Unusual Metal to Marginal-Metal Transition in Two-Dimensional Ferromagnetic Electron Gases
UUnusual Metal to Marginal-Metal Transition in Two-Dimensional Ferromagnetic Electron Gases
Weiwei Chen, C. Wang, ∗ Qinwei Shi, Qunxiang Li, and X. R. Wang
3, 4, † Hefei National Laboratory for Physical Sciences at the Microscale & Synergetic Innovation Center of Quantum Information and Quantum Physics,University of Science and Technology of China, Hefei, Anhui 230026, China School of Electronic Science and Engineering and State Key Laboratory of Electronic Thin Films and Integrated Devices,University of Electronic Science and Technology of China, Chengdu 610054, China Department of Physics, The Hong Kong University of Science and Technology (HKUST), Clear Water Bay, Kowloon, Hong Kong HKUST Shenzhen Research Institute, Shenzhen 518057, China (Dated: December 5, 2019)Two-dimensional ferromagnetic electron gases subject to random scalar potentials and Rashba spin-orbitinteractions exhibit a striking quantum criticality. As disorder strength W increases, the systems undergo a tran-sition from a normal di ff usive metal consisting of extended states to a marginal metal consisting of critical statesat a critical disorder W c , . Further increase of W , another transition from the marginal metal to an insulator oc-curs at W c , . Through highly accurate numerical procedures based on the recursive Green’s function method andthe exact diagonalization, we elucidate the nature of the quantum criticality and the properties of the pertinentstates. The intrinsic conductances follow an unorthodox single-parameter scaling law: They collapse onto twobranches of curves corresponding to di ff usive metal phase and insulating phase with correlation lengths diverg-ing exponentially as ξ ∝ exp[ α/ √| W − W c | ] near transition points. Finite-size analysis of inverse participationratios reveals that the states within the critical regime [ W c , , W c , ] are fractals of a universal fractal dimension D = . ± .
02 while those in metallic (insulating) regime spread over the whole system (localize) with D = D = ff usive metals,marginal metals, and the Anderson insulators. Anderson localization is a long-lasting fundamental con-cept in condensed matter physics [1–9] and keeps bringing ussurprising, especially in its critical dimensionality of two [10–12]. In the early time, the orthodox view is the absence ofdi ff usion of an initially localized wave packet at an arbitraryweak disorder in one- and two-dimensional electron gases(2DEGs) while metallic states and Anderson localization tran-sitions (ALTs) can occur in 3D [13, 14]. Later more carefulrenormalization group calculations [15] and numerical sim-ulations [16–20], together with experiments [11], show thatintrinsic degrees of freedom can alter the results in 2D: Half-integer spin particle systems can also support ALTs when thespin rotational symmetry is broken through spin-orbit interac-tions (SOIs), regardless whether the time reversal symmetryis preserved (symplectic class) [15–20] or not (unitary class)[21–23]. The most unquestionable examples would be quan-tum Hall e ff ects of both non-interacting [24–26] and interact-ing [27, 28] 2DEGs in strong perpendicular magnetic fields.On the other hand, all states of disordered non-interactinginteger-spin particle systems must be localized [29, 30].However, recent numerical studies [21–23, 31–34] showedthat the current understanding of ALTs in non-interacting2DEGs is far from completed when SOIs are involved. Forexample, in contrast to the predictions based on the nonlinear σ model that claims only localized states are allowed in theunitary ensemble [15], a band of extended states together withan ALT or critical states usually accompanied by a Kosterlitz-Thouless (KT) transition could exist in 2DEGs without time-reversal symmetry, depending on the form of SOIs and thestrength of magnetic field [21–23]. In this work, we ob-serve an anomalous phase transition from a normal metal toa marginal metal [35], consisting of a band of metallic critical states, in a ferromagnetic 2DEG on a square lattice subject toa Rashba SOI and random on-site potentials. The statementsare supported by argumentations based on two independenthighly-accurate numerical approaches: the finite-size scalinganalysis of two-terminal conductances and the inverse partici-pation ratios (IPRs) analysis of wave functions obtained fromthe exact diagonalization. The unaccustomed marginal metal(MM) phase exists between a di ff usive metal (DM) phaseat weak disorders and an Anderson insulator (AI) phase atstrong disorders. Scaling analyses of IPRs show that wavefunctions of states in the MMs are of fractals of dimension D = . ± .
