Abstract

The Kitaev chain model exhibits topological order that manifests as topological degeneracy, Majorana edge modes and Z2 topological invariant of the bulk spectrum. This model can be obtained from a transverse field Ising model(TFIM) using the Jordan–Wigner transformation. TFIM has neither topological degeneracy nor any edge modes. Topological degeneracy associated with topological order is central to topological quantum computation. In this paper, we explore topological protection of the ground state manifold in the case of Majorana fermion models which exhibit Z2 topological order. We show that there are at least two different ways to understand this topological protection of Majorana fermion qubits: one way is based on fermionic mode operators and the other is based on anti-commuting symmetry operators. We also show how these two different ways are related to each other. We provide a very general approach to understanding the topological protection of Majorana fermion qubits in the case of lattice Hamiltonians. We then show how in topological phases in Majorana fermion models gives rise to new braid group representations. So, we give a unifying and broad perspective of topological phases in Majorana fermion models based on anti-commuting symmetry operators and braid group representations of Majorana fermions as anyons.

Highlights

  • In this paper, we take an approach to understanding the topological protection of Majorana fermions qubits based on Majorana zero mode operators that are odd normalized zero modes

  • Similar to even mode operators in many-body localization( MBL), we show that odd zero-mode operators are integrals of motion and the emergent symmetry operators of the Hamiltonian

  • We have explored the topological protection based on the fermionic mode operators which lead to the emergent symmetries of the Majorana fermion Hamiltonians

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Summary

Introduction

We take an approach to understanding the topological protection of Majorana fermions qubits based on Majorana zero mode operators that are odd normalized zero modes. We have explored Majorana zero mode operators which are an important indicator and manifestation of the topological phases in one-dimensional Hamiltonians like the Kitaev Chain model. Using dualities and a bond-algebra approach and holographic symmetries, there is already an understanding of how topological order in the Kitaev chain model is related to Landau order in the corresponding spin model [11]. We go ahead to establish the relation between emergent Majorana mode operators and the Z2 topological order for the general case of odd-numbered Majorana fermion models. This unification is important because it opens the ways to understand the topological protection from symmetries of the lattice models.

Kitaev p-Wave Chain
Algebra of Majorana Doubling
Emergent Supersymmetry in Majorana Fermions Models
Majorana Zero Modes and Γ Operator
Zero-Mode Operators for a General Case
Interactions
Symmetry Algebra of Topological Protection
Topological Order and Yang-Baxter Equation
Summary
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