# Topological ordering in the Majorana Toric Code

## TL;DR

This paper investigates the topological phases of a Majorana toric code system, characterizing topological order using lattice gauge theory and Wilson-loop order parameters, and analyzing the effects of Cooper pair tunneling.

## Contribution

It introduces a lattice gauge theory framework and uses a generalized Wilson-loop order parameter to identify topological phases in the Majorana toric code.

## Key findings

- Toric code phase exists in Mott insulator and charge-2e superconductor phases.
- Toric code phase disappears in charge-e superconductor phase.
- Cooper pair tunneling stabilizes the topological gap of the toric code.

## Abstract

At zero temperature, a two-dimensional lattice of Majorana zero modes on mesoscopic superconducting islands exhibits a topologically-ordered toric code phase. Recently, a Landau field theory was used to describe the different phases of the aforementioned system and the phase-transitions separating them. While the field theory provides details on the properties of the system close to the phase-transitions, signatures of topological ordering in the different phases have not been computed. This is the primary goal of the current work. We describe a lattice gauge theory of the Majorana toric code in terms of $\mathrm{U}(1)$ matter fields coupled to an emergent $\mathbb{Z}_2$ gauge field. Subsequently, we use a generalized Wilson-loop order-parameter, the equal-time Fredenhagen-Marcu order parameter, to characterize the topological ordering in the different phases. Our computation provides evidence of the toric code phase both in the Mott insulator and the charge-$2e$ superconductor phases, while showing that the toric code phase disappears in the charge-$e$ superconductor phase. In addition, we perturbatively analyze the influence of Cooper pair tunneling on the topological gap of the toric code in the limit of strong charging energy and show that the toric code phase is, in fact, stabilized by the Cooper pair tunneling. Our results are relevant for experimental realizations of the Majorana toric code.