Table Of Content

Anomalous Hall Effect on the surface of topological Kondo insulators E. J. König,1 P. M. Ostrovsky,2,3 M. Dzero,4 and A. Levchenko1,5 1Department of Physics, University of Wisconsin-Madison, Madison, Wisconsin 53706, USA 2Max Planck Institute for Solid State Research, Heisenbergstr. 1, 70569 Stuttgart, Germany 3L. D. Landau Institute for Theoretical Physics RAS, 119334 Moscow, Russia 4Department of Physics, Kent State University, Kent, OH, 44242, USA 5Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824, USA (Dated: January 2, 2016) WecalculatetheanomalousHallconductivityσ ofthesurfacestatesincubictopologicalKondo xy insulators. We consider a generic model for the surface states with three Dirac cones on the (001) 6 surface. TheFermivelocity,theFermimomentumandtheZeemanenergyindifferentDiracpockets 1 may be unequal. The microscopic impurity potential mediates mixed intra and interband extrinsic 0 scatteringprocesses. Ourcalculationofσ isbasedontheKubo-Stredadiagrammaticapproach. It 2 xy includesdiffractiveskewscatteringcontributionsoriginatingfromtheraretwo-impuritycomplexes. n Remarkably,thesecontributionsyieldanomalousHallconductivitythatisindependentofimpurity a concentration,andthusisofthesameorderasotherknownextrinsicsidejumpandskewscattering J terms. We discuss various special cases of our results and the experimental relevance of our study 2 in the context of the recent hysteretic magnetotransport data in SmB samples. 6 ] PACSnumbers: 72.10.Fk,72.25.-b,73.23.-b,75.20.Hr l l a h Topological Kondo Insulators. Topological insu- - s lators [1–3] remain a vibrant field of research in present e daycondensedmatterphysics. Themainthrustsforthis m extraordinaryscientificinterestincludethevastpotential . t technologicalapplicationsinthefieldsofnanoelectronics a andquantumcomputationaswellasthefascinatinginno- m vativerealizationoffundamentalconceptsfromquantum - d field theory and differential geometry. n Among the various realizations of topological phases o of matter, topological Kondo insulators (TKIs) [4, 5], c FIG. 1: Dispersion relation of surface states in a TKI with takeaspecialplace. Theirtopologicallyprotectedmetal- [ cubic symmetry. We chose experimentally realistic parame- lic surface states emerge as a result of the hybridiza- ters for the case of SmB [15]: Fermi velocities of (cid:39) 72 (26) 1 6 tionbetweenweakly correlatedconductionelectrons and meV·nm at the Γ (X and Y) points, an offset E ≈39meV v Γ 7 strongly correlated states. In particular, theories [6–9] of the central Dirac cone, an ellipticity (cid:112)vX,y/vX,x ≈ 1.2, 9 describing states on the (001) surface suggest a low en- a reciprocal lattice constant 2π/a = 15 nm−1. Further, we 1 ergy Hamiltonian with three Dirac bands located at Γ, assumedagapof10meV(20meV)attheΓ(XandY)points. 0 X and Y points of the surface Brillouin zone (BZ), see 0 Fig. 1. Main experimentally distinguishing characteris- . 1 tics of the TKIs are the saturation of resistivity at very It has been also proposed that the surface states in 0 low temperatures, pronounced temperature dependence SmB may be of conventional type [21, 22], i.e. they 6 6 ofthemagneticsusceptibilityacrossawiderangeoftem- have quadratic dispersion modified by the presence of 1 peratures and a fairly narrow insulating gap [10, 11]. In- strong spin-orbit coupling. Note, that in this scenario, : v triguingrecentexperimentalevidencefortheTKIphysics the conduction states will remain decoupled from the i X was reported in SmB6 samples revealing the predicted Sm moments on the surface via the Kondo breakdown surface dominated transport directly [12], by thickness mechanism,sothatferromagneticorderingofthesamar- r a independent resistivity measurements showing the viola- ium f-electrons would still be possible. This controversy tion of Ohm’s law [13], and 2D (surface) weak antilocal- – Dirac vs. conventional surface states – motivates us ization data [14]. Furthermore, characteristics of Dirac to study the magnetotransport properties of the surface electronswererevealedusingARPES[15–17]andtorque states on the background of induced nonzero magnetiza- magnetometry [18]. In addition, hysteretic magneto- tion. Specifically, in this paper we calculate the anoma- transport measurements have been reported by several lous Hall conductivity for a cubic topological Kondo in- groups [19, 20]. This effect can be attributed to ferro- sulator with three Dirac surface bands. Our results for magnetic domains formed on the surface by unscreened theanomalousHallconductivityallowustoelucidateex- samarium magnetic moments or samarium sesquioxide perimentally distinguishable characteristics of the Dirac (Sm O ) impurities. electrons and