Disorder and magnetic transport in tilted Weyl semimetals
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Disorder and magnetic transport in tilted Weyl semimetals
Yi-Xiang Wang , School of Science, Jiangnan University, Wuxi 214122, China. Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA
PACS – Disordered solids
PACS – Landau levels
PACS – Fermion systems and electron gas
Abstract – We investigate the effect of disorder on the Landau levels (LLs) in tilted three-dimensional Weyl semimetals (WSMs) when a magnetic field is present. Based on the minimumlattice model and by using the exact diagonalization and Kubo’s formula, we numerically calculatethe Hall conductivity and the density of states (DOS), from which several striking signatures arefound to distinguish type-I WSMs from type-II WSMs: the first is the response of the Hall con-ductivity to the Fermi energy around the band center in clean limit, the second is the performanceof the Hall conductivity to disorder, where in type-I WSMs, the robustness of the low-energy LLsis broken successively from the higher LLs to the lower ones and can be understood with the sink down picture, and the third is the behavior of the DOS at zero energy to disorder. Theimplications of our results are discussed.
Introduction. –
In the past few years, the study oftopological phases of matter has been extended beyondthe gapped states and now also includes various gaplessnodal systems. Three-dimensional (3D) Weyl semimetals(WSMs) are the prime examples of such gapless system[1], whose characteristic properties lead to the protectedFermi arcs surface states and novel chiral anomaly in re-sponse to the external electromagnetic fields [2]. As thestrict symmetries of free space do not necessarily hold ina crystal, new types of fermions may emerge in condensedmatter background. In high-energy physics, the Weyl conetilting is usually forbidden by the Lorentz symmetry, butit can generically appear in a linearized low-energy the-ory around an isolated twofold band-crossing point in acrystal [3]. If the tilting is weak that the Fermi surface re-mains pointlike, the system is classified as a type-I WSM(WSM1). While if the tilting becomes strong, the Fermisurface may instead consist of the electron and hole pock-ets. In this case, the system is classified as a type-IIWSM (WSM2) [4]. It is likely that some materials canundergo the transitions from WSM1 to WSM2 with dop-ing or under pressure. The experimental evidences for theWSM2 state have been reported recently in MoTe [5–8],WTe [9], and the alloy Mo x W − x Te [10, 11]. Besidesthe Fermi arcs, WSM2 also owns an additional class ofthe so-called track states [12], which are nontopologicalbut degenerate with the Fermi arcs and have been demon- strated in experiment [5].When the WSM system is disordered, a common view-point dominates that disorder plays an indispensable rolein determining the properties of the system and may evendrive the phase transition into the diffusive metal state[13–16]. The previous studies show that when there isno tilting, the vanishing density of states (DOS) at Weylnodes can persist up to a finite value of disorder strength,beyond which a metallic state is driven by disorder [17–19].The critical properties of the semimetal-metal transitionhave also been analyzed [20–23]. In Ref. [24], it wasdemonstrated that the Weyl Fermi arcs are not topologi-cally protected when disorder sets in, but the surface chiralvelocity is robust and survives in the presence of strongdisorder.One significant feature of WSMs is that the system canreach the quantum limit where the Landau levels (LLs)are formed even in a weak magnetic field [25–29]. Con-sequently, WSMs provide an ideal platform to investigatethe 3D Hall physics. The studies of magneto-optical prop-erties revealed that the absorption peaks of optical con-ductivity can give evident signals of WSM2 [30–32]. Toour knowledge, the effects of disorder on the LLs and mag-netic transport properties in 3D tilted WSMs are less stud-ied. In this paper, based on the minimum lattice model,we try to investigate the interplay between disorder andLLs in tilted WSMs.p-1 a r X i v : . [ c ond - m a t . d i s - nn ] J a n i-Xiang Wang , As the quantum Hall effect (QHE) is a hallmark intwo-dimensional (2D) electron system, 3D system nor-mally does not exhibit the QHE. This is due to the dis-persive bands along the direction of the magnetic field,which can smear the energy gap between the LLs. Inrecent experiments, due to the delicate quantum confine-ments, the QHE in 3D Dirac semimetal Cd As thin filmshas been successfully observed [33–37], but its origin isin debate and no consensus has been reached yet. Sev-eral mechanisms were proposed for the QHE that it maybe attributed to the quantum confinement induced bulksubbands [33, 34], the Weyl orbits that connect the op-posite surfaces via bulk Weyl nodes [35, 36], as well asthe topological-insulator-type surface states [37]. Here weonly consider the underlying 3D LL physics as the stacked2D quantum Hall system.In this paper, by using the exact diagonalization andKubo’s formula, we calculate the Hall conductivity and theDOS in tilted WSMs. Several striking signatures are foundto distinguish WSM1 and WSM2. The first is the responseof the Hall conductivity to the Fermi energy around theband center in clean limit: the slope of the Hall conductiv-ity vs the Fermi energy is vanishingly small in WSM1, butwill jump to be much large in WSM2. The second is thatin WSM1, the Hall conductivity at low energy shows cer-tain robustness to weak disorder and the robustness will bebroken successively from the higher LLs to the lower oneswhen disorder increases, which can be understood by the sink down picture. While in WSM2, the Hall conductivityat low energy is more fragile to disorder and the robust-ness is absent. The third is the behavior of the DOS atzero energy, where in WSM1, it remains to be quite smalluntil disorder increases to the critical value, beyond whichthe system is driven into the diffusive metallic state, butin WSM2, it decreases continuously with disorder. Theseobservations demonstrate that the low-energy LLs in theenergy window are robust to disorder, but the high-energyLLs outside the energy window are not. Our work mayhelp deepen the understanding of the interplay betweendisorder and magnetic fields in 3D tilted WSMs. Model. –
We start from the minimum model Hamil-tonian that supports a pair of Weyl fermions [13–15,31,38], H =2 t (sin k y σ y + sin k z σ z ) + ( m − t cos k x ) σ x + m (2 − cos k y − cos k z ) σ x + 2 t z sin k z σ . (1)Here σ are the Pauli matrices acting on the spin space, t is the hopping integral and the wave vector k i is mea-sured by 1 /a with a being the lattice spacing. m con-trols the positions of Weyl nodes and m denotes theWilson mass term as to open a finite energy windowto avoid the band overlapping. The last term specifiesthe Weyl cone tilting in z − direction. When the tiltingis absent, t z = 0, H preserves the inversion symmetrybut breaks the time-reversal-symmetry (TRS) of the sys-tem. For | m | < t , the two Weyl nodes are given as - 2 - 1 0 1 2- 3- 2- 10123 e /t k Z e /t k Z energy window - 3 - 2 - 1 0 1 2 3- 2- 1012 -1 .0 -0 .5-0 .50 .00 .5 e /t k z e /t k Z ( a ) ( b ) Fig. 1: (Color online) The LL energy spectra of WSMs areobtained from the discretized lattice model with magnetic flux p = 60, (a) t z = 0 and (b) t z = 1 .
5. The chiral n = 0 LLs arelabeled with red color and the energy window is also specified.The arrows in the above point out the Van-Hove singularitieswhere the bands are very dense and the DOS are divergent.The insets show that the LL splittings exhibit the same char-acteristics as they cross the energy window even if the Weylcones are overtilted. K η = (cid:16) η arccos( m / t ) , , (cid:17) , with η = ± . The intro-duction of the tilting term in z − direction that is perpen-dicular to the distance between the Weyl nodes breaksthe inversion symmetry. More importantly, it can easilytune the Weyl cone tilting to feature the electron and holepockets. Around Weyl node K η , the Hamiltonian H isexpanded to yield a low-energy continuous description, H η ( k ) = (cid:126) v ( ηk x σ x + k y σ y + k z σ z ) + (cid:126) v z k z , (2)with the velocities v = 2 t/ (cid:126) and v z = 2 t z / (cid:126) . In the follow-ing, t is set as the unit of energy, and the mass parametersare taken as m = 0 and m = 2.We concentrate on a magnetic field along the tiltingdirection of the Weyl cones. Such a magnetic field canlead to the dispersive LLs only along the z − direction andthe flat LLs in the x-y plane. If the magnetic field actsin the x − y plane, the chiral LLs are missing [31]. Themagnetic field B = (0 , , B ) can be included by the Peierlssubstitution, p → p − e A , with the vector potential in theLaudau gauge of A ( r ) = ( − yB, , n − LL is obtained as ε n ≥ ( k z ) = sgn( n ) (cid:115) (cid:126) v k z + 2 (cid:126) v | n | l B + (cid:126) v z k z , (3) ε n =0 ,η ( k z ) = (cid:126) ( ηv + v z ) k z , (4)with the magnetic length l B = (cid:112) (cid:126) /eB .Eq. (3) shows that n ≥ n = 0 chiral LLs areclearly distinguishable in two Weyl cones and are inde-pendent of the magnetic field. The Weyl cone tilting candrastically change the properties of the chiral n = 0 LLs[31]. If t z < t and the system lies in WSM1, the two chiralp-2isorder and magnetic transport in tilted Weyl semimetals n = 0 modes in different Weyl cones own opposite veloci-ties and are counter-propagating, as shown in fig. 1(a). If t z increases to t z > t , the Weyl cones are overtilted andthe system lies in WSM2, the two chiral modes acquirethe velocities in the same direction, as in fig. 1(b).The tight-binding model can be discretized on a cubiclattice with the periodic boundary conditions in all direc-tions. In the lattice model, the magnetic field is written as B = φ/a · h/e with the help of the dimensionless quantity φ , which gives the magnetic flux penetrating a unit cell in x − y plane and is in unit of the elementary flux quantum h/e [41]. For convenience, we take φ = 1 /p in which p is an integer so that the magnetic flux will be commen-surate with the lattice structure. We take the model asa cubic lattice of L x = L y = L z = p . fig. 1 shows theLL energy spectra of WSMs obtained from the discretizedlattice model with the magnetic flux p = 60.In this paper, for WSM1, we term the LLs in the energywindow as the “low-energy LLs”, as specified in fig. 1(a),and those outside the energy window as the “high-energyLLs”. When the Weyl cones are overtilted, the topolog-ical properties of the LLs will keep unchanged, includingthe characteristic Chern number and the splittings whenthe LLs cross the energy window (see the insets in fig. 1).So we still use the low-energy and high-energy LLs to dis-tinguish the different LLs in WSM2. It has been checkedthat the low-energy LLs in fig. 1 are consistent with thecontinuous model, as given in eqs. (3) and (4).The disorder effect is introduced by adding the ran-dom on-site potential H dis [13–16] to the discretized latticemodel, H dis = (cid:88) x,y,k z ,s (cid:15) x,y,k z ,s c † x,y,k z ,s c x,y,k z ,s , (5)where x/y and s are the coordinate and spin index, respec-tively. (cid:15) is a random number uniformly distributed in therange [ − W/ , W/ W being the disorder strength.As the disorder configurations do not preserve the TRS,both the charge and magnetic disorder are included. Tomake the exact-diagonalization calculations tractable, weconsider the quasi-disorder case, i.e. , the disordered sys-tem keeps the translational symmetry along z − direction.Such a quasi-disorder case can represent the completelyrandom disorder to some extent and was previously usedto investigate the one-parameter scaling behavior at theDirac point in graphene [39] and the disorder-inducedphase transitions from WSM1 to WSM2 [14]. In next twosections, we will calculate the Hall conductivity and theDOS. As disorder sets in the system, we need to do certainconfiguration average with the number N c = 10 ∼ toensure the convergence of the numerical results. More de-tails can be found in the Appendix. Hall conductivity. –
