Anomalous Hall Effect due to Non-collinearily in Pyrochlore Compounds: Role of Orbital Aharonov-Bohm Effect
aa r X i v : . [ c ond - m a t . s t r- e l ] S e p Anomalous Hall Effect due to Non-collinearityin Pyrochlore Compounds: Role of Orbital Aharonov-Bohm Effect
Takeshi Tomizawa and Hirhoshi Kontani
Department of Physics, Nagoya University, Furo-cho, Nagoya 464-8602, Japan. (Dated: August 20, 2018)To elucidate the origin of spin structure-driven anomalous Hall effect (AHE) in pyrochlore com-pounds, we construct the t g -orbital kagome lattice model and analyze the anomalous Hall con-ductivity (AHC). We reveal that a conduction electron acquires a Berry phase due to the complex d -orbital wavefunction in the presence of spin-orbit interaction. This “orbital Aharonov-Bohm (AB)effect” produces the AHC that is drastically changed in the presence of non-collinear spin structure.In both ferromagnetic compound Nd Mo O and paramagnetic compound Pr Ir O , the AHC givenby the orbital AB effect totally dominates the spin chirality mechanism, and succeeds in explainingthe experimental relation between the spin structure and the AHC. Especially, “finite AHC in theabsence of magnetization” observed in Pr Ir O can be explained in terms of the orbital mechanismby assuming small magnetic order of Ir 5 d -electrons. PACS numbers: 72.10.-d, 72.80.Ga, 72.25.Ba
I. INTRODUCTION
Recently, theory of intrinsic anomalous Hall effect(AHE) in multiband ferromagnetic metals has been de-veloped intensively from the original work by Karplusand Luttinger (KL) [1]. The anomalous Hall conduc-tivity (AHC) σ AH ≡ j x /E y due to intrinsic AHE showsthe almost material-specific value that is independent ofthe relaxation time. The intrinsic AHE in heavy fermioncompounds [2], Fe [3], and Ru-oxides [4–6] had been stud-ied intensively based on realistic multiband models. Also,large spin Hall effect (SHE) observed in Pt and otherparamagnetic transition metals [7], which is analog to theAHE in ferromagnets, is also reproduced well in terms ofthe intrinsic Hall effect [8–10]. The intrinsic AHE andSHE in transition metals originate from the Berry phasegiven by the d -orbital angular momentum induced by thespin-orbit interaction (SOI), which we call the “orbitalAharonov-Bohm (AB) effect” [11].In particular, AHE due to nontrivial spin structureattracts increasing attention, such as Mn oxides [12]and spin glass systems [13]. The most famous exam-ple would be the pyrochlore compound Nd Mo O [14–17]. Here, Mo 4 d electrons are in the ferromagnetic statebelow T c = 93K, and the tilted ferromagnetic state inFig. 1 is realized by the non-coplanar Nd 4 f magneticorder below T N ≈ d - f exchange in-teraction. Below T N , the AHC is drastically changedby the small change in the tilting angle θ of Mo spin; FIG. 1: Tilted ferromagnetic state in the kagome lattice. Bluecircles are Mo ions. Arrows at Mo sites are the tilted ferro-magnetic exchange field. θ < − in the neutron-diffraction study [16]. This be-havior strongly deviates from the KL-type conventionalbehavior σ AH ∝ M z ∝ − θ . Moreover, the AHCgiven by the spin chirality mechanism [19, 20], whichis proportional to the solid angle s A · ( s B × s C ) ∝ θ subtended by three spins, is also too small to explainexperiments. Moreover, h θ i takes the minimum valueunder H ∼ H . Thus, the origin of the unconventional AHEin Nd Mo O had been an open problem for a long time.Very recently, this problem was revisited by the presentauthors by considering the d -orbital degree of freedomand the atomic SOI [21], and found that a drastic spinstructure-driven AHE emerges due to the orbital AB ef-fect, in the presence of non-collinear spin order. Sincethe obtained AHC is linear in θ , it is much larger thanthe spin chirality term for | θ | ≪
1. In Ref. [21], we con-structed the t g orbital kagome lattice model based onthe spinel structure ( X Mo O ): Although Mo atoms in X Mo O and X Mo O are equivalent in position andforms the pyrochlore lattice, positions of O atoms in X Mo O are much complicated.In this paper, we construct the t g kagome lattice tight-biding model based on the pyrochlore structure, by tak-ing the crystalline electric field into account. We findthat the orbital AB effect causes large θ -linear AHC, re-sulting from the combination of the non-collinear spinorder (including orders with zero scalar chirality) andatomic SOI. The realized AHC is much larger than thespin chirality term due to non-coplanar spin order, andit explains the salient features of spin structure-drivenAHE in Nd Mo O . We also study another pyrochlorecompound Pr Ir O , and find that the orbital AB effectalso gives the dominant contribution: We show that im-portant features of the unconventional AHE in Pr Ir O ,such as highly non-monotonic field dependence and resid-ual AHC in the absence of magnetization, are well repro-duced by the orbital AB effect. FIG. 2: Pyrochlore structure. Blue (white) circles are Mo (O)ions. The Mo ions on the [111] plane form the kagome lattice.FIG. 3: Configurations of Mo A − D and O − . The x Ξ y Ξ z Ξ -coordinate is defined by the surrounding O tetrahedron. The paper is organized as follows: In Sec. II, we intro-duce the pyrochlore-type t g orbital tight-binding modeland the Hamiltonian. We give the general expressionsfor the intrinsic AHC in Sec. III, and explain the orbitalAharonov-Bohm effect in Sec. IV. The numerical resultsfor Nd Mo O and Pr Ir O are presented in Sec. Vand VI, respectively. In Sec. VII, we make comparisonbetween theory and experiment. II. MODEL AND HAMILTONIAN
