Equivalent topological invariants for one-dimensional Majorana wires in symmetry class D

TL;DR

This paper demonstrates the equivalence of two topological invariants used to characterize one-dimensional Majorana superconductors in symmetry class D, linking the Pfaffian sign and the Zak-Berry phase through phase winding analysis.

Contribution

It explicitly shows the equivalence between the Pfaffian-based invariant and the Zak-Berry phase invariant in 1D Majorana wires by relating both to phase winding of transformation matrices.

Findings

Pfaffian sign and Zak-Berry phase are equivalent topological invariants.

Both invariants can be related to phase winding of transformation matrices.

Provides a unified understanding of topological invariants in 1D Majorana systems.

Abstract

Topological superconductors in one spatial dimension exhibiting a single Majorana bound state at each end are distinguished from trivial gapped systems by a Z_2 topological invariant. Originally, this invariant was calculated by Kitaev in terms of the Pfaffian of the Majorana representation of the Hamiltonian: The sign of this Pfaffian divides the set of all gapped quadratic forms of Majorana fermions into two inequivalent classes. In the more familiar Bogoliubov de Gennes mean field description of superconductivity, an emergent particle hole symmetry gives rise to a quantized Zak-Berry phase the value of which is also a topological invariant. In this work, we explicitly show the equivalence of these two formulations by relating both of them to the phase winding of the transformation matrix that brings the Majorana representation matrix of the Hamiltonian into its Jordan normal form.