Upper critical dimension for dirty Weyl semimetal-to-metal quantum phase transitions

Weyl or Dirac fermions with the iconic linear energy-momentum relation and average density of states (ADOS) ρ(E) about E d-1 at energy E in d spatial dimensions, constitute a unique setup to study the disorder-driven semimetal-to-metal quantum phase transition (QPT) between ballistic (realized for weak disorder) and diffusive (stabilized at stronger disorder) quasiparticles. Such a QPT takes place only for d>2 and falls beyond the realm of the Anderson metal-to-insulator transition. From numerically computed ADOS (using the kernel polynomial method) in dirty Weyl systems in d=2 to 6, here we show that d=2 and d=4 are the lower (d) and upper (d u) critical dimensions for such an unconventional QPT, respectively, as suggested from the solution of quasiparticle lifetime within the self-consistent Born approximation. Consequently, for d at least 4 the associated correlation length exponent is found to be ν about 0.5 (within numerical accuracy). However, the dynamic scaling exponent at the quantum critical point is pinned close to z about d/2 (numerically) for any d at least 3, which is shown to be an exact result from a field-theoretic renormalization group calculation. Therefore, Weyl semimetal-to-metal QPTs can be studied field theoretically around both d and d u.