First, we introduce the crystal structure of the py-rochlore oxide A B O : It has the face centered cubicstructure, in which two individual 3-dimensional net-works of the corner-sharing A and B tetrahedron areformed. In this paper, we mainly discuss the AHE inNd Mo O , and Pr Ir O is also discussed in section VI.Figure 2 represents the Mo ions (Blue circles) and O ions TABLE I: Coordinates of Mo and O in pyrochlore structureas shown in Fig. 3 in the xyz -coordinate.Ion Site CoordinateMo A (1/4 ,0, 0)B (0, 1/4, 0)C (0, 0, 1/4)D (1/4, 1/4, 1/4)O 1 (1/8, 1/8, -1/16)2 (1/8, -1/16, 1/8)3 (-1/16, 1/8, 1/8)4 (5/16, 1/8, 1/8)5 (1/8, 5/16, 1/8)6 (1/8, 1/8, 5/16) (White circles) in the pyrochlore structure. The [111] Molayer forms the kagome lattice. The Mo 4 d -electrons giveitinerant carriers while the Nd 4 f -electrons form localmoments.We construct pyrochlore type t g -orbital tight bindingmodel in the kagome lattice for Mo 4 d electrons, wherethe unit cell contains three sites A, B and C in Fig. 3.The coordinates of Mo and O are shown in Table I [22],and the quantization axis for the Mo d -orbital is fixed bythe surrounding O octahedron. To describe the d -orbitalstate, we introduce the ( xyz ) Ξ -coordinate for Ξ = A , B , Csites shown by Fig.3. The ( xyz ) Ξ -coordinate is definedby the surrounding O ions. In the case of the ( xyz ) A -coordinate, we choose x A , y A and z A axes as Mo A → O ,Mo A → O and Mo A → O direction, respectively, in Fig.3. We also choose the ( xyz ) B - and ( xyz ) C -coordinates inthe same way.Moreover, we introduce the XY Z -coordinate on thekagome layer shown in Fig. 4(b). We choose X axisas Mo A → Mo B direction and Y axis is perpendicularto X axis on the kagome layer. A vector ( n x , n y , n z ) Ξ inthe ( xyz ) Ξ -coordinate is transformed into [ n X , n Y , n Z ] inthe XY Z -coordinate as ( n x , n y , n z ) Ξ = [ n X , n Y , n Z ] ˆ O Ξ ,where the coordinate transformation matrix ˆ O Ξ is givenby ˆ O A = 13 √ √ √ √ − −√ −√ √ , (1a)ˆ O B = 13 √ −√ − √ −√ − −√ −√ √ , (1b)ˆ O C = 13 √ − √ √ − − − −√ −√ √ . (1c)Arrows in Fig. 4(a) represents the local effective mag-netic field at Mo sites, which is composed of the ferro-magnetic exchange field for Mo 4 d -electrons and the ex-change field from Nd 4 f electrons. Under the magneticfield parallel to [111] direction below T N , the direction FIG. 4: (a) Umbrella like locale exchange field at Mo sitesrepresented by arrows. A unit cell contains sites A, B, C. (b)Kagome lattice. a ij ( i, j = A, B, C ) is a half Bravais vector.(c) First Brillouin zone in a hexagon shape. of the local exchange fields at sites A, B and C in the
XY Z -coordinate are ( φ A = π/ , θ ), ( φ B = 5 π/ , θ ) and( φ C = − π/ , θ ), respectively. In Nd Mo O , the tiltingangle θ changes from negative to positive as H increasesfrom +0 Tesla, corresponding to the change in the spin-ice state at Nd sites [16, 23].Now, we explain the Hamiltonian. The Hamiltonianfor the t g -orbital kagome lattice model is given by H = X iα,jβ,σ t iα,jβ c † iα,σ c jβ,σ − X iα,σσ ′ h i · [ µ e ] σ,σ ′ c † iα,σ c iα,σ ′ + λ X iαβ,σσ ′ [ l ] α,β · [ s ] σ,σ ′ c † iα,σ c jβ,σ (2) where c † is a creation operator for 4 d -electron on Moions while the field h arises from the ordered Nd mo-ments, which are treated as a static, classical background.( i, j ), ( α, β ) and ( σ, σ ′ ) represent the sites, t g -orbitalsand spins, respectively. Hereafter, we denote the t g -orbitals ( xy, yz, zx ) as (1 , ,
3) for simplicity. The firstterm in eq. (2) describes electrons hopping. t iα,jβ is thehopping integrals between ( i, α ) and ( j, β ). The direct d - d hopping integrals are given by the Slater-Koster (SK)parameters ( ddσ ), ( ddπ ) and ( ddδ ) [24]. In the presentmodel, however, SK parameter table given in Ref. [24]is not available since the d -orbitals at each site are de-scribed in the different coordinate as shown in Fig. 3.In Appendix A, we will derive the hopping integral be-tween the sites with different coordinates. The secondterm in eq. (2) represents the Zeeman term, where h i isthe local exchange field at site i . µ e ≡ − s is the mag-netic moment of an electron. Here, we put µ B =1. Thethird term represents the SOI, where λ is the spin-orbitcoupling constant, and l and s are the d -orbital and spinoperators, respectively.The Hamiltonian in Eq. (2) is rewritten in the mo-mentum space as H = X k C † k ˆ H k C k , (3)where k summation is over the first Brillouin zone in Fig.4(c), and C † k = ( a † k , ↑ , a † k , ↓ , b † k , ↑ , b † k , ↓ , c † k , ↓ , c † k , ↑ a † k , ↑ , a † k , ↓ , b † k , ↑ , b † k , ↓ , c † k , ↓ , c † k , ↑ a † k , ↑ , a † k , ↓ , b † k , ↑ , b † k , ↓ , c † k , ↓ , c † k , ↑ ) . (4)Here and hereafter, we denote the creation operators atsites A, B and C as a † k ,ασ , b † k ,ασ and c † k ,ασ , respectively.ˆ H k is given by 18 ×
