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 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−γ 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 γ 0 = 0.62 ± 0.07. 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 ± 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 ξ ∝ |E| −ν with ν = 0.89 ± 0.05. In the ergodic limit, the level number variance Σ 2 around Weyl nodes increases linearly with the average level number N, Σ 2 = χN, where χ = 0.2 ± 0.1 is independent of system sizes and disorder strengths.Crystal Weyl semimetals (WSMs), characterized by the linear crossings of their conduction and valence bands at Weyl nodes (WNs) and topologically protected surface states, have attracted enormous attention in recent years because of their exotic properties and potentials in applications [1][2][3][4][5][6][7][8]. There is little doubt that WSMs are a new state of matter in nature. However, the characteristics of WSMs for a crystal do not apply to a disordered system since the lattice momentum is not a good quantum number. To fully establish the WSMs as a genuine state of matter, one needs to find their fingerprints in disorders that inevitably exist in all materials. It was originally believed that disordered WSMs are featured by vanishing density of states (DOS) at WNs [9][10][11][12][13]. The divergence of the bulk state localization length at the Weyl-semimetalto-diffusive-metal (WSM-to-DM) transition was also conjectured [14][15][16][17][18]. Nonetheless, these features were challenged in many recent studies [19][20][21][22]. So far, a simple working criterion for disordered WSMs is still lacking. This study aims to search the fingerprints from the random matrices that describe the disordered WSMs.Random matrices have broad applications in many fields of physics [23][24][25][26][27]. In the condensed matter physics, random matrix theory can be used to distinguish different types of metals. The distribution P(s) of nearest-neighbour level spacing s (in the unit of the mean level spacing) of diffusive metals (normal dirty metals) and the level number variance Σ 2 (∆E) = n 2 − n 2 in a given energy range are governed by the Wigner-Dysion distributions [28]. For example, P(s) follows the Wigner surmises, P β (s) = C 1 s β exp[−C 2 s 2 ] (C 1 and C 2 are determined by the probability normalization and the u...

Level statistics of extended states in random non-Hermitian Hamiltonians

“…If there exists an ALT from extended states to localized states when disorder strength W varies for a fixed E, the correlation length ξ diverges at the critical value W c as ξ(W) ∝ |W − W c | −ν . p 2 near W c satisfies the following oneparameter scaling function [55][56][57]…”

Disorder-induced quantum phase transitions in three-dimensional second-order topological insulators

“…1, there are two more phases: the WSMs characterized by paired Weyl nodes in the clean limit and the normal gapped insulators. Noticeably, the phase boundary of the disordered WSMs is still an issue under debate, e.g., whether the WSMs can exist in finite disorders [52] and whether there is a direct WSM-to-DM transition without the intermediated Chern insulator phase [47,48,53] or two quantum phase transitions of WSM-to-CI-to-DM with increasing disorders [55]. However, this challenging problem is not the focus of this paper.…”

“…To investigate the nature of this Anderson localization transition and its associated universality class, we compute the PR p 2 (E = 0, W ), which measures how many lattice sites are occupied by the wave function of E = 0 [49][50][51]. Near the critical disorder W c3 of the Anderson localization transitions, p 2 satisfies the one-parameter scaling function [52,53],…”

“…D is the fractal dimension of critical wave functions which occupy a subspace of dimensionality smaller than the embedded space dimension d = 3. By defining Y L (W ) = p 2 L −D − CL −y , we use the following criteria to identify an Anderson localization transition [53]: enters a WSM at M = 1 − B, reenters into the 3DSOTI at M = 1 + B, becomes a WSM again for 3 − B < M < 3 + B, and is a normal gapped insulator for M > 3 + B where there is no band inversion for surface states. Remarkably, there are two pairs of one-dimensional helical edge channels (hinge states) for M < 1 − B, indicating the occurrence of band inversions for two Dirac cones.…”

Duality between disordered nodal semimetals and systems with power-law hopping

“…In semimetals, the instantonic contribution ρ smooth to the DoS may be exponentially suppressed by the small deviation |d − d c | of the dimension d from the critical dimension d c = 2γ of the transition or the number of the particle flavours [17]. Even in the absence of small parameters, various numerical studies of 3D Weyl and Dirac semimetals have found this contribution to be rather small or unobservable [17,[54][55][56][65][66][67][68][69], which allows one to use the DoS, to a good approximation, as an order parameter for the transition. It has also been suggested [14] that the instantonic contribution may broaden criticality, thus converting the transition to a sharp crossover.…”