Emergent space-time supersymmetry at disordered quantum critical points

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