Kinetic equations for massive Dirac fermions in electromagnetic field with non-Abelian Berry phase
aa r X i v : . [ h e p - t h ] A p r USTC-ICTS-13-17
Kinetic equations for massive Dirac fermions in electromagnetic field withnon-Abelian Berry phase
Jiunn-Wei Chen, Jin-yi Pang,
1, 2
Shi Pu,
1, 2 and Qun Wang Department of Physics, National Center for Theoretical Sciences,and Leung Center for Cosmology and Particle Astrophysics,National Taiwan University, Taipei 10617, Taiwan Interdisciplinary Center for Theoretical Study and Department of Modern Physics,University of Science and Technology of China, Hefei 230026, China
We derive a semi-classical effective action and the kinetic equation for massive Dirac fermionsin electromagnetic fields. The non-Abelian Berry phase structure emerges from two helicity statesof massive fermions with positive energy. The classical spin emerges as a vector in SU(2) helicityspace. The continuity equations for the fermion number and the classical spin are derived. Thefermion number is conserved while the spin charge is not conserved by anomaly. Previous resultsabout the coefficients of the chiral magnetic effect for the fermion and axial currents in the chirallimit can be reproduced after including the anti-fermion contributions. This provides an examplefor the emerging spin and non-Abelian Berry phase of Dirac fermions arising from the fermion mass.
I. INTRODUCTION
Berry phase is the phase factor acquired by the energy eigenstate of a quantum system when the parameters of theHamiltonian undergo a cyclic change [1]. The Berry potential can be regarded as the induced gauge field in parameterspace. If the energy eigenstates are degenerate, there is an internal symmetry in the Hilbert space spanned by thedegenerating states. This will lead to the non-Abelian Berry phase or non-Abelian gauge field [2–4]. Ever since itsdiscovery, the Berry phase structure has permeated through all branches of physics especially in condensed matterphysics, see, e.g., [5] for a review.Kinetic theory is an important tool to study the phase space dynamics of non-equilibrium systems. In such aclassical description, it is difficult to accommodate quantum effects such as chiral anomaly which has no counterpartin classical dynamics. Recently it was proposed that the Abelian Berry potential can be introduced into the actionof chiral (massless) fermions in electromagnetic fields to accommodate the axial anomaly in semi-classical dynamics[6, 7]. It have been shown by some of us that the Covariant Chiral Kinetic Equation (CCKE) can be derived inquantum kinetic theory from the Wigner function for massless fermions in electromagnetic fields [8]. The CCKEprovides a semi-classical description of the phase space dynamics of chiral (massless) fermions with the 4-dimensionalAbelian Berry monopole and is connected to the axial anomaly. The chiral kinetic equation for massless fermions in3-dimension [6, 7, 9] can be obtained from the CCKE by integrating out the zero component of the momentum andturning off the vortical terms.In this paper we try to generalize the previous work [8] by some of us about massless fermions by investigating thefermion mass effects on the kinetic equation and the Berry phase. Our motivation is to make connection to spintronicsfor massive fermions as what Ref. [10, 11] did. However, Ref. [10, 11] derived single fermion semi-classical equationsof motion but the anomaly and finite temperature and finite density effects were not addressed. In our work, we coverthese areas by including the axial anomaly and develope the kinetic theory. We use the path integral approach whichis also different from Ref. [10, 11].As in Ref. [10, 11], we focus on massive Dirac fermions of positive energy in background electromagnetic fields.Non-Abelian Berry potentials emerge in the effective action as the result of two degenerate positive energy states. Asthe consequence of the non-Abelian feature, the classical spin emerges as a degree of freedom in phase space [12–15].Different from Ref. [10, 11], we use helicity states as the basis for spin states. We can recover the spin state basisof Ref. [10, 11] by an effective gauge transformation, through which our Berry curvature and equations of motionare consistent to Ref. [10, 11]. The equation of motions for all phase space variables including the classical spincan be derived from the effective action. The continuity equations for the fermion number and the classical spincan be obtained from the kinetic equation. The anomalous source term in the fermion number continuity equationsis vanishing for massive fermions, which means the fermion number conservation. However the conservation of theclassical spin is broken by the anomalous source term. In the massless limit, the Berry phase becomes Abelian. Thereare always anomalous source terms in the spin continuity equation for massive and massless fermions. In the masslesslimit, when including anti-fermions, the spin continuity equation becomes that of axial anomaly. This provides anexample for the emerging spin and non-Abelian Berry phase of Dirac fermions arising from the fermion mass.In order for readers to understand the nature of the problem with the kinetic equation for massive Dirac fermions,we need to make some remarks about it. • The Hamiltonian for a massless fermion in the Weyl bases is diagonal, H = diag( − σ · p , σ · p ) , where σ denotes Pauil matrices and p the vector momentum. The upper/lower block is the effective Hamiltonian forthe left/right-handed fermion. It is natural to consider a single block, e.g. H = σ · p for the right-handedfermion, to derive the kinetic equation. The generalization to the left-handed fermion is straightforward andtrivial. However, the above treatment does not work for a massive fermion whose Hamilton has the fermionmass as off-diagonal elements, H ( p ) = (cid:18) − σ · p mm σ · p (cid:19) . (1)For positive (or negative) energy solutions there are two degenerate eigenstates with opposite helicities. This willlead to a SU(2) non-Abelian Berry potential in the effective action. The fermion mass breaks chiral symmetryexplicitly, so the axial current is not conserved at the classical level and is not appropriate for kinetic theory.We need to find a better quantity which