From Majorana Fermions to Topological Order

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.

4+4 pages, 5 figures. v2 has a new reference and a few new comments. In v3: yet another new reference and Supplementary Material is renamed Appendix. In v4: several typos are corrected, to appear in Phys. Rev. Lett

References in corpus (8)

• Superconducting proximity effect and Majorana fermions at the surface of a topological insulator
• Signatures of Majorana fermions in hybrid superconductor-semiconductor nanowire devices
• Majorana Fermions and a Topological Phase Transition in Semiconductor-Superconductor Heterostructures
• Topological fault-tolerance in cluster state quantum computation
• Coulomb-assisted braiding of Majorana fermions in a Josephson junction array
• Adiabatic manipulations of Majorana fermions in a three-dimensional network of quantum wires
• Anyonic interferometry without anyons: How a flux qubit can read out a topological qubit
• Perturbative approach to an exactly solved problem: the Kitaev honeycomb model

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