02, a feature reminiscent of a band of criticalstates in the random SU(2) model subject to strong magneticfields [21].Our main result is the new (marginal metallic) phase whose β -function (symbols and black line) defined as β (ln g L ) = d ln g L / d ln L describes an unconventional DM-MM-AI tran-sition, as summarized in Fig. 1. In comparison, β for othertypes of phase transitions in 2D are also included (elucidatein the caption). Here g L and L are dimensionless conduc-tance and system size, respectively. For a large (small) con-ductance, i.e., g L > g c , ( g L < g c , ), β is positive (nega-tive), indicating a metallic (insulating) phase. While betweentwo critical conductances g c , and g c , , β = ff erent from an ALT at a fixed point g c [16–20, 22, 23], the MM phase between [ g c , , g c , ] is a fixedline in which the system does not flow away when its size isscaled. Furthermore, near both DM-MM and MM-AI tran-sition points, correlation lengths ξ locating on the DM andAI sides, respectively, diverge with disorder strength W as ξ ( W ) ∝ exp[ α/ √| W − W c | ], a similar finite-size scaling law a r X i v : . [ c ond - m a t . m e s - h a ll ] D ec L = 1 2 8 L = 1 9 2 L = 2 5 6 L = 3 8 4 L = 5 1 2 l n g c , l n g c , b l n g L l n g c FIG. 1. β (ln g L ) of various 2D materials. Black solid line standsfor the numerical results (symbols in Fig. 2(a) for di ff erent systemsizes) presented in this work, which displays the coexistence of DM( β > β = β < β = in KT transitions (green line) [21, 31, 35].Our model is a tight-binding Hamiltonian on a square latticeof size L with unit lattice constant, H = (cid:88) i c † i (cid:15) i c i − (cid:88) (cid:104) i , j (cid:105) c † i R i j c j , (1)where c † i = ( c † i , ↑ , c † i , ↓ ) and c i are, respectively, the single elec-tron creation and annihilation operators on site i = ( x i , y i ) with x i , y i being integers and 1 ≤ x i , y i ≤ L . (cid:104) i j (cid:105) denotes i and j asthe nearest-neighbor sites. The first term stands for on-siteenergy: (cid:15) i = (cid:15) σ − ∆ σ z + V i σ . (2)Here σ and σ x , y , z are the 2-by-2 identity and the Pauli ma-trices respectively. (cid:15) is a constant energy term. − ∆ σ z in-troduces a ferromagnetic term that breaks the time-reversalsymmetry with ∆ quantifying the mean-field exchange split-ting [40]. Disorders are modelled by V i σ with uncorrelatedrandom numbers V i following the normal distribution of zeromean and the variance of W . Thus, W measures the degreeof randomness. A Rashba SOI is encoded in the hopping ma-trices R i j , parameterized by matrices R x and R y along the x − and y − directions, respectively, R x =
12 ( t σ + i ˜ ασ y ) and R y =
12 ( t σ − i ˜ ασ x ) . (3) where t is the energy unit and ˜ α measures the strength of SOIs.In the clean limit, model (1) can be blocked diagonalized inthe momentum space as H = (cid:80) k c † k h ( k ) c k with h ( k ) = ( (cid:15) − cos k x − cos k y ) σ − ∆ σ z + ˜ α (sin k x σ y − sin k y σ x ) . (4)Hereafter we fix (cid:15) = ff ective k · p Hamilto-nian near the band edge reads p + ˜ α ( p × σ ) · ˆ z . This formof Hamiltonians has been widely employed to enlighten theintrinsic and extrinsic mechanism of the anomalous Hall ef-fect, and possible physical realizations of model (1) include alarge family of ferromagnetic semiconductors such as GaAsand other III-V host materials [40].To investigate the localization properties of states of model(1), we employ the Landauer formula to calculate the dimen-sionless conductance of a disordered sample between twoclean semi-infinite leads at a given Fermi level E , ˜ g L = Tr[
T T † ], where T is the transmission matrix [41]. To excludethe contribution from contact resistances, we define the di-mensionless conductance g L as, 1 / g L = / ˜ g L − / N c , where N c is the number of propagating modes in the leads at Fermienergy E [42]. The determination of quantum phase transi-tions is based on the following criteria: (1) For the Fermi levelin the DM (AI) phase, dg L / dL is positive (negative), while inthe MM phase, g L is independent of L . (2) In the vicinity ofphase transition points, there exists the one-parameter scalinghypothesis [13] such that g L of di ff erent system sizes mergeinto a universal smooth scaling function f ( x ), i.e., g L = f ( L /ξ ) , (5)with the correlation length ξ diverging at the transition points. g L WW c =0.53 W c , =0.43 g L L=128L=192L=256L=384L=512W c , =0.38(a) -26 -16 g L L ξ (b) W -17 -7 g L L ξ W (c) FIG. 2. (a) g L as a function of W for L = E = . ∆ = .
