Defects and degeneracies in supersymmetry protected phases

TL;DR

This paper studies 1D lattice models with supersymmetry, identifying defects linked to parafermion conformal field theories that cause protected ground state degeneracies, similar to quantum Hall quasi-holes.

Contribution

It introduces a classification of defects in supersymmetric lattice models and connects them to parafermion CFTs, revealing their fusion rules and degeneracy protection.

Findings

Defects correspond to spin fields in parafermion CFTs.

Injected defects lead to supersymmetry-protected ground state degeneracies.

Fusion rules are Ising type for k=2 and Fibonacci type for k=3.

Abstract

We analyse a class of 1D lattice models, known as M$_k$ models, which are characterised by an order-$k$ clustering of spin-less fermions and by ${\cal N}=2$ lattice supersymmetry. Our main result is the identification of a class of (bulk or edge) defects, that are in one-to-one correspondence with so-called spin fields in a corresponding $\mathbb{Z}_k$ parafermion CFT. In the gapped regime, injecting such defects leads to ground state degeneracies that are protected by the supersymmetry. The defects, which are closely analogous to quasi-holes over the fermionic Read-Rezayi quantum Hall states, display characteristic fusion rules, which are of Ising type for $k=2$ and of Fibonacci type for $k=3$.