18 matrix:ˆ H k = ˆ H k ˆ H k ˆ H k ˆ H k ˆ H k ˆ H k ˆ H k ˆ H k ˆ H k , (5)where ˆ H k αβ is a 6 × , σ ).Here, we divide the Hamiltonian (3) into four parts:ˆ H k αβ = ˆ H t k αβ + ˆ H Ze αβ + ˆ H λαβ + ˆ H CEF αβ , (6)where we added the crystalline electric field potentialterm ˆ H CEF αβ to Eq. (2). The kinetic term ˆ H t k αβ is givenbyˆ H t k αβ = β A ↑ β A ↓ β B ↑ β B ↓ β C ↑ β C ↓ α A ↑ p A α, B β p A α, C β α A ↓ p A α, B β p A α, C β α B ↑ p B α, A β p B α, C β α B ↓ p B α, A β p B α, C β α C ↑ p C α, A β p C α, B β α C ↓ p C α, A β p C α, B β , (7)where p iα,jβ = 2 t iα,jβ cos( k · a ij ) and a ij is a half Bravaisvector in Fig. 4(b).The Zeeman term ˆ H Ze αβ is given by [20]ˆ H Ze αβ = h δ αβ β A β B β C α A ˆ R θ,π/ α B 0 ˆ R θ, π/ α C 0 0 ˆ R θ, − π/ , (8)ˆ R θ,φ = (cid:18) ↑ ↓↑ cos θ sin θe − iφ ↓ sin θe iφ − cos θ (cid:19) , (9)where h = | h i | , and δ αβ is a Kronecker’s delta.Now, we consider the SOI term ˆ H λ in Eq. (6). For con-venience in calculating the AHC, we take the Z -axis forthe spin quantization axis. Then, 2 s = 2[ s X , s Y , s Z ] isgiven by the Pauli matrix vector in the XY Z -coordinate.To derive the ˆ H λαβ , however, we have to express the spin operator in the ( xyz ) Ξ -coordinate, which is given by therelationship ( s Ξ x , s Ξ y , s Ξ z ) = [ s X , s Y , s Z ] ˆ O Ξ and Eqs. (1a)-(1c). In the ( xyz ) Ξ -coordinate, the nonzero matrix ele-ments of l are given as h | l x | i = h | l y | i = h | l z | i = i and their Hermite conjugates [10, 25]. Thus, the ma-trix elements ( ˆ H λα,β ) Ξ σ, Ξ ′ σ ′ ≡ h Ξ ασ | ˆ H λ | Ξ βσ ′ i · δ Ξ , Ξ ′ for( α, β ) = (3 ,
1) are given as( ˆ H λ , ) A σ, A σ ′ = iλ √ h σ | (4 √ s X − s Y − √ s Z ) | σ ′ i , ( ˆ H λ , ) B σ, B σ ′ = iλ √ h σ | ( −√ s X + 7 s Y − √ s Z ) | σ ′ i , ( ˆ H λ , ) C σ, C σ ′ = iλ √ h σ | ( − √ s X − s Y − √ s Z ) | σ ′ i . Thus, the 3-1 component of the third term in Eq. (6)becomesˆ H λ = iλ √ × ↑ ↓ ↑ ↓ ↑ ↓ ↑ −√ √ i ↓ √ − i √ ↑ −√ −√ − i ↓ −√ i √ ↑ −√ − √ i ↓ − √ − i √ . (11)The 1-2 and 2-3 components are calculated in a similar way. The obtained results are given byˆ H λ = iλ √ × ↑ ↓ ↑ ↓ ↑ ↓ ↑ −√ √ − i ↓ √ i √ ↑ −√ − √ i ↓ − √ − i √ ↑ −√ √ i ↓ √ − i √ , (12)ˆ H λ = iλ √ × ↑ ↓ ↑ ↓ ↑ ↓ ↑ √ √ − i ↓ √ i − √ ↑ √ −√ − i ↓ −√ i − √ ↑ √ i ↓ − i − √ . (13)Finally, we consider the crystalline electric field Hamil-tonian ˆ H CEF αβ , which describes the splitting of t g levelinto two levels a g (non-degeneracy) and e ′ g (two-fold de-generacy) by the trigonal deformation of MoO octahe-dron. The crystalline electric field Hamiltonian in thiscase is given byˆ H CEF αβ = E (1 − δ α,β ) · ˆ1 (14)The eigenvalues of ˆ H CEF at each site are 2 E for a g state; | a g i = √ ( | xy i + | yz i + | zx i ), and − E for e ′ g states; | e g i = √ ( | yz i − | zx i ) and | e g i = √ (2 | xy i −| yz i − | zx i ). Thus, the crystalline electric field splittingbetween a g and e ′ g is 3 | E | . III. ANOMALOUS HALL CONDUCTIVITY
In this section, we propose the general expressions forthe intrinsic AHC based on the linear-response theory.The Green function is given by a 18 ×
18 matrix: ˆ G k ( ǫ ) =(( ǫ + µ )ˆ1 − ˆ H k ) − , where µ is the chemical potential.According to the linear response theory, the AHC is givenby σ AH = σ IAH + σ IIAH [26]: σ IAH = 12 πN X k Tr h ˆ j X ˆ G R ˆ j Y ˆ G A i ǫ =0 (15) σ IIAH = − πN X k Z µ −∞ dǫ Tr " ˆ j X ∂ ˆ G R ∂ǫ ˆ j Y ˆ G A − ˆ j X ˆ G R ˆ j Y ∂ ˆ G A ∂ǫ − h R → A i . (16)Here, ˆ G R ( A ) k ( ǫ ) ≡ ˆ G k ( ǫ + ( − ) iγ ) is the retarded (ad-vanced) Green function, where γ ( >
0) is the quasiparticledamping rate. ˆ j k µ ≡ − e∂ ˆ H k /∂k µ = − e ˆ v µ ( µ = X, Y )is the charge current, where − e is the electron charge.Since all the matrix ˆ j k µ is odd with respect to k , thecurrent vertex correction due to local impurities vanishesidentically [8, 10]. Thus, we can safely neglect the cur-rent vertex correction in calculating AHC in the presentmodel. In the band-diagonal representation, eqs. (15)and (16) are transformed into σ IAH = 12 πN X k ,l = m j mlX j lmY µ − E l k + iγ )( µ − E m k − iγ ) , (17) σ IIAH = i πN X k ,l = m Z µ −∞ dǫj mlX j lmY Im " ǫ − E l k + iγ ) ( ǫ − E m k + iγ ) − ǫ − E l k + iγ )( ǫ − E m k + iγ ) . (18) TABLE II: Phases for t g orbitals ψ Ξ xy ψ Ξ yz ψ Ξ zx Ξ=A,B,C Φ xy + q θ zx + q θ at zero temperature. Here, l and m are the band indices,and we dropped the diagonal terms l = m since their con-tribution vanishes identically. We perform the numericalcalculation for the AHC using Eqs. (17) and (18) in latersection. σ IAH and σ IIAH are called the Fermi surface term and theFermi sea term, respectively. According to Refs. [10, 26], σ IIAH can be uniquely divided into σ IIaAH and the Berrycurvature term σ IIbAH . The intrinsic AHC is given by σ IIbAH when γ → σ IAH + σ IIaAH = 0. In general cases,however, the total AHC is not simply given by σ IIbAH since σ IAH + σ IIaAH is finite when γ = 0 or γ l /γ m = 1. Therefore,we calculate the total AHC σ AH = σ IAH + σ IIAH in thispaper.