01. Inset is the same plot but for ∆ = W c , . Inset: Scaling function f ( x = L /ξ ) obtained by collapsing data for W < W c , into a singlecurve. (c) Same as (b) but for the regime near W c , . Two typical examples are shown in Fig. 2(a) and its insetthat plot g L as a function of W for E = . α = .
2, and ∆ = .
01 and 0 (inset), respectively. Clearly,in the absence of the ferromagnetic coupling ∆ = W c . dg L / dL is positive (negative) for W smaller(larger) W c . These features are concrete evidence of an ALTat W c . Finite-size scaling analysis [43] shows that g L for dif-ferent sizes L collapse to a single smooth scaling curve, and ξ diverges as | E − E c | − ν with ν = . ± .
2, consistent withprevious estimates [18–20].Strikingly, once systems enter the unitary class by turn-ing on the ferromagnetic coupling, say ∆ = .
01, we ob-serve a MM phase in the window of W ∈ [ W c , , W c , ] ( W c , = . ± .
01 and W c , = . ± .
01) within which dg L / dL = L , while states for W < W c , and W > W c , are extendedand localized, respectively. Two phase transitions are evidentin Figs. 2 (b) and (c), which illustrate the enlargements near W c , and W c , , respectively. The MM-AI transition at W c , , inaddition to the zero-plateau of β function shown in Fig. 1, ishighly evocative of the KT criticality arising in another unitaryensemble with random SOIs [21] and the graphene with ran-dom fluxes or long-range impurities [31, 35]. Nonetheless, theDM-MM transition at W c , from a band of extended states toa band of critical states is highly nontrivial since both of themare of metallic phases in the sense that their wave functionsspread over the whole lattice (illustrate later). To the best ofour knowledge, this kind of disorder-driven metal-metal tran-sitions has never been observed before in 2D materials, but in3D semimetals [44–48].To substantiate the validity of one-parameter scaling hy-pothesis, we show that all curves in Figs. 2(b,c) collapse intotwo smooth functions f ( x = L /ξ ) shown in the insets of thefigures [43], which o ff er direct verifications of quantum phasetransitions at W c , and W c , . On the insulating side and near thephase transition point W c , , the correlation length is expectedto diverge as ξ ∝ exp[ α / (cid:112) | W − W c , | ] with α = . ± . ff erently,there are no reliable analytical and numerical estimates forthe divergence law near the DM-MM transition W c , . Scal-ing analysis also suggests ξ ∝ exp[ α / (cid:112) | W − W c , | ] with theexponent α = ±
3. Besides, a power-law divergence ofcorrelation lengths for ALTs, i.e., ξ ∝ | W − W c | − ν , also fitthe numerical data with ν =
32. However, the obtained ν ismuch larger than any known critical exponents in disordered2D systems [11, 12]. We thus argue that at the DM-MM tran-sition the correlation lengths show the same scaling behavioras those for KT transitions, rather than the power-law diver-gence for ALTs.So far, we have provided one example of the DM-MM-AItransition in model (1). Needless to say, many questions arise,and, among them, the most important one may be the proof ofthe universality of such a quantum phase transition. In Sup-plemental Materials [43], we show indications of universal-ity by substantiating emergences of the three phases at dif-ferent Fermi energies with the same divergence law of ξ , i.e., ξ ∝ exp[ α/ √| W − W c | ] at critical points. The same physicsis observed if we choose E as the scaling variable at a fixeddisorder. Furthermore, simulations for a di ff erent form of dis- orders and a distinct SOI due to the Dresselhaus e ff