A generic phase between disordered Weyl semimetal and diffusive metal

Quantum phase transitions of three-dimensional (3D) Weyl semimetals (WSMs) subject to uncorrelated on-site disorder are investigated through quantum conductance calculations and finite-size scaling of localization length. Contrary to previous claims that a direct transition from a WSM to a diffusive metal (DM) occurs, an intermediate phase of Chern insulator (CI) between the two distinct metallic phases should exist due to internode scattering that is comparable to intranode scattering. The critical exponent of localization length is ν≃\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\,\simeq $$\end{document} 1.3 for both the WSM-CI and CI-DM transitions, in the same universality class of 3D Gaussian unitary ensemble of the Anderson localization transition. The CI phase is confirmed by quantized nonzero Hall conductances in the bulk insulating phase established by localization length calculations. The disorder-induced various plateau-plateau transitions in both the WSM and CI phases are observed and explained by the self-consistent Born approximation. Furthermore, we clarify that the occurrence of zero density of states at Weyl nodes is not a good criterion for the disordered WSM, and there is no fundamental principle to support the hypothesis of divergence of localization length at the WSM-DM transition.

where k is the lattice momentum. Thus, h(k) must take a form of ε σ , where I, σ α , and h α (α = x, y, z) are respectively the 2 × 2 identity matrix, Pauli matrices, and functions of k characterizing materials. The two bands cross each other at a WN of k = K when h α (K) = 0. This can happen in three dimensions (3D) because three conditions match with three variables, and the level repulsion principle can at most shift the WNs. Moreover, WNs must come in pairs with opposite chirality according to the no-go theorem 13 , and the band inversion occurs between two paired WNs, resulting in the topologically protected surface states and accompanying Fermi arcs on crystal surfaces. The only way to destroy a WSM is the merging of two WNs of opposite chirality or via superconductivity 11 .
How does the above picture based on the lattice translational symmetry change when disorders are presented and the lattice momentum is not a good quantum number anymore? This is an important question that has been investigated intensively with conflicting results [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31] . Disorder can greatly modify electronic structures, resulting in the well-known Anderson localization. One expects that disorder has much more interesting effects to a WSM than that to a normal metal. For example, electrons with linear dispersion relations around the WNs (Dirac nodes) are governed by the effective Weyl (massless Dirac) equation. Weyl electrons cannot be confined by any potential due to the Klein paradox 32 . Early theoretical studies ignored internode scattering and predicted that the WSM phase featured by vanishing density of states (DOS) at WNs is robust against weak disorder and undergoes a direct quantum phase transition to the diffusive metal (DM) phase as disorder increases [14][15][16][17][18][19] . The divergence of the bulk state localization length at the WSM-DM transition was conjectured 16,20 and was used in recent numerical studies [26][27][28] to support disordered WSMs in a wide range of disorder and direct WSM-DM transitions. Strangely, the evidences of the transition resemble the conventional Anderson localization transitions at which the localization lengths of different sample sizes cross at the same point, and the uncorrelated on-site disorder is used in these studies so that internode scattering is comparable to intranode scattering and should be significantly important. However, a real WSM has at least two WNs of opposite chirality, and disorder can mix two nodes by internode scattering so that the Anderson localization can happen as shown in the disordered graphene 33 . Therefore, the applicability of the direct WSM-DM transition conjectured by theories of a single WN [14][15][16][17][18][19] for real disordered WSMs is questionable. The predicted vanishing DOS at WNs have also attracted many numerical studies 20,25,27,31 , and recent works concluded that zero DOS cannot exist at nonzero disorder due to rare region effects and no WSM phase is allowed at an arbitrary weak disorder if zero DOS at WNs is demanded 21,31 .
Strictly speaking, because the lattice momentum is not a good quantum number in a disordered WSM, k-space is only an approximate language although the concepts of band and DOS are still accurate. Thus, the validity of DOS ρ(E) ∝ E 2 from 3D linear dispersion relations as a signature of disordered WSMs is doubtful. The distinct property of a WSM is the existence of topologically protected surface states that do not necessarily rely on the linear crossing of two bands and zero DOS at WNs, and should be robust against disorder, at least against the weak one. Therefore, a disordered WSM is defined as a bulk metal with topologically protected surface states in this work. Since both the WSM and DM are bulk metals, bulk states of them are extended and no theoretical basis supports the hypothesis of the divergence of localization length at the WSM-DM transition. Focusing on the previously proposed quantum critical point between the WSM and DM phases [26][27][28] , we show that the so-called direct WSM-DM transition actually corresponds to two quantum phase transitions and a narrow Chern insulator (CI) phase (which is also called the 3D quantum anomalous Hall phase in ref. 26 ) exists between the two distinct metallic phases. The critical exponent of localization length takes the value of 3D Gaussian unitary ensemble of the conventional Anderson localization transition [34][35][36][37] . Nontrivial topological nature of the CI phase is confirmed by Hall conductance calculations that show well-defined quantized plateaus in the bulk insulating phase. Furthermore, the disorder-induced various plateau-plateau transitions between different quantized values of Hall conductance can be well explained by the self-consistent Born approximation (SCBA).

