  We study the effect of disorder on the spacetime supersymmetry that is
proposed to emerge at the quantum critical point of pair density wave
transition in (2+1)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, while 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 for weak random
potentials. Our work will pave the way for exploration supersymmetry in
realistic condensed matter systems.