IV. ORBITAL AHARONOV-BOHM EFFECT
Before proceeding to the numerical calculation for theAHE, we present an intuitive explanation for the un-conventional AHE induced by the non-collinear local ex-change field h i . For this purpose, we assume the strongcoupling limit where the Zeeman energy is much largerthan the kinetic energy and the SOI [21]. The t g energylevels are split into the two triply-degenerate states bythe Zeeman effect, as shown in Fig. 5. Its eigenstate for − h / | α i = sin θ | α ↑i + e iφ cos θ | α ↓i , (19)where α = xy, yz, zx . In addition, we assume the SOIis much larger than the kinetic energy. Since µ e = − s ,the SOI term at site i is replaced with ( − λ/ l · n i , where n i ≡ h i / | h i | . Its eigenenergies in the t g space are 0 and ± λ/
2, as shown in Fig. 5. The corresponding eigenstatesare given by [21] | n i = n z | xy i + n x | yz i + n y | zx i , (20a) | n ± i = 1 q n y + n z ) [ − ( n x n z ± in y ) | xy i + ( n y + n z ) | yz i − ( n x n y ∓ in z ) | zx i ] , (20b)where n = ( n x , n y , n z ) Ξ in the ( xyz ) Ξ -coordinate is givenby [sin θ cos φ, sin θ cos φ, cos θ ] ˆ O Ξ . In the complex wave-function | n − i , the phase of each d -orbital within the θ -linear term is given in Table II.Here, we explain that the θ -dependence of the d -orbital wavefunction | n − i gives rise to a prominent spinstructure-driven AHE [21]. Figure 6 shows the motionof an electron: (a) moving from | B; yz i to | C; zx i , (b)transferring form | C; zx i to | C; yz i at the same site, and(c) moving from | C; yz i to | A; zx i . Here, we assumethat the electron is in the eigenstate | n − i at each site.The total orbital phase factor for the triangle path alongA → B → C → A is given by the phase of the followingamplitude: T orb = h A; n − | ˆ H t | C; n − ih C; n − | ˆ H t | B; n − i×h B; n − | ˆ H t | A; n − i , (21)where ˆ H t is the kinetic term in the Hamiltonian.For simplicity, we take only the following largest hop-ping t = h A; zx | ˆ H t | B; yz i = h B; zx | ˆ H t | C; yz i = h C; zx | ˆ H t | A; yz i , and assume that t is real. Consider-ing that | Ξ; n − ih Ξ; n − | ∋ | Ξ; yz ih Ξ; zx | exp( − i (Φ zx + q θ )) for Ξ=A, B, and C, the hopping amplitude isexpressed as T orb ∼ | T orb | e − i π Φ orb / Φ , (22)where Φ = 2 π ~ / | e | is the flux quantum, and2 π Φ orb / Φ = 3Φ zx + (9 / p / θ is “the effective ABphase” induced by the complex d -orbital wavefunction.The large θ -linear term in Φ orb gives rise to the largespin structure-driven AHE in Nd Mo O .Note that h B; n − | ˆ H t | A; n ′− i is not actually a real num-ber if n = n ′ , since the rotation of the spin axis inducesthe phase factor; see Eq. (19). This fact gives rise to theeffective flux due to the spin rotation Φ spin ; − π Φ spin isgiven by the solid angle subtended by n A , n B and n C [19]. Thus, the total flux is given by Φ tot = Φ orb + Φ spin .However, Φ spin ∝ θ is negligible for | θ | ≪ tot ≈ Φ orb , the orbitalAB effect induces prominent spin structure-driven AHEin Nd Mo O . V. NUMERICAL STUDY
In this section, we perform numerical calculation forthe AHC using Eqs. (17) and (18), using realistic modelparameters. We use two SK parameters between thenearest neighbor Mo sites as SK( − . , . , − .
1) andSK( − . , . , − .
1) where we represent the set of SK pa-rameters as SK(( ddσ ) , ( ddπ ) , ( ddδ )). Hereafter, we putthe unit of energy | ( ddσ ) | = 1, which corresponds to2000K in real compound. The spin-orbit coupling con-stant for Mo 4 d is λ = 0 . N = 6 (1/3-filling) for Nd Mo O sincethe valence of Mo ion is 4+. We choose | h i | to reproducethe magnetization of Mo ion 1 . µ B in Nd Mo O [16].Figure 7 shows the total and partial density of states(DOS) for SK( − . , . , − .
1) at (a) E = 0 and (b) E = −
2, with the damping rate γ = 0 .
05. For E = 0( E = −
2) we set | h i | = 0 . | h i | = 3 .
0) to reproduce themagnetization of Mo ion 1 . µ B [16]. The crystalline elec-tric field splitting for E = − FIG. 5: Eigenenergies for t g electron under the exchange field h and the SOI ( − λ/ n · l ; see Ref. [21].FIG. 6: Orbital AB phase given by the complex t g orbitalwavefunction at site C. The electron acquires the phase dif-ference between zx - and yz -orbitals (orbital AB phase) viathe movement (a) → (b) → (c). to 3 | E | ∼ | a g ↓i and | e ′ g ↑i giveslarge partial DOS near the Fermi level.Figure 8 shows the 3rd-12th bands from the lowest.Nine bands near the Fermi level( N = 6) are composedof | a g ↓i and | e ′ g ↑i as understood from Fig. 7(b). Asshown in Fig. 9, the band structure and the Fermi surfaceare hardly changed by varying θ by 3 degrees.Here, we present the numerical results of the AHCfor two SK parameter sets; SK( − . , . , − .
1) andSK( − . , . , − . E , | h i | ) = ( − , .
0) or(0 , . . µ B [16]. We also put the damping rate γ = 0 .
001 (clean limit) unless otherwise noted. Here-after, the unit of the conductivity is e /ha , where h isthe Plank constant and a is the lattice constant. If weassume a = 4˚A, e /ha ≈ Ω − cm − . In the numericalstudy, we use 512 k -meshes.Figure 10 shows the obtained AHC forSK( − . , . , − .
1) at θ = 0 and ± ◦ , for (a) awide range of µ ( N = 4 . −
12) and (b) a narrow range of µ ( N = 5 . − . θ , a very small θ causes a prominentchange in the AHC although the Fermi surfaces arehardly changed (see Fig. 9). The µ -dependence of theAHC for other SK parameter SK( − . , . , − .
1) isshown by Fig. 11 for N = 4 . − .
6. A remarkablechange of the AHC is also caused by small change in θ .Therefore, large θ -linear term in the AHC is obtained byusing general SK parameters.The finite AHC at θ = 0 is nothing but the conven- −5 0 502460246 updown totala e’ g Energy D O S E =0 N=6 h =0.8 γ =0.05(a) −10 −5 0 502460246 updown totala e’ g Energy D O S N=6 E =−2h =3.0 γ =0.05(b) FIG. 7: Total and partial DOS for (a) E = 0 and (b) E = − − . , . , − . θ = 0. FIG. 9: Fermi surface for SK( − . , . , − . θ = 0, and(b) θ = 3 ◦ . −2 −1 0−10123−1.7 −1.5 −1.30123 E =−2 N=6.0N=4.8 N=12 µ A HC ( e / ha ) θ =−3 o θ =0 o θ =3 o γ =0.001 SK(−1.0, 0.6, −0.1) (a) µ A HC ( e / ha ) N=5.0 N=6.7N=6.0 (b)
FIG. 10: µ -dependence of AHC for SK( − . , . , − . N = 4 . −
12 and (b) N = 5 . − . tional KL-term. However, obtained θ -linear AHC devi-ates from the conventional KL-term that is proportionalto the magnetization M z ∝ θ . We stress that the large θ -linear term in Figs. 10 and 11 cannot be simply under-stood as the movement of Dirac points (or band crossingpoints) across the Fermi level, since the change in theband structure by θ = ± ◦ is very tiny as recognized inFig. 9. Thus, the origin of the θ -linear term should beascribed to the orbital AB phase [21] discussed in Sec.IV. −1.5 −1.4 −1.3 −1.2−202 E =−2 N=6.0 µ A HC ( e / ha ) θ =−3 o θ =0 o θ =3 o γ =0.001 SK(−1.0, 0.4, −0.1)N=5.0 N=6.7
FIG. 11: µ -dependence of AHC for SK( − . , . , − .
