Confinement-deconfinement transitions in the Majorana Toric Code

A two-dimensional lattice of Majorana zero modes on mesoscopic superconducting islands at zero temperature can give rise to the Z₂ toric code spin liquid. The latter is one of the most promising candidates for performing quantum error-correction, a crucial primitive for realization of a fault-tolerant quantum computer. Depending on the strength of the charging energy, single-electron tunneling rate and the Cooper-pair tunneling rate, field theory analysis predicts a rich phase diagram for the model comprising first and second order phase transition lines as well as tricritical points, with the toric code phase persisting in the Mott insulating and the charge-2e superconducting phase. However, quantitative analysis of the phase diagram including signatures of the topological order has never been performed. In this work, we provide unbiased numerical evidence of the topological stability of the toric code phases, as well as the nature of the quantum phase transitions. This is done by mapping this model onto a modified Bose-Hubbard model on the square lattice, which we then analyze through a sign-problem-free continuous-time Quantum-Monte-Carlo approach using a modified Worm algorithm. We compute analytical properties of the ground state energy, two-point correlation functions as well as Wilson loop operators to probe the signatures of the topological order in the different phases.