Kitaev Chain with a Fractional Twist

TL;DR

This paper introduces a fractional twist to the Kitaev chain by employing a generalized Dirac operator with a central extension, leading to a topological phase with a rational winding number that remains robust under weak disorder.

Contribution

It develops a novel fractional topological phase in the Kitaev chain using rational operator powers, extending topological classification to non-integer indices.

Findings

The phase diagram includes a pseudo-metallic topological phase with rational winding number.

A mode remains extended despite weak disorder, indicating topological robustness.

The rational winding number aligns with projective Dirac operators lacking spin$^ ext{C}$ structure.

Abstract

The topological non-triviality of insulating phases of matter is by now well understood through topological K-theory where the indices of the Dirac operators are assembled into topological classes. We consider in the context of the Kitaev chain a notion of a generalized Dirac operator where the associated Clifford algebra is centrally extended. We demonstrate that the central extension is achieved via taking rational operator powers of Pauli matrices that appear in the corresponding BdG Hamiltonian. Doing so introduces a pseudo-metallic component to the topological phase diagram within which the winding number is valued in $\mathbb Q$. We find that this phase hosts a mode that remains extended in the presence of weak disorder, motivating a topological interpretation of a non-integral winding number. We remark that this is in correspondence with Ref.[J. Differential Geom. 74(2):265-292] demonstrating that projective Dirac operators defined in the absence of spin$^\mathbb C$ structure have rational indices.