1) and N = 4 . − . −3 −2 −1 −4 −2 E =−2 γ A HC ( e / ha ) | σ AH (3 o )− σ AH (−3 o )| σ AH ( θ =0 o )SK(−1.0, 0.4, −0.1)<0 >0 γ −2 |2 σ AH (3 o )|<0 (i)(ii)(iii) λ =0 spin FIG. 12: γ -dependence of AHCs for SK( − . , . , − .
1) and N = 6. The AHC starts to decrease for γ ∼ .
02, whichcorresponds to ρ = 0 . Next, we discuss the γ -dependence of the AHC. As γ increases from 0 . γ exceeds theband-splitting ∆, proved by using tight-binding models[2, 6, 28] or local orbitals approach [28]. Figure 12 showsthe γ -dependence of the AHC in the present model. Line(i) represents the total AHC for θ = 0; σ AH ( θ = 0),and line (ii) represents the variation of the AHC from θ = − ◦ to 3 ◦ ; | σ AH (3 ◦ ) − σ AH ( − ◦ ) | . We also calculatethe AHC for λ = 0, which represents the spin chiralitydriven AHC σ spinAH . Note that σ spinAH ( θ ) is an even functionof θ , and σ spinAH (0)=0. In Fig. 12, we plot | σ spinAH (3 ◦ ) | as line (iii). The variation of the AHC from θ = − ◦ to 3 ◦ due to the orbital mechanism is 100 times largerthan the spin chirality term in the clean limit. Note thatthe intrinsic AHC follows an approximate scaling relation σ AH ∝ ρ [2, 6, 28] in the “high-resistivity regime”. InFig. 12, we see that | σ AH (3 ◦ ) − σ AH ( − ◦ ) | also followsthe relation ρ similarly. −90 −60 −30 0 30 60 90−202 A HC ( e / ha ) θ (degree) γ =0.001 E =0 λ =0, E =−2 E =−2 SK(−1.0, 0.6, −0.1)
FIG. 13: θ -dependence of AHCs for SK( − . , . , − . N =6 and γ = 0 . E , λ ) = ( − , . E , λ ) =(0 , . E , λ ) = ( − , −3 −2 −1 0 1 2 301 A HC ( e / ha ) θ (degree) γ =0.001 λ =0 E =−2 SK(−1.0, 0.4, −0.1)
0T 3T 6T
X (−100)SK(−1.0, 0.6, −0.1)
SK(−1.0, 0.4, −0.1)
FIG. 14: θ -dependence of AHCs for | θ | ≤ ◦ at γ = 0 . λ = 0) is very small. Next, we analyze the overall θ -dependence of the AHC,by ignoring the experimental condition | θ | ≪
1. Figure13 shows the AHCs as functions of θ . Solid and dashedlines represent the AHCs for E = 0 and −
2, respectively.They have large θ -linear terms for θ ∼
0, and they takefinite values even if θ = ± π/ θ -dependence of the AHC is insensitive tothe value of E . The AHCs for θ = 0 corresponds tothe conventional KL-type AHE. Dotted line in Fig. 13shows the AHC for λ = 0, which gives the spin chiral-ity term σ spinAH . It is proportional to θ for small θ , andbecomes zero when θ = ± π/
2. Finally, we analyze the θ -dependence of the AHC more in detail for | θ | ≤ ◦ inFig. 14. In the case of SK( − , . , − . λ = 0 . θ ∼ ◦ due to the orbital ABeffect, and it is more that 100 times larger than the AHCfor the spin chirality term ( λ = 0). VI. Pr Ir O In the previous section, we discussed the unconven-tional AHE in the pyrochlore Nd Mo O . Here, wediscuss other pyrochlore Pr Ir O . Unlike Mo 4 d elec-trons in Nd Mo O , Ir 5 d electrons are in the paramag-netic state. Below θ W = 1 . f electronsform non-coplanar spin-ice magnetic order. Under themagnetic field along [111], the non-coplanar structureof Pr Ising moments are expected to change from “2in2out”( H ∼ . H > . H > . Ir O , the tilted ferromagnet stateshown in Fig. 1 is also realized. In Pr Ir O , however,the ferromagnetic exchange interaction is absent, and thelocal exchange field h i on Ir ion is composed of only theexchange field from the Pr moment; ∼ J df . Since h i isparallel to the sum of the nearest Pr momenta, θ of Irspin is much larger than the θ of Mo spin in Nd Mo O .Therefore, the tilted ferromagnetic state with large θ andsmall | h i | is realized in Pr Ir O .Now, we explain the local exchange field on Ir sitesgiven by Pr tetrahedron. Details of the derivation ofthese local exchange field are presented in Appendix B.In the strong magnetic field along [111] ( >> . φ A , φ B , φ C ) = ( − π/ , − π/ , π/
2) and θ = 29 . ◦ in Fig.15. We denote this Ir spin structure as [3 ↓ ↑ ]. In thissection, we promise that 0 ≤ θ ≤ π and − π ≤ φ ≤ π .In the intermediate field ( ∼ . φ A , φ B , φ C ) = (2 π/ , π/ , π/ θ A,B = 58 . ◦ ,and θ C = 29 . ◦ in Fig. 15. We denote this Ir spin struc-ture as [2 ↓ ↑ ]. In real compounds, domain structuresof three “2in 2out” structures are expected to be formed,and the total magnetization is parallel to Z -axis. Thetotal AHC will be insensitive to the domain structuresince σ AH ’s due to three [2 ↓ ↑ ] structures are equiva-lent. If we take average of three “2in 2out” structures,the local exchange fields belongs to the 120 ◦ -structurewith θ = 14 . ◦ , as shown in Fig. 15. We denote this Irspin structure as [2 ↓ ↑ ]. As a result, the Ir spin struc-ture changes as [2 ↓ ↑ ] (or [2 ↓ ↑ ]) → [3 ↓ ↑ ] withincreasing the field from ∼ Ir O . We put the atomic SOI as λ = 3000K, whichis slightly smaller than the atomic value for Ir [10]. Thenumber of electrons per unit cell is N = 15 for Pr Ir O .We set | h i | = 20K since | h i | ∼ J df is estimated to belarger than 14K experimentally [29]. We also put thedamping rate γ = 0 .
