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

Abstract

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 ${P}_{c}(s)={C}_{1}{s}^{2}exp[\ensuremath{-}{C}_{2}{s}^{2\ensuremath{-}{\ensuremath{\gamma}}_{0}}]$ 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 ${\ensuremath{\gamma}}_{0}=0.62\ifmmode\pm\else\textpm\fi{}0.07$. In contrast to the Bloch wave functions of clean WSMs that uniformly distribute over the whole space of $(D=3)$ at large length scale, the wave function of disordered WSMs at a WN occupies a fractal space of dimension $D=2.18\ifmmode\pm\else\textpm\fi{}0.05$. Away from the WN, wave function is a fractal at a length scale below a correlation length diverging at the WN as $\ensuremath{\xi}\ensuremath{\propto}{|E|}^{\ensuremath{-}\ensuremath{\nu}}$ with $\ensuremath{\nu}=0.89\ifmmode\pm\else\textpm\fi{}0.05$. Beyond the length scale, the wave function is homogeneous. In the ergodic limit, the level number variance ${\mathrm{\ensuremath{\Sigma}}}_{2}$ around Weyl nodes increases linearly with the average level number $N,{\mathrm{\ensuremath{\Sigma}}}_{2}=\ensuremath{\chi}N$, where $\ensuremath{\chi}=0.2\ifmmode\pm\else\textpm\fi{}0.1$ is independent of system sizes and disorder strengths.