should help to resolve the controversy dis- 2 3 2 ^j pX pΨ (a) y (b) AB 0.6AB 0.4 ^jy R1 0.4 ^jx R2 R1 ^jx R2 0.2 R1 0.2 R R1 (c) ^jy (d) ^jy R2 2 4 6 82 10 p |R -R | F 1 2 -0.2 + 2 4 6 8pF |R1 -R2 | FIG. 3: Diffractive skew scattering: Spatially-resolved scat- ^j R ^j R tering probabilities pX and pΨ for intraband scattering. x 1 x 1 AB AB R R 2 2 FIG. 2: Diagrammatic representation and real space trajec- theAHErelyonrareimpurityconfigurations(seeFig.2). toriesforextrinsiccontributionstoσxy. Quantumcomplexes Therefore, both crossed and non-crossed diagrams are of responsiblefortheAHEareshownbyanellipsewithfocuses the same order and suppressed by λ /l as compared to F inpointsR andR . Inthenoncrossingapproximation,both 1 2 the diagonal conductance. Diagrams with more than a skew-scattering (a) and side jump (b) contributions rely on single cross are even smaller [34]. These qualitative ar- coherent interband scattering between opposite branches of theDiracspectrum. Thecorrespondingvirtualstatesaswell guments are fully supported by the microscopic compu- as off-shell excitations entering crossed X and Ψ diagrams, tation which we present in the remainder of the paper. (c) and (d) respectively, are marked by a yellow arrow in ex- Tomakethediffractiveanalogytransparentwepresent emplary positions of the diagrams. Due to the uncertainty in Fig. 2 examplary electron trajectories in real space. principle,thetypicalextensionofaquantumcomplexisthus The probability p =|(cid:80) A |2 =(cid:80) A A∗ of an elec- of the order of Fermi wavelength |R −R |∼λ . AB i i ij i j 1 2 F tron reaching a point r from r is the square of the B A sum of the amplitudes for all paths i,j. In Fig. 2, A i and A∗ are represented by different colors and opposite cussed above. j orientationofarrows. Inthenoncrossingapproximation, Anomalous Hall effect. Electron transport in fer- p(nc) =(cid:80) |A |2, and all interference terms are omitted. romagnetshasalonghistorygoingbacktoE.Hall’s1881 TAhBe crosseid Xi-diagrams contribute pX = (cid:80)(cid:48) A A∗, discovery of the anomalous Hall effect (AHE) [23], i.e. of AB i(cid:54)=j i j where the sum includes pairs of nonequal trajectories a transverse conductivity σ generated by the magne- xy equivalent to Fig. 2 (c). An analogous expression holds tization (Zeeman coupling) rather than by orbital cou- for pΨ . The interference pattern becomes apparent pling to a magnetic field. To account for this effect, two AB in the plots of spatially-resolved scattering probabilities equally appropriate techniques are commonly employed. pX,Ψ off two-impurity complexes, see Fig. 3. The lat- First, in the semiclassical approach [24] different terms AB ter exhibit pronounced Fraunhofer oscillatory interfer- in σ stem from distinct physical mechanisms of intrin- xy ence patterns. The novel extrinsic contributions [Fig. sic [25], skew scattering [26] and side jump [27] contri- 2 (c,d)] constitute inherent parts of the skew scattering butions. Second, σ can be directly calculated using xy and should necessarily be included to properly compute Kubo-Streda diagrammatic response theory [28]. Semi- the transverse conductivity. These terms can be distin- classics appear to be more intuitive, while the diagram- guished from previously studied processes [Fig. 2 (a,b)] matic treatment is more systematic. Notably, an ad- by means of their diffractive nature. ditional skew scattering mechanism was discovered only Model and Assumptions. We employ the dia- very recently with the help of diagrams [29–31]. Phys- grammatic approach to calculate the anomalous Hall re- ically, it originates from diffractive skew scattering off sponse on the surface of 3D TKIs with cubic symmetry two impurities residing about one Fermi wavelength λ F taking into account all the diagrams to the leading order from each other. Diagrammatically, these processes can inimpurityconcentration. Forthispurpose,considerthe be understood by considering crossed impurity lines in following low energy Hamiltonian the conductivity bubble [see Fig. 2 (c), (d)], and can be equivalently treated in the semiclassical approach pro- (cid:88) H = [v σ·p+m σ +E ]Π . (1) vided that crossed impurity lines are included into the 0 K K z K K K full scattering amplitude. At first glance, this observa- tionseemstobeinsharpcontrastwithconventionalwis- Throughout the paper, matrices in the space of Dirac dom of the impurity diagrammatic technique [32] that pockets (DPs) are denoted by an underscore and have dictates that a single cross of impurity lines implies a indices K,K(cid:48) ∈ {Γ,X,Y}. The symbol Π denotes a K rare disorder configuration and thus smallness in the pa- projector on Kth DP. Rotational C symmetry imposes 4 rameter λ /l(cid:28)1, where l is the elastic mean free path. v = v ≡ v, m = m ≡ m. We count energies from F X Y X Y However, it should be stressed that even previously dis- theDiracpointoftheX pocket, E =E ≡0, andmo- X Y cussed[24–27,29,33]contributionsofweakimpuritiesto menta in X (Y) pocket relative to Q =(π/a)eˆ X(Y) X(Y) 3 (a is the lattice spacing). For simplicity we omit the ν ((cid:15) )=θ((cid:15)2 −m2 )|(cid:15) |/2πv2 isthedensityofstates. K K K K K K anisotropy of X and Y pockets and set (cid:126) = 1 in the Furthermore we introduced the matrix intermediate formulas (we restore Planck’s constants in   W W W thefinalexpressionsforσ ). TheHamiltonianinEq.