Non-vanishing Density of States in Disordered Weyl Semimetals

There is disagreement in the literature concerning whether the nodal point of a three-dimensional Weyl semimetal survives weak short-range disorder: instanton calculations including fluctuations claim that the zero-energy density of states $ρ(0)$ vanishes, while exact numerics find a systematically finite number. We settle this disagreement with a full saddle-point calculation, including both instanton and fluctuations. Within the supersymmetric formulation for disorder, the instanton obeys a nonlinear Weyl equation whose solution is an exact $j=1/2$ hedgehog whose full functional form we numerically compute. We then compute the fluctuations about this saddle point, carefully identifying all zero modes, and computing the reduced superdeterminant as a convergent Fredholm determinant. No fermionic zero mode beyond a Kramers doublet exists, and at finite disorder $w$, the fluctuations become a finite one-loop prefactor, $ρ(0)=\mathcal{A} w^{-4} \exp(-s^*/w^2)(1+O(w^2))$, in normalized units with $s^* = 6.4163(2)$ and $\mathcal{A} = 27.47(8)$ for Gaussian-correlated disorder. This expression matches exact numerics on a single Weyl cone over four orders of magnitude with no fitted parameters (the amplitude is $\times 0.76$ the one-loop value), establishing that the density of states is finite for any disorder strength.