Entangled end states with fractionalized spin projection in a time-reversal-invariant topological superconducting wire

TL;DR

This paper investigates the unique fractionalized spin projections and entangled end states in a time-reversal-invariant topological superconductor wire, revealing novel spin distributions and their potential experimental observability.

Contribution

It provides analytical and numerical analysis of end-state spin properties and excitations in TRITOPS wires, highlighting fractionalization and entanglement phenomena.

Findings

Spin projection at ends is fractionalized as ±1/4

Energy of subgap excitations relates to tunneling and entanglement

Localized end states exhibit even-odd site alternation

Abstract

We study the ground state and low-energy subgap excitations of a finite wire of a time-reversal-invariant topological superconductor (TRITOPS) with spin-orbit coupling. We solve the problem analytically for a long chain of a specific one-dimensional lattice model in the electron-hole symmetric configuration and numerically for other cases of the same model. We present results for the spin density of excitations in long chains with an odd number of particles. The total spin projection along the axis of the spin-orbit coupling $S_z= \pm 1/2$ is distributed with fractions $\pm 1/4$ localized at both ends, and shows even-odd alternation along the sites of the chain. We calculate the localization length of these excitations and find that it can be well approximated by a simple analytical expression. We show that the energy $E$ of the lowest subgap excitations of the finite chain defines tunneling and entanglement between end states.We discuss the effect of a Zeeman coupling $\Delta_Z$ on one of the ends of the chain only. For $\Delta_Z<E$, the energy difference of excitations with opposite spin orientation is $\Delta_Z/2$, consistent with a spin projection $\pm 1/4$. We argue that these physical features are not model dependent and can be experimentally observed in TRITOPS wires under appropriate conditions.