Topological Vortices in Chiral Gauge Theory of Graphene
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J un Topological Vortices in Chiral Gauge Theory ofGraphene
Xin Liu ∗ and Ruibin ZhangSchool of Mathematics and Statistics, University of SydneyNSW 2006, Australia Abstract
Generation mechanism of energy gaps between conductance and valence bandsis at the centre of the study of graphene material. Recently Chamon, Jackiw, etal. proposed a mechanism of using a Kekul´e distortion background field ϕ and itsinduced gauge potential A i to generate energy gaps. In this paper various vortexstructures inhering in this model are studied. Regarding ϕ as a generic backgroundfield rather than a fixed Nielson-Oleson type distribution, we have found two newtypes of vortices on the graphene surface — the velocity field vortices and themonopole-motion induced vortices — from the inner structure of the potential A i .These vortex structures naturally arise from the motion of the Dirac fermions insteadof from the background distortion field.PACS number(s): 71.10.-w, 11.15.-q, 02.40.HwKeyword(s): Graphene; Energy Gaps; Topological Vortex Structures. ∗ Author to whom correspondence should be addressed. Electronic address: [email protected]. Introduction
Graphene is a lately realized single-atomically thin 2-dimensional sheet of carbon. Its non-Bravais honeycomb lattice arrangement of atoms gives rise to a new many-body problemthat single-particle quantum mechanics is relativistic, but interactions are non-relativisticinstantaneous Coulomb interactions. Ever since the breakthrough of its manufacturingtechnique in 2004 [1] graphene has attracted enormous interest because of the funda-mental new physics exhibited and the highly unusual electronic, mechanical and opticalproperties. These properties lead to numerous potential application and make graphenea new material of growing technological importance [2].Graphene has a remarkable electronic property of ballistic transport at room tem-perature under electrical and chemical doping. This makes graphene a candidate to bethe next dominant semiconductor material in electronics after silicon. However, a majorobstacle for this application is that the energy spectrum of monolayer graphene remainsmetallic at the Dirac points because of the relativistic linear dispersion relation, i.e., thevanishing gap between the conductance and valence bands in the energy spectrum. Manymethods have been developed for generating a non-zero gap, such as spatial confinementand lateral super-lattice potentials. In this paper we mainly consider the generation mech-anism proposed by Chamon, et al. [3, 4, 5, 6, 7, 8, 9] and aim to find new topologicalvortex structures originating from the singular configurations of the Dirac fermion field.The Hamiltonian for the honeycomb lattice of monolayer graphene is given by H = − t X x ∈ A X i =1 (cid:2) a † ( x ) b ( x + s i ) + b † ( x + s i ) a ( x ) (cid:3) , (1)where a and b are respectively the fermion operators acting on the sublattice A and B ,and t the uniform hopping strength. In the momentum space Eq.