Avoided quantum criticality in exact numerical simulations of a single disordered Weyl cone

Existing theoretical works differ on whether three-dimensional Dirac and Weyl semimetals are stable to a short-range-correlated random potential. Numerical evidence suggests the semimetal to be unstable, while some field-theoretic instanton calculations have found it to be stable. The differences go beyond method: the continuum field-theoretic works use a single, perfectly linear Weyl cone, while numerical works use tight-binding lattice models which inherently have band curvature and multiple Weyl cones. In this work, we bridge this gap by performing exact numerics on the same model used in analytic treatments, and we find that all phenomena associated with rare regions near the Weyl node energy found in lattice models persist in the continuum theory: The density of states is nonzero and exhibits an avoided transition. In addition to characterizing this transition, we find rare states and show that they have the expected behavior. The simulations utilize sparse matrix techniques with formally dense matrices; doing so allows us to reach Hilbert space sizes upwards of ${10}^7$ states, substantially larger than anything achieved before.

Figure 1

(a) Density of states $\rho \left(\epsilon \right)$ as a function of energy $\epsilon$ for various different values of disorder $W$ ranging from $W=0.2$ to $W=1.6$ in steps of 0.2. The red curve is for $W=0.9$ and is close to the avoided transition. Far enough away from zero energy this behaves as $\rho \left(\epsilon \right)\sim \left|\epsilon \right|$, consistent with avoided criticality. (b) Depiction of the fcc momentum space lattice and the rhombic dodecahedron Brillouin zone.

Figure 2

Properties of a rare wave function computed with the fcc model with $L=25\sqrt{3}$, $\mathrm{\Lambda }=\frac{\pi }{\sqrt{2}}$, and a disorder strength below the AQCP $W=0.7<W_c\left(\mathrm{\Lambda }\right)\approx 0.9$. The energy of the state is $\epsilon =0.0168$ but can made to pass smoothly through zero energy with a small perturbation of the disorder potential [39]. Left: The probability density of a rare wave function for a cut through the real-space bcc lattice where ${\widehat{\mathbf{a}}}_1$ $\left[{\widehat{\mathbf{a}}}_2\right]$ is in the $(1,-1,1)$ $\left[\right(-1,1,1\left)\right]$ direction and $\mathbf{r}=n_1{\mathbf{a}}_1+n_2{\mathbf{a}}_2+22{\mathbf{a}}_3$. Right: A scatter plot of the wave function as a function of the distance to its maximum value demonstrating a clear power-law decay with $\left|\psi \left(\mathbf{r}\right)\right|\sim 1/r^{1.94}$ for this rare state.

Figure 3

Demonstrating the avoided transition and finite density of states. (a) From lightest to darkest each curve represents $N_C=2^8$ to $2^{13}$ in multiples of two ($\mathrm{\Lambda }=\pi /\sqrt{2}$ and $L=160\sqrt{3}$). Other measures of the avoided criticality put $W_c=0.9$ but we see that there is no saturation of $N_C{\rho }_{N_C}\left(0\right)$ as we would expect from scaling with $N_C$ at the quantum critical point. (b) We fit ${\rho }_{N_C}\left(0\right)$ to Eq. (10) to extract $\rho \left(0\right)$ and ${\rho }^{″}\left(0\right)$ in the $N_C\to \infty$ limit [39]. We find for each cutoff studied, a peak in ${\rho }^{″}\left(0\right)$ saturated in system size (all gray curves are smaller system sizes $L\mathrm{\Lambda }\sqrt{2}/\pi =32\sqrt{3},64\sqrt{3},128\sqrt{3}$ and for $\mathrm{\Lambda }=\pi /\sqrt{2}$ we include $L=160\sqrt{3}$). (b) Inset: Normalized with respect to the peak value ($W_c=0.8,0.9,1.2$ left to right), the peaks line up well with each other, displaying a weak cut-off dependence. (c) We find $\rho \left(0\right)$ is well fit by the rare-region form in Eq. (12) over approximately four orders of magnitude (gray lines). The data as plotted are converged in system size, as the gray curves indicate in (b) and as expanded upon in the Supplemental Material for (c) [39]. The inset shows the same data on a linear scale.

Figure 4

Convergence of the peak of ${\rho }_{N_C}^{″}\left(0\right)$ at the AQCP. (a) $N_C$ dependence of the second derivative of the density of states for the fcc model. (b) A plot of the peak saturated ${\rho }_{N_C}^{″}\left(0\right)$ data for both fcc and cubic data. Data at fixed $N_C$ are converged in system size [39]. We see that they do not match the scaling we would expect from a true quantum critical point ${\rho }_{N_C}^{″}\left(0\right)\propto N_C$ (gray line). The fcc data has $\mathrm{\Lambda }=\pi /\sqrt{2}$ and $L=160\sqrt{3}$ while the cubic data has $\mathrm{\Lambda }=\pi$ and $L=160$.