  The toric code based on Majorana fermions on mesoscopic superconducting
islands is a promising candidate for quantum information processing. In the
limit of vanishing Cooper-pair tunneling, it has been argued that the phase
transition separating the topologically ordered phase of the toric code from
the trivial one is in the universality class of (2+1)D-XY. On the other hand,
in the limit of infinitely large Cooper-pair tunneling, the phase transition is
in the universality class of (2+1)D-Ising. In this work, we treat the case of
finite Cooper-pair tunneling and address the question of how the continuous XY
symmetry breaking phase transition turns into a discrete $\mathbb{Z}_2$
symmetry breaking one when the Cooper-pair tunneling rate is increased. We show
that this happens through a couple of tricritical points and first order phase
transitions. Using a Jordan-Wigner transformation, we map the problem to that
of spins coupled to quantum rotors and subsequently, propose a Landau field
theory for this model that matches the known results in the respective limits.
We calculate the effective field theories and provide the relevant critical
exponents for the different phase transitions. Our results are relevant for
predicting the stability of the topological phase in realistic experimental
implementations.