Results
Model. In order to compare directly with previous studies, we consider a tight-binding Hamiltonian on a cubic lattice of unity lattice constant that was used in refs 2,26 , x x y y . The dispersion relation of the Hamiltonian is ε . In this study, = . m t 2 1 0 , identical to that in ref. 26 , is used. Model parameter m z is the tunable variable to control different phases. The energy band gap closing requires In order to study the disorder effect, a spin-resolved on-site disorder is included in the model, with the bar denoting ensemble average over different configurations. According to the Fermi golden rule, the internode and intranode scattering around the WNs have the same rate of where ρ E ( ) F is the DOS at Fermi energy and ρ ≠ (0) 0 for nonzero disorder (see methods). Therefore the two kinds of scatterings are equally important in the disordered WSM. Moreover, because ρ E ( ) F is an increasing function of | | E F around WNs, the scattering rates increases as the Fermi energy shifts away from the WNs. z 0 in the WSM phase since it was reported that the system undergoes a WSM-DM transition as disorder increases 26 , the normalized localization length λ Λ = M M ( )/ versus W for various M is shown in Fig. 1(a). Very similar to early studies [26][27][28] , two phase transition points b and c of Λ = d dM / 0 seem appear. Zooming in on these transition regions, the normalized localization length are shown in Fig. 1(b,c) for b and c, respectively. Apparently, the normalized localization length curves of different M cross at a single critical disorder W c in Fig. 1 2 , we employ the finite size scaling analysis for these bulk state localization lengths. For the transition at b, the single-parameter scaling hypothesis is applied as ξ W W c diverges at the transition point. The scaling functions from both metallic (upper branch) and insulating (lower branch) sides are shown in Fig. 1(d). The perfect collapse of the data points in Fig. 1(b) into the smooth curves supports our claim of the quantum phase transition. The analysis yields = . ± . W t / 21 81 0 02 c and ν = . ± . 1 31 0 02, consistent with the previous numerical and experimental results [34][35][36][37] for 3D Gaussian unitary ensemble. For the quantum phase transitions at critical points W c1 and W c2 shown in Fig. 1(c), the crossing of different curves is less perfect as it often happens in 3D sys- 6.0 6.5 7.0 (e,f) The scaling functions obtained from the corrections to the single-parameter scaling ansatz by collapsing data points around the critical points W c1 and W c2 in (c) into the smooth curves, respectively. tems when the system size is limited by the computer resources. We therefore follow the more accurate analysis used in ref. 40 to include the contributions of the most important irrelevant parameter to the scaling function where ψ is the relevant scaling variable with ν > 0 and φ is the irrelevant scaling variable with µ < 0. Using ν = .
1 30 for the 3D Gaussian unitary class and by minimizing χ 2 , we fit the data points around the two transition points shown in Fig. 1(c) to the scaling function Eq. (4) (see methods). Indeed, the perfect scaling curves in Fig. 1(e,f) with = . ± . and 0.94, quite satisfactory numbers. We also calculate the localization length for various m z (see Fig. 2) and E F (see Figs 3 and 4) in the WSM phase. It is shown that the insulating phase between the two distinct metallic phases is generic, as shown in Fig. 2 and will be discussed below. As E F increases from zero energy (see Fig. 3), the intermediate insulating phase expands initially (see Fig. 4) since the internode scattering rate increases with E F according to Eq. (3). Further increase of E F , the linear dispersion relation fails and the system becomes a conventional 3D metal with Fermi energy deep inside the conduction band, as shown in Fig. 4. Moreover, in a recent work, it was shown that the intermediate CI phase becomes more apparent for tilted WNs 41 . This is consistent with our scattering analysis, since tilting WNs increases the density of states around WNs so that the internode scattering rate is enhanced.
Quantum transport. In order to investigate the chiral surface states and topological nature of the intermediate insulating phase identified above, we calculate the quantum conductance of a four-terminal Hall bar of size × × 80 40 8 marked by blue color in Fig. 5(a)

,4) labels allowed k z within the first Brillouin zone (BZ). For
Thus, a chiral surface state must exist for each allowed ∈ −| | | | k K K ( , ) z z z , and contribute a quantized Hall conductance of e h / 2 . Therefore, the total Hall conductance from the surface states is  where T ij is the transmission coefficient from lead j to lead i, and current I i in lead i is given by the where the voltage on lead i is V i 43,44 . For the clean system, the Hall conductance as a function of m m / z 0 is shown in Fig. 5  , the clean system is a WSM whose Hall conduction at WNs is from the surface states and is quantized at a value determined by m z as mentioned early. Interestingly, at a fixed m z (along a vertical line in Fig. 5(c)), the Hall conductance can jump from one quantized value into another as disorder increases.

Self-consistent Born approximation.
In order to understand these transitions, we use the SCBA to see how the disorder modifies the model parameters 26,45,46 . The self-energy at the Fermi energy due to the disorder is , one has σ Σ = Σ z z since  has the particle-hole symmetry 47  . Equation (6) Fig. 5(c). Only those m z , at which the clean system is in the WSM phase and was reported to undergo the WSM-DM transition as disorder increases 26 , are considered. The two green curves are the boundaries of the DM/ CI phases (upper line) and CI/WSM phases (lower line). The narrow CI phase region separates the WSM phase   from the DM phase. The CI phase is inferred from the fact that all bulk states are localized according to the localization length calculations while the Hall conductance of a finite bar is nonzero and takes several quantized values (red for 5, blue for 3, and green for 1 in units of e h / 2 ), as shown in Fig. 5(c). The WSM phase is defined as bulk metallic states (extended wavefunctions) with surface conducting channels while the DM phase has bulk metallic states without surface conducting channels. Both the CI and WSM phases can have well quantized Hall conductance (red, blue, and green regions in Fig. 5(c)) while quantized Hall conductance is absent in the DM phase.

Discussion
The generality of the no direct WSM-DM transition can be understood from the following reasoning. In order to have a direct WSM-DM transition, WNs and topologically protected surface states should be destroyed simultaneously. However, the two events are not exactly the same although they are related. The topologically protected surface states are due to nonzero band Chern numbers of two-dimensional slices between the two WNs. In general, disorder pushes the two WNs away from each other and towards the BZ boundary (as elaborated by the SCBA) where they can merge. As a result, the WNs are destroyed while the nonzero band Chern numbers of two-dimensional slices survive, resulting in the intermediate CI phase. Whether disorder can pull two paired WNs together and towards the BZ center so that the WNs and band Chern numbers can simultaneously be destroyed is an open question.
In conclusion, we show that the claimed direct transition from a WSM to a DM do not exist under uncorrelated on-site disorder due to non-negligible internode scattering. Instead, there exists a intermediate CI phase that separates a WSM phase from a DM phase. Namely, there are actually two quantum phase transitions between the disordered WSM and the DM: One is from the WSM to the CI, and the other is from the CI to the DM. The critical exponent of ν .  expands at weak disorder as the Fermi energy slightly shifts away from the WNs. Our results do not dependents on specific choices of lattice model since the analysis based on low-energy effective Weyl Hamiltonians is general.

Methods
Internode and intranode scattering rates.
x y z , ,

H
where the Fermi velocities are . The energy bands of the Weyl To be concrete and without losing generality, we fix the Fermi energy in the conduction band = E E F q as shown in Fig. 6, and the eigenstates with the Fermi energy are In the presence of disorder, the total scattering processes consist of two parts: the internode scattering and intranode scattering that are schematically shown in Fig. 6. According to the Fermi golden rule, the internode and intranode scattering rates are , , is the uncorrelated on-site disorder in Eq. (2) and the bar denotes ensemble average over different configurations. Therefore, we can conclude that the internode and intranode scattering rates are identical in Weyl semimetals subject to uncorrelated on-site disorder. Moreover, the scattering rates increase with | | E F since the density of states is an increasing function of | | E F .
Correction to the single-parameter scaling hypothesis. Following the more accurate analysis used in ref. 40  Data availability. All data generated or analysed during this study are included in this published article.