corresponds to a conserved quantum number. • One possible way is to define a spin state | s i in the basis of two eigenstates of σ z in the positive energy solutionof the Dirac equation. Then one can obtain the average spin s a = h s | σ a | s i . The effective action and equationsof motion (including the precession equation for the averaged spin) have been obtained with such a spin statein time-dependent variational method for the wave-packet evolution in condensed matter physics [10, 11]. • In this paper, we choose another representation of | s i . Instead of using two eigenstates of σ z as a basis, we use twohelicity states as the basis for spin states. We know that the helicity operator Σ = diag { σ · p , σ · p } is conservedin absence of external electromagnetic fields, [Σ , H ( p )] = 0 . The average spin (classical spin) s a = h s | σ a | s i inthe helicity basis can be well defined and the spin precession equation or the Bargmann–Michel–Telegdi equation[16] can be obtained. In the classical or weak field limit, the spin current is conserved for the Dirac fermions ofpositive energy. Therefore, in the semi-classical kinetic theory, the classical spin can be regarded as an additionalphase space variable. In the massless limit, the classical spin which is a continuous variable becomes the helicitywith two discrete values ± / , and the helicity current becomes the axial current j µ . For other spin state basesthan the helicity one, such as the one used in Ref. [10, 11], it is less obvious to see such a connection betweenthe spin and the axial current. Due to the above advantages of the helicity basis, we will use it for spin statesand the non-Abelian Berry potential in the path integral formulation of the effective action. • Although the SU(2) Berry phase has already been explored in Ref. [10, 11], to our knowledge, such aspects orcomponents as the anomaly, the path integral approach, the helicity baisis, transparent transition to masslesslimit and the conituity equations have not been addressed in literatrue before. • Since we only consider the positive energy fermions, it is implied that m ≫ eB so that no antifermionscan be involved. This is called the weak field condition. We will see that from the phase-space measure, √ γ = (1 + e Ω · B ) , in Eq. (38), we can also obtain the same condition by the requirement e | Ω · B | ≪ to ensurethat √ γ should not vanish and then invalidate the Hamiltonian dynamics. A weak magnetic field is also neces-sary from the consideration of the lowest Landau level. In presence of a strong magnetic field, particles will allstay at the lowest Landau level and the classical description fails. We see that the fermiom mass provides anatural scale for the magnetic field.Our current work is closely related to the Chiral Magnetic and Vortical Effect (CME and CVE) [17–19]. TheCME and related topics have been extensively studied in several approaches, including AdS/CFT correspondence[20–36], relativistic hydrodynamics [37–40], kinetic theories [6–8, 41–43], lattice simulations [44–48], and quantumfield theory or other effective theories [18, 49–63]. Recently there have been some developments related to the CMEin Weyl semi-metals [64–66]. For a recent review of the CME/CVE and related topics, see e.g. [67]. The path integralformulation of the effective action for chiral (massless) fermions with Berry phase has been given in Ref. [7]. Thecanonical approach from the Hamiltonian can also be used [9]. The effective action with the Berry phase can alsobe written as the Wess-Zumino-Witten form [63, 68]. In these approaches, the off-diagonal elements of the Berrypotentials have been neglected under the adiabatic expansion such that the Berry potential is Abelian.The paper is organized as follows. In Sec. II, we introduce the derivation of the effective action for chiral (massless)fermions in electromagnetic fields via conventional path integral approach in some approximations. In Sec. III, weconstruct an improved and more rigorous path integral approach to the effective action. The non-Abelian Berrypotentials and the classical spin emerge in the formalism. The kinetic equations and continuity equations will bederived in Sec. IV. We make conclusions in Sec. V. II. CHARGED CHIRAL (MASSLESS) FERMION IN CONVENTIONAL PATH INTEGRALQUANTIZATION
We consider a particle moving in a background electromagnetic field. We denote the canonical momentum as p c and the mechanical momentum as p , they are related by p c = p + e A ( x ) . The Hamiltonian can be expressed in termsof the canonical momentum p c , H = ǫ ( p c − e A ) + eφ ( x ) , (2)where ǫ ( p ) is the particle energy. The equation of motion can be derived by the Hamilton equations, x = − ∂H/∂ p c and p c = − ∂H/∂ x , as ˙ x = ∇ p ǫ p ≡ v , ˙ p = e E + e v × B , (3)where E = − ∇ φ and B = ∇ × A .The path integral quantization is based on the Hamiltonian (2). We can write the transition matrix element inpath integral, K fi = h x f | e − iH ( t f − t i ) | x i i = ˆ [ D x ( t )][ D p c ( t )] P exp (cid:20) i ˆ t f t i dt ( p c · ˙ x − H ) (cid:21) . (4)Note that the starting and end points of the path are fixed at x ( t i ) = x i and x ( t f ) = x f . Then we have h x | p c i = exp( i p c · x ) . (5)After completing the path integral, we resume the use of p (writing p c in terms of p ) and obtain the Euler-Lagrangeformulation of quantum mechanics, h x f | e − iHt | x i i = ˆ [ D x ( t )][ D p ( t )] exp( iS ) , (6)where the action is S = ˆ dt [ e A ( x ) · ˙ x − eφ ( x ) + p · ˙ x − ǫ ( p )] . (7)From the Euler-Lagrange equation we can also obtain Eq. (3).We now consider a massless and charged fermion moving in the background electromagnetic field. The Hamiltoniancan be written as H = σ · [ p c − e A ( x )] + eφ ( x ) , (8)where σ are Pauli matrices σ = ( σ , σ , σ ) and we emphasize that the quantum Hamiltonian should be constructedfor the canonical momentum p c instead of the mechanical momentum p = p c − e A . The transition amplitude in pathintegral in is given by Eq. (4) with the Hamiltonian (8).Since the Hamiltonian (8) is a matrix, it is necessary to diagonalize it at each point of the path. To this end weuse a unitary matrix U ( p c − e A ) for diagonalization U † ( p c − e A ) HU ( p c − e A ) = (cid:20) | p c − e A | + eφ ( x ) 00 −| p c − e A | + eφ ( x ) (cid:21) = σ ǫ ( p c − e A ) + eφ ( x ) , (9)where ǫ ( p ) ≡ | p | . One can easily verify that U p = ( χ + , χ − ) = (cid:18) e − iϕ cos θ − e − iϕ sin θ sin θ cos θ (cid:19) ,U † p = (cid:18) χ † + χ †− (cid:19) , (10)where θ and φ are polar angles of p as p = | p | (sin θ cos φ, sin θ sin φ, cos θ ) , and χ ± are eigenstates of σ · p satisfying Hχ ± = ± χ ± . The transition amplitude is K fi = lim N →∞ ˆ N Y j =1 d x j d p cj d x h x f | x N i× N Y j =1 h x j | e − iH ∆ t | p cj ih p cj | e − iH ∆ t | x j − i h x | x i i . (11)So we can finally obtain the action with Berry connection from the path integral quantization for massless and chargedfermions in electromagnetic field (see Appendix A), S = ˆ dt [ p · ˙ x + e A ( x ) · x − σ ǫ ( p ) − eφ ( x ) − A ( p ) · ˙ p ] , (12)In deriving the above action, we have used the definition A ( p ) = − iU † p ∇ p U p with U p given by Eq. (10), whoseexplicit form is A ( p ) = − | p | (cid:18) e φ cot θ e φ − i e θ e φ + i e θ e φ tan θ (cid:19) . (13)Note that the action (12) is in a matrix form. We can expand A ( p ) in terms of Pauli matrices (including theunit matrix) and obtain each component as A = − / (2 | p | sin θ ) e φ , A = − / (2 | p | ) e φ , A = − / (2 | p | ) e θ and A = − cot θ/ (2 | p | ) e φ .The action (12) is based on the condition (A4). Note that due to off-diagonal elements of A ( p ) in Eq. (13), thecommutator of σ and A is non-vanishing, [ σ , A ( p )] = − | p | (cid:18) e φ − i e θ − e φ − i e θ (cid:19) . (14)If we neglect the off-diagonal elements of A ( p ) , the condition (A4) is satisfied automatically and leads to the action(12). This can be made diagonal into the positive and negative helicity components [6, 7], S ± = ˆ dt [ p · ˙ x + e A ( x ) · x ∓ ǫ ( p ) − eφ ( x ) − a ± ( p ) · ˙ p ] , (15)where we used a ± ( p ) = A / ( p ) . Those helicity changing process are neglected in an adiabatic expansion treatment.The Hamilton equations can be derived with the Abelian Berry phase which modify Eq. (3) but keep the symplecticstructure [69]. III. ACTION FOR MASSIVE FERMIONS: IMPROVED PATH INTEGRAL APPROACH
In this section, we will formulate the action for massive fermions with non-Abelian Berry phase structure. For freemassive fermions, the positive energy eigenstate has the degeneracy two which corresponds to two opposite helicities.It means the system has a SU(2) symmetry, which can be shown to lead to a SU(2) non-Abelian Berry phase [2–4].To deal with the action of the matrix form in the path integral in the previous section, we expand the state space byintroducing the classical spin (sometimes we call it spin for short) degree of freedom in phase space which is a vectorin the SU(2) space.We now consider a transition from an initial state | x i , s i i to a final state | x i , s f i , where s i and s f denote the initialand final spin states, respectively. We treat the helicity space to be an internal symmetry space, which is independentof coordinate and momentum states, | x , s i = | x i| s i and | p , s i = | p i| s i . A spin state in the Dirac space in the helicitybasis is defined as | s λ i = g | λ i , (16)where g is an element of the SU(2) ⊕ SU(2) representation in the Dirac space from doubling the fundamental represen-tation in dimension 2 (i.e. it is a × matrix). The reference spin states | λ i along an arbitrary direction with positive( λ = + ) and negative ( λ = − ) polarizations are 4-dimensional vectors and satisfy the orthogonal and completenessrelations: h λ | λ ′ i = δ λλ ′ and P λ = ± | λ i h λ | = 1 . One can check that the same relations also hold for | s λ i due to gg † = 1 .For example, the form of g can be chosen as follows, g ( ξ ) = exp( iξ Σ ) exp( iξ Σ ) exp( iξ Σ ) , (17)where Σ = diag( σ , σ ) with σ = ( σ , σ , σ ) being Pauli matrices and ξ = ( ξ , ξ , ξ ) are three Euler angles. But inlater discussions, we do not adopt any concrete form of g ( ξ ) . Eq. (16) shows that the spin state | s λ i can be labeledby ξ . From the completeness relation for the spin states | s λ i , we can use the following shorthand notation for theintegral over the phase space ξ , ˆ d ξ | s i h s | ≡ ˆ d ξ X λ | s λ i h s λ | = ˆ d ξ = const .. (18)So in the path integral we can insert ´ d ξ | s i h s | at different space-time points along the path.The transition amplitude from an initial state | x i , s i i to a final state | x i , s f i is given by K fi = h x f , s f | e − iHt | x i , s i i , (19)where H is the Hamiltonian for Dirac fermions with mass m given by H = α · p + βm, (20)with α = (cid:18) σσ (cid:19) , β = (cid:18) − (cid:19) . (21)In the Hamiltonian we did not include the electromagnetic field just for simplicity, we will consider it later. TheHamiltonian H can be diagonalized by βE p = U p HU † p = U p ( α · p + βm ) U † p , (22)where E p = p | p | + m and U p and U † p are unitary × matrices with U p U † p = 1 , U † p is given by, U † p = ( u + , u − , v + , v − )= N r (cid:18) χ + χ − a p χ + a p χ − a p χ + − a p χ − − χ + χ − (cid:19) , (23)with N r = p ( E p + m ) / (2 E p ) and a p = | p | / ( E p + m ) . Here u e ( e = ± ) are positive energy eigenstates of the Diracequation, while v e are negative energy eigenstates. The helicity states are denoted by χ e which satisfy σ · ˆ p χ e = eχ e .For the path integral, we can insert complete sets of coordinate and spin states at N ( N → ∞ will be taken in theend) time points along the space-time path, then the transition amplitude (19) becomes K fi = lim N →∞ ˆ N Y j =1 [ d x j ][ d ξ j ] h x f , s f | x N , s N i× N − Y j =1 h x j +1 , s j +1 | e − iH ∆ t | x j , s j i h x , s | x i , s i i . (24)Each of the amplitudes between two states can be evaluated as (see Appendix B for the details of the derivation) I j +1 ,j = h x j +1 , s j +1 | e − iH ∆ t | x j , s j i = ˆ [ d p ′ ][ d x ′ ][ d p ′ ] exp [ i p ′ · ( x j +1 − x ′ )] exp [ i p ′ · ( x ′ − x j )] × e − iβE p ′ ∆ t Tr[ λ j +1 ,j ( g j +1 ) − U † p ′ U p ′ g j ] . (25)The trace in the last line can be re-written as Trace ≈ − Tr[ λ j +1 ,j ( g j +1 ) − ( g j +1 − g j )] − Tr[ λ j +1 ,j ( g j +1 ) − ( p ′ − p ′ ) · U † p ′ ∇ p ′ U p ′ g j +1 ] ≈ exp (cid:8) − ∆ t Tr[ λ j +1 ,j ( g j +1 ) − ˙ g j ] − i ∆ t Tr[ λ j +1 ,j ( g j +1 ) − ˙ p · A ( p ) | p = p ′ g j ] o (26)where we have defined A ( p ) = − iU † p ∇ p U p . Combining Eqs. (24,25,26) we derive the final form of the amplitude K fi = ˆ [ D x ( t )][ D p ( t )][ D ξ ] exp( iS ) , (27)with the action S = ˆ t dt (cid:8) p · ˙ x − βE p + i Tr [ λg − ( d/dt − ˙ p · A ) g ] (cid:9) , (28)and boundary conditions x (0) = x i , s (0) = s i , x ( t ) = x f , and s ( t ) = s f . Note that we have neglected all irrelevantconstants in Eq. (27) and suppress the subscripts of λ in the action (28).In the adiabatic expansion, we neglect negative energy eigenstates or anti-fermions. In this case, g is × matricesand given by Eq. (17) with replacement Σ → σ , and one keeps the upper-left × block of the matrix A ( p ) . As a × matrix, we can expand A ( p ) as A = A a σ a / with a = 0 , , , and σ = . The action (28) becomes S = ˆ dt (cid:2) p · ˙ x − E p + i Tr ( λg − ˙ g ) − s a A a ( p ) · ˙ p (cid:3) , (29)where s a is the average of σ a over a spin state, s a = 12 h s | σ a | s i = 12 Tr ( λg − σ a g ) . (30)We see that s = 1 / and s a ( a = 1 , , ) are functions of ξ . Note that the Lagrangian is given by the content ofthe square bracket in the action (29) and is a functional of ( x , p , ξ , ˙ x , ˙ p , ˙ ξ ) . Using Eq. (23) and the definition for theBerry connection or potential A ( p ) = − iU † p ∇ p U p , we obtain A = − | p | θ e φ , A = mE p | p | e φ , A = − mE p | p | e θ , A = − | p | cot θ e φ , (31)where φ and θ are spherical angles of ˆ p = p / | p | , e φ and e θ are associated univectors, we have e θ × e φ = ˆ p . If we set m = 0 , i.e. the massless or chiral fermion case, we recover A and A (up to a factor 2 from the definition of s a inEq. (30)) for the chiral fermion in Eq. (13). The difference between Eq. (13) and (31) is: A , = 0 from Eq. (31)but A , = 0 from Eq. (13). In the case of Eq. (31), we really have an Abelian Berry potential. This difference isrooted in different bases of spinors used in Eq. (13) and (31). The better way is to use the bases in Eq. (23) for thepositive energy which leads to Eq. (31).In the presence of background electromagnetic fields, we use the canonical momentum p c = p + e A to label amomentum state | p c i instead of | p i . Then all p in the above should be replaced by p c − e A , and the conjugaterelation becomes h p c | x i = e − i p c · x . Following the same procedure as in Sect. II and recovering back to p in the end,we finally obtain the action for massive fermions in electromagnetic fields S = ˆ dt (cid:2) i Tr ( λg − ˙ g ) + p · ˙ x + e A · ˙ x − eφ − E p − s a A a · ˙ p (cid:3) . (32)The first term of the Lagrangian can also be written in such a form i Tr ( λg − ˙ g ) = i Tr ( λg − ∂g∂ξ a ) ˙ ξ a = − G − ba s b ˙ ξ a , (33)where G − ba is defined in Eq. (D4).The effective action (32) can also be derived directly from the Lagrangian for Dirac fermions by separating the fastmodes from the slow ones, see Appendix C. IV. KINETIC EQUATION WITH NON-ABELIAN BERRY PHASE
The equations of motion from the above action read (see Appendix D for detailed derivation) ˙ x = v p + ˙ p × s a Ω a ˙ p = e E + e ˙ x × B , ˙ s a = ǫ abc ( ˙ p · A b ) s c , (34)where we have implied a = 0 , , , and a, b, c = 1 , , in the first and last line respectively, we have defined v p ≡ ∇ p E p = p /E p , ǫ abc is the anti-symmetric tensor with ǫ = 1 . We observe ˙ s a s a = d ( s a ) /dt = 0 , so s = s + s + s is a constant. In absence of external electromagnetic fields, p and s a are constants of motion. TheBerry curvature in Eq. (34) are given by Ω ≡ ∇ p × A , Ω a ≡ ∇ p × A a − ǫ abc A b × A c . (35)From the Berry connection (31), we obtain Ω = , Ω = mE p e θ , Ω = mE p e φ , Ω = 1 E p ˆ p . (36)We see that the a = 0 component does not appear in the first line of equations of motion (34) due to the vanishingof Ω . We define ρ a which we will use in the continuity equations later, ρ a = ( δ ab ∇ p + ǫ abc A c ) · Ω b . (37)Substituting Eq. (35) into the above we obtain ρ a = 0 ( a = 1 , , ) for p = , where we can check ∇ p · Ω a = − ǫ abc A c · Ω b = − ǫ abc ǫ ijk ( ∂ i A bj ) A ck . On the other hand, one can use explicit expressions in Eqs. (31,36) to obtain ρ = ρ = 0 . For ρ , we get ∇ p · Ω = 2 m / ( | p | E p ) and ǫ bc Ω b · A c = − m / ( | p | E p ) for p = . One can also verify ´ d p ρ = 0 . Therefore we finally obtain ρ = 0 . If we consider massless fermions, the only non-vanishing componentsare A , Ω = 0 and the Berry phase is Abelian, one can check ρ = ρ = 0 and ρ = ∇ p · Ω = 4 πδ (3) ( p ) . Theappearance of the delta-function is because there is a singularity at zero momentum p = 0 in the Berry curvature formassless fermions.The first two equations of (34) can be simplified as √ γ ˙ x = v p + e E × Ω + e B ( v p · Ω ) , √ γ ˙ p = e E + v p × e B + e ( E · B ) Ω , (38)where we have defined Ω ≡ s a Ω a ( a = 1 , , ) and √ γ = (1 + e Ω · B ) as the phase space measure. Note that we onlyconsider the positive energy solution, it is implied that m ≫ eB so that no antifermions can be involved. With thiscondition, the phase-space measure √ γ cannot vanish and invalidate the Hamiltonian dynamics since e | Ω · B | ≪ .Equation (38) has a dual symmetry under the interchange of ˙ x ↔ ˙ p , v p ↔ e E , Ω ↔ e B . (39)We have to extend the the phase space by including the spin vector s . The phase space distribution is denoted by f ( t, x , p , s ) and we assume it satisfies the collisionless Boltzmann equation, dfdt = ∂f∂t + ˙ x i ∂f∂x i + ˙ p i ∂f∂p i + ˙ s a ∂f∂s a = 0 . (40)The invariant phase space volume element is then d Γ = √ γ π ) S d x d p d s . (41)Since s = s + s + s = 1 / , there are only two independent variables, so we add a delta-function δ ( s − s a s a ) inthe phase space integral and denote d s ≡ d s δ ( s − s a s a ) . Note that S = π is a normalization constant from thecondition S − ´ d s = 1 . We define n ( t, x ) and J ( t, x ) as the fermion number density and current respectively, n ( t, x ) = ˆ d p d s (2 π ) S √ γf ( t, x , p , s ) , J ( t, x ) = ˆ d p d s (2 π ) S √ γ ˙ x f ( t, x , p , s ) . (42)Then the continuity equation for the fermion number is (see Appendix E for the derivation) ∂n∂t + ∇ x · J = e ( E · B ) ˆ d p d s (2 π ) S ρ a s a f. (43)We see that the source term in the continuity equation (43) is proportional to the anomaly quantity E · B and involves ρ a . For massive fermions, the source term is vanishing due to ρ a = 0 ( a = 1 , , ), so the fermion number is conserved.The physical reason for the vanishing ρ a is that the Berry phase is non-Abelian and Berry curvature is analytic atthe zero momentum due to non-zero fermion mass. This reflects the fact that the classical spin for a massive fermionvaries in time and the total fermion number with all spin orientations is conserved.However, for massless fermions the only non-vanishing component of the Berry potentials or curvatures is alongthe third direction, so the Berry phase is Abelian. The only spin component s is a constant in time from Eq.