As we consider the 3D LLphysics as the stacked 2D quantum Hall system, theHamiltonian H ( k ) is regarded as a superposition of the 2Dslices H ( k x , k y ) in z -direction, with the mass term m ( k z ). e / t s H (e2/h) p = 4 0 p = 6 0 p = 8 0 p = 4 0 p = 6 0 p = 8 0 a n a ly t ic a l g t t z / t W S M 1 W S M 2 e / t s H (e2/h) t z = 0 . 2 t t z = 0 . 5 t t z = 1 . 3 t t z = 1 . 5 t ( a ) ( b ) Fig. 2: (Color online) The Hall conductivity σ H vs the Fermienergy ε around the band center for WSMs under the magneticfield. (a) is for different magnetic flux p and no tilting. (b) isfor different tilting factors t z with p = 60. The inset in (a)shows the slope γ vs t z around zero energy, where in WSM1,the numerical results are in good agreement with the analyticalresults. Then σ H is expressed as a function of the Fermi energy ε [40], σ H ( ε ) = (cid:90) π − π dk z π σ DH ( ε, k z ) . (6)Here the 2D Hall conductivity σ DH at zero temperature iscalculated with the Kubo’s formula [41, 42].First of all, we investigate the Hall conductivity σ H inthe disorder-free WSM system. Because the LLs are dis-persive in 3D WSMs and almost indiscernible near theband edge, we focus on σ H around the band center. Thenumerically calculated Hall conductivity on the latticemodel is shown in fig. 2 with the given parameters.Actually, when t z < t and the Fermi energy ( ε >
0) liesin the low-energy region, σ H is given analytically as σ H = e h vπ (cid:126) ( v − v z ) (cid:104) ε + (cid:88) n ≥ θ ( ε > ε vn ) (cid:113) ε − ε v, n (cid:105) , (7)here ε vn = (cid:126) /l B · (cid:112) n ( v − v z ) is the vertex energy ofthe dispersive n − LL and θ ( x ) is the step function. Ineq. (7), the first term is due to the contributions from n = 0 LLs, which for specific k z carry Chern number 1,and the second term comes from n ≥ k z carry Chern number 2 due to the degeneracyof the LLs in the two Weyl nodes. As the particle-holesymmetry is preserved, similar results for σ H with negativeFermi energy can also be obtained.Eq. (7) tells us that σ H increases with ε as the LLs oflarger momentum region are occupied by the electronicstates. Each time the Fermi energy crosses the vertex en-ergy ε vn of the higher n − LL, the n − LL begins to contributeto σ H . This makes σ H seem like folding lines, as demon-strated by the numerical results in fig. 2(a) with t z = 0.Note that the larger p corresponds to the smaller mag-netic flux, so more folding lines in σ H appear. Comparingthe different lines in fig. 2(a), we can see that when p in-creases, σ H also increases, as more LL states are occupiedfor certain Fermi energy ε . This observation agrees withp-3i-Xiang Wang , previous works, as in fig. 1(b) of Ref. [43] and in fig. 10of Ref. [44]. It is worth emphasizing that the study of σ H here lies in the quantum oscillation regime, or regime II ofRef. [43]. In the classical picture, when the magnetic fieldis strong enough and the temperature is very low, the Hallconductivity is given as σ H = − neB , with n denoting theelectron density. This result is a consequence of the com-pletely disorder-free limit that we consider here, in whichthe scattering time is infinite. For a finite scattering time,one retrieves the usual behavior σ H ∝ B in the small- B field limit.When the Weyl cones are tilted as t z >
0, if the tilting isweak, t z < t , the multiple folding lines still exist in σ H , asshown in fig. 2(b). If the Weyl cones are overtilted, t z > t ,the system enters WSM2 and σ H includes the contribu-tions from the low-energy LLs as well as the high-energyones. As the Fermi energy increases, the contributions to σ H from the lower Weyl cone are decreasing, while thecontributions from the upper Weyl cone are increasing,leading to an approximate linear relationship of σ H with ε , as shown in fig. 2(b). When ε increases to the lowestVan-Hove singularity of a specific LL, σ H reaches its max-imum and then decreases rapidly to the negative value.The positions of the Van-Hove singularities are shown bythe arrows in fig. 1, where the LLs are very dense and theDOS diverge. In the extreme case that the Weyl cones areheavily overtilted, t z >
