  In the presence of randomness, a relativistic semimetal undergoes a quantum
transition towards a diffusive phase. A standard approach relates this
transition to the $U(N)$ Gross-Neveu model in the limit of $N \to 0$. We show
that the corresponding fixed point is infinitely unstable, demonstrating the
necessity to include fluctuations beyond the usual Gaussian approximation. We
develop a functional renormalization group method amenable to include these
effects and show that the disorder distribution renormalizes following the
so-called porous medium equation. We find that the transition is controlled by
a nonanalytic fixed point drastically different from that of the $U(N)$
Gross-Neveu model. Our approach provides a unique mechanism of spontaneous
generation of a finite density of states and also characterizes the scaling
behavior of the broad distribution of fluctuations close to the transition. It
can be applied to other problems where nonanalytic effects may play a role,
such as the Anderson localization transition.
