Supersymmetry in the Fractional Quantum Hall Regime

TL;DR

This paper proposes a condensed matter realization of supersymmetry at the edge of Read-Rezayi quantum Hall states, revealing a hidden topological structure and robust zero-modes related to SUSY, with implications for well-known quantum Hall states.

Contribution

It introduces a supersymmetric framework for the edge states of Read-Rezayi quantum Hall states, connecting SUSY to topological invariants and zero-modes in condensed matter systems.

Findings

Edge hosts $k+1$ protected zero-modes.

SUSY remains stable under generic perturbations.

Results apply to the $ u=1/3$ Laughlin state, showing robust SUSY properties.

Abstract

Supersymmetry (SUSY) is a symmetry transforming bosons to fermions and vice versa. Indications of its existence have been extensively sought after in high-energy experiments. However, signatures of SUSY have yet to be detected. In this manuscript we propose a condensed matter realization of SUSY on the edge of a Read-Rezayi quantum Hall state, given by filling factors of the form $\nu =\frac{k}{k+2}$, where $k$ is an integer. As we show, this strongly interacting state exhibits an $\mathcal{N}=2$ SUSY. This allows us to use a topological invariant - the Witten index - defined specifically for supersymmetric theories, to count the difference between the number of bosonic and fermionic zero-modes in a circular edge. In our system, we argue that the edge hosts $k+1$ protected zero-modes. We further discuss the stability of SUSY with respect to generic perturbations, and find that much of the above results remain unchanged. In particular, these results directly apply to the well-established $\nu=1/3$ Laughlin state, in which case SUSY is a highly robust property of the edge theory. These results unveil a hidden topological structure on the long-studied Read-Rezayi states.