(1) ΓX ΓX xy containsjusttwoessentialingredientsforthefiniteAHE, W =WΓX W WXY  (5) W W W namely spin-orbit coupling and magnetization (time re- ΓX XY versal symmetry breaking), which is implicit in the mass with entries W = n |u˜(0)|2, W = n |u˜(π/a)|2, term of the Dirac fermions. √imp ΓX imp and W =n |u˜( 2π/a)|2, where u˜(q) is the Fourier Our model also contains uniformly distributed scalar XY imp transform of u(r). impurities with isotropic potential u(|r|) which is short- ranged on the scale of the smallest Fermi wavelength WenowturnourattentiontotheHallresponse. When minK(p−F,1K) with vKpF,K = (cid:112)(cid:15)2K −m2K where (cid:15)K = the Fermi energy lies in the gap, |(cid:15)K| < |mK|, the con- (cid:15) − E and (cid:15) is the Fermi energy. Our calculation is tribution of the Kth DP to the Hall response is K controlled in the parameter n /n (cid:28) 1 with n imp min imp sgn(m )e2 aanndd nthmeinca=rrmieirndKe(nnsKit)yboefitnhgetlheeasitmppoupruitlyatceodncpeonctkreattiroen- σxy|K =− 2 K h, (6) spectively. We assume weak impurities and treat them and stems from σII, only. The half-integer quantization in the leading Born approximation. xy isaconsequenceoffermionnumberfractionalization[35– Technically, the anomalous Hall response involves vir- 37]. Thisresultcanbeunderstoodintermsoftheintrin- tualstates(off-shellcontributions)residingwithinthera- sicmechanism,socalledanomalous velocity contribution dius ∆p=3max (p ) around Γ, X, Y and M points K F,K to AHE, as originally introduced by Karplus and Lut- of the BZ. As a consequence, the minimal three-band tinger [25]. Its topological origin was realized much later model Eq. (1) is applicable only for sufficiently small [38], and can be equivalently understood in terms of the Fermi momenta max (p ) (cid:28) π/a (see yellow plane K F,K Berry curvature that acts as an effective magnetic field in Fig. 1). Furthermore, the contribution to σ origi- xy for electron wave-packet motion in parameter space of nating from the states in the vicinity of the M point is √ momenta. Indeed, Hall conductivity can be presented negligible provided 2mM∆(cid:29)minK(pF,K), where mM as σ | = −(e2v2 /2πh)(cid:82) Ω (k)d2k, where the Berry (∆)istheeffectivemass(excitationgapofcloseststates) xy K K xy curvature for a single gapped Dirac cone is given explic- at the M point [35]. itly by Ω (k) = m /2(m2 + v2 k2)3/2 so that upon Calculation and results. It is common to distin- xy K K K momentum integration Eq. (6) follows. guish the following two contributions to the anomalous In contrast, outside the gap the contribution of σII Hall conductivity σ =σI +σII: xy xy xy xy is subleading in n /n (cid:28) 1. We therefore now fo- imp min σxIy = eh2 (cid:68)Tr(cid:104)ˆjxGRˆjyGA(cid:105)(cid:69), σxIIy =ec(cid:88) ∂∂nBK(cid:12)(cid:12)(cid:12)(cid:12) . rceupsreosnentthaeticoonntinribDuPtiosnpaocfeσaxnIyd.fiWnde switch to a matrix K B=0 The angular brackets denote disorder average in t(h2is) σI =2e2vF (cid:2)b+a[x+ψ]a(cid:3)FTvT. (7) xy h expression. The bare current operators are ˆj = µ diag(v σ ,vσ ,vσ ), while the clean Green’s functions In this expression, the bare velocity vertex is v = Γ µ µ µ at the Fermi energy (cid:15) are GR/A =((cid:15)±i0−H )−1. (vΓ,v,v) and the trace over spin space was already per- 0 0 formed. Thevariouscontributionshavethefollowingori- Thedisorderaverageleadstoafiniteself-energyenter- gin [cf. Fig. 2]: F is the noncrossed vertex correction, in ing the Green’s function. We find in momentum space, which at each stringer a of the ladder only contributions GR(p,(cid:15))=(cid:88) (cid:15)+K +vKσ·p+m−Kσz Π (3) whichareon-shellwerekept;bisalsopartoftheladderin ((cid:15)+)2−[(v p)2+(m−)2] K diagram,butinvolvescontributionsawayfromtheFermi K K K K surface; finally x [ψ] originates from the central part of with (cid:15)± = (cid:15) ±iΓ , m± = m ±iΓ(m). Here we also diagrams (c) and (d). All the elements of Eq. (7) are K K K K K K introduced the total scattering rates in the pocket K derived explicitly in Ref. [35] in terms of the microscopic