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The Berry phase in ferromagnetic spin systems and anomalous Hall Effect 7 0 B. Basu∗ and P. Bandyopadhyay† 0 Physics and Applied Mathematics Unit 2 Indian Statistical Institute n Kolkata-700108 a J Wehaveshownthatthestudyoftopologicalaspectsoftheunderlyinggeometryinaferromagnetic 8 spinsystemgivesrisetoanintrinsicBerryphase. ThisrealspaceBerryphasearisesduetothespin 1 rotations of conducting electrons which can be manifested as a further contribution in anomalous Hall effect. ] l PACSnumbers: 03.65.Ud,03.65.Vf l a h - In ferromagnetic metals with a spontaneous broken time reversal symmetry, besides the ordinary s e Hall effect linear dependence of the off diagonal resistivity on the appilied magnetic field, there exists m a contribution proportional to the magnetization of the ferromagnet. The transverse resistivity ρ in H ferromagnets consists of the two contributions, . t a m ρH = R0B+RSM (1) - whereB,M,R andR aremagneticinduction, magnetization,ordinaryHallcoefficientandanomalous d 0 S n Hall coefficient. The second term, which is proportional to the magnetization, represents the anomalous o Halleffect(AHE).Conventionally,theAHEisascribedtospin-orbitinteractionwhichinvolvesacoupling c of orbital motion of electrons to the spin polarization of conduction electrons [1] or to the asymmetric [ skew scattering of conduction electrons by the fluctuation of localised moments [2]. 1 In a recent paper Haldane, [3] has pointed out that the intrinsic AHE in metallic ferromagnets is v controlledbyBerryphasesaccumulatedbyadiabaticmotionofquasiparticlesontheFermisurfaceinthe 0 presence of broken inversion or time -reversalsymmetry. This theory was in support of a report of Fang 4 et. al[4] where they have shown that the AHE is related to the Berry phase acquired by the Bloch wave 4 functions when the magnetic ”monopole” is in the crystal momentum space. The topological features 1 associated with the AHE have also been studied by other authors [5, 6]. It has been realized that under 0 7 appropriate conditions a chirality contribution shows up in AHE [5, 6]. In case of canonical spin glass 0 chirality contribution to the AHE was examined by Tatara and Kawamura [6, 7]. However, in order to / geta net topologicalfield (chirality)the spin-orbitcoupling must be invoked. Besides,we haveproblems t a relatedtothe2DKagomelatticeor3DPyrochlorelatticewherethenettopologicalfieldvanishesthough m a nonvanishing topological Hall effect may be obtained [6, 8]. Breno, Dugaev and Taillefumier [9]have - shownthata nettopologicalfieldcanbe obtainedbymeans ofsome externalparameter. The analysisof d this topologicalHalleffect does not depend onspin-orbitcoupling but arisessolely fromthe Berryphase n acquired by an electron moving in a smoothly varying magnetization. o Recently, the AHE has been studied in a seriesof AuFe samples [10]. It has been observedthat below c : a criticalFe concentrationthe alloysarespinglasseswhile for higherconcentrationsthe alloyshavebeen v dubbed ‘re-entrant’; on cooling one first encounters a ferromagnetic ordering temperature T and then a i c X second canting temperature T which is signalled by a dramatic drop in the low field ac susceptibility. k r Below Tk the system still has anoverallferromagneticmagnetizationbut the individualFe spins become a canted locally with respect to the global magnetization axis. The experimental data demonstrate that the degree of canting strongly modifies the AHE. This is a physically distinct Berry phase contribution occurring in real space when the spin configuration is topologically nontrivial. The present paper is an attempt to address some aspects associated with the geometry of this type of ferromagnetic system and show that the contribution of AHE in this system is a topological effect which arises due to the Berry curvature accumulated by spin rotations of moving electrons. We start with a model which represents the effective ferromagnetic interaction between electrons on ∗Electronicaddress: [email protected] †Electronicaddress: [email protected] 2 different sites. The Hamiltonian is given by J H = tc†c − H S .