2, the Van-Hove singularities willmove to be around zero energy, leading to the negativeHall conductivity (not shown here).According to the above analysis, around the band cen-ter, the linear relationship can be fit between σ H and ε as σ H ( t z , ε ) = e h γ ( t z ) ε. (8)The slope γ vs the tilting factor t z for different magneticflux p is plotted in the inset of fig. 2(a). It shows that inWSM1, the slope γ is vanishingly small, while in WSM2, γ suddenly jumps to a much larger value. This is indeedexpected as one has a large DOS at the Fermi level ofWSM2, in contrast to the vanishing DOS of WSM1. Wecan also see that in WSM1, the slope γ is independent ofthe magnetic flux p , because the energies of n = 0 LLsare independent of p , as in eq. (4). The numerical resultson the lattice model are in good agreement with the ana-lytical results in eq. (7), from which γ = v/ [ π (cid:126) ( v − v z )].In WSM2, γ increases with p , as more LLs are occupiedbelow the Fermi energy. The different responses of σ H tothe Fermi energy around the band center can be used as asignature to distinguish WSM1 from WSM2. To observethe characteristic variance of Hall conductivity in exper-iment, one needs a clean WSM sample where the Fermienergy can be modulated by doping the system.Next we turn to the effect of random disorder on theHall conductivity in WSMs. The results of σ H for differ-ent disorder strength W are plotted in fig. 3. We noticethat σ H always vanishes at zero energy for all W , as the s H (e2/h)
W = 0 W = 0 . 2 t W = 0 . 5 t W = t W = 2 t W = 3 t W = 4 t W = 5 t e / t e / t W = 0 W = 0 . 2 t W = 0 . 5 t W = t W = 2 t W = 3 t W = 4 t s H (e2/h) s H (e2/h) e / t ( a ) ( b ) Fig. 3: (Color online) The Hall conductivity σ H around theband center for different disorder strength W . The tilting pa-rameter is t z = 0 in (a) and t z = 1 . p = 60. The inset in (a) shows an expandedview of σ H around zero energy. particle-hole symmetry is still preserved in a disorderedsystem. In fig. 3(a) with t z = 0, it shows when W in-creases, σ H is suppressed successively from the higher LLsto the lower ones. At W = 0 . t , the contributions to σ H from n = 0 , , , n ≥ W = 2 t , the contributionsfrom n (cid:54) = 0 LLs are suppressed. Finally, at strong disor-der W = 5 t , σ H due to n = 0 LLs is suppressed, withthe expanded view of σ H around zero energy being shownin the inset of fig. 3(a). These numerical results suggestthat in the low-energy region, the higher LLs are moresusceptible to disorder while the lower LLs exhibit certainrobustness. The effects of disorder on σ H are also validfor weakly tilted WSM1.These results can be explained as follows. We label thewavefunction of one specific low-energy LL state as ψ ε ,k z and the wavefunction of another specific state in the high-energy Van-Hove singularity regime as ψ ε ,k z . As the twostates own opposite Chern numbers, they will be annihi-lated if they meet, only when two conditions are satisfied:the momenta between the two states are equal k z = k z and the energy difference ∆ = ε − ε is compensatedby the disorder-induced scatterings. The compensation ofthe larger energy difference requires the stronger disorderstrength W . So the contributions to σ H are suppressedfrom the higher LLs at first, as the energy difference issmall, and then with the increasing of disorder, graduallyto the lower LLs and finally to n = 0 LLs. Thus for spe-cific k z , it seems like that the LLs carrying negative-Chernnumber coming from the higher energy region sink down and are moving continuously towards the band center withincreasing disorder strength. When they cross the Fermienergy at the critical disorder strength W c , they annihi-late with the LLs carrying positive Chern number and thecorresponding Hall conductivity is suppressed. These ar-guments remind us of the float up picture in 2D quantumHall system [45–47], which explains the collapse of the Hallplateau through the annihilation of positive-Chern num-ber LLs around the band center and the float up negative-Chern number LLs from the Van-Hove singularities duep-4isorder and magnetic transport in tilted Weyl semimetals e / t s H(W)/ s H(W=0) s H(W)/ s H(W=0) e / t W = 3 t W = 0 . 5 t 0 . 0 0 . 5 1 . 0 1 . 501 e / t p = 2 0 p = 4 0 p = 6 0 p = 8 0 s H(W)/ s H(W=0) ( a ) ( b ) ( c ) W = 3 tp = 6 0p = 6 0
