Supersymmetry in the fractional quantum Hall regime

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 paper we study a condensed matter realization of SUSY on the edge of a Read-Rezayi quantum Hall state, given by filling factors of the form $\ensuremath{\nu}=\frac{k}{k+2}$, where $k$ is an integer. As we show explicitly, 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 this 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 $\ensuremath{\nu}=1/3$ Laughlin state, in which case SUSY is a robust property of the edge theory. These results unveil a hidden topological structure on the long-studied Read-Rezayi states.