(1) is written as H = X k (cid:2) τ ( k ) a † ( k ) b ( k ) + τ ∗ ( k ) b † ( k ) a ( k ) (cid:3) , (2)where ( a ( k ) b ( k ) ) = P x e − i k · x ( a ( x ) b ( x ) ) and τ ( k ) = − t P i =1 e i k · s i . The single-particleenergy spectrum contains two Dirac points K ± = ± (cid:16) π √ ℓ , (cid:17) at the zone boundaries, ℓ being the length of lattice, satisfying τ ( K ± ) = 0. In the neighborhood of K ± , k is writtenas k = K ± + p and H is linearized as H = X p h τ + ( p ) a † + ( p ) b + ( p ) + τ ∗ + ( p ) b † + ( p ) a + ( p )+ τ − ( p ) a †− ( p ) b − ( p ) + τ ∗− ( p ) b †− ( p ) a − ( p ) i , (3)2here a ± ( p ) = a ( K ± + p ), b ± ( p ) = b ( K ± + p ) and τ ± ( p ) = τ ( K ± + p ) = ± v F ( p x ± ip y ), v F being the Fermi velocity, v F = 1. H has the following matrix form: H = Ψ † α i p i Ψ , i = 1 , . (4)Here Ψ ( x ) is a Dirac 4-spinor, Ψ ( x ) = (cid:16) ψ b + ( x ) , ψ a + ( x ) , ψ a − ( x ) , ψ b − ( x ) (cid:17) T , with ( a ± ( p ) b ± ( p ) ) = R d x ′ e i p · x ′ ( ψ a ± ( x ′ ) ψ b ± ( x ′ ) ) . The α i , i = 1 , , is given by α i = γ γ i , accompa-nied by the definition of the Dirac matrices in quantum electrodynamics: γ k = − σ k σ k ! , k = 1 , , γ = β = II ! ; γ = iγ γ γ γ = I − I ! , σ k the Pauli matrices. The γ -matrices are Clifford algebraic 1-vectors, satisfying γ a γ b + γ b γ a = 2 δ ab I , a, b = 1 , , , γ a ∂ a Ψ = 0 , a = 1 , , , (5)which is recognized to be the massless Dirac equation in (2 + 1)-dimensions. The energydispersion of (5) is a linear relation: ε ( p ) = ± | p | , which has vanishing gaps between theconductance and valence bands in the energy spectrum.For the purpose of generating energy gaps, Chamon, Hou and Mudry [4, 3] introducedto the monolayer graphene a complex scalar background field ∆ ( r ). The ∆ ( r ) describesa Kekul´e distortion of the lattice and leads to a small variation δt r ,i over the unit hoppingstrength t : δt r ,i = − (cid:2) ∆ ( r ) e i K + · s i e i G · r + ∆ ∗ ( r ) e − i K + · s i e − i G · r (cid:3) , G = K + − K − , (6) δt r ,i yielding chiral mixing between K + and K − . With (6) the Hamiltonian (1) is modifiedas H = − X r ∈ A X i =1 ( t + δt r ,i ) (cid:2) a † ( r ) b ( r + s i ) + b † ( r + s i ) a ( r ) (cid:3) , (7)and the matrix form (4) becomes H = Ψ † K Ψ = Ψ † [ α i p i + gβ ( ϕ Re − iγ ϕ Im )] Ψ , i = 1 , , (8)where the kernel K is K = − i∂ z ∆ ( r ) 0 − i∂ ¯ z r )∆ ∗ ( r ) 0 0 2 i∂ z ∗ ( r ) 2 i∂ ¯ z . (9)3n (8) one denotes ∆ ( r ) = gϕ , where g is a coupling strength describing the dimension of∆ ( r ), and ϕ = ϕ Re + iϕ Im a complex scalar, ϕ Re and ϕ Im being the real and imaginary partsrespectively. ϕ can alternatively be expressed via its modulus and phase as ϕ = | ϕ | e iχ .The energy level equation derived from (8) reads[ α i p i + gβ ( ϕ Re − iγ ϕ Im )] Ψ = E Ψ . (10)It is pointed out [4, 5] that χ can be removed by a gauge transformation of ϕ : χ → χ ′ = χ + 2 ω, Ψ → Ψ ′ = e iωγ Ψ . (11)Thus (8) yields ( γ a ∂ a + ig | ϕ | ) Ψ ′ = 0. This gives rise to a mass in the dispersion relation, ε ( p ) = ± q p + | ∆ | , and hence generates a desired energy gap between the conductanceand valence bands.For the purpose of keeping the kinetic portion of (10) invariant against the localgauge transformation (11), Jackiw and Pi [5] extended (10) by introducing a U (1) gaugepotential A i : [ α i ( p i − γ A i ) + gβ ( ϕ Re − iγ ϕ Im )] Ψ = E Ψ , i = 1 , , (12)where A i transforms as A i → A i + ∂ i ω under (11). (In (12) the chemical potential is notconsidered [6].) This new model (12) is an equation