ect areboth in qualitative agreement with Fig. 2. All these featuresindicate that the MM phase prevails in ferromagnetic 2DEGswith SOIs and favors the exponential divergence of ξ at criti-cal points.Having established the universality of DM-MM-AI transi-tions, we further consider the nature of wave functions in threephases, especially the fractal structure of wave functions inMM phase. Wave functions at an isolated critical point ofan ALT or in the critical band are known to have multifrac-tal structures characterizing by a set of anomalous dimensionsmeasuring how their moments scale with sizes [12, 21, 35].Among them, the fractal dimension plays a pre-eminent role,which is related to the IPR defined as p ( E , W ) = (cid:88) i | ψ i ( E , W ) | (6)with ψ i ( E , W ) being the renormalized wave functions of en-ergy E and disorder W at site i for a specific realization. Forlarge enough systems, the average IPR scales with size L as[49–51] p ( E , W ) ∝ L − D (7)with D being the fractal dimension. If the state is extended(localized), D = d ( D = d = L is expected, i.e., D ∈ [0 , d ]. Thus, we expect that states in theMM phase have a universal fractal dimension such that theirwave functions occupy much sparse space like fractals. A I -
D MM M D W( b ) D W FIG. 3. (a) −(cid:104) ln p (cid:105) vs ln L for E = .
2. Several disorders in di ff erentphases are chosen. DM: W = .
08 (squares) and 0.28 (circles); MM: W = .
42 (up-triangles); AI: W = .
56 (down-triangles) and 0.66(diamonds). Solid lines are linear fits of numerical data. (b) D as afunction of W for E = . D = .
90. The three phases, colored by magenta (DM), blue (MM),and green (AI), are identified according to data in Fig. 2.
Numerically, we use the exact diagonalization to find theeigenfunctions of model (1). In our scenario, we constructthe tight-binding Hamiltonian by the Kwant package [52] inPython and solve the eigenequation H ψ = E ψ by the Scipylibrary [53] for L varying from 200 to 300 . The averagelogarithms of IPR as a function of ln L for W = .
08, 0.28,0.42, 0.56, and 0.66 at E = . W from the DM phase to the AI phase,and D = . ± .
02 for W c , < W = . < W c , in the MMphase.We further authenticate the universality of the fractal natureby displaying D ( W ) at E = . D = .
90 is observed in the MM phase determined by data inFig. 2(a), indicating that the fractal dimension of the fixed linedoes not depend on W . For W < W c , (DM), wave functionsare not a fractal any more since D (cid:39) d , while, for W (cid:29) W c , ,IPRs are found to be independent of L , i.e., D =
0, see theinset of Fig. 3(b), a typical feature for AIs. Noticeably, wavefunctions near MM-AI transitions and on the localized side,for example W ∈ [0 . , . > W c , , have fractal structures aswell, which can be contributed to the finite-size e ff ect, ratherthan the criticality, since D will decrease if we evaluate it byusing larger system sizes [43]. D M M M W D A IA L T
FIG. 4. Schematic phase diagram in the ∆ − W plane: DM (ma-genta), MM (blue), and AI (green) at E = .
2. Only an iso-lated critical level exists at ∆ = ξ ∝ | W − W c | − ν and ν = .
8. For ∆ (cid:44)
0, the MM phase ex-ists within a window of [ W c , ( ∆ ) , W c , ( ∆ )], and ξ in the vicinity of W c , (black squares) and W c , (black circles) diverges exponentiallyas ξ ∝ exp[ α/ √| W − W c | ]. It is also enlightening to compare the fractal dimensions ofthe MM phase in model (1) with those of critical states in other2D materials. The fractal dimension of isolated critical levelsfor ALTs in the symplectic ensemble is found to be 1 . ± .
