Unconventional localization transition in high dimensions

We study noninteracting systems with a power-law quasiparticle dispersion ${\xi }_{\mathbf{k}}\propto k^{\alpha }$ and a random short-range-correlated potential. We show that, unlike the case of lower dimensions, for $d>2\alpha$, there exists a critical disorder strength (set by the bandwidth), at which the system exhibits a disorder-driven quantum phase transition at the bottom of the band that lies in a universality class distinct from the Anderson transition. In contrast to the conventional wisdom, it manifests itself in, e.g., the disorder-averaged density of states. For systems in symmetry classes that permit localization, the striking signature of this transition is a nonanalytic behavior of the mobility edge, which is pinned to the bottom of the band for subcritical disorder and grows for disorder exceeding a critical strength. Focusing on the density of states, we calculate the critical behavior (exponents and scaling functions) at this novel transition, using a renormalization group, controlled by an $\varepsilon =d-2\alpha$ expansion. We also apply our analysis to Dirac materials, e.g., Weyl semimetals, where this transition takes place in physically interesting three dimensions.

Figure 1

Critical behavior of weak short-correlated disorder in materials with power-law quasiparticle dispersion ${\xi }_{\mathbf{k}}\propto k^{\alpha }$ in dimension $d$. In low dimensions $d<2\alpha$, the effects of disorder grow at small momenta (strong-disorder regime), while in high dimensions $d>2\alpha$, there is a quantum phase transition between the strong-disorder and weak-disorder regimes. The insets show the RG flow of the dimensionless measure of the disorder strength relative to kinetic energy $\gamma \left(K\right)\sim {\left[K\ell \left(K\right)\right]}^{-1}$ with decreasing the characteristic momentum $K$, where $\ell \left(K\right)$ is the mean free path.

Figure 2

The energy ($E$, in the conduction band) vs disorder strength $\left(\varkappa \right)$ phase diagram for a semiconductor in the orthogonal symmetry class (permitting Anderson localization) above the critical dimension, $d>2\alpha$. The disorder-averaged density of states in different regimes is indicated. The parameter $t=\varkappa /{\varkappa }_c-1$ is the deviation of the disorder strength $\varkappa$ from the critical value ${\varkappa }_c$. The hatched region corresponds to localized states if $d>2$. The mobility threshold is shown as the blue curve. The dotted curve indicates a crossover from the critical to the effective disorder-free regime.

Figure 3

The low-energy density of states in a disordered semiconductor in a dimension $d$ above $2\alpha$ for subcritical disorder strength $\left(\varkappa <{\varkappa }_c\right)$.

Figure 4

Mobility threshold $E^{*}\left(\varkappa \right)$ showing a nonanalytic behavior [given by Eq. (1.8) close to the critical point] as a function of the disorder strength $\varkappa$ in higher dimensions $\left(d>2\alpha \right)$. In the blue (grey) wedgelike region, the critical behavior is that of the Anderson universality class with the localization length (1.7), and outside it is determined by the high-dimensional critical point $(\varkappa ={\varkappa }_c,E=0)$ studied here.

Figure 5

The phase diagram for a disordered Weyl semimetal $(d=3,\alpha =1)$ illustrating weak-to-strong-disorder phase transition at $E=0$. Unlike a semiconductor in the orthogonal symmetry class (Fig. 2), in Weyl semimetals there are no localized states for the sufficiently smooth disorder potential under consideration. The values of the exponents in the density of states $\rho \left(E\right)$ are calculated using a perturbative one-loop RG scheme for Dirac quasiparticles, controlled by an $\varepsilon =2-d$ expansion.

Figure 6

The renormalized density of states in a disordered Weyl semimetal $(d=3,\alpha =1)$ near the Dirac point for subcritical disorder. It illustrates the crossover from the linear-in-$E$ form (controlled by the 3D critical point close to $\varkappa ={\varkappa }_c$ and $E=0$) to disorder-free quadratic $E^2$ form at lowest energies, with a universal prefactor enhanced by disorder.

Figure 7

Impurity line.

Figure 8

The leading-order diagrams for the renormalization of the impurity line due to scattering through states with large momenta. Large momentum $\mathbf{K}$ significantly exceeds the other incoming and outgoing momenta. Diagrams (a)–(d) have equal values.

Figure 9

Diagram corresponding to the renormalization of the quadratic part of the Lagrangian.