about the Dirac fermion wavefunctionΨ and the background field ϕ and its induced gauge potential A i . In [5] Ψ is solved outby fixing A i and ϕ within the Nielsen-Olesen (N-O) vortex configuration. Very recentdevelopments based on the above model include the study of the superconducting vorticesby P. Ghaemi, S. Ryu and D.H. Lee [7] and P. Ghaemi and F. Wilczek [8], and the studyof the fractionally charged vortices by J.K. Pachos, M. Stone and K. Temme [9].In this paper we will present another analysis for ϕ and A i . Firstly, our starting pointis that the background field ϕ permits more generic distributions on the graphene surfacerather than a fixed N-O type, since ϕ is an externally given distortion field. Correspond-ingly, we regard A i and Ψ as two unknown quantities that can be determined by thegoverning equation (12) together with supplementary restraints, where the restraints arenaturally provided by the subsequent study of topological vortices. Secondly, two othertypes of topological vortex structures can be derived from the physical equation (12) —the velocity field vortices and the monopole-motion induced vortices. These vortices arisefrom the singular configurations of the spinor field Ψ instead of from the background ϕ .Thirdly, moreover, it will be shown that the potential A i contains a term which describesthe direct coupling of ϕ and Ψ fields and can be expressed formally by an SO (4) gaugepotential.This paper is arranged as follows. In Sect.2 the inner structure of the gauge potential A i in terms of the Dirac fermion spinor wavefunction Ψ is studied. In Sect.3 two types of4opological vortex structures in the graphene system are studied. In Sect.4. the paper issummarized. In following it is shown that introducing the A i in (12) implies formally introducing one SO (4) potential and two U (1) potentials.Consider the zero-energy level of the model (12): γ i ( ∂ i − iγ A i ) Ψ + g ( iϕ Re + γ ϕ Im ) Ψ = 0 , i = 1 , . (13)Left-multiplying (13) with Ψ † γ j γ there isΨ † γ j γ γ i ∂ i Ψ + iA i Ψ † γ j γ i Ψ − igϕ Re Ψ † γ γ j Ψ + gϕ Im Ψ † γ j Ψ = 0 . (14)(14) minus its complex conjugate gives a decomposition expression for A i in terms of thespinor wavefunction Ψ [11, 12]: A i = A (1) i + A (2) i + A (3) i , (15)where A (1) i ( x ) = g (cid:20) ϕ Re Ψ † γ γ i ΨΨ † Ψ + iϕ Im Ψ † γ i ΨΨ † Ψ (cid:21) , (16) A (2) i ( x ) = (cid:2) Ψ † γ ∂ i Ψ − ∂ i Ψ † γ Ψ (cid:3) i Ψ † Ψ , (17) A (3) i ( x ) = ∂ j (cid:2) Ψ † γ I ji Ψ (cid:3) Ψ † Ψ , (18) I ij being the SO (4) generator, I ij = i [ γ i , γ j ] . Eq.(15) is a sufficient but not necessarycondition for the original equation (13). In this section and the next the three terms A (1) i , A (2) i and A (3) i will be investigated respectively for their topological and gauge fieldtheoretical essence. A (1) i contains the coupling between Ψ and the background distortion field ϕ . Thediscussion is twofold:Firstly, A (1) i can be formally expressed by an SO (4) gauge potential. Indeed, singlingout the A (1) i part in (13) leads to γ i ∂ i Ψ − iγ i γ A (1) i Ψ + igϕ Re Ψ + gϕ Im γ Ψ + iγ γ i (cid:16) A (2) i + A (3) i (cid:17) Ψ = 0 , i = 1 , , (19)which can be compared to the following equation γ i ∂ i Ψ − γ a ω a Ψ + igϕ Re Ψ + gϕ Im γ Ψ + iγ γ i (cid:16) A (2) i + A (3) i (cid:17) Ψ = 0 , a = 1 , , , . (20)5ere ω a = ω abc I bc is an SO (4) potential, ω abc being anti-symmetric for the b c indices. Our aim is to express (19) by (20), hence we write ω abc in three parts: ω abc = ω Aabc + ω S abc + ω S abc , where ω Aabc is fully anti-symmetric for a b c , ω S abc symmetricfor a b , and ω S abc symmetric for a c : ω Aabc = ( ω abc + ω bca + ω cab ), ω S abc = ( ω abc + ω bac ), ω S abc = ( ω abc + ω cba ). Defining ¯ ω b = 2 ω aba , ˜ ω a = ǫ abcd ω Abcd , one obtains after simple Clif-ford algebra: ω Aabc γ a γ b γ c = iγ a γ ˜ ω a , ω S abc γ a γ b γ c = − ¯ ω c γ c , ω S abc γ a γ b γ c = − ¯ ω b γ b . Then(20) becomes γ i ∂ i Ψ − γ a ( iγ ˜ ω a − ¯ ω a ) Ψ + igϕ Re Ψ + gϕ Im γ Ψ + iγ γ i (cid:16) A (2) i + A (3) i (cid:17) Ψ = 0 . (21)Thus comparing (21) with (19) one arrives at a conclusion that if the evaluation¯ ω a = 0; ˜ ω i = 4 A (1) i , i = 1 ,
2; ˜ ω = ˜ ω = 0 (22)is taken, the 2-component potential A (1) i , i = 1 , , can be formally expressed by the 4-component SO (4) potential ω a , a = 1 , , , , via the expression (20). This is the gaugefield theoretical essence of the potential A (1) i . It is addressed in addition that since A (1) i is real and γ i anti-Hermitian, the SO (4) potential ω a is required to be Hermitian, whichis different from ordinary anti-Hermitian gauge potentials in physics. For the discussionof Hermitian gauge potentials see [13].Secondly, A i is the induced gauge potential of the background distortion ϕ , and A (1) i the only term of A i which explicitly contains ϕ . Our second observation for A (1) i is that ϕ enters the potential A i by coupling to the Clifford algebraic odd vectors [10]. Go backto (13) and write the A (1) i part explicitly γ i ∂ i Ψ + igϕ Re Ψ † γ γ i ΨΨ † Ψ γ γ i Ψ + gϕ Im Ψ † γ i ΨΨ † Ψ γ i γ Ψ+ iγ γ i (cid:16) A (2) i + A (3) i (cid:17) Ψ + igϕ Re Ψ + gϕ Im γ Ψ = 0 , i = 1 , . (23)Keep in mind that γ i is a Clifford algebraic 1-vector and γ γ i a 3-vector (or pseudo 1-vector). Then introducing the projection operator ˆ P = ΨΨ † Ψ † Ψ , Eq.(23) reads γ i ∂ i Ψ+ g (cid:16) iϕ Re ˆ P γ γ i + ϕ Im ˆ P γ i γ (cid:17) Ψ+ iγ γ i (cid:16) A (2) i + A (3) i (cid:17) Ψ+ igϕ Re Ψ+ gϕ Im γ Ψ = 0 . (24)Here ˆ P γ γ i = T r (cid:16) ˆ P γ γ i (cid:17) γ γ i and ˆ P γ i = T r (cid:16) ˆ P γ i (cid:17) γ i are the Clifford vectorial componentsof ˆ P respectively along the 3-vector γ γ i and the 1-vector γ i . Hence it is concluded from(24) that ϕ enters the potential A i by coupling to the Clifford odd vectors.Moreover, it can be checked that A (1) i and A (3) i are invariant under the gauge transfor-mations (11) of ϕ , whereas A (2) i undergoes a U (1)-type gauge potential transformation: A (1) i → A (1) ′ i = A (1) i , A (3) i → A (3) ′ i = A (3) i ; A (2) i → A (2) ′ i = A (2) i + ∂ i ω. Topological Vortex Structures
In this section the A (2) i and A (3) i terms of (15) will be investigated. It will be shown thatthey lead to different types of topological vortices on the graphene surface. Vortices Arising from A (2) i With respect to the chiral γ matrix one considers the self-dual and anti-self-dual partof Ψ separately: Ψ = Ψ + + Ψ − , with Ψ ± = (1 ± γ ) Ψ satisfying γ Ψ ± = ± Ψ ± . Theexplicit representation of Ψ ± is: Ψ + = (cid:16) Φ T + (cid:17) T , Ψ − = (cid:16) T − (cid:17) T , where Φ + ( x )and Φ − ( x ) are two Pauli 