(34), so we have s = ± / which correspond to the positive/negative helicity. Hence the spins are not continuousvariables of phase space anymore. We denote the distribution functions for positive/negative helicity fermions as f ± ( t, x , p ) ≡ f ( t, x , p , s = ± / . We can replace the integral over s with a sum over s , i.e. S − ´ d s → P s = ± / .The only non-vanishing component of ρ a is ρ = ∇ p · Ω = 4 πδ (3) ( p ) , which is singular and behaves like a monopole atthe zero momentum. Both the Berry phase and anomaly take their roles in the non-vanishing source of the continuityequation as follows ∂n∂t + ∇ x · J = e π ( E · B )[ f + ( t, x , p = ) − f − ( t, x , p = )] . (44)For equilibrium Fermi-Dirac distribution at zero temperature and finite chemical potential, we have f ± ( t, x , p = ) =1 , then fermion number is conserved. If there are only positive or negative helicity fermions in the system, the abovecontinuity equation becomes ∂n ± ∂t + ∇ x · J ± = ± e π ( E · B ) f ± ( t, x , p = ) . (45)which is identical to Eq. (23) of Ref. [8]. Here n ± and J ± are the fermion number densities and currents forpositive/negative helicity fermions respectively, which are obtained by integration over momenta for √ γf ± ( t, x , p ) and √ γ ˙ x f ± ( t, x , p ) respectively with √ γ = 1 ± ( e/ Ω · B . Note that the fermions we are considering in this paperhave positive energies. If we include the contribution from anti-particles and assume equilibrium distributions withdifferent chemical potentials for positive and negative helicities, the continuity equation (44) becomes the conservationequation for fermion number in the chiral (massless) limit or Eq. (24) of Ref. [70], ∂n∂t + ∇ x · J = 0 , (46)where n and J are the net fermion number density and current (fermion minus anti-fermions). Note that the sourceterm is vanishing because we have made the following replacement f + → f + + ¯ f − = f R + ¯ f R → p = ) ,f − → f − + ¯ f − = f L + ¯ f L → p = ) , (47)where ¯ f ± denote the the distributions for anti-fermions with positive/negative (right-handed/left-handed) helicity,and f R/L and ¯ f R/L denote those for fermions and anti-fermions with right-handed/left-handed (positive/negative)chirality respectively. They are related in the massless limit by ¯ f ± = ¯ f L/R . We can also reproduce the CME currentfrom Eq. (42), namely, J = ξ B B , where ξ B is the CME coefficient in Eq. (22) of Ref. [70]. A systematic way ofincluding fermions and anti-fermions is to work in the full Dirac space with 4-dimensional Dirac spinors.Furthermore, we can define the spin density and current, n a ( t, x ) = ˆ d p d s (2 π ) S √ γs a f ( t, x , p , s ) , J a ( t, x ) = ˆ d p d s (2 π ) S √ γs a ˙ x f ( t, x , p , s ) . (48)Note that we have chosen two helicity states of the positive energy solution in Eq. (23) as the basis for the spin statesand the Berry potential in our formulation. Therefore the state | s i and the vector s a are defined in the helicity basis.We can also show the physical meaning of the spin density and current by a transformation to another spin statebasis used in Ref. [10, 11], U † p → U ′† p = V p U † p , (49)where V p = diag( R, ( σ · ˆ p ) R ) with R defined by R = (cid:18) e − iϕ cos θ sin θ − e iϕ sin θ cos θ (cid:19) , Here θ, ϕ are polar angles of p . Under the transformation of the spin state bases in (49), the spin state, the Berrypotential and Berry curvature transform as | s i → | s ′ i = V p | s i , A = − iU † p ∇ p U p → A ′ = V p A V † p − iV p ∇ p V † p ,σ a Ω a → σ a Ω ′ a = V p σ a Ω a V † p , (50)where the quantities with prime denote those in Ref. [10, 11]. The above formula show that the transformation issimilar to an ordinary gauge transformation in coordinate space. We can further prove that the Ω = s a Ω a is invariantunder such a gauge transformation, Ω → Ω ′ = Tr ( | s ′ i h s ′ | σ a Ω ′ a ) = Ω . (51)The fermion number (or vector current) conservation (43) still holds after the transformation. But the spin densityand current in Eq. (48) is gauge dependent. By this gauge transformation, our results including the equation ofmotions in (38) are consistent to Ref. [10, 11]. However, it is not obvious that the spin current defined in the basis ofRef. [10, 11] reproduces the axial current in the massless limit. The advantage of our spin current (48) in the helicitybasis is that it naturally recovers the axial current in the massless limit.We then derive the continuity equation for the spin current, ∂n a ∂t + ∇ x · J a = e ( E · B ) ˆ d p d s (2 π ) S ρ b s b s a f + ˆ d p d s (2 π ) S √ γ ˙ s a f, (52)see Appendix E for the details of the derivation. We see that there are two source terms in the continuity equation(52). The first term is vanishing for massive fermions. The second term is from the time derivative of the spin, whichcan be simplified by using Eqs. (34,38), S − ˆ d s √ γ ˙ s a = ǫ abc S − ˆ d s ( √ γ ˙ p · A b ) s c = 112 e ( E · B ) Ω c · A b ǫ abc S − ˆ d s . (53)If we focus on the a = 3 component, the continuity equation (52) becomes ∂n ∂t + ∇ x · J = 16 e ( E · B ) m ˆ d p d s (2 π ) S E p | p | f ( t, x , p , s ) , (54)Although the source term is proportional to m superficially, the above is non-vanishing in the massless limit becausethe integral is singular and behaves as /m .0Now let us look at the alternative way of taking the massless limit, i.e. we take the limit in Eq. (52), so only s isnon-vanishing as a constant of time. Then the second source term of the continuity equation (52) is vanishing. Thefirst source term leads to ∂n ∂t + ∇ x · J = e π ( E · B )[ f + ( t, x , p = ) + f − ( t, x , p = )] . (55)From Eq. (45) we have n = ( n + − n − ) / and J = ( J + − J − ) / . Including the anti-fermion distributions and usingthe replacement (47), we obtain in the massless limit, ∂n ∂t + ∇ x · J = e π ( E · B ) , (56)which is actually the continuity equation for the axial current with anomaly, and n and J are just the chiral densityand current respectively in this limit. This can be seen from the fact that the axial current can be derived fromEq. (48), namely, J = J / ξ B B / , where ξ B is the CME coefficient for the axial current in Eq. (23) of Ref.[70]. We note that a systematic way of including fermions and anti-fermions is to work in the full Dirac space with4-dimensional Dirac spinors. V. CONCLUSIONS