Fig. 4: (Color online) The normalized Hall conductivity bythe zero disorder limit σ H ( W ) σ H ( W =0) . (a) and (b) are for differentdisorder W in WSM1 with t z = 0 and WSM2 with t z = 1 . p in WSM1 with t z = 0. to the disorder-induced scatterings. Note here we con-sider the positive Fermi energy, compared with the nega-tive Fermi energy in their works [45–47].For WSM2, in fig. 3(b) with t z = 1 . t , it shows that σ H decreases gradually with disorder. When the disorderstrength reaches W ∼ t , σ H tends to vanish completely.These numerical results suggest that σ H in WSM2 aremore fragile to disorder. We can attribute this to the factthat the overtilted Weyl nodes are concealed in the high-energy LLs. As the high-energy LLs are not robust andeasily broken by the disorder-induced scatterings (demon-strated by the DOS in the next section), it leads to theevident suppression of σ H in WSM2. Thus the differentresponses of σ H to disorder can be used as the secondsignature to distinguish WSM1 from WSM2.To further clarify how the difference between WSM1and WSM2 depends on energy, we calculate the normal-ized Hall conductivity by the zero disorder limit σ H ( W ) σ H ( W =0) .In figs. 4(a) and (b), when ε is small, we compare the nor-malized σ H for different disorder W in WSM1 and WSM2,respectively. From these figures, the qualitatively differentbehaviors can be seen that the normalized σ H exhibits thenon-monotonous behavior with disorder for WSM1, butdecreases monotonously for WSM2. Note that in WSM2, σ H is vanishing at ε = t even if disorder sets in the sys-tem, so there exist large fluctuations of the normalized σ H around ε = t .We also investigate the influence of the magnetic fluxon the Hall conductivity in disordered WSM1. In fig. 4(c),we plot the normalized σ H for different magnetic flux p ,with a fixed disorder W = 3 t . It shows that with thedecreasing of p , the normalized σ H (cid:39) n = 0LLs. When the Fermi energy crosses the vertex of n = 1LL, the robustness of the normalized σ H will be broken bydisorder. This is because the smaller p means the largerenergy of n = 1 LL, as in eq. (3). Then with the helpof the above sink down picture, the normalized σ H that iscontributed by n = 0 LLs for smaller p can keep unaffectedto a larger Fermi energy. Wc/t t z /t3 D H a ll s ta te r (0) W / t t z = 0 t z = 0 . 4 t t z = 1 . 5 t · - 3 e / t W = 0 W = 0 . 2 t W = 0 . 5 t W = t W = 2 t W = 3 t W = 4 t W = 5 t r ( e ) · - 3 ( a ) ( b ) Fig. 5: (Color online) (a) The DOS ρ ( ε ) with t z = 0 around theband center for different disorder strength W . (b) The DOS ρ (0) vs disorder strength W for different t z . The inset in (b)gives the critical disorder strength W c vs the weak t z , whichseparates the 3D Hall state from the diffusive metal state. Weset the magnetic flux p = 60. Density of states. –
To further investigate the effectof disorder on LLs, we calculate the normalized DOS intilted WSMs, ρ ( ε ) = 1 N N (cid:88) i =1 δ ( ε − ε i ) , (9)with N = 2 L x L y L z / ( a ) being the total number of eigen-values in the system. When t z < t and ε > ρ ( ε ) = 12 πl B v (cid:126) ( v − v z ) (cid:104) (cid:88) n ≥ θ ( ε > ε vn ) ε (cid:112) ε − ε v, n (cid:105) , (10)here ε vn is the same as in eq. (7). The first term in eq. (10)comes from the chiral n = 0 LLs and the second term isfrom n ≥ t z = 0 are plot-ted in fig. 5. In fig. 5(a) when disorder is absent, the DOSexhibits a sawtooth shape with the square-root singulari-ties at the vertex energy ε = ε vn from the one-dimensionaldispersion of each LL. With increasing disorder, the widthof these peaks is broaden and their height is reduced. If W is beyond the critical value W c , the corresponding peakin the DOS will disappear. This also happens successivelyfrom the higher LLs at first and then to the lower onesand is consistent with the Hall conductivity analysis inthe previous section.We focus on the DOS at zero energy ρ (0). In WSM1 andwith no disorder, ρ (0) is due to the chiral n = 0 LLs, whichis a constant and rather small. When disorder increases, ρ (0) remains to be unaffected, as shown in fig. 5(b) of t z =0. Further increasing disorder to beyond the critical value W c ∼ . ± . ρ (0) will become finite and the systemis driven into the diffusive metal state. The observationalso holds for t z = 0 .