parameters of the model. The anomalous Hall response, (cid:88) (cid:88)π Γ = Γ = ν ((cid:15) )[W] , (4a) Eqs. (7), constitutes the main result of our work. Un- K K→K(cid:48)→K 2 K(cid:48) K(cid:48) K(cid:48)K K(cid:48) K(cid:48) like previous calculations of the AHE in multiband sys- Γ(m) =(cid:88)Γ(m) =(cid:88)Γ mK(cid:48) , (4b) tems [39, 40], we included diagrams with crossed impu- K K→K(cid:48)→K K→K(cid:48)→K (cid:15) rity lines as they equally contribute to the leading order K(cid:48) K(cid:48) K(cid:48) approximation. The common physical origin of crossed as a function of intra- (K(cid:48) =K) and interpocket (K(cid:48) (cid:54)= diagramsmanifestsitselfinthecomplementarycontribu- K) scattering rates Γ and Γ(m) . Here, tions from the off-diagonal terms in x and ψ . K→K(cid:48)→K K→K(cid:48)→K KK(cid:48) KK(cid:48) 4 -σ Discussion. We now analyze our general result xy Eq. (7) in various simplifying cases. We consider both particularlimitsoftheimpuritypotentialu(x),andspe- 3 cial values of theparameters entering the clean Hamilto- 2 nian (1). 1 (i) Smooth disorder potential. We first consider the case when u(x) is smooth on the scale of the lattice con- 1 stant a. Interband scattering is negligible and we obtain 2 ϵ σxy =σx(0y)(|(cid:15)Γ|/mΓ)+2σx(0y)(|(cid:15)|/m). (8) -5 0 5 10 15 m The anomalous Hall conductivity of a single Dirac cone FIG. 4: Comparison of the AHE in the cases of a smooth is [30] disorder potential (dotted) and point like scatterers (solid). Weassumedm =m>0andadditionallyimposedv =vin (cid:18)|(cid:15)|(cid:19) e2 (cid:20)16|(cid:15)|m3θ((cid:15)2−m2) (cid:21) the case of shoΓrt ranged impurities. For the blue curΓves, we σ(0) =− +θ(m2−(cid:15)2) . xy m 2h ((cid:15)2+3m2)2 set (cid:15)Γ =(cid:15) while the red curves are obtained for (cid:15)Γ =(cid:15)−7m. (9) Itshouldbenotedthatinthiscasethereisnocontribu- theweak-antilocalizationeffect[41]inSmB samples[14, tionfromtheΨskewscatteringdiagrams[seeFig.2(d)] 6 20] suggests that this limit is most important for present as they vanish. This pecularity is accidental and specific day experiments. tothesingleDiracconelimit. Itcanbetracedbacktothe The explicit formula of σ ((cid:15)/m,(cid:15) /m) for the case destructive interference of scattering from two-impurity xy Γ v = v, and m = m has been relegated to Ref. [35]. If complexes as evidenced from the plot of the probability Γ Γ (cid:15) =(cid:15) this result further simplifies to pΨ in Fig. 3. The result for smooth disorder potential Γ AB is plotted in Fig. 4 using dotted curves. e28|(cid:15)|m(cid:0)(cid:15)2+8m2(cid:1) (ii) Fermi momentum in the gap. In what follows we σ =− . (11) xy h 3((cid:15)2+3m2)2 restore the possibility of nonzero interpocket scattering. When (cid:15)2 <m2, i.e. when the Fermi energy is in the gap Agraphicalcomparisonbetweenthecasesofsmoothdis- of X and Y pockets, the problem essentially simplifies order and point like impurities is shown in Fig. 4. It to a single Dirac cone and Eq. (8) holds again (using should be noted, that in the limit of short range scatters 2σx(0y)(|(cid:15)|/m)=−e2/h). our result for the Hall conductance ceases to be a con- Further, when (cid:15)2Γ < m2Γ (Fermi energy in the gap of tinuousfunction: itdisplaysdiscontinuitiesat(cid:15)K =mK. the Γ pocket) the problem simplifies to two equal Dirac A similar behavior was recently found in the Bychkov- cones. Surprisingly, the resulting Hall conductivity is Rashba model [31]. This is an artifact of an approxima- again given by Eq. (8) (using σx(0y)(|(cid:15)Γ|/mΓ) = −e2/2h) tionthatexploitsthebasicassumptionnimp (cid:28)nmin. As and is independent on the ratio W /W. a consequence, our result is not applicable in the energy XY (iii)EqualDPs. Wenextconsiderthesituationwhen window |(cid:15) −m | (cid:46) Γ. A more elaborate calculation K K the three DPs are equal, i.e. E = 0, v = v and m = should reveal smoothening of the discontinuities in the Γ Γ Γ m. The general expression for the Hall conductance is immediate vicinity of the band edges. presented in Ref. [35]. We note that for smooth disorder Conclusion and outlook. We have derived the potential, the effect of intraband scattering enters only anomalousHallresponseEq.