Emergent space-time supersymmetry at disordered quantum critical points

We study the effect of disorder on the space-time supersymmetry that is proposed to emerge at the quantum critical point of pair density wave transition in $(2+1)$-dimensional (D) Dirac semimetals and $(3+1)$D Weyl semimetals. In the $(2+1)$D Dirac semimetal, we consider three types of disorder, including random scalar potential, random vector potential and random mass potential, whereas the random mass disorder is absent in the $(3+1)$D Weyl semimetal. Via a systematic renormalization-group analysis, we find that any type of weak random disorder is irrelevant due to the couplings between the disorder potential and the Yukawa vertex. The emergent supersymmetry is, thus, stable against weak random potentials. Our paper will pave the way for exploration supersymmetry in realistic condensed-matter systems.