We have formulated a semi-classical kinetic description of Dirac fermions in background electromagnetic fields. Wehave shown that the non-Abelian Berry phase structure and the classical spin emerge in such a kinetic description. Wework in the path integral approach to derive the effective action for Dirac fermions of positive energy in electromagneticfields. We start from the Hamiltonian for the Dirac fermions in electromagnetic fields and calculate the transitionamplitude between the initial and final states of the spin and coordinate. The degenerate positive energy states withopposite helicities are chosen as the basis for spin states. The spin states enter the formalism and finally make thedynamical variables in the action. The phase space has to be enlarged by joining of the classical spin. The non-AbelianBerry potentials in momentum space appear in the action from diagonalization of the Hamiltonian. We also providesan alternative and much simpler approach to the effective action from the Dirac Lagrangian. We separate the fast andslow modes of the positive energy fermionic field and then integrate out the fast modes. The emerging non-AbelianBerry potentials in the effective Lagrangian are given by the fast mode spinor wave functions, while the emergingspins are determined by the slow mode wave functions.The equation of motions for Dirac fermions can be obtained from the effective action which involve electromagneticfields and non-Abelian Berry potentials and curvatures. Besides the equation of motions for x and p , the equationof motion for the spin precession, the Bargmann–Michel–Telegdi equation, can also be derived, whose time variationis controlled by the Berry potentials. We have observed a dual symmetry in the equation of motions for x and p byinterchanges ˙ x ↔ ˙ p , v p ↔ e E and Ω ↔ e B . Since the classical spin is conserved in absence of external fields andanomaly, we can also define a spin current. The continuity equations for the fermion number and the classical spin canbe derived from the equations of motions and the kinetic equation for distribution functions. Anomalous source termsproportional to E · B appear in continuity equations and involve integrals of Berry magnetic charges and the spin.The anomalous source term in the continuity equation for the fermion number is vanishing for massive fermions, whileit is present in the continuity equation for the spin current. We can reproduce the result of Ref. [10, 11] by a gaugetransformation of the spin basis. For massless fermions, the Berry phase becomes Abelian and the spin becomes thehelicity which is not a continuous phase space variable anymore. In this case, the fermion number is conserved whentaking anti-fermions into account, while the chiral charge is not conserved by the anomaly. The CME coefficientsfor the fermion and axial currents can be obtained after including the anti-fermion contributions, same as previousresults.QW thanks P. Horvathy for a helpful discussion about classical equations of motion with the Berry phase. Thiswork is supported by the NSFC under grant No. 11125524 and 11205150. JWC and SP are supported in part by theNSC, NTU-CTS, and the NTU-CASTS of R.O.C. Appendix A: Conventional path integral for chiral fermions
In this appendix, we will present the conventional path integral quantization of charged massless fermions inelectromagnetic. We will derive the action (12) with the Berry phase. The transition amplitude is given by (4). Wecan evaluate the amplitudes inside the parenthesis by inserting complete set of states. For the first amplitude in theparenthesis of Eq. (11) we evaluate as1 h x j | e − iH ∆ t | p cj i = ˆ " Y i =1 d x j ( i ) d p cj ( i ) ×h x j | U p c − e A | p cj ih p cj | e − i ( σ ǫ + eφ )∆ t | x j ih x j | U † p c − e A | p cj i×h p cj | U p c − e A | x j ih x j | e − i ( σ ǫ + eφ )∆ t | p cj ih p cj | U † p c − e A | x j i×h x j | U p c − e A | p cj ih p cj | e − i ( σ ǫ + eφ )∆ t | x j ih x j | U † p c − e A | p cj i = ˆ " Y i =1 d x j ( i ) d p cj ( i ) exp " i ∆ t X i =1 p cj ( i ) · x j ( i +1) − x j ( i ) ∆ t ! × U ( x j , p cj ) exp (cid:2) − i ∆ t ( σ ǫ + eφ )( x j , p cj ) (cid:3) U † ( x j , p cj ) U ( x j , p cj ) × exp (cid:2) − i ∆ t ( σ ǫ + eφ )( x j , p cj ) (cid:3) U † ( x j , p cj ) U ( x j , p cj ) × exp (cid:2) − i ∆ t ( σ ǫ + eφ )( x j , p cj ) (cid:3) U † ( x j , p cj ) exp( i p cj · x j ) , (A1)where we have used the notation x j ≡ x j . We can use the following formula to simplify the above equation, U † ( x j , p cj ) U ( x j , p cj ) ≈ exp (cid:2) − i A ( x j , p cj ) · ( p cj − p cj ) (cid:3) ,U † ( x j , p cj ) U ( x j , p cj ) ≈ exp (cid:2) i A ( x j , p cj ) · ( e A ( x j ) − e A ( x j )) (cid:3) , (A2)where A ( p ) ≡ − iU † p ∇ p U p are called Berry connection. So the amplitude in Eq. (A1) becomes h x j | e − iH ∆ t | p cj i = ˆ " Y i =1 d x j ( i ) d p cj ( i ) U ( x j , p cj ) exp " i ∆ t X i =1 p cj ( i ) · x j ( i +1) − x j ( i ) ∆ t ! × exp n − i ∆ t [( σ ǫ + eφ )( x j , p cj ) − A ( x j , p cj ) · ( e ˙ A ( x j ) − e ˙ A ( x j ))] o × exp (cid:8) − i ∆ t [( σ ǫ + eφ )( x j , p cj ) + A ( x j , p cj ) · ( ˙ p cj − ˙ p cj )] (cid:9) × exp (cid:2) − i ∆ t ( σ ǫ + eφ )( x j , p cj ) (cid:3) U † ( x j , p cj ) exp( i p cj · x j ) . (A3)Here we have used exp( i ∆ tCσ ) exp( i ∆ tC ′ A ) = exp (cid:26) i ∆ t ( Cσ + C ′ A ) −