[c †~σc ] (2) i j 2 i i j X X i,j i where (i,j) runs the nearest neighbour sites, c (c†) is the annihilation (creation) operator at the site i i i and S is the classical spin localised at the site i and ~σ are Pauli matrices. This describes an electron i hopping fromsite i to site j coupled to a spin at eachsite with a Hund coupling J . When J is strong H H enough the spin of the hopping electron is forced to align parallel to S and S at each site. i j For a ferromagnetic system which allows a ferromagnetic ordering below T as well as canting of local c spins below T we shall focus only on the position dependent magnetization direction k M(r) n= (3) M with a fixed magnitude |M|=M. The magnetization direction n(r,t) satisfies the equation of motion ∂ n(r,t)=−γn(r,t)×H (r) (4) t eff where γ is (minus) the gyromagnetic ratio H (r) is the so-called effective magnetic field depending eff on the magnetization distribution M(r). Indeed, we can use a gauge transformation U(r) which makes the quantization axis oriented along the vector n(r) at each point so that we can write U†(r)[~σ.n(r)]U(r)=σ (5) z In generalized spin systems we may assume that the direction of the vector n(r) is arbitrary at different sites. To view the rotation of the magnetization vector such that the vector n(r) is oriented along the quantization axis we may consider the vector n(r) is rotating with an angular velocity ω around the 0 z-axis under an angle ν so that at any instant of time, the magnetization vector is at a position which makesanarbitraryangleν withthe quantizationaxis. Thenwe canwrite the unitvectorintermsofthe variables ν and t as n(r)→n(ν,t) (6) where sinν cos(ω t) 0 n(ν,t)= sinν sin(ω t)  (7) 0 cosν   It may be added here that there may be an additional phase factor in a multi-spin system where cos(ω t)should be replaced by cos(ω t+φ). However as this will not change the physics we are con- 0 0 sidering, we may omit it here. The instantaneous eigenstates of a spin operator in direction n(ν,t) expanded in the σ -basis are given by z ν ν |χ(↑)) ;t> = cos |↑ >+ sin eiω0t|↓ > n z z 2 2 (8) ν ν |χ(↓)) ;t> = −sin |↑ >+ cos eiω0t|↓ > n z z 2 2 2π For the time evolution from t = 0 to t = τ where τ = each eigenstate will pick up a geometric ω 0 phase (Berry phase) apart from the dynamical phase which is of the form |χ(↑) ;t=0>→|χ(↑) ;t=τ >=eiγ+(ν) eiθ+|χ(↑) ;t=0> n n n |χ(↓) ;t=0>→|χ(↓) ;t=τ >=eiγ−(ν) eiθ−|χ(↓) ;t=0> (9) n n n where γ is the Berry phase which is half of the solid angle 1Ω swept out by the magnetization vector ± 2 and θ is the dynamical phase. The dynamical phase can be eliminated by the spin-echo method [11]. ± 3 However, in the context of AHE we are only concerened with the Berry phase. Geometrically we may get the explicit values of the Berry phase given by γ as ± γ (ν) = −π(1−cosν) + (10) γ (ν) = −π(1+cosν)=−γ (ν)−2π − + This equation helps us to note that in any arbitrary spin system the Berry phase aquired by a spin up eigenstate in direction n(ν) is given by γ (ν) =−π(1−cosν) (11) + Itisknownthat afreepolarizedspincanbe representedby atwo-componentspinorwhichis achieved when a scalar particle is attached with one magnetic flux quantum. Now in unit of magnetic flux quantum hc the correspondingmagnetic(monopole)strengthisgivenby|µ|= 1. Sothe phaseacquired |e| 2 by a free polarized spin encircling a closed path may be represented by the phase acquired by a scalar particle encircling a magnetic flux quantum which is given by ei2πµ with µ=1/2. Indeed this phase eiπ corresponds to the phase acquired by a fermion after 2π rotation. From eqn.