2-spinors and have their respective SU (2) sub-group spaces(hereafter marked as SU (2) ± subspaces). Thus (17) reads A (2) i = cos θ Φ † + ∂ i Φ + − ∂ i Φ † + Φ + i Φ † + Φ + − sin θ Φ †− ∂ i Φ − − ∂ i Φ †− Φ − i Φ †− Φ − , (25)where cos θ = Ψ † + Ψ + Ψ † Ψ and sin θ = Ψ †− Ψ − Ψ † Ψ , θ ( x ) called the duality rotation [11].As mentioned in the introductory section, in the present paper the gauge potential A i and the spinor wave function Ψ are regarded as two unknowns determined by the governingequation (12), with the background field ϕ being a generic externally given distortion forthe graphene system. (12) provides 8 restraints but A i and Ψ have totally 10 unknowncomponents. Hence for determining the unknowns one needs two extra supplementaryrestraints. In this regard one employs the following normalization conditions of Ψ to playthe role of the required supplementary restraints:sin θ = cos θ = 12 , Ψ † Ψ = 1 . (26)Eq.(26) means that there is no priority between the self- and anti-self-dual subspaces,hence (26) is a natural choice. Thus A (2) i = 12 X ± ± A (2) i ± , A (2) i ± = 1 i h Φ †± ∂ i Φ ± − ∂ i Φ †± Φ ± i . (27)The bilinear form A (2) i + ( x ) [resp. A (2) i − ( x )] is a U (1) gauge potential in the SU (2) + [resp. SU (2) − ] subspace, and has the physical meaning of the velocity field of spinning quantumfluid, up to a mass constant.Topologically singular property of the configuration of Φ ± is studied by consideringthe first Chern class, a topological characteristic class on the 2-dimensional graphenesurface [14]. The first Chern class constructed from A (2) i ± is C (2)1 ± = π f (2) ij ± dx i ∧ dx j , where f (2) ij ± is the gauge field strength defined as f (2) ij ± = ∂ i A (2) j ± − ∂ j A (2) i ± . The physical meaningof C (2)1 ± is the vorticity of the velocity field distributed on the graphene surface. It is7nown in hydrodynamics that non-vanishing vorticity possesses vortices; this fact can begeometrically shown here for C (2)1 ± . Indeed, a spinning quantum fluid such as the Helium-3superfluid has the so-called Mermin-Ho relation [15] ∂ i A (2) j ± − ∂ j A (2) i ± = 12 ǫ abc n a ± ∂ i n b ± ∂ j n c ± d x, (28)where n a ± , a = 1 , ,
3, denote the spin unit vectors respectively in the SU (2) ± subspaces, n a ± ( x ) = Φ †± σ a Φ ± Φ †± Φ ± . From the geometric point of view, the RHS of (28) is the pullback ofa surface element on the hypersphere S ± , where S ± is formed by n a ± in the group spaceof SU (2) ± . The S ± tangentially intersects the graphene surface at the studied point x = ( x , x ), hence n a ± is locally perpendicular to the graphene surface at x . Meanwhile,it is known that the S ± surface element can be geometrically given by a U (1)-type gaugepotential w ± as ǫ abc n a ± dn b ± dn c ± = dw ± . (29)Here w ± is defined as w ± = ~e ± · d~e ± , where ~e and ~e [resp. ~e − and ~e − ] are a pair of 2-dimensional unit vectors normal to n a + [resp. n a − ] on S [resp. S − ]. Namely, ( ~e , ~e , ~n + )[resp. ( ~e − , ~e − , ~n − )] forms an orthogonal frame: ~e ± · ~e ± = ~e ± · ~n ± = ~e ± · ~n ± = 0, ~e ± · ~e ± = ~e ± · ~e ± = ~n ± · ~n ± = 1. The w ± has the physical meaning of the so-calledWu-Yang potential [16]. Comparing (28) and (29) one knows that locally A (2) i ± = w i ± = ~e ± · ∂ i ~e ± (30)up to a removable phase angle. Hence in following one can use (30) to investigate thetopological structures arising from A (2) i ± .For convenience we use one unique 2-component vector to replace the pair ( ~e ± , ~e ± ).This vector, denoted as ~ξ ± = ( ξ ± , ξ ± ), is required to reside in the plane spanned by ~e ± and ~e ± and satisfy e A ± = ξ A ± k ξ ± k , e A ± = ǫ AB ξ B ± k ξ ± k , with k ξ ± k = ξ A ± ξ A ± and A, B = 1 ,
