Fractional Topology in Interacting 1D Superconductors

TL;DR

This paper explores the topological phases of interacting 1D superconducting wires, identifying fractional topological phases with Majorana modes and analyzing their stability under various interactions and couplings.

Contribution

It introduces measurable topological markers for interacting 1D superconductors and characterizes the stability of fractional and integer topological phases using numerics and field theory.

Findings

The double critical Ising phase is a fractional topological phase with gapless Majorana modes.

The phase diagram remains stable under inter-wire hopping at certain length scales.

Large inter-wire hopping leads to integer topological phases with edge modes.

Abstract

We investigate the topological phases of two one-dimensional (1D) interacting superconducting wires and propose topological markers directly measurable from ground state correlation functions. These quantities remain powerful tools in the presence of couplings and interactions. We show with the density matrix renormalization group that the double critical Ising (DCI) phase discovered in [1] is a fractional topological phase with gapless Majorana modes in the bulk, and a one-half topological invariant per wire. Using both numerics and quantum field theoretical methods, we show that the phase diagram remains stable in the presence of an inter-wire hopping amplitude $t_{\bot}$ at length scales below $\sim 1/t_{\bot}$. A large inter-wire hopping amplitude results in the emergence of two integer topological phases, stable also at large interactions. They host one edge mode per boundary shared between both wires. At large interactions, the two wires are described by Mott physics, with the $t_{\bot}$ hopping amplitude resulting in a paramagnetic order.