4, with the critical disorder W c ∼ . ± .
2. So ρ (0) can serve as an order parameter tocharacterize the continuous phase transition from the 3DHall state to the diffusive metal state. This is similar top-5i-Xiang Wang , the previous studies about the disordered WSMs withouta magnetic field [17, 18], which revealed that for t z = 0,the critical disorder strength for ρ (0) from vanishing tononvanishing is around W c ∼ .
55 [17] and W c ∼ . W c vs t z < W c arises in determining theexact position of the phase transition. It shows that W c decreases with t z and tends to zero when t z →
1. This canbe attributed to the competition between the Weyl conetilting and the disorder that more electronic states fromthe high-energy regime can be scattered into the energywindow when the Weyl cone tilting increases. Note thatthe critical W c should also depend on the magnetic fieldand it may be scaled with the energy separation betweenthe LLs, which will be left for the future work.For comparison, in fig. 5(b) for WSM2 with t z = 1 . ρ (0) is sufficiently large when W = 0.When disorder increases, the high-energy LLs are broken,leading to the gradually decreasing of ρ (0). In the senseof renormalization group, sufficiently weak disorder is anirrelevant perturbation in WSM1 [18] and a marginallyrelevant perturbation in WSM2. Here through ρ (0), wefurther demonstrate that the low-energy LLs are robust todisorder, while the high-energy LLs are easily broken bydisorder. The performance of ρ (0) vs disorder can be usedas the third signature to distinguish WSM1 from WSM2. Conclusions and Discussions. –
To summarize, inthis paper we have investigated the effect of disorder onthe LLs and magnetic transport in tilted WSMs. By cal-culating the Hall conductivity and DOS, we find severalobservable signatures to distinguish WSM1 from WSM2.The broken of the low-energy LLs in WSM1 by disorderfrom the higher ones to the lower ones can be understoodby the sink down picture. We suggest that our results canbe generalized to WSMs with two pairs and even multiplepairs of Weyl nodes.Here we have used the quasi-disorder model, while thecompletely random disorder potential in 3D WSMs may beeffectively treated by using the kernel polynomial method[24, 48], which needs more work in the future. Thereare also some open questions, such as the critical prop-erties of disorder-induced phase transitions in WSM1 un-der the magnetic field and the effect of disorder on thequantum Hall effect in WSMs, with the physical mech-anism originating from the Weyl orbits [35, 36], or fromthe topological-insulator-type surface states [37]. The in-terplay between disorder, magnetic field and topologicalWeyl and Dirac semimetals will open up new research intotopological phenomena and device applications in three di-mensions beyond 2D electron system.
Acknowledgments. –
We would like to thank Fux-iang Li, Linghua Wen and W. Vincent Liu for manyhelpful discussions. This work was supported by NSFC r ( e ) e = N c · -4 s H ( e ) c e = ( a ) ( b ) Fig. 6: (Color online) (a) The Hall conductivity σ H ( ε ) for dif-ferent energy ε vs configuration number N c , with t z = 0 and W = 4. (b) The DOS ρ ( ε ) for different energy ε vs the con-figuration number N c , with t z = 0 . W = 3 .
4. We set themagnetic flux p = 60. (Grant No. 11804122) and China Scholarship Council (No.201706795026). Appendix. –
Here we provide some evidences for theconvergence of the results after the configuration number N c = 10 ∼ is performed for the disordered WSMsystem. The Hall conductivity σ H and the DOS ρ fordifferent energy ε vs N c are shown in figs. 6(a) and (b),respectively. We can see that at the beginning, there arecertain fluctuations in both physical quantities, but σ H exhibit good convergence when N c > ρ showsconvergence when N c > σ H iscontributed from all the electronic states below the energy ε while ρ is only due to the electronic states around ε . As σ H includes much more electronic states than ρ , so σ H converges more quickly. In this paper, we need to takeproper configuration number N c in the calculations as toachieve more precise results of σ H and ρ . REFERENCES[1]
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