Majorana end states in an interacting quantum wire

TL;DR

This paper models a quantum wire with strong spin-orbit interaction and electron repulsion, revealing a topological phase with Majorana zero modes at the edges, protected by magnetization parity.

Contribution

It introduces a simple interacting 1D model showing a symmetry-protected topological phase with Majorana end states, highlighting the role of long-range string operators.

Findings

Ground state is unique in ring geometry, two-fold degenerate with open boundaries.

Localized zero-energy Majorana states appear at the wire ends.

Long-range correlations persist at the edges despite bulk decay.

Abstract

We propose and investigate a simple one-dimensional model for a single-channel quantum wire hosting electrons that interact repulsively and are subject to a significant spin-orbit interaction. We show that an external Zeeman magnetic field, applied at the right angle to the Rashba spin-orbit axis, drives the wire into a correlated spin-density wave state with gapped spin and gapless charge excitations. By computing the ground-state degeneracies of the model with either (anti-)periodic or open boundary conditions, we conclude that the correlated spin-density state realizes a gapless symmetry-protected topological phase, as the ground state is unique in the ring geometry while it is two-fold degenerate in the wire with open boundaries. Microscopically the two-fold degeneracy is found to be protected by the conservation of the magnetization parity. Open boundaries induce localized zero-energy (midgap) states which are described, at the special Luther-Emery point of the model, by Majorana fermions. We find that spin densities at the open ends of the wire exhibit unusual long-ranged correlations despite the fact that all correlations in the bulk of the wire decay in a power-law or exponential fashion. Our study exposes the crucial importance of the long-ranged string operator needed to implement the correct commutation relations between spin densities at different points in the wire. Along the way we rederive the low-energy theory of Galilean-invariant electron systems in terms of current operators.