Topological aspects of $\pi$ phase winding junctions in superconducting wires

TL;DR

This paper explores the topological properties of $ ext{pi}$ phase winding Josephson junctions in superconducting wires, analyzing their unique features, differences from traditional $ ext{pi}$ junctions, and proposing a topological field theory to describe them.

Contribution

It introduces and analyzes phase winding $ ext{pi}$-junctions, highlighting their distinct topological characteristics and differences from conventional $ ext{pi}$-junctions, and proposes a minimal topological field theory for such systems.

Findings

Phase winding junctions lack protected zero energy modes.

Comparison between topological ($p$-wave) and trivial ($s$-wave) superconducting wires.

Proposed a topological field theory for $ ext{pi}$-junctions in superconducting wires.

Abstract

We theoretically investigate Josephson junctions with a phase shift of $\pi$ in various proximity induced one-dimensional superconductor models. One of the salient experimental signatures of topological superconductors, namely the fractionalized $4 \pi$ periodic Josephson effect, is closely related to the occurrence of a characteristic zero energy bound state in such junctions. We make a detailed analysis of a more general type of $\pi$-junctions coined "phase winding" junctions where the phase of the order parameter rotates by an angle $\pi$ while its absolute value is kept finite. Such junctions have different properties, also from a topological viewpoint, and there are no protected zero energy modes. We compare the phenomenology of such junctions in topological ($p$-wave) and trivial ($s$-wave) superconducting wires, and briefly discuss possible experimental probes. Furthermore, we propose a topological field theory that gives a minimal description of a wire with defects corresponding to $\pi$-junctions. This effective theory is a one-dimensional version of similar theories describing Majorana bound states in half-vortices of two-dimensional topological superconductors.