05 [54] while D = .
75 for the quantum Hall type criticality[55]. Thus, wave functions in the MM phase with D = . ± .
02 occupy a larger space than those critical states of random SU(2) model under strong magnetic fields [21].A more inclusive picture is procured by executing exhaus-tive simulations for di ff erent ∆ at E = .
2. The fixed line per-sists at finite ∆ as expected such that the three phases coexist,and model (1) always experiences the DM-MM-AI phase tran-sitions at ∆ − dependent transition points W c , and W c , . While,for ∆ =
0, there is only one quantum critical state at which thesystem undergoes a normal ALT. Furthermore, it is numeri-cally justified that the correlation lengths ξ always diverge as ξ ∝ exp[ α/ √| W − W c | ] near W c , (squares) and W c , (circles)in Fig. 4, see clarifications in Supplemental Materials [43].In conclusion, analyses of the dimensionless conductanceand the IPR provide substantial evidence to the existenceof an unusual marginal metallic phase between the di ff usivemetal and the Anderson insulator in ferromagnetic 2DEGswith SOIs. Such systems undergo a DM-MM-AI transition aseither disorder strength or Fermi level varies. Near the tran-sition points, the conductance can be described well by theone-parameter scaling hypothesis. The criticality of the DM-MM-AI transitions is consistent with universality descriptionof a quantum phase transition in the sense that correlationlengths diverge as an exponential of an inverse square rootof | W − W c | for all critical points. Besides, eigenfunctions inthe MM phase are of fractals of dimension D = .
90, whileextended states in the DM phase spread over the entire lattice.A schematic phase diagram in the ∆ − W plane is presented.This work is supported by the National Key Research & De-velopment Program of China (Grants No. 2016YFA0200604),the National Natural Science Foundation of China (GrantsNo. 11774296, 11704061, 21873088, and 11874337), andHong Kong RGC (Grants No. 16301518 and 16300117).C.W. is supported by UESTC and the China Postdoc-toral Science Foundation (Grants No. 2017M610595 and2017T100684). C.W. also acknowledges the support fromPeng Yan and Xiansi Wang. ∗ Corresponding author: [email protected] † Corresponding author: [email protected][1] P. W. Anderson, Absence of Di ff usion in Certain Random Lat-tices, Phys. Rev. , 1492 (1958).[2] S. John, Electromagnetic Absorption in a Disordered Mediumnear a Photon Mobility Edge, Phys. Rev. Lett. , 2169 (1984).[3] P. Sheng and Z.-Q. Zhang, Scalar-Wave Localization in a Two-Component Composite, Phys. Rev. Lett. , 1879 (1986).[4] D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, Lo-calization of light in a disordered medium, Nature , 671(1997).[5] M. St¨orzer, P. Gross, C. M. Aegerter, and G. Maret, Observationof the Critical Regime Near Anderson Localization of Light,Phys. Rev. Lett. , 063904 (2006).[6] A. A. Chabanov, M. Stoytchev, A. Z. Genack, Statistical signa-tures of photon localization, Nature , 850 (2000).[7] H. Hu, A. Strybulevych, J. H. Page, S. E. Skipetrov, and B. A.van Tiggelen, Localization of ultrasound in a three-dimensionalelastic network, Nat. Phys. , 945 (2008). [8] T. Schwartz, G. Bartal, S. Fishman, M. Segev, Transport andAnderson localization in disordered two-dimensional photoniclattices, Nature , 52 (2007).[9] C. Wang, Y. Cao, X. R. Wang, and P. Yan, Interplay of wavelocalization and turbulence in spin Seebeck e ff ect, Phys. Rev. B , 144417 (2018).