2. Thezero points of ~ξ ± are the singular points of ~e ± and ~e ± . Then in terms of ~ξ ± one rewrites A (2) i ± = ǫ AB ξ A ± k ξ ± k ∂ i ξ B ± k ξ ± k , and then obtains C (2)1 ± = π ǫ ij ǫ AB ∂ i ξ A ± k ξ ± k ∂ j ξ B ± k ξ ± k d x . According to [17]it can be proved that C (2)1 ± = δ (cid:16) ~ξ ± (cid:17) D ( ξ ± x ) d x, (31)where D ( ξ ± /x ) = ǫ ij ǫ AB ∂ i ξ A ± ∂ j ξ B ± is the Jacobian determinant. Eq.(31) shows that non-vanishing C (2)1 ± occurs only at the zero-points of ~ξ ± . Hence in order to find the expectedtopological vortices on the graphene surface one should study the zero point equationof ~ξ ± : ξ A ( x ) = 0, A = 1 ,
2. The implicit function theory [18] declares that under theregular condition D ( ξ ± /x ) = 0 the general solutions of the zero point equations are afinite number of 2-dimensional isolated points: x i ± = x ik ± , k ± = 1 , , ..., L ± , (32)8here L ± denotes the number of the isolated points. Thus these points are the predictedtopological singularities in the configuration of the Φ ± field on the graphene surface. Inthe scenario of hydrodynamics they are called the velocity field vortices, the topologicalvortices arising from non-vanishing vorticity.The δ (cid:16) ~ξ ± (cid:17) in (31) can be expanded onto these vortex points as δ (cid:16) ~ξ ± (cid:17) = P L ± k ± =1 W k ± δ (cid:0) ~x − ~x k ± (cid:1) , where W k ± is the winding number of the k ± -th zero-point, playing the role of the topological charge of that zero-point. Therefore the firstChern number given by the Chern class C (2)1 ± is c (2)1 ± = Z C (2)1 ± = L ± X k ± =1 W k ± . (33) c (2)1 ± has the physical meaning of quantized vorticity. Vortices Arising from A (3) i Another kind of vortex structures originate from the third term of Eq.(15), A (3) i = ∂ j (cid:2) Ψ † γ I ji Ψ (cid:3) .Defining the dual tensor of I ij as ∗ I ij = γ I ij = ǫ ijkl I kl ( i, j = 1 , k, l = 3 , A (3) i = ∂ j (cid:2) Ψ † ∗ I ji Ψ (cid:3) . (34)The RHS of (34) can be expressed by a Maxwell-type U (1) field tensor. Indeed, accordingto [12] a Maxwell-type U (1) electromagnetic field strength F µν has a Dirac spinor repre-sentation F µν = Ψ † I µν Ψ , µ, ν = 1 , , , , provided that Ψ is non-singular. Therefore, for(34) one can extend the studied base space from the 2-dimensional to the 4-dimensional,and consider only the weak solutions for Ψ. Here the weak solutions refer to the case thatthe Ψ field has only a countable number of isolated singular points and is well-definedalmost everywhere except at the singular points. Then the RHS of (34) can be expressedby a dual electromagnetic field strength as A (3) i = ∂ j ∗ F ji , where ∗ F ij = ǫ ijkl F kl is thedual tensor of F ij .The second Maxwell equation reads ∂ j ∗ F ji = − π ∗ J i , where ∗ J i is the current ofmonopoles. When ∗ J i = 0 , namely there is no existence of monopole-type excitations inthe system, there is ∂ j ∗ F ji = 0 , which corresponds