Topological Ordering in the Majorana Toric Code

At zero temperature, a two-dimensional lattice of Majorana zero modes on mesoscopic superconducting islands has a Z2 topologically-ordered toric code phase. Recently, a Landau field theory was proposed for this system that describes its phases and the different phase-transitions separating them. The system is in the toric code phase as a Mott insulator and a charge-2e superconductor. However, the topological ordering is absent in the charge-e superconducting phase. While the field theories for the different phase-transitions were obtained in the earlier work, the signatures of topological ordering in the different phases were not investigated in detail. This is the goal of the current work. We describe a lattice gauge theory of the Majorana toric code in terms of a U(1) matter field coupled to an emergent Z2 gauge field. Subsequently, we use a generalized Wilson-loop order-parameter, namely, the equal-time Fredenhagen-Marcu order parameter, to distinguish between the different phases. Furthermore, we calculate perturbatively the energy gap of the toric code in the presence of Cooper-pair tunneling. Our results are relevant for the current efforts to experimentally realize the Majorana toric code.