaking 3D-Ising transition through a couple of tricritical points
and first order transitions. Using the proposed field theory, we analyze the
charge-response of the system at the different continuous phase-transitions. We
calculate the universal conductivity at the 3D-XY transitions and the change in
the superconducting density at the Ising transition using 1/N expansion.
Furthermore, by computing a one-loop correction to the field theory, we show
that an additional tricritical point is likely to be present in the
phase-diagram. Finally, we provide a mean-field calculation that supports the
earlier proposed field theory.

Mesoscale and Nanoscale Physics;Quantum Physics