Supersymmetric quantum criticality with discrete symmetry

Abstract

Supersymmetry, originally proposed in high-energy physics, can emerge as a remarkable low-energy structure in condensed matter systems. While emergent supersymmetry at quantum critical points is widely discussed in models with continuous symmetries, real materials are constrained by microscopic discrete symmetries. To address this, we investigate (2+1)-dimensional Gross-Neveu-Yukawa theories coupling Dirac fermions to a complex order parameter with discrete $Z_n$ anisotropy. Using the functional renormalization group, we find that for $n>3$, the anisotropic perturbations are irrelevant at the fixed point, yielding a $\mathcal{N}=2$ Wess-Zumino supersymmetric critical point. In the ordered phase, this dangerously irrelevant anisotropy gives rise to a second characteristic length scale, $ξ'$, alongside the usual correlation length, $ξ$. By tracking mass thresholds along symmetry-broken renormalization group trajectories, we extract the exponents $ν'$ and $ν$ without imposing prior scaling assumptions. For the $Z_4$, $Z_5$, and $Z_6$ models, our results support the scaling relation $ν'/ν= 1+|y_n|/p$ with $p=2$ in the isotropic framework used here.