to the Bianchi identity. In this case, A (3) i = 0; When ∗ J i = 0, A (3) i = − π ∗ J i . One needs to consider the vortex structuresarising from the motion of monopoles on the 2-dimensional graphene surface.There are various models for magnetic monopoles in literature [14, 19], dependingon different choices for the symmetry of the monopole field or the gauge potential. Infollowing we simply consider the complex scalar field model for the monopoles, and letthe monopole current and density take the following form ∗ J i = 12 iψ ∗ ψ ( ψ ∗ ∂ i ψ − ∂ i ψ ∗ ψ ) , i = 1 , ρ = ψ ∗ ψ, (35)9here ψ is a complex scalar, ψ = φ + iφ . (35) leads to ∗ J i = ǫ αβ (cid:16) φ α | φ | (cid:17) ∂ i (cid:16) φ β | φ | (cid:17) with | φ | = φ α φ α , α, β = 1 ,
2. The first Chern class induced by A (3) i is C (3)1 = ǫ αβ ǫ ij ∂ i (cid:16) φ α | φ | (cid:17) ∂ j (cid:16) φ β | φ | (cid:17) . Since only the case of weak solutions is taken for Ψ, C (3)1 can be expressed in the δ -functionform as C (3)1 = δ (cid:16) ~φ (cid:17) D (cid:0) φx (cid:1) , similarly to the last subsection. Apparently non-vanishing C (3)1 occurs only at the zero-points of ~φ , so the zero point equation φ α ( x ) = 0, α = 1 , D ( φ/x ) = 0 the general solu-tions of the zero-point equations are a finite number of 2-dimensional isolated points: x i = x iℓ , ℓ = 1 , , ..., N, (36)where N denotes the number of the isolated points. These points are topological singular-ities in the configuration of the φ field on the graphene surface; they are vortex structuresdue to the motion of monopoles. Furthermore, the δ (cid:16) ~φ (cid:17) can be expanded onto the vor-tex points as δ (cid:16) ~φ (cid:17) = P Nℓ =1 W ℓ δ ( ~x − ~x ℓ ), where W ℓ is the winding number of the ℓ -thzero-point, playing the role of the topological charge of that zero-point. Correspondinglythe first Chern number given by the Chern class C (3)1 is c (3)1 = Z C (3)1 = N X ℓ =1 W ℓ . (37) The massive Dirac equation (13) proposed in Ref.[5] is an important mechanism for gener-ating energy gaps between conductance and valence bands of graphene material. We startfrom this equation and proposed that the distortion field ϕ is permitted to take a genericdistribution rather than a fixed Nielson-Oleson (N-O) configuration on the graphene sur-face. The ϕ -induced gauge potential A i and the Dirac fermion wavefunction Ψ are de-termined by (13) and the normalization condition (26). In this paper we focus on theinner spinor structure of A i and study its three terms A (1) i , A (2) i and A (3) i . Emphasis ison two new types of topological vortices arising from A (2) i and A (3) i — the velocity fieldvortices and the monopole-motion induced vortices. These vortices are different from theN-O vortices, as they both originate from the singularities of the wavefunction Ψ insteadof from the background field ϕ . Moreover it is shown that the A (1) i term can be formallyexpressed via an SO (4) gauge potential ω a . The author X.L. is indebted to Profs. J. Keller and W.A. Rodrigues Jr. for the discus-sion on the equivalence between the Maxwell and Dirac equations. X.L. was financially10upported by the USYD Postdoctoral Fellowship of the University of Sydney. R.Z. wasfinancially supported by the Australian Research Council.
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