# Topological degeneracy and pairing in a one-dimensional gas of spinless Fermions

## TL;DR

This paper explores how strong interactions in a one-dimensional spinless fermion system induce a transition from a Luttinger liquid to a strong pairing phase, revealing topological degeneracies and emergent Majorana modes.

## Contribution

It demonstrates the emergence of a strong pairing phase with topological degeneracies and Majorana modes using a microscopic model and numerical DMRG calculations.

## Key findings

- Identification of a strong pairing phase with gapped single fermion excitations.
- Observation of increased central charge at the critical point indicating Majorana modes.
- Confirmation of topological degeneracies at phase interfaces.

## Abstract

We revisit the low energy physics of one dimensional spinless fermion liquids, showing that with sufficiently strong interactions the conventional Luttinger liquid can give way to a strong pairing phase. While the density fluctuations in both phases are described by a gapless Luttinger liquid, single fermion excitations are gapped only in the strong pairing phase. Smooth spatial Interfaces between the two phases lead to topological degeneracies in the ground state and low energy phonon spectrum. Using a concrete microscopic model, with both single particle and pair hopping, we show that the strong pairing state is established through emergence of a new low energy fermionic mode. We characterize the two phases with numerical calculations using the density matrix renormalization group. In particular we find enhancement of the central charge from $c=1$ in the two Luttinger liquid phases to $c=3/2$ at the critical point, which gives direct evidence for an emergent critical Majorana mode. Finally, we confirm the existence of topological degeneracies in the low energy phonon spectrum, associated with spatial interfaces between the two phases.