[10] P. A. Lee and T. V. Ramakrishnan, Disordered electronic sys-tems, Rev. Mod. Phys. , 287 (1985).[11] B. Kramer and A. MacKinnon, Localization: theory and exper-iment, Rep. Prog. Phys. , 1355 (2008).[13] E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ra-makrishnan, Scaling Theory of Localization: Absence of Quan-tum Di ff usion in Two Dimensions, Phys. Rev. Lett. , 673(1979).[14] D. Friedan, Nonlinear Models in 2 + (cid:15) Dimensions, Phys. Rev.Lett. , 1057 (1980).[15] S. Hikami, Localization, Nonlinear σ Model and String Theory,Prog. Theor. Phys. Suppt. , 213 (1992).[16] S. N. Evangelou, Anderson Transition, Scaling, and LevelStatistics in the Presence of Spin Orbit Coupling, Phys. Rev.Lett. , 2550 (1995).[17] R. Merkt, M. Janssen, and B. Huckestein, Network model fora two-dimensional disordered electron system with spin-orbitscattering, Phys. Rev. B , 4394 (1998).[18] Y. Asada, K. Slevin, and T. Ohtsuki, Anderson Transition inTwo-Dimensional Systems with Spin-Orbit Coupling, Phys.Rev. Lett. , 256601 (2002).[19] P. Markoˇs and L. Schweitzer, Critical regime of two-dimensional Ando model: relation between critical conduc-tance and fractal dimension of electronic eigenstates, J. Phys.A , 3221 (2006).[20] G. Orso, Anderson Transition of Cold Atoms with SyntheticSpin-Orbit Coupling in Two-Dimensional Speckle Potentials,Phys. Rev. Lett. , 105301 (2017).[21] C. Wang, Y. Su, Y. Avishai, Y. Meir, and X. R. Wang, Band ofCritical States in Anderson Localization in a Strong MagneticField with Random Spin-Orbit Scattering, Phys. Rev. Lett. ,096803 (2015).[22] Y. Su, C. Wang, Y. Avishai, Y. Meir, and X. R. Wang, Absenceof localization in disordered two-dimensional electron gas atweak magnetic field and strong spin-orbit coupling, Sci Rep , 104204 (2017).[24] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. denNijs, Quantized Hall Conductance in a Two-Dimensional Peri-odic Potential, Phys. Rev. Lett. , 405 (1982).[25] A. M. M. Pruisken, Universal Singularities in the Integral Quan-tum Hall E ff ect, Phys. Rev. Lett. , 1297 (1988).[26] B. Huckestein, Scaling theory of the integer quantum Hall ef-fect, Rev. Mod. Phys. , 357 (1995).[27] D. C. Tsui, H. L. Stormer, and A. C. Gossard, Two-DimensionalMagnetotransport in the Extreme Quantum Limit, Phys. Rev.Lett. , 1559 (1982).[28] R. B. Laughlin, Anomalous Quantum Hall E ff ect: An Incom-pressible Quantum Fluid with Fractionally Charged Excita-tions, Phys. Rev. Lett. , 1395 (1983).[29] R. Sepehrinia, Universality of Anderson transition in two-dimensional systems of symplectic symmetry class, Phys. Rev.B , 045104 (2010).[30] Y. Su and X. R. Wang, Role of spin degrees of freedom in An- derson localization of two-dimensional particle gases with ran-dom spin-orbit interactions, Phys. Rev. B , 224204 (2018).[31] X. C. Xie, X. R. Wang, and D. Z. Liu, Kosterlitz-Thouless-TypeMetal-Insulator Transition of a 2D Electron Gas in a RandomMagnetic Field, Phys. Rev. Lett. , 3563 (1998).[32] G. Xiong, S.-D. Wang, Q. Niu, D.-C. Tian, and X. R. Wang,Metallic Phase in Quantum Hall Systems due to Inter-Landau-Band Mixing, Phys. Rev. Lett. , 216802 (2001).[33] Y. Avishai and Y. Meir, New Spin-Orbit-Induced UniversalityClass in the Integer Quantum Hall Regime, Phys. Rev. Lett. ,076602 (2002).[34] J. Bang and K. J. Chang, Localization and one-parameter scal-ing in hydrogenated graphene, Phys. Rev. B , 193412 (2010).[35] Y.