(11) we note that in a multi-spin system, the Berry phase acquired by a spin-up eigenstate in direction n(ν) can be written in terms of µ as eiγ+(ν) =ei2πµ(1−cosν) (12) with µ=1/2 This helps us to note that in a spin system there is a deviation of the phase factor acquired by a spin eigenstate in direction n(ν)from that of a free polarized spin. The deviation of the phase factor | ∆γ | from the free spin case is given by 1 |∆γ |= cosν (13) 2 Now for a ferromagnetic system where all the spins are aligned along the quantization axis (ν = 0) we have the intrinsic phase factor acquired by a spin eigenstate 1 |∆γ |= (14) 2 Thisintrinsicphasefactoracquiredbythespineigenstateinaferromagneticsystem,isthemanifestation of some inherent magnetic field in these types of systems. This fictitous magnetic field is present due to the effectofthe Berryphaseofthe localisedspinsandconductionelectronsmoveinthis fieldwhenthere is strong Hund coupling. Now from eqn.(13)we observe that we will have variation of |∆γ| depending on the angle ν which is associated with the degree of canting of the local spins. In fact, the system will still have an overall ferromagnetic magnetization but the degree of canting of individual spins will now change the inherent magnetic field associated with |∆γ| inducing a modification of the AHE as observed in experinents [10] We introduce a gauge potential corresponding to this magnetic field A(r,t)=−2πiφ U†(r,t)∂ U(r,t) (15) 0 i where φ = hc is the flux quantum and i=x,y . Now if we neglect the spin-flip transitions and assume 0 |e| that the systemis in the spinup (or, spindown)subspace only, we can substitute the matrix gaugefield by an Abelian gauge field a(r,t). In fact, we can choose a(r,t) as a(r,t)=i<n(r,t)|∇n(r,t)> (16) The Berry phase for a closed path Γ is given by Γ=expi a(r)dr (17) I Γ 4 If we assume conduction electrons as to represent 2D electron gas we can consider the continuum limit. Writing n ∂ n −n ∂ n a (r,t)= x i y y i x (18) i 1+n z the corresponding field B takes the form 1 B = ǫ n (∂ n )(∂ n ) (19) µνλ µ x ν y λ 4π The integralof this topologicalfield over an area enclosed by an arbitrarycontour is proportionalto the Berry phase. The topological current can be defined as 1 J = ǫ n(∂ n×∂ n) (20) µ µνλ ν λ 8π In terms of the vector potential a the topological current is given by µ J =ǫ ∂ a (21) µ µνλ ν λ This is generated due to the Berry phase accumulated by the spin rotations of moving electrons when the background magnetization is not uniform in space. In terms of the Abelian gauge field, the Hamiltonian for a 2D electron gas may be expressed as ~2 ∂ H =− [ −iea(r)]2−gMσ (22) 2m ∂r z and we can define the covariantmomentum operator ∂ π =−i −ea (r) (23) µ µ ∂x µ This leads to the noncomutativity of the momentum components [π ,π ]=ie(∂ a −∂ a )=ieB (24) x y x y y x z This (B ) is the inherent magnetic field we were talking about. We may add here that a new form of z non-commuatative space can be formulated where noncommutativity in the momentum space induces a singular type of magnetic field in the real space [12, 13]. To conclude, we may say that for a spatially varying magnetization in a ferromagnetic spin system a topologicalcurrentisgeneratedduetothetopologicalpropertiesassociatedwiththeunderlyinggeometry ofthe system. The inherentmagnetic type ofbehavioris causedbythe Berrycurvaturewhicharisesdue to the spin rotations of conducting electrons and is the effect of noncommutativity in momentum space. [1] R.Karpusand J. M. Luttinger,Phys. Rev. 95, 1154 (1954) [2] J. Kondo, Prog. Theo. Phys. (Kyoto), 27, 772 (1962) [3] F.D. M. Haldane, Phys. rev.Lett. 93, 206602 (2004) [4] Z. Fanget. al. Science 302, 92 (2003) [5] J. Yeet. al., Phys.Rev. Lett.83, 3737 (1999) [6] G. Tatara and H.Kawamura, J.Phys. Soc. (Japan) 71, 2613 (2002) [7] H.Kawamura, Phys.Rev.Lett. 90, 047202 (2003) [8] S.Onoda and N. Nagaosa, Phys. 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