001 (clean limit). Figure 16 (a)shows the AHC in Pr Ir O with SK( − . , . , − . FIG. 15: The spin configurations in Pr tetrahedron are shownin the first line. Three “2in 2out” states are realized under H . . ◦ -structure, ( φ A , φ B , φ C ) = ( − π/ , − π/ , π/ ◦ -structure, ( φ A , φ B , φ C ) = (2 π/ , π/ , π/ Each line represents the AHC for the 120 ◦ -structure inFig. 15. The line with “ λ = 0” represents the spinchirality term. In the case of [2 ↓ ↑ ], the AHC inthe present model is 10 times larger than the AHC for λ = 0. Thus, the orbital AB effect dominates the chiral-ity mechanism. The variation of the AHC for [2 ↓ ↑ ] (or[2 ↓ ↑ ]) → [3 ↓ ↑ ] can explain the experimental results,ignoring the sign of the AHC. For example, the sign ofthe AHC is changed if J df is negative.In Fig. 16 (b), we put SK( − . , . , − x ) with x =0 . , . .
5. Although the KL term at θ = 0 de-creases from negative to positive with x , the overall θ -dependence of the AHC is not very sensitive to x .Recently, Ref. [30] reports that the AHC in Pr Ir O shows a hysteresis behavior under the magnetic field be-low T H ≈ Ir O . In termsof the spin chirality mechanism, the authors claimed theexistence of a long-period magnetic (or chirality) order ofPr sites with 12 original unit cells [30]. However, there isno theoretical justification for this complex state. Even ifit is justified, the origin of the hysteresis behavior is un-clear. In addition, the magnetic susceptibility χ s ≈ χ s Pr does not show anomaly at T H experimentally.Here, we propose an alternative explanation for theresidual AHE based on the orbital AB effect: In Ln Ir O with Ln =Nd, Sm, and Eu, the Ir 5 d -electronsshow magnetic order at T IrN =36 K, 117 K, and 120 K,respectively [31]. Thus, T IrN monotonically decreases asthe radius of Ln ion increases. Since Pr is on the left-hand-side of Nd in the periodic table, one may expecta finite T IrN ( ∼ Ir O . We stress that smallamount of impurities could induce the magnetic orderin the vicinity of magnetic quantum-critical-point [32].0 A HC ( e / ha ) θ (degree) o θ =14 o γ =0.001SK(−1, 0.6, −0.4), λ =0 [2 ↓ ↑ ] [3 ↓ ↑ ][2 ↓ ↑ ] (a) λ =0 θ =71 o [2 ↓ ↑ ] H → +0 [2 ↓ ↑ ] A HC ( e / ha ) θ (degree) o θ =14 o γ =0.001SK(−1, 0.6, −x), x=0.4x=0.3 (b) [2 ↓ ↑ ] x=0.5 FIG. 16: θ -dependence of AHC in Pr Ir O for (a)SK( − . , . , − .
4) and (b) SK( − . , . , − x ) where x =0 . ∼ .
5. Each line represents the AHC for the 120 ◦ -structure in Fig. 15. (c) Ir spin structure below T IrN under theweak exchange field h i due to “2in 2out” Pr spin-ice order;[2 ↓ ↑ ] H → +0 . The total magnetization of the Ir tetrahedronis zero since S C = − S D k CD , where site D is the apical Irsite. Note that [2 ↓ ↑ ] H →− spin structure is the reverse of[2 ↓ ↑ ] H → +0 . Here, we analyze the Ir spin structure below T IrN , consid-ering the classical Heisenberg model for Ir tetrahedronunder the exchange field h i by Pr spins (see in AppendixB): E = J ( X i =1 S i ) − X i =1 h i · S i , (23)where S i is the i -th Ir spin, and the positive J ( ∼ T IrN ) isthe antiferromagnetic interaction between Ir spins. When J ≪ | h i | , then S i is parallel to h i . When J ≫ | h i | , wehave to find the spin configuration to minimize eq. (23) under the constraint P i =1 S i = 0.Under the exchange field by one of ‘2in 2out” Pr order,the obtained Ir spin structure for J ≫ | h i | is shown inFig. 16. For J ≫ | h i | under H = +0 Tesla, the Ir spinstructure is changed to the 120 ◦ -structure with θ = 70 . ◦ in Fig. 15 (c), which we denote [2 ↓ ↑ ] H → +0 structure.The obtained AHC under this spin structure is − . H = − T H ∼ ↓ ↑ ] H →± structure, since S A · ( S B × S C ) = 0.Also, the Ir spins under the exchange field by the av-eraged “2in 2out” Pr order for J ≫ | h i | show the 120 ◦ -structure in Fig. 15 with θ = 70 . ◦ : We denote thisstructure as [2 ↓ ↑ ] H → +0 . The total magnetization iszero since the Ir spin on the apical site (not shown) isantiparallel to the Z -axis. In this case, we can also ex-plain the “hysteresis behavior of the AHC” below T H ∼ ↓ ↑ ] H → +0 is dif-ferent from that for [2 ↓ ↑ ] under the positive H . VII. DISCUSSIONA. Comparison between theory with experiments
First, we compare the theory with experiments forNd Mo O [14–16, 23] in detail. Under H || [111] be-low T N , σ AH monotonically decreases with H from0Tesla ( θ ≈ − . ◦ ) to 6Tesla ( θ ≈ . ◦ ). This mono-tonic decreasing in AHC can be explained by the θ -linear term in the present model. The relation ρ H ∼ πR s M Mo Z +4 πR ′ s M Nd Z describes the experimental resultswell, where M Mo Z ( M Nd Z ) and R s ( R ′ s ) are the magnetiza-tion and anomalous Hall coefficients for the Mo(Nd) mo-ment [14–16]. In this equation, the first term representsthe conventional AHE that is recognized as the KL mech-anism. In contrast, the second term is highly unusual inthat Nd electrons are totally localized; it represents theunconventional AHE due to the non-collinear spin config-uration. As the magnetic field increases from 0 Tesla to6 Tesla, M Nd Z increases from negative to positive. Since θ ∝ M Nd Z [21], the second term corresponds to the θ -linear term given by