Non-Wigner-Dyson level statistics and fractal wavefunction 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 wavefunction around Weyl nodes of disordered WSMs. The nearest-neighbor level spacing follows a new universal distribution $P_c(s)=C_1 s^2\exp[-C_2 s^{2-\gamma_0}]$ 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 $\gamma_0=0.62\pm0.07$. In contrast to the Bloch wavefuntions of clean WSMs that uniformly distribute over the whole space of ($D=3$) at large length scale, the wavefunction of disordered WSMs at a Weyl node occupies a fractal space of dimension $D=2.18\pm 0.05$. The finite size scaling of the inverse participation ratio suggests that the correlation length of wavefunctions at Weyl nodes ($E=0$) diverges as $\xi\propto |E|^{-\nu}$ with $\nu=0.89\pm0.05$. In the ergodic limit,the level number variance $\Sigma_2$ around Weyl nodes increases linearly with the average level number $N$, $\Sigma_2=\chi N$, where $\chi= 0.2\pm0.1$ is independent of system sizes and disorder strengths.