12 (∆ t ) CC ′ [ σ , A ] (cid:27) ≈ exp [ i ∆ t ( Cσ + C ′ A )] (A4)where C and C ′ are constants. We will see that [ σ , A ] = 0 , so we have assumed the (∆ t ) term is much smaller thanthe ∆ t terms in Eq. (A4).For the second amplitude in the parenthesis of Eq. (11), we evaluate as h p cj | e − iH ∆ t | x j − i = ˆ " Y i =1 d x ′ j ( i ) d p ′ cj ( i ) ×h p cj | U p c − e A | x ′ j ih x ′ j | e − i ( σ ǫ + eφ )∆ t | p ′ cj ih p ′ cj | U † p c − e A | x ′ j i×h x ′ j | U p c − e A | p ′ cj ih p ′ cj | e − i ( σ ǫ + eφ )∆ t | x ′ j ih x ′ j | U † p c − e A | p ′ cj i×h p ′ cj | U p c − e A | x ′ j ih x ′ j | e − i ( σ ǫ + eφ )∆ t | p ′ cj ih p ′ cj | U † p c − e A | x j − i = ˆ " Y i =1 d x ′ j ( i ) d p ′ cj ( i ) exp( − i p cj · x ′ j ) U ( x ′ j , p cj ) × exp " i ∆ t X i =1 p ′ cj ( i ) · x ′ j ( i ) − x ′ j ( i − ∆ t ! × exp (cid:8) − i ∆ t [( σ ǫ + eφ )( x ′ j , p ′ cj ) + A ( x ′ j , p ′ cj ) · ( p ′ cj − p ′ cj ) / ∆ t ] (cid:9) × exp (cid:8) − i ∆ t [( σ ǫ + eφ )( x ′ j , p ′ cj ) − A ( x ′ j , p ′ cj ) · ( e A ( x ′ j ) − e A ( x ′ j )) / ∆ t ] (cid:9) × exp (cid:2) − i ∆ t ( σ ǫ + eφ )( x ′ j , p ′ cj ) (cid:3) U † ( x j − , p ′ cj ) (A5)2where we have denoted x ′ j (0) = x j − . We have also used (A4).We observe that U † ( x j , p cj ) exp( i p cj · x j ) of Eq. (A3) and exp( − i p cj · x ′ j ) U ( x ′ j , p cj ) of Eq. (A5) can be combinedas, h x j | e − iH ∆ t | p cj ih p cj | e − iH ∆ t | x j − i→ exp (cid:2) − i ∆ t ( σ ǫ + eφ )( x j , p cj ) (cid:3) × U † ( x j , p cj ) exp( i p cj · x j ) exp( − i p cj · x ′ j ) U ( x ′ j , p cj )= exp (cid:20) i ∆ t p cj · x j − x ′ j ∆ t (cid:21) × exp (cid:8) − i ∆ t [( σ ǫ + eφ )( x j , p cj ) − A ( x j , p cj ) · ( e A ( x j ) − e A ( x ′ j )) / ∆ t ] (cid:9) (A6)Finally taking the limit N → ∞ , we can write the amplitude (11) into a compact form, K fi = ˆ D x D p c U ( x f , p c f ) P exp (cid:26) i ˆ t f t i dt [ p c · ˙ x − σ ǫ ( p c − e A ) − eφ ( x ) − A ( p c − e A ) · ( ˙ p c − e ˙ A ) io U † ( x i , p c i )= ˆ D x D p U ( x f , p c f ) P exp (cid:26) i ˆ t f t i dt [ p · ˙ x + e A ( x ) · x − σ ǫ ( p ) − eφ ( x ) − A ( p ) · ˙ p ] } U † ( x i , p c i ) . (A7)We can read out the action (12) from above amplitude. Appendix B: Transition amplitude in path integral for Dirac fermions
In this appendix, we give the derivation of Eq. (25). We can insert the complete set of states as follows I j +1 ,j = h x j +1 , s j +1 | e − iH ∆ t | x j , s j i = h x j +1 , s j +1 | U † p e − iβE p ∆ t U p | x j , s j i = ˆ [ d p ′ ][ d ξ ′ ][ d x ′ ][ d ξ ′ ][ d p ′ ][ d ξ ′ ] ×h x j +1 , s j +1 | U † p | p ′ , s ′ i×h p ′ , s ′ | e − iβE p ∆ t | x ′ , s ′ ih x ′ , s ′ | U p | p ′ , s ′ ih p ′ , s ′ | x j , s j i . (B1)Note that coordinate and momentum states are decoupled from the spin states, i.e. | x , s i = | x i| s i and | p , s i = | p i| s i .Then we can combine the spin states and obtain I j +1 ,j = ˆ [ d p ′ ][ d ξ ′ ][ d x ′ ][ d ξ ′ ][ d p ′ ][ d ξ ′ ] × h x j +1 | p ′ i h s j +1 | U † p ′ | s ′ i h p ′ | x ′ i e − iβE p ′ ∆ t h s ′ | s ′ i× h x ′ | p ′ i h s ′ | U p ′ | s ′ i h s ′ | s j i h p ′ | x j i = ˆ [ d p ′ ][ d ξ ′ ][ d x ′ ][ d ξ ′ ][ d p ′ ][ d ξ ′ ] × exp [ i p ′ · ( x j +1 − x ′ )] exp [ i p ′ · ( x ′ − x j )] e − iβE p ′ ∆ t ×h s j +1 | U † p ′ | s ′ i h s ′ | s ′ i h s ′ | U p ′ | s ′ i h s ′ | s j i . (B2)Here we have used the fact that β is commutable with g ( because [ β, Σ ] = 0 ) in evaluating the amplitude of e − iβE p ∆ t ,so we have h p ′ , s ′ | e − iβE p ∆ t | x ′ , s ′ i = h p ′ | x ′ i e − iβE p ′ ∆ t h s ′ | s ′ i . [ d ξ ′ ][ d ξ ′ ][ d ξ ′ ] to remove intermediate spin states by using Eq. (18). Here weneglect constants from the integral ´ d ξ . Then we have I j +1 ,j ≈ ˆ [ d p ′ ][ d x ′ ][ d p ′ ] exp [ i p ′ · ( x j +1 − x ′ )] exp [ i p ′ · ( x ′ − x j )] × e − iβE p ′ ∆ t h s j +1 | U † p ′ U p ′ | s j i = ˆ [ d p ′ ][ d x ′ ][ d p ′ ] exp [ i p ′ · ( x j +1 − x ′ )] exp [ i p ′ · ( x ′ − x j )] × e − iβE p ′ ∆ t h λ j +1 | ( g j +1 ) − U † p ′ U p ′ g j | λ j i = ˆ [ d p ′ ][ d x ′ ][ d p ′ ] exp [ i p ′ · ( x j +1 − x ′ )] exp [ i p ′ · ( x ′ − x j )] × e − iβE p ′ ∆ t Tr[ λ j +1 ,j ( g j +1 ) − U † p ′ U p ′ g j ] (B3)where we have used Eq. (16) in the second equality and h λ j +1 | C | λ j i = Tr( λ j +1 ,j C ) in the last one. Appendix C: Action for massive fermions: separation of fast and slow modes
In this appendix, we try to derive the action (32) directly from the Lagrangian for Dirac fermions by separating thefast modes from the slow ones. We can rewrite the Lagrangian for massive fermions in the electromagnetic field as L = ¯ ψ [ iγ µ ( ∂ µ + ieA µ ) − m ] ψ = ψ † ( i∂ t − H ) ψ, (C1)where the Hamiltonian is given by H = α · [ − i ∇ − e A ] + mγ + eφ. (C2)The wave function for the positive energy can be written in the form ψ = X e = ± C e ( t ) e − iEt + i p · x u e ( p ) , (C3)where u e are the positive energy solutions given in Eq. (23). Here the phase factor and u e ( p ) correspond to fastmodes, while C e ( t ) describe slow modes. We assume that C e satisfy the normalization condition, | C + | + | C − | = 1 .We assume that p depends on t . Substituting the above into the Lagrangian (C1), we obtain L = X d,f = ± C ∗ d ( t ) u † d ( p )( i∂ t − ˙ p · x + e α · A − eφ ) C f ( t ) u f ( p )= X d,f = ± n C ∗ d ( t ) u † d ( p ) u f ( p )( i∂ t − ˙ p · x ) C f ( t )+ C ∗ d ( t ) C f ( t ) u † d ( p )( i∂ t ) u f ( p ) + C ∗ d ( t ) C f ( t ) u † d ( p )( e α · A − eφ ) u f ( p ) o = C † ( t )[ i∂ t − ˙ p · x + e v p · A − eφ − ˙ p · A ( p )] C ( t ) (C4)where we have used u † d ( p ) u f ( p ) = δ df , u † d ( p ) α u f ( p ) = v p δ df , ( α · p + mγ ) u d ( p ) = Eu d ( p ) and − ˙ p · x = i∂ t ( e i p · x ) ,we have also used the notation C ( t ) ≡ ( C + ( t ) , C − ( t )) T . We can further rewrite Eq. (C4) as L = Tr (cid:8) C ( t ) C † ( t )[ i∂ t − ˙ p · x + e v p · A − eφ − ˙ p · A ( p )] (cid:9) = Tr n C ( t ) C † λC C † ( t )[ i∂ t − ˙ p · x + e v p · A − eφ − ˙ p · A ( p )] o → Tr (cid:8) λg − [ i∂ t − ˙ p · A ( p )] g (cid:9) + p · ˙ x + e v p · A − eφ, (C5)where we have dropped in the last line the complete time derivative term d ( x · p ) /dt . We have inserted a constant C † λC = 1 between C ( t ) and C † ( t ) , where C ( t ) is an arbitrary normalized column vector with C † ( t ) C ( t ) = 1 and λ is an arbitrary matrix with trace 1. We have assumed g = C ( t ) C † and g † = C C † ( t ) , one can check that g is unitary,i.e. gg † = 1 . The Lagrangian (C5) gives the action (32).4 Appendix D: Derivation of equations of motion