(7)onthesurfaceofacubic to second order ((cid:15)2 >m2) topological Kondo insulator. We investigated a surface state model with three Dirac fermions plus an incipient σ =σ(0)(cid:18)|(cid:15)|(cid:19)(cid:20)3+(cid:18) (cid:15)2 −1(cid:19)WΓ2X +2WΓXWXY (cid:21). forth M-band in the generic case allowing for unequal xy xy m m2 W2 Fermi and Zeeman energies as well as unequal Fermi ve- (10) locities. We have analyzed several limiting cases of our While the single band Hall conductivity, Eq. (9), de- general result Eq. (7). As a byproduct of this analysis, cays as |(cid:15)|−3, for finite WΓX we find a term which de- we found that a system of two equal Dirac cones (as it cays only as |(cid:15)|−1. Thus interband scattering strongly occurs in Graphene) displays an AHE which is universal enhances anomalous Hall conductivity. This effect per- and independent of details of the scattering potential. sists to the case of arbitrary interband scattering. The Inasmuch experiments on TKIs are concerned, our Hall conductance (10) continuously approaches the gap mostimportantconclusionisthatthemagnetizationand value σxy =−3sgn(m)e2/2h. gate voltage dependence of the AHE can be used to (iv)Pointlikescatterers. Whentheimpuritypotential gaininformationaboutthemicroscopicnatureofsurface is short ranged on the scale of a, the matrix W = statesandimpurities. Indeed,theanalysisofvariouslim- K,K(cid:48) n |u(0)|2 for all K,K(cid:48). The experimental analysis of itingcasesofthethree-bandDiracmodelrevealsthatthe imp 5 large energy asymptote of the anomalous Hall response [5] M. Dzero, J. Xia, V. Galitski, and P. Coleman, preprint scales as (m/|(cid:15)|)3 in the case of smooth impurity poten- arXiv:1506.05635 (2015). tial while σ ∼ m/|(cid:15)| for short range scatterers. This [6] F. Lu, J. Zhao, H. Weng, Z. Fang, and X. Dai, Phys. xy Rev. Lett. 110, 096401 (2013). behaviorpersistsinthegenericresult. Incontrast,inthe [7] V. Alexandrov, M. Dzero, and P. Coleman, Phys. Rev. Bychkov-Rashba model σ ∼m/(cid:15)2 [31]. xy Lett. 111, 226403 (2013). As mentioned in the introduction, present day experi- [8] M. Ye, J. Allen, and K. Sun, preprint arXiv:1307.7191 mental samples are believed to host a multitude of large (2013). ferromagnetic domains. In our theory, smooth fluctua- [9] B.Roy,J.D.Sau,M.Dzero,andV.Galitski,Phys.Rev. tions of the magnetization can be taken into account by B 90, 155314 (2014). [10] A. Menth, E. Buehler, and T. H. Geballe, Phys. Rev. averaging the final result. Even after this procedure, the Lett. 22, 295 (1969). asymptotics allow to distinguish smooth and sharp im- [11] J. W. Allen, B. Batlogg, and P. Wachter, Phys. Rev. B purity potentials in the described manner. Up to now 20, 4807 (1979). magnetotransport experiments on TKIs concentrated a [12] S.Wolgast,C. Kurdak,K.Sun,J.W.Allen,D.-J.Kim, hystereticbehaviorinthelongitudinalconductance. Sys- and Z. Fisk, Phys. Rev. B 88, 180405 (2013). tematic investigation of the transverse conductance is [13] D. J. Kim, S. Thomas, T. Grant, J. Botimer, Z. Fisk, still needed. and J. Xia, Scientific Rep. 3, 3150 EP (2013). [14] S. Thomas, D. Kim, S. Chung, T. Grant, Z. Fisk, and In this paper we analyzed the semiclassical AHE and X. J. (2013), arXiv1307.4133. uncovered the importance of diffractive skew scattering [15] M. Neupane, N. Alidoust, S. Xu, T. Kondo, Y. Ishida, in the context of topological Kondo insulators. In a par- D.-J. Kim, C. Liu, I. Belopolski, Y. Jo, T.-R. Chang, allel vein, our results have further rich consequences for et al., Nat. Comm. 4 (2013). anomaloustransportphenomenainothermultibandma- [16] N. Xu, X. Shi, P. K. Biswas, C. E. Matt, R. S. Dhaka, Y.Huang, N.C.Plumb, M.Radović, J.H.Dil, E.Pom- terial systems such as Weyl semimetals [42] and chiral jakushina, et al., Phys. Rev. B 88, 121102 (2013). p-wave superconductors [43–45]. Quantum effects, such [17] J.Jiang,S.Li,T.Zhang,Z.Sun,F.Chen,Z.Ye,M.Xu, asinteractionandlocalizationcorrectionstotheconduc- Q. Ge, S. Tan, X. Niu, et al., Nat. Comm. 4 (2013). tivity tensor, the quantum AHE [46–48] and the surface [18] G. Li, Z. Xiang, F. Yu, T. Asaba, B. Lawson, P. Cai, state quantum Hall effect [37, 49–51] on TKIs remain a C. Tinsman, A. Berkley, S. Wolgast, Y. S. Eo, et al., theoretical and experimental challenge for the future. Science 346, 1208 (2014). [19] S.Wolgast,Y.S.Eo,T.O¨ztu¨rk,G.Li,Z.Xiang,C.Tins- Acknowledgements. We thank A. Andreev, I. man, T. Asaba, B. Lawson, F. Yu, J. W. Allen, K. Sun, Dmitriev, M. Khodas, L. Li, J. Sauls, K. Sun and M. L. Li, Ç. Kurdak, D.-J. Kim, and Z. Fisk, Phys. Rev. B Titov for important discussions. P.M.O. and E.J.K. ac- 92, 115110 (2015). knowledge hospitality by the Department of Physics and [20] Y. Nakajima, P. S. Syers, X. Wang, R. Wang, and AstronomyatMichiganStateUniversity,andbytheDe- J. Paglione, preprint arXiv:1312.6132 (2013). partment of Physics at University of Michigan (E.J.K.). [21] Z.