From Majorana Fermions to Topological Order

TL;DR

This paper demonstrates how a 2D network of Majorana fermions on superconducting islands can exhibit topological order, with implications for topological quantum computation, by mapping the system to known models and analyzing phase transitions.

Contribution

It introduces a model of Majorana fermions on superconducting islands that shows topological order and maps it to 2D Ising models, providing insight into non-perturbative regimes and phase transitions.

Findings

Kitaev's toric code emerges in the model at fourth-order perturbation.

The system can be mapped onto signed 2D Ising models in a transverse field.

The model exhibits a phase transition to a non-topological phase.

Abstract

We consider a system consisting of a 2D network of links between Majorana fermions on superconducting islands. We show that the fermionic Hamiltonian modeling this system is topologically-ordered in a region of parameter space. In particular we show that Kitaev's toric code emerges in fourth-order perturbation theory. By using a Jordan-Wigner transformation we can map the model onto a family of signed 2D Ising models in a transverse field where the signs (FM or AFM) are determined by additional gauge bits. Our mapping allows an understanding of the non-perturbative regime and the phase transition to a non-topological phase. We discuss the physics behind a possible implementation of this model and argue how it can be used for topological quantum computation by adiabatic changes in the Hamiltonian.