In this appendix, we will derive the equations of motion (34) from the action (32). We treat the Lagrangian as thefunction of ( x , p , ξ , ˙ x , ˙ p , ˙ ξ ) . We will use the notation, for example, x i for the i -th component of the vector x . TheEuler-Lagrange equation for x is derived as ddt ∂L∂ ˙ x i = ˙ p i + e ∂A i ∂t + e ∂A i ∂x j ˙ x j ,∂L∂x i = e ∂A j ∂x i ˙ x j − e ∂A ∂x i , → ˙ p i = − e ∂A i ∂t − e ∂A ∂x i + e (cid:18) ∂A j ∂x i − ∂A i ∂x j (cid:19) ˙ x j , = eE i + eǫ ijk ˙ x j B k , (D1)which is the second line of Eq. (34). For the Euler-Lagrange equation for p , we obtain ddt ∂L∂ ˙ p i = − ds a dt A ai − s a ∂ A ai ∂p j ˙ p j ∂L∂p i = ˙ x i − ∂E p ∂p i − s a ∂ A aj ∂p i ˙ p j → ˙ x i = p i E p + s a ˙ p j (cid:18) ∂ A aj ∂p i − ∂ A ai ∂p j (cid:19) − ds a dt A ai . (D2)We will evaluate ds a /dt using the equation of motion for ξ . Since s = 1 / , there is no ds /dt = 0 term in the lastline of Eq. (D2). In order to derive the equation for ξ , we need to define ∂g ( ξ ) /∂ξ a properly. We define ξ ( θ ) as exp (cid:18) i θ a σ a (cid:19) g ( ξ ) = g [ ξ ( θ )] , ξ (0) = ξ , (D3)with a = 1 , , . Taking derivative on θ a and setting θ a = 0 we obtain i σ a g ( ξ ) = ∂g ( ξ ) ∂ξ b N ba ,G ba ≡ ∂ξ b ( θ ) ∂θ a (cid:12)(cid:12)(cid:12)(cid:12) θ =0 . (D4)One can prove det( G ) = 0 so the matrix N is invertible. The the Euler-Lagrange equation for ξ is derived as ddt ∂L∂ ˙ ξ a = i ddt Tr (cid:18) λg − ∂g∂ξ a (cid:19) = − G − ca ddt Tr (cid:16) λg − σ c g (cid:17) = − G − ca ds c dt ,∂L∂ξ a = i Tr (cid:18) λ ∂g − ∂ξ a ∂g∂ξ b ˙ ξ b (cid:19) + i Tr (cid:18) λg − ∂∂ξ a ∂g∂ξ b ˙ ξ b (cid:19) − ∂∂ξ a Tr (cid:16) λg − σ a g (cid:17) A a · ˙ p = iG − ca G − db Tr (cid:16) λg − h σ c , σ d i g (cid:17) ˙ ξ b + iG − ca Tr (cid:16) λg − h σ c , σ a i g (cid:17) A a · ˙ p = − G − ca ǫ cad ( A a · ˙ p ) s d , (D5)which leads to ds a dt = ǫ abc ( A b · ˙ p ) s c , (D6)5which is just the third line of Eq. (34). Note that s does not appear in Eq. (D6), so we have implied a, b, c = 1 , , .Substituting Eq. (D6) back into Eq. (D2), we obtain the last line of Eq. (34), ˙ x i = p i E p + s c ˙ p j (cid:18) ∂ A cj ∂p i − ∂ A ci ∂p j − ǫ abc A ai A bj (cid:19) + s ˙ p j ∂ A j ∂p i − ∂ A i ∂p j ! = p i E p + ǫ ijk ˙ p j Ω ck s c , (D7)where Ω c with c = 0 , , , are given by Eq. (35). Appendix E: Derivation of continuity equations
In this appendix, we will derive continuity equations (43,52) for the fermion number and the classical spin. We usethe notation, for example, x i for the i -th component of the vector x . To derive the continuity equation (43) for thefermion number, we start from taking the time derivative of n ( t, x ) in Eq. (42) and using Eq. (40), ∂n ( t, x ) ∂t = ˆ d p d s (2 π ) S ∂ √ γ∂t f + ˆ d p d s (2 π ) S √ γ ∂f∂t = ˆ d p d s (2 π ) S ∂ √ γ∂t f − ˆ d p d s (2 π ) S √ γ (cid:20) ˙ x i ∂f∂x i + ˙ p i ∂f∂p i + ˙ s a ∂f∂s a (cid:21) . (E1)where we have used d s ≡ d s δ ( s − s a s a ) and S = π . Using the partition formula for integrals, we can rewrite thesecond term into the following form I = − ˆ d p d s (2 π ) S √ γ (cid:20) ˙ x i ∂f∂x i + ˙ p i ∂f∂p i + ˙ s a ∂f∂s a (cid:21) = − ∂∂x i ˆ d p d s (2 π ) S √ γ ˙ x i f + ˆ d p d s (2 π ) S ∂ ( √ γ ˙ x i ) ∂x i f − ˆ d p d s (2 π ) S ∂ ( √ γ ˙ p i f ) ∂p i + ˆ d p d s (2 π ) S ∂ ( √ γ ˙ p i ) ∂p i f − ˆ d p d s (2 π ) S ∂ [ √ γδ ( s − s b s b ) ˙ s a f ] ∂s a + ˆ d p d s (2 π ) S δ ( s − s b s b ) ∂ ( √ γ ˙ s a ) ∂s a f = − ∂J i ∂x i + ˆ d p d s (2 π ) S ∂ ( √ γ ˙ x i ) ∂x i f + ˆ d p d s (2 π ) S ∂ ( √ γ ˙ p i ) ∂p i f + ˆ d p d s (2 π ) S ∂ ( √ γ ˙ s a ) ∂s a f, (E2)We have dropped the complete derivatives for the momentum and the classical spin whose integrals are vanishing.Note that in the fourth line, we have recovered δ ( s − s a s a ) because these terms are related to ∂/∂s a and should behandled with care, and one should pay special attention to the the second term: we have pulled the delta functionout of ∂/∂s a since the partial derivative of the delta function gives a s a which will combine with ˙ s a and vanishes.Applying the equation of motion (38), we obtain the following formula to further evaluate I , ∂ √ γ∂t + ∂ ˙ x i √ γ∂x i = e ˙ B · Ω + e ( ∇ x × E ) · Ω = 0 ,∂ ( √ γ ˙ p i ) ∂p i = e ( E · B ) ∇ p · Ω ,∂ ( √ γ ˙ s a ) ∂s a = ∂ ( √ γ ˙ p i ) ∂s a ǫ abc A bi s c = e ( E · B ) ǫ abc ( Ω a · A b ) s c , (E3)where we have used the Maxwell equations ∇ x · B = 0 and ∇ x × E + ˙ B = 0 . Finally we arrive at the continuityequation (43) for the fermion number from Eq. (E1).6Now we give the derivation of the continuity equation (52) for the classical spin. To this end we follow the sameprocedure by taking the time derivative of the spin density, ∂n a ( t, x ) ∂t = ˆ d p d s (2 π ) S ∂ √ γ∂t s a f + ˆ d p d s (2 π ) S √ γs a ∂f∂t = ˆ d p d s (2 π ) S ∂ √ γ∂t s a f − ˆ d p d s (2 π ) S √ γs a (cid:20) ˙ x i ∂f∂x i + ˙ p i ∂f∂p i + ˙ s b ∂f∂s b (cid:21) . (E4)The second term is evaluated as I = − ∂∂x i ˆ d p d s (2 π ) S √ γs a ˙ x i f + ˆ d p d s (2 π ) S ∂ ( √ γs a ˙ x i ) ∂x i f + ˆ d p d s (2 π ) S ∂ ( √ γs a ˙ p i ) ∂p i f + ˆ d p d s (2 π ) S ∂ ( √ γ ˙ s b ) ∂s b s a f + ˆ d p d s (2 π ) S √ γ ˙ s a f (E5)Using Eq. (E3) to further simplify I , we obtain the continuity equation (52) for the classical spin. [1] M. V. Berry, Proc.Roy.Soc.Lond. A392 , 45 (1984).[2] F. Wilczek and A. Zee, Phys.Rev.Lett. , 2111 (1984).[3] J. Moody, A. D. Shapere, and F. Wilczek, Phys.Rev.Lett. , 893 (1986).[4] H. Lee, M. A. Nowak, M. Rho, and I. Zahed, Annals Phys. , 175 (1993), hep-ph/9301242.[5] D. Xiao, M.-C. Chang, and Q. Niu, Rev.Mod.Phys. , 1959 (2010), 0907.2021.[6] D. T. Son and N. Yamamoto, Phys.Rev.Lett. , 181602 (2012), 1203.2697.[7] M. Stephanov and Y. Yin, Phys.Rev.Lett. , 162001 (2012), 1207.0747.[8] J.-W. Chen, S. Pu, Q. Wang, and X.-N. Wang, Phys.Rev.Lett. , 262301 (2013), 1210.8312.[9] D. T. Son and N. Yamamoto, Phys.Rev. D87 , 085016 (2013), 1210.8158.[10] M.-C. Chang and Q. Niu, J. Phys.: Condens. Matter , 193202 (2008).[11] C.-P. Chuu, M.-C. Chang, and Q. Niu, Solid State Communication , 533 (2010).[12] A. Balachandran, P. Salomonson, B.-S. Skagerstam, and J.-O. Winnberg, Phys.Rev. D15 , 2308 (1977).[13] A. Balachandran, S. Borchardt, and A. Stern, Phys.Rev.
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