-Y. Zhang, J. Hu, B. A. Bernevig, X. R. Wang, X. C. Xie, andW. M. Liu, Localization and the Kosterlitz-Thouless Transitionin Disordered Graphene, Phys. Rev. Lett. , 106401 (2009).[36] We cannot use the numerical data at hand to depict β ( g L ) forlarge g L . Instead, at this stage, we use an analytical formula β ( g ) = f g − + f g − + o ( g − ) based on the non-linear σ model[15] with f and f being fitting parameters. The exhaustive il-lustration of β ( g ) in the large g limit surely deserves furthere ff orts, which, however, is not the main focus in this work.[37] Y. Asada, K. Slevin, and T. Ohtsuki, Numerical estimation ofthe β function in two-dimensional systems with spin-orbit cou-pling, Phys. Rev. B , 035115 (2004).[38] J. H. Bardarson, J. Tworzydło, P. W. Brouwer, and C. W.J. Beenakker, One-Parameter Scaling at the Dirac Point inGraphene, Phys. Rev. Lett. , 106801 (2007).[39] Z. Xu, L. Sheng, R. Shen, B. Wang, and D. Y. Xing, Kosterl-itzThouless transition in disordered two-dimensional topologi-cal insulators, J. Phys. Condens. Matter , 065501 (2013).[40] N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N. P.Ong, Anomalous Hall e ff ect, Rev. Mod. Phys. , 1539 (2010).[41] A. MacKinnon, The calculation of transport properties and den-sity of states of disordered solids, Z. Phys. B , 385 (1985).[42] S. Datta, Electronic Transport in Mesoscopic Systems (Cam-bridge University Press, Cambridge, England, 1995).[43] See Supplemental Materials at http: // link.aps.org / supplemental.[44] P. Goswami and S. Chakravarty, Quantum criticality betweentopological and band insulators in 3 + , 196803 (2011).[45] K. Kobayashi, T. Ohtsuki, K. T. Imura, and I. F. Herbut, Densityof states scaling at the semimetal to metal transition in three di-mensional topological insulators, Phys. Rev. Lett. , 016402(2014).[46] C.-Z. Chen, J. Song, H. Jiang, Q. Sun, Z. Wang, and X. C. Xie,Disorder and Metal-Insulator Transitions in Weyl Semimetals,Phys. Rev. Lett. , 246603 (2015).[47] B. Fu, W. Zhu, Q. Shi, Q. Li, J. Yang, and Z. Zhang Accu-rate Determination of the Quasiparticle and Scaling PropertiesSurrounding the Quantum Critical Point of Disordered Three-Dimensional Dirac Semimetals, Phys. Rev. Lett. , 146401(2017).[48] B. Roy and M. S. Foster, Quantum Multicriticality near theDirac-Semimetal to Band-Insulator Critical Point in Two Di-mensions: A Controlled Ascent from One Dimension, Phys.Rev. X , 011049 (2018).[49] X. R. Wang, Y. Shapir, and M. Rubinstein, Analysis of mul-tiscaling structure in di ff usion-limited aggregation: A kineticrenormalization-group approach, Phys. Rev. A , 5974 (1989).[50] M. Janssen, Statistics and scaling in disordered mesoscopicelectron systems, Phys. Rep. , 1 (1998).[51] J. H. Pixley, P. Goswami, and S. Das Sarma, Anderson Local-ization and the Quantum Phase Diagram of Three Dimensional Disordered Dirac Semimetals, Phys. Rev. Lett. , 076601(2015).[52] C. W. Groth, M. Wimmer, A. R. Akhmerov, and X. Waintal,Kwant: a software package for quantum transport, New J. Phys. , 063065 (2014).[53] E. Jones, E. Oliphant, P. Peterson, et al. , SciPy: Open SourceScientific Tools for Python, [Online; accessed 2019-05-24].[54] H. Obuse and K. Yakubo, Anomalously localized states and multifractal correlations of critical wave functions in two-dimensional electron systems with spin-orbital interactions,Phys. Rev. B , 125301 (2004).[55] F. Evers, A. Mildenberger, and A. D. Mirlin, Multifractality atthe spin quantum Hall transition, Phys. Rev. B67