the orbital AB effect. Moreover, theAHC for [2 ↓ ↑ ] H → +0 in Fig. 16 is finite, irrespectiveof the absence of magnetization.Next, we compare the present theory with experimentsfor Pr Ir O [29]. Under the magnetic field along [111],the observed AHC increases in proportion to the magne-tization with field from 0 Tesla, whereas it decrease with H above 0 . − cm − . The AHCs in Fig.16 (a) are -11Ω − cm − for [2 ↓ ↑ ], and it is doubled if1 FIG. 17: Examples of the second-order diagrams for spinstructure-driven AHCs: (a) orbital mechanism and (b) spinchirality mechanism. The conventional AHE (KL-term) isgiven by the diagrams (a) by replacing λl + s − and h − s + with λl z s z and h z s z , respectively. we put h i → h i . Thus, the variation of the AHC for[2 ↓ ↑ ] → [3 ↓ ↑ ] in Fig. 16 (a) can explain the exper-imental field dependence. The obtained AHC is mainlygiven by the orbital AB effect, and the spin chirality termis too small to reproduce experimental values. B. Second-order-perturbation theory for spinstructure-driven AHCs
Here, we discuss the spin structure-driven AHC basedon the second-order-perturbation theory with respect to λ and h . The present weak-coupling analysis togetherwith the strong-coupling analysis in Sec. IV will provideus useful complementary understanding. Since their ex-pressions in the present model are too complicated, weshow only some examples of the the second order dia-grams for the spin structure-driven AHCs in Fig. 17: (a) σ orbAH due to the orbital mechanism, and (b) σ spinAH due tothe spin chirality mechanism. In (a), the spin of conduc-tion electron is flipped by x, y -components of the Zeemanterm, h ± s ∓ , and the SOI term, λl ± s ∓ , and the obtainedSHC is σ orbAH ∼ h ± λ/ ∆, where ∆ is the band splittingnear the Fermi level. We stress that this term vanisheswhen C rotational symmetry along Z -axis exists [21]:In the present model, the C rotational symmetry of thesimple kagome lattice in Fig. 4 (b) is violated by the exis-tence of oxygen atoms. In Fig. 17 (b), the spin is flippedby h ± s ∓ twice, and it is given by σ spinAH ∼ h ± / ∆. Thus, σ orbAH and σ spinAH are proportional to sin θ and sin θ , respec-tively. (We note that the conventional AHE (KL-term)is given by replacing λl + s − and h − s + in Fig. 17 (a)with λl z s z and h z s z , and σ KLAH ∼ h z λ/ ∆.)In Nd Mo O , the relation | θ | ∼ O (10 − ) is realized.Thus, we obtain σ spinAH /σ orbAH ∼ θ since λ ∼ h ∼ σ orbAH is about 100 times larger than σ spinAH inNd Mo O . This result is recognized in the present nu-meraical calculation in Fig. 14 and in Ref. [21].In Pr Ir O , the relation | θ | ∼ O (1) is realized. Thus,we obtain σ spinAH /σ orbAH ∼ h /λ ∼ O (10 ) since h ∼ λ ∼ σ AH is only 10 ∼
20 times larger than σ spinAH , asshown in Fig. 16. This discrepancy originates from thehigher-order correction of large λ on the band-splitting∆: In fact, σ orbAH starts to decrease for λ > λ .Finally, we comment that σ orbAH is not suppressed bylarge crystalline electric field. Since ˆ l ± ˆ s ∓ mixes the states | a g , σ i and | e ′ g , − σ i , σ orbAH will be large if these two statesoccupy large portion of the DOS at the Fermi level. Thissituation is actually realized the presence of crystallineelectric field, as shown in Fig. 7 (b). For this reason,large θ -linear spin structure-driven AHE is realized for E = − C. Summary
In summary, we studied the AHE in the pyrochloretype t g -orbital model in the presence of non-collinearmagnetic configurations and the crystalline electric field.Thanks to the SOI, the complex d -orbital wave function ismodified by the tilting angle θ , and the resultant orbitalAB phase gives large θ -linear AHC. This orbital term, σ orbAH , dominates the AHE in Nd Mo O since the spinchirality term, σ spinAH , is proportional to θ ( ≪ Ir O , σ orbAH also dominates σ spinAH since the SOIfor Ir 5 d -electron ( λ ∼ d - f exchange interaction ( J df ∼ Ir O below T H ≈ d -electrons at T H . Infact, the AHC under the Ir spin structure in Fig. 16 (c),which would be realized below T IrN under weak exchangefield from “2in 2out” Pr order, is finite as shown in Fig.16 (a). The total AHC will be insensitive to the forma-tion of domain structure with three “2in 2out” Pr ordersin Fig. 15, since σ AH ’s due to three [2 ↓ ↑ ] structuresare equivalent. The AHC obtained in the present studyis expected to give a major part of the AHC observed inthree dimensional compounds, as discussed in AppendixC.Since σ orbAH in the present model is nonzero unless n A || n B || n C , the realization condition for the orbitalmechanism is just the “non-collinearity of the spin struc-ture”, which is much more general than that for σ spinAH .The orbital mechanism might be the origin of interest-ing spin structure-driven AHE in Fe Sn [33, 34] andPdCrO [35].2 Acknowledgments
The authors are grateful to M. Sato, Y. Yasui, D. S.Hirashima, Y. Maeno, H. Takatsu, S. Nakatsuji and Y.Machida for fruitful discussions. This work has beensupported by a Grant-in-Aid for Scientific Research onInnovative Areas “Heavy Electrons” (No. 20102008) ofThe Ministry of Education, Culture, Sports, Science, andTechnology, Japan.
