Disorder-Driven Quantum Transition in Relativistic Semimetals: Functional Renormalization via the Porous Medium Equation

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.

Figure 1

Fixed point solution to the functional renormalization group equation for $d=3$ expressed as a backward self-similar solution $Z(\zeta )$ to the porous medium equation, Eq. (5), with $\delta =1$. The inset shows the integral curve representing the BSS in the phase plane $(Z,Y)$, which clarifies the nature of nonanalyticity of the FP.

Figure 2

Analytical continuation of the backward self-similar solution, Eq. (6), for $t<T$ to the forward self-similar solution [Eq. (10)] for $t>T$ which provides a nonanalytic mechanism for the DOS generation at the nodal point. The generated DOS is related to the nonzero $\tilde{F}(0)$. Blue solid line is the BSS function $F(\zeta )$ for $\delta =1$, green solid line is the corresponding FSS $\tilde{F}(\tilde{\zeta })$ with $\tilde{F}(0)>0$. Both solutions have the same asymptotic behavior at large $\zeta$, which ensures the matching between $u(r,T^{-})$ and $u(r,T^{+})$. Red dashed line is the GN FP $F(\zeta )={\zeta }^2/8$ shown for comparison.