-H. Zhu, A. Nicolaou, G. Levy, N. P. Butch, P. Syers, This work was financially supported in part by NSF X. F. Wang, J. Paglione, G. A. Sawatzky, I. S. Elfimov, and A. Damascelli Phys. Rev. Lett. 111, 216402 (2013). Grant No. DMR-1506547 (M.D.), and NSF Grants No. [22] P. Hlawenka, K. Siemensmeyer, E. Weschke, DMR-1606517 and ECCS-1560732 (E.J.K. and A.L.). A. Varykhalov, J. Snchez-Barriga, N. Y. Shitseval- SupportforthisresearchattheUniversityofWisconsin- ova, A. V. Dukhnenko, V. B. Filipov, S. Gabni, MadisonwasprovidedbytheOfficeoftheViceChancel- K. Flachbart, O. Rader, E. D. L. Rienks, preprint lor for Research and Graduate Education with funding arXiv:1502.01542 (2015). from the Wisconsin Alumni Research Foundation. Sup- [23] N. Nagaosa, J. Sinova, S. Onoda, A. MacDonald, and port for this research at Michigan State University was N. Ong, Rev. Mod. Phys. 82, 1539 (2010). [24] N. A. Sinitsyn, J. of Phys.: Cond. Matt. 20, 023201 provided by the Institute for Mathematical and Theo- (2008). retical Physics with funding from the office of the Vice [25] R. Karplus and J. M. Luttinger, Phys. Rev. 95, 1154 President for Research and Graduate Studies. (1954). [26] J. Smit, Physica 21, 877 (1955). [27] L. Berger, Physica 30, 1141 (1964). [28] P. Streda, J. Phys. C 15, L717 (1970). [29] N. Sinitsyn, A. MacDonald, T. Jungwirth, V. Dugaev, [1] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 and J. Sinova, Phys. Rev. B 75, 045315 (2007). (2010). [30] I.A.Ado,I.A.Dmitriev,P.M.Ostrovsky,andM.Titov, [2] X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 Europhys. Lett. 111, 37004 (2015). (2011). [31] I.A.Ado,I.A.Dmitriev,P.M.Ostrovsky,andM.Titov, [3] B. Bernevig and T. Hughes, Topological Insulators and preprint arXiv:1511.07413 (2015). TopologicalSuperconductors (PrincetonUniversityPress, [32] A. A. Abrikosov, L.P. Gor’kov, and I. E. Dzyaloshin- 2013), ISBN 9780691151755. skii, Quantum Field Theoretical Methods in Statistical [4] M. Dzero, K. Sun, V. Galitski, and P. Coleman, Phys. Physics, (Pergamon, 1965). Rev. Lett. 104, 106408 (2010). 6 [33] N. A. Sinitsyn, J. E. Hill, H. Min, J. Sinova, and A. H. MacDonald, Phys. Rev. Lett. 97, 106804 (2006). [34] Maximallycrosseddiagramscanformanexception,since they lead to weak localization corrections that are en- hanced logarithmically. [35] Supplementary materials to this article. [36] R. Jackiw and C. Rebbi, Phys. Rev. D 13, 3398 (1976). [37] E.Ko¨nig,P.Ostrovsky,I.Protopopov,I.Gornyi,I.Bur- mistrov,andA.Mirlin,Phys.Rev.B90,165435(2014). [38] Ming-Che Chang and Qian Niu, Phys. Rev. Lett. 75, 1348 (1995). [39] A. A. Kovalev, Y. Tserkovnyak, K. Vy`borny`, and J. Sinova, Phys. Rev. B 79, 195129 (2009). [40] A. A. Kovalev, J. Sinova, and Y. Tserkovnyak, Phys. Rev. Lett. 105, 036601 (2010). [41] M.Dzero,M.G.Vavilov,K.Kechedzhi,andV.Galitski, Phys. Rev. B 92, 165415 (2015). [42] A. A. Burkov, Phys. Rev. Lett. 113, 187202 (2014). [43] Roman M. Lutchyn, Pavel Nagornykh, Victor M. Yakovenko, Phys. Rev. B 80, 104508 (2009). [44] E.Tailor,C.Kallin,Phys.Rev.Lett.108,157001(2012). [45] SongciLi,A.V.Andreev,B.Z.Spivak,Phys.Rev.B92, 100506 (2015). [46] H. Weng, R. Yu, X. Hu, X. Dai, and Z. Fang, Adv. in Phys. 64, 227 (2015). [47] C.-X. Liu, S.-C. Zhang, and X.-L. Qi, preprint arXiv:1508.07106 (2015). [48] C.-Z. Chang and M. Li, preprint arXiv:1510.01754. [49] X.-L. Qi, R. Li, J. Zang, and S.-C. Zhang, Science 323, 1184 (2009). [50] C. Bru¨ne, C. X. Liu, E. G. Novik, E. M. Hankiewicz, H. Buhmann, Y. L. Chen, X. L. Qi, Z. X. Shen, S. C. Zhang, and L. W. Molenkamp, Phys. Rev. Lett. 106, 126803 (2011). [51] R. Yoshimi, A. Tsukazaki, Y. Kozuka, J. Falson, K. S. Takahashi, J. G. Checkelsky, N. Nagaosa, M. Kawasaki, and Y. Tokura, Nat. Comm. 6 (2015). 1 Supplementary Materials For Anomalous Hall Effect on the surface of topological Kondo insulators SUMMARY OF RESULTS FROM SUPPLEMENTARY MATERIAL Expressions entering Eq. (7) of the main text We here express all quantities of Eq. (7) of the main text in terms of microscopic parameters. The contributions from non-crossing diagrams are F = (1−aW)−1, (S1a) a = (cid:88) ((cid:15)2K −m2K)πνK((cid:15)K) Π , (S1b) 4(cid:15) (cid:16)(cid:15) Γ +m Γ(m)(cid:17) K K K K K K K b = −(cid:88)(ΓKmK +(cid:15)KΓ(Km))πνK((cid:15)K)Π , (S1c) 2(cid:15) (cid:16)(cid:15) Γ +m Γ(m)(cid:17) K K K K K K K while diagrams (c) and (d) of Fig. 2 involve (cid:88)W2 x = KK(cid:48)c d , (S1d) KK v v(cid:48) (K,K(cid:48)) (K,K(cid:48)) K(cid:48) K K ψ = 2(cid:88)WKKWKK(cid:48)c d˜ f , (S1e) KK v v(cid:48) (K,K(cid:48)) (K,K(cid:48)) (K,K(cid:48)) K(cid:48) K K x K(cid:54)==K(cid:48) θ((cid:15)K(cid:15)(cid:48)K)[c d˜ (1−f )]W W +K ↔K(cid:48), (S1f) KK(cid:48) v2 (K,K(cid:48)) (K,K(cid:48)) (K(cid:48),K) KK KK(cid:48) K ψ K(cid:54)==K(cid:48) θ(−(cid:15)K(cid:15)(cid:48)K)[c d˜ (1−f )]W W +K ↔K(cid:48). (S1g) KK(cid:48) v2 (K,K(cid:48)) (K,K(cid:48)) (K(cid:48),K) KK KK(cid:48) K Note the complementary contributions of x and ψ . We introduced the following dimensionless functions KK(cid:48) KK(cid:48) c = 2πν ((cid:15) )m /p2 , (S2a) (K,K(cid:48)) K(cid:48) K(cid:48) K(cid:48) F,K d = [(cid:15) /(cid:15) +m /m ], (S2b) (K,K(cid:48)) K K(cid:48) K K(cid:48) d˜ = [(cid:15) /(cid:15) +v /v ], (S2c) (K,K(cid:48)) K K(cid:48) K K(cid:48) f = θ(p −p )[1−p2 /p2 ]. (S2d) (K,K(cid:48)) F,K F,K(cid:48) F,K(cid:48) F,K Anomalous Hall effect for equal Dirac pockets In this section we present the general formula for the AHE in the case of equal Dirac pockets, i.e. E =0, v =v Γ Γ and m =m: Γ 4Ab σ (|(cid:15)|/m) = 3σ(0)(|(cid:15)|/m)+ xy xy (A(2W2 −W2−WW )+(W2+3WW +WW +2W2 +2W W ))2 ΓX XY ΓX XY ΓX ΓX XY (cid:104) (cid:32) (cid:0)4A2+A−2(cid:1)W2 12(A+1)WW (cid:33) × 4WW W (W +W )+W2 2W2− XY − XY ΓX XY XY ΓX (A−1)2 A−1 W3 (cid:0)2(A(4A+7)+10)W −4(cid:0)A2+A−2(cid:1)W(cid:1) (A(2A+11)+2)W4 (cid:105) + ΓX XY + ΓX (S3) (1−A)2 (1−A)2 2 Here we introduced the notation (cid:15)2−m2 A = , (S4a) 2((cid:15)2+m2) |(cid:15)|m b = − . (S4b) 2((cid:15)2+m2) These expressions are the origin of Eq. (10) of the main text. Anomalous Hall response in the limit of point-like scatterers We here present the formula for the anomalous Hall response in the case of equal velocities v = v and equal Γ Zeeman field m = m in Γ, X and Y pockets. Then the Hall conductivty is a function of two parameters (cid:15)¯= (cid:15)/m Γ and (cid:15)¯ =(cid:15) /m, only. Outside of any gap we obtain Γ Γ (cid:104) (cid:105) σ =sgn((cid:15)) θ((cid:15)2−(cid:15)2)σ[1,sgn((cid:15)(cid:15)Γ)]+θ((cid:15)2 −(cid:15)2)σ[2,sgn((cid:15)(cid:15)Γ)] (S5) xy Γ xy Γ xy with σ[1,+] =−4e2[8(cid:15)¯6(cid:15)¯ +4(cid:15)¯5(cid:0)5(cid:15)¯2 +4(cid:1)+2(cid:15)¯4(cid:15)¯ (cid:0)7(cid:15)¯2 +38(cid:1)+(cid:15)¯3(cid:0)−(cid:15)¯4 +242(cid:15)¯2 +63(cid:1)+2(cid:15)¯2(cid:15)¯ (cid:0)2(cid:15)¯4 +82(cid:15)¯2 +261(cid:1) xy h Γ Γ Γ Γ Γ Γ Γ Γ Γ +(cid:15)¯(cid:0)7(cid:15)¯6 +38(cid:15)¯4 +315(cid:15)¯2 +288(cid:1)+2(cid:15)¯ (cid:0)(cid:15)¯6 +2(cid:15)¯4 +9(cid:15)¯2 +72(cid:1)]/[4(cid:15)¯3(cid:15)¯ +4(cid:15)¯2(cid:0)(cid:15)¯2 +2(cid:1)+(cid:15)¯(cid:15)¯ (cid:0)(cid:15)¯2 +23(cid:1)+5(cid:15)¯2 +27]2, Γ Γ Γ Γ Γ Γ Γ Γ Γ Γ Γ Γ (S6a) σ[1,−] =−4e2[−8(cid:15)¯6(cid:15)¯ −4(cid:15)¯5(cid:15)¯2 −2(cid:15)¯4(cid:15)¯ (cid:0)7(cid:15)¯2 +10(cid:1)+(cid:15)¯3(cid:0)33(cid:15)¯4 +26(cid:15)¯2 −3(cid:1)−2(cid:15)¯2(cid:15)¯ (cid:0)5(cid:15)¯4 +12(cid:15)¯2 −16(cid:1) xy h Γ Γ Γ Γ Γ Γ Γ Γ Γ −5(cid:15)¯(cid:0)(cid:15)¯2 −2(cid:1)(cid:0)(cid:15)¯2 −1(cid:1)2+2(cid:15)¯ (cid:0)(cid:15)¯2 −1(cid:1)3]/(cid:2)(cid:15)¯ (cid:0)(cid:15)¯(cid:0)((cid:15)¯ −2(cid:15)¯)2+7(cid:1)−3(cid:15)¯ (cid:1)+3(cid:3)2, (S6b) Γ Γ Γ Γ Γ Γ Γ σ[2,+] =−4e2[8(cid:15)¯7+8(cid:15)¯6(cid:15)¯ −2(cid:15)¯5(cid:0)(cid:15)¯2 −8(cid:1)+4(cid:15)¯4(cid:15)¯ (cid:0)3(cid:15)¯2 +19(cid:1)+(cid:15)¯3(cid:0)19(cid:15)¯4 +242(cid:15)¯2 +63(cid:1)+2(cid:15)¯2(cid:15)¯ (cid:0)4(cid:15)¯4 +82(cid:15)¯2 +261(cid:1) xy h Γ Γ Γ Γ Γ Γ Γ Γ Γ +(cid:15)¯(cid:0)(cid:15)¯6 +38(cid:15)¯4 +315(cid:15)¯2 +288(cid:1)+2(cid:15)¯ (cid:0)2(cid:15)¯4 +9(cid:15)¯2 +72(cid:1)]/(cid:2)4(cid:15)¯3(cid:15)¯ +4(cid:15)¯2(cid:0)(cid:15)¯2 +2(cid:1)+(cid:15)¯(cid:15)¯ (cid:0)(cid:15)¯2 +23(cid:1)+5(cid:15)¯2 +27(cid:3)2, Γ Γ Γ Γ Γ Γ Γ Γ Γ Γ Γ (S6c) σ[2,−] =−4e2[8(cid:15)¯7−8(cid:15)¯6(cid:15)¯ +(cid:15)¯5(cid:0)8−10(cid:15)¯2(cid:1)−4(cid:15)¯4(cid:15)¯ (cid:0)3(cid:15)¯2 +2(cid:1)+(cid:15)¯3(cid:0)25(cid:15)¯4 +18(cid:15)¯2 +1(cid:1)−2(cid:15)¯2(cid:15)¯ (cid:0)5(cid:15)¯2 (cid:0)(cid:15)¯2 +4(cid:1)−18(cid:1) xy h Γ Γ Γ Γ Γ Γ Γ Γ Γ +(cid:15)¯(cid:0)(cid:15)¯6 +20(cid:15)¯4 −29(cid:15)¯2 +10(cid:1)−2(cid:0)(cid:15)¯5 −(cid:15)¯3 +(cid:15)¯ (cid:1)]/(cid:2)(cid:15)¯ (cid:0)(cid:15)¯(cid:0)((cid:15)¯ −2(cid:15)¯)2+7(cid:1)−3(cid:15)¯ (cid:1)+3(cid:3)2. (S6d) Γ Γ Γ Γ Γ Γ Γ Γ Γ SUPPLEMENTARY MATERIAL: CALCULATION OF σI xy In this supplementary material we present details regarding the calculation of Eq. (7) and (S1) of the main text. Average Green’s function WefirstpresentthecalculationoftheaverageGreen’sfunctioninthelimitn (cid:28)n . Inthislimittheretarded imp. min self-energy approximately becomes ΣˆR((cid:15))(cid:39)(cid:88)tˆR ((cid:15)). (S7) Rj Rj Here, {R } are the impurity positions and the T-matrix tR of a single impurity is calculated at the level of Born j Rj approximation, see Fig. S1. Upon disorder average denoted by angular brackets we obtain (cid:68) (cid:69) (cid:68) (cid:69) (cid:28)(cid:90) (cid:29) [tR ((cid:15))] = u (p−p(cid:48)) + u (p−p(cid:48)(cid:48))GR(p(cid:48)(cid:48),(cid:15))u (p(cid:48)(cid:48)−p(cid:48)) Rj p,p(cid:48) Rj Rj 0 Rj p(cid:48)(cid:48) (cid:88) (cid:90) Π = u(p+Q −p(cid:48)(cid:48)−Q )GR (p(cid:48)(cid:48),(cid:15))u(p(cid:48)(cid:48)+Q −p(cid:48)−Q ) K(2π)2δ(p−p(cid:48)) (S8) K K(cid:48) K(cid:48),0 K(cid:48) K V K,K(cid:48) p(cid:48)(cid:48) 3 We introduced the symbol (cid:82) = (cid:82) d2p/(2π)2. Momentum conservation (reobtained after impurity average) forbids p off-diagonalmatrixelementsinDiracpocket(DP)space. Weabsorbtherealpartoftheself-energyintoaredefinition of E and m . The imaginary part of the self-energy leads to the scattering rates presented in Eqs. (4) of the main K K text. In our calculations, we formally do not only include the Born approximation diagram, Fig. S1, but also the resum- mation of rainbow diagrams (self consistent Born approximation) and diagrams with a single intersection of impurity lines. To the leading order, this does not alter the result for the scattering rates, Eqs. (4). Current vertex and non-crossing approximation We first consider the dressed velocity vertices. In view of momentum conservation it is useful to represent the left (right) current vertices as row (column) vectors in DP space, and, at the same time, as vertices in spin space, i.e. in formulas ˆjL =jσ and ˆjR =jTσ (S9) µ µ µ µ where, on the bare level, j →v =(v ,v,v). Γ Fortheresummationofimpuritypotentialsattheleftvertex, wewillneedthefollowingintegral(whichisamatrix in spin space and a matrix element of the diagonal matrices IL,R in DP space) µ (cid:90) [IL] := GA(p,(cid:15))σ GR(p,(cid:15))(cid:39)σ [a] +iσ σ [b] . (S10) µ KK K µ K µ KK µ z KK p Similarly, we also need for the right vertex the object (cid:90) [IR] := GR(p,(cid:15))σ GA(p,(cid:15))(cid:39)σ [a] +iσ σ [b] . (S11) µ KK K µ K µ KK z µ KK p The diagonal matrices a and b are presented in Eq. (S1b) and (S1c) of the main text. We readily find ˆjL (cid:39)vFσ +iσ σ vFbWF (S12) µ µ µ z and analogously ˆjR (cid:39)(vF)Tσ +iσ σ (vFbWF)T. (S13) µ µ z µ The result for σI in the non-crossing approximation immediately follows xy e2 e2 [σ ] = trσ[ˆjLIRvT]= 2vFbFTvT. (S14) xy nc h x y h X and Ψ diagrams We will distinguish diagonal (“intraband”) and off-diagonal (“interband”)parts of the matrices X = 2axa and Ψ = 2aψa introduced in Eq. (7) of the main text. For the evaluation of X and Ψ diagrams we define the following quantities : N (p)=(cid:15) +m σ +v p·σ. (S15) K K K z K R R R xj xj xj disorder p-p' p-p'' p''-p' average [t R (ϵ)] = + +... Rj p,p' pp p' p p'' p' p p'' p FIG. S1: T-matrix of a single impurity at the level of Born approximation, before and after disorder average. 4 p1,K1,R p3,K3,R p2,K2,R p1,K1,Rp3,K3,Rp4,K4,R p2,K2,R p1,K1,R p2,K2,R [σxy]X = jx p-p1+3-QQKK13 p-p2+3-QQKK23 jy [σxy]Ψ = jx jy+ jx jy p1,K1,A p,K,A p2,K2,A p1,K1,A p2,K2,A p1,K1,Ap,K,Ap,K,Ap2,K2,A 4 4 3 3 4 4 FIG. S2: The X and Ψ- diagrams labelled with momentum variables, pocket index K and retarded/advanced (R/A) labels. Momentum conservation is analyzed in Eqs. (S18) and (S19). and (assuming (cid:15)2 >m2 ) K K 1 GR/A(p) = , (S16a) K ((cid:15) ±i0)2−(v p)2−m2 K K K (cid:90) 1 GR/A(r) = eiprGR/A(p)= [Y (p r)∓isgn((cid:15) )J (p r)], (S16b) K K 4v2 0 F,K K 0 F,K p K Here, J (x) and Y (x) are the zeroth Bessel functions of first and second kind. 0 0 In the evaluation of diagrams, we use σ = −σ and the fact that only the symmetric part of 3×3 matrices X xy yx and Ψ enters the Hall conductance. This section of the supplementary material will be devoted to the calculation of those matrices. The matrices X and Ψ are diagrammatically represented by diagrams analogous to Fig. 2, (b)-(d), of the main text, but the vertex correction should be omitted and is incorporated separately in the final result. Momentum conservation We label X and Ψ-diagrams with momentum variables, see Fig. S2. We readily conclude from momentum conservation that for the X diagram p +Q +p +Q =p +Q +p +Q . (S17) 1 K1 2 K2 3 K3 4 K4 Since we assume distant Fermi surfaces, p (cid:28)π/a, this implies F,K Q +Q = Q +Q , (S18a) K1 K2 K3 K4 p +p = p +p . (S18b) 1 2 3 4 Analogously we analyze the Ψ diagram. As compared to the X diagram, momentum conservation and the assump- tion of distant Fermi surfaces now imply Q −Q = Q −Q , (S19a) K1 K2 K3 K4 p −p = p −p . (S19b) 1 2 3 4 Note that Eqs. (S18a) and (S19a) are to be understood modulo a reciprocal lattice vector (Umklapp scattering). In particular, there is an Umklapp Ψ diagram in which K =K (cid:54)=K =K . 1 4 2 3 Integrals over Bessel functions The evaluation of X and Ψ diagrams involves various integrals over Bessel functions. These are (cid:90) ∞ 1 (cid:20) θ(r−1)(cid:21) I(1)(r) = dsJ2(rs)J (s)Y (s)= θ(1−r)+ , (S20a) A 1 1 0 2π r2 0 (cid:90) ∞ I(1)(r) = dsJ2(rs)J (s)Y (s)=−1/π+I(1)(r), (S20b) B 1 0 1 A 0 (cid:90) ∞ I(1)(r) = dsJ (s)J (rs)J (s)Y (rs)=−I(1)(r), (S20c) C 1 1 0 1 A 0 (cid:90) ∞ 1(cid:104) (cid:105) I(2)(r) = dsJ(cid:48)(s)J (s)Y (rs)J (rs)= I(1)(1/r)+I(1)(1/r) . (S20d) B 1 1 0 0 2 A B 0 Details on the evaluation of such integrals can be found in Ref. [S31].