Appendix A: Hopping integral between the siteswith the different coordinates
In this Appendix, we derive the hopping integrals be-tween the sites with the different d -orbital coordinates asshown in Fig. 3. Here, we represent the five d -orbitals xy , yz , zx , x − y and 3 z − r as 1, 2, 3, 4 and 5. Thewavefunctions in the d -orbitals are given by φ = 1 √ i ( Y − Y − ) = A xyr φ = − √ i ( Y + Y − ) = A yzr φ = − √ Y − Y − ) = A zxr φ = 1 √ Y + Y − ) = 12 A x − y r φ = Y = √ A z − r r where Y ml is the spherical harmonics and A = p / π .We consider the coordinate transformation matrixˆ O AB , which transforms ( n x , n y , n z ) B in the ( xyz ) B -coordinate into ( n x , n y , n z ) A in the ( xyz ) A -coordinate as( n x , n y , n z ) A ˆ O AB = ( n x , n y , n z ) B . It is given byˆ O AB = 19 − − − − − − . (A1)Since ˆ O BC and ˆ O CA are equivalent to ˆ O AB , we have toderive only the hopping integral between sites A and B.Using r B l ′ = r A l O AB ll ′ where l, l ′ = x, y, z , the wavefunc-tion for orbital β at site B can be expressed as linearcombination of the wavefunction for orbital γ at site A.Thus, φ B β = X γ a AB ( β, γ ) φ A γ , (A2)where a AB (1 ,
1) = O AB11 O AB22 + O AB12 O AB21 ,a AB (1 ,
2) = O AB12 O AB23 + O AB13 O AB21 ,a AB (1 ,
3) = O AB11 O AB23 + O AB13 O AB22 ,a AB (1 ,
4) = 2 O AB11 O AB21 + O AB13 O AB23 ,a AB (1 ,
5) = √ O AB13 O AB23 , a AB (2 ,
1) = O AB21 O AB32 + O AB22 O AB31 ,a AB (2 ,
2) = O AB22 O AB33 + O AB23 O AB32 ,a AB (2 ,
3) = O AB21 O AB33 + O AB23 O AB31 ,a AB (2 ,
4) = 2 O AB21 O AB31 + O AB23 O AB33 ,a AB (2 ,
5) = √ O AB23 O AB33 ,a AB (3 ,
1) = O AB11 O AB32 + O AB12 O AB31 ,a AB (3 ,
2) = O AB12 O AB33 + O AB13 O AB32 ,a AB (3 ,
3) = O AB11 O AB33 + O AB13 O AB31 ,a AB (3 ,
4) = 2 O AB11 O AB31 + O AB13 O AB33 ,a AB (3 ,
5) = √ O AB13 O AB33 . Therefore, the hopping integral t B β, A α ( R AB ) = h B β, R B | H | A α, R A i is given by t B β, A α ( R AB ) = X γ a AB ( β, γ )˜ t A γ, A α ( R AB ) , (A3)where ˜ t A γ, A α ( R AB ) = h A γ, R B | H | A α, R A i is the usualhopping integral between the equivalent coordinates,which is given by the SK parameter table in Ref. [24]. Appendix B: Local effective field from Pr tetrahedron In this Appendix, we derive the local effective field atIr sites induced by the spin structure of Pr tetrahedron.Sites A, B, C and D of Pr tetrahedron are located at(1 / , , , / , , , /
4) and (1 / , / , / xyz -coordinate as shown in Table I, andthe center of the tetrahedron is located at (1 / , / , / >> . ∼ D are given by P ↓ ↑ A = ( − , , / √ , P ↓ ↑ B = (1 , − , / √ , P ↓ ↑ C = (1 , , − / √ , P ↓ ↑ D = (1 , , / √ . In the intermediate field( ∼ . P ↓ ↑ C inthe “3in 1out” structure. That is, the configuration ofthe Pr spins in this “2in 2out” structure is given by P ↓ ↑ A = P ↓ ↑ A , P ↓ ↑ B = P ↓ ↑ B , P ↓ ↑ C = − P ↓ ↑ C and P ↓ ↑ D = P ↓ ↑ D . We also consider another case wherethree “2in 2out” structure are averaged. Then, the Prmoments at sites A, B and C are given by 1 / P ↓ ↑ A = P ↓ ↑ A / P ↓ ↑ B = P ↓ ↑ B / P ↓ ↑ C = P ↓ ↑ C / P ↓ ↑ D = P ↓ ↑ D .The effective magnetic fields at Ir sites are obtained bysumming six Pr spins: Ir i atom is surrounded by two Pr j ,two Pr k and two Pr l , where we represent { i, j, k, l } as apermutation of sites { A , B , C , D } . Therefore, the localexchange fields at Ir sites are given by h n ↓ m ↑ i = − J df X j = i P n ↓ m ↑ j . (B1)where J df is d - f exchange interaction. Hereafter, we as-sume J df >
0. We calculate the local exchange field h using above equation. The obtained results in each Prstructure are as follows: In the “3in 1out” structure case, h ↓ ↑ A = − ˜ J (3 , , , h ↓ ↑ B = − ˜ J (1 , , , h ↓ ↑ C = − ˜ J (1 , , . h ↓ ↑ D = − ˜ J (1 , , . where ˜ J = 2 J df / √
3. In the “2in 2out” structure case, h ↓ ↑ A = − ˜ J (1 , − , , h ↓ ↑ B = − ˜ J ( − , , , h ↓ ↑ C = − ˜ J (1 , , . h ↓ ↑ D = − ˜ J ( − , − , . In the averaged “2in 2out” structure case, h ↓ ↑ A = − ˜ J (5 / , , , h ↓ ↑ B = − ˜ J (1 , / , , h ↓ ↑ C = − ˜ J (1 , , / . h ↓ ↑ D = − ˜ J (1 , , / . Next, we rewrite the obtained h ’s in the XY Z -coordinateusing [ n X , n Y , n Z ] = ( n x , n y , n z ) ˆ O − , where the trans-formation matrix ˆ O − is given byˆ O − = 1 √ −√ − √ √ − √
20 2 √ . (B2)In the “3in 1out” structure case, h ↓ ↑ A = − ˜ J [ − √ , − , √ / √ , h ↓ ↑ B = − ˜ J [2 √ , − , √ / √ , h ↓ ↑ C = − ˜ J [0 , , √ / √ , h ↓ ↑ D = − ˜ J [0 , , √ / √ . In spherical coordinates, the direction of the local ex-change fields at site A, B and C are θ = 29 . ◦ and ( φ A , φ B , φ C ) = ( − π/ , − π/ , π/ h ↓ ↑ A = − ˜ J [ − √ , , √ / √ , h ↓ ↑ B = − ˜ J [2 √ , , √ / √ , h ↓ ↑ C = − ˜ J [0 , , √ / √ , h ↓ ↑ D = − ˜ J [0 , , √ / √ , that is, θ A = θ B = 31 . ◦ , θ C = 58 . ◦ , and ( φ A , φ B , φ C ) =(2 π/ , π/ , π/ h ↓ ↑ A = − ˜ J [ − √ , − , √ / √ , h ↓ ↑ B = − ˜ J [2 √ , − , √ / √ , h ↓ ↑ C = − ˜ J [0 , , √ / √ , h ↓ ↑ D = − ˜ J [0 , , √ / √ , that is, θ = 14 . ◦ and ( φ A , φ B , φ C ) =( − π/ , − π/ , π/ Appendix C: AHC in three dimensional compounds
In this paper, we have studied the AHE in the kagomelattice model, which represents the two-dimensional Moor Ir network in the pyrochlore compounds. In the pres-ence of “3in 1out” or “2in 2out” of Nd or Pr spin-iceorder, it was shown that prominent spin structure-drivenAHE are induced on the kagome lattice on the [1,1,1]plane. However, other three kagome layers on the [1,1,-1],[1,-1,1] and [-1,1,1] planes, which are not perpendicularto the magnetic field, also give finite contribution to theAHC.In this section, we shortly discuss the total AHCinduced by four kagome lattices, assuming that theselattices are independent. Here, we put the magneticfield parallel to the [1 , ,
1] plane, which is given bythe ABC plane in Fig. 4 (a), and apply the electricfield along Y axis. Then, the AHC due to the [1,1,1]plane, σ [1 , , , is given in the present study. Consid-ering the relative angles and positions of other threekagome lattices, it is easy to show that the total AHCdue to the electric field on the [1 , ,
1] plane is given by σ totAH = σ [1 , , + ( σ [1 , , − + σ [1 , − , + σ [ − , , ) /
3. Notethat [1 , , − , − , − , ,
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