Non-Wigner-Dyson level statistics and fractal wave function of disordered Weyl semimetals

Finding fingerprints of disordered Weyl semimetals (WSMs) is an unsolved task. Here we report such findings in the level statistics and the fractal nature of electron wave function around Weyl nodes (WNs) of disordered WSMs. The nearest-neighbor level spacing follows a new universal distribution $P_c\left(s\right)=C_1{s}^{2}exp[-C_2{s}^{2-{\gamma }_0}]$ originally proposed for the level statistics of critical states in the integer quantum Hall systems or normal dirty metals (diffusive metals) at metal-to-insulator transitions, instead of the Wigner-Dyson distribution for diffusive metals. Numerically we find ${\gamma }_0=0.62\pm 0.07$. In contrast to the Bloch wave functions of clean WSMs that uniformly distribute over the whole space of $\left(D=3\right)$ at large length scale, the wave function of disordered WSMs at a WN occupies a fractal space of dimension $D=2.18\pm 0.05$. Away from the WN, wave function is a fractal at a length scale below a correlation length diverging at the WN as $\xi \propto {\left|E\right|}^{-\nu }$ with $\nu =0.89\pm 0.05$. Beyond the length scale, the wave function is homogeneous. In the ergodic limit, the level number variance ${\mathrm{\Sigma }}_2$ around Weyl nodes increases linearly with the average level number $N,{\mathrm{\Sigma }}_2=\chi N$, where $\chi =0.2\pm 0.1$ is independent of system sizes and disorder strengths.

Figure 1

(a) $ln{p}_2(E=0)$ vs $lnL$ for $W=4$. The solid line is the linear fit of slope $D=2.18\pm 0.05$. (b) $D$ vs $W$ for $2\le W\le 9$ and $18\le W\le 24$ (inset). (c) $Y_L$ vs $E$ for $L=42,40,38,\cdots ,14$ (from up to down) for $W=4$. (d) Scaling function $f[x={log}_{10}(L/\xi )]$. Black dashed line connects points to guide the eyes. Inset: $\xi (E,W=4)$ vs $E$. Red solid line is the fit of $\xi ={\xi }_0{\left|E\right|}^{-\nu }$. Dashed lines locate $-{\epsilon }_c\left(L\right)$ and ${\epsilon }_c\left(L\right)$. Error bars for all data points are smaller than symbol sizes in (c) and (d).

Figure 2

$\nu$ (obtained by the same parameters as those in the Fig. 1) as a function of $W$. The black dashed line is $\nu =0.89$.

Figure 3

(a) $P\left(s\right)$ for different energy windows at $W=4$. The cyan and slate-blue lines are Eq. (1) and $P_{\beta =2}\left(s\right)$, respectively. (b) $P\left(s\right)$ for different $W$ in a fixed energy window $E\in [-0.03,0.03]$. Here $L=30$.

Figure 4

$P\left(s\right)$ for $L=20$ (black squares), 26 (red circles), 30 (blue up-triangles), and 36 (orange down-triangles). The disorder is fixed at $W=3$. Every point is averaged over more than 2000 samples. The cyan line is $P_c\left(s\right)$, and the slate-blue line is $P_{\beta =2}\left(s\right)$.

Figure 5

(a) $P\left(s\right)$ for $W=5.1$ and 5.9 at $L=30$ and $\left|E\right|<0.03$. Cyan and slate-blue lines are the same as those in Fig. 3. (b) ${\gamma }_0$ vs $W$ for $L=20,24,30,32$. (c) Scaling function $g\left(x\right)$ of ${\gamma }_0(W,L)$.

Figure 6

${\gamma }_0$ vs $W$ for $L=20$ (red circles), 24 (blue squares), 30 (orange triangles), and 32 (dark green diamonds).

Figure 7

$P\left(s\right)$ for two different models: Model (7) (red circles) and model (2) (blue squares). For model (7), the energy interval is $[-0.01t,0.01t]$, disorder strength is $W=2$, and system length is $L=30$. The cyan line is $P_c\left(s\right)$, and the slate-blue line is $P_{\beta =2}\left(s\right)$.

Figure 8

$P\left(s\right)$ for two different types of disorders. The energy interval is $[-0.01,0.01]$. The disorder strength is $W=2$. The system length is $L=30$. The cyan line is $P_c\left(s\right)$, and the slate-blue line is $P_{\beta =2}\left(s\right)$.

Figure 9

${\mathrm{\Sigma }}_2$ vs $N$ for $W=4$ (orange circles) and 8 (purple squares). Here $L=30$.