Effects of geometric frustration in Kitaev chains

TL;DR

This paper investigates how geometric frustration, introduced by long-range hopping in a Kitaev chain, affects its topological phase transitions and Majorana modes, revealing complex phase diagrams and persistent frustration effects.

Contribution

It generalizes the transfer matrix approach to analyze topological phases in frustrated Kitaev chains and explores the effects in both non-translational and translational invariant models.

Findings

Frustration causes alternating topological phases.

Topological bulk invariant remains valid despite frustration.

Frustration effects persist even with restored translational invariance.

Abstract

We study the topological phase transitions of a Kitaev chain in the presence of geometric frustration caused by the addition of a single long-range hopping. The latter condition defines a legged-ring geometry (Kitaev tie) lacking of translational invariance. In order to study the topological properties of the system, we generalize the transfer matrix approach through which the emergence of Majorana modes is studied. We find that geometric frustration gives rise to a topological phase diagram in which non-trivial phases alternate with trivial ones at varying the range of the extra hopping and the chemical potential. Frustration effects are also studied in a translational invariant model consisting of multiple-ties. In the latter system, the translational invariance permits to use the topological bulk invariant to determine the phase diagram and bulk-edge correspondence is recovered. It has been demonstrated that geometric frustration effects persist even when translational invariance is restored. These findings are relevant in studying the topological phases of looped ballistic conductors.