Vortex description of quantum Hall ferromagnets
aa r X i v : . [ c ond - m a t . s t r- e l ] N ov Vortex Description of Quantum HallFerromagnets
Taro Kimura ∗ Department of Basic Science, University of Tokyo,Meguro-ku, Komaba, Tokyo 153-8902, Japan
Abstract
We study particle states of quantum Hall ferromagnet from the viewpoint of theincompressible fluid description. It is shown that phase space of Chern-Simons matrixtheory which is an effective theory for the incompressible fluid is equivalent to modulispace of vortex theory. According to this correspondence, elementary excitations invortex theory are identified as particle states in quantum Hall ferromagnet, and thuswe propose that a pure electron state is absent from the strong coupling region butonly a composite particle state is present. ∗ E-mail: [email protected] Introduction
The quantum Hall effect is one of the most remarkable phenomena in condensed matterphysics, and gives a rich mathematical structure[1]. It is well known that electrons in a lowenergy region behave as incompressible fluid. An important property of the incompressiblefluid is that it possesses no dynamical degree of freedom and the residual degree of free-dom comes from geometry of the fluid, which is related to area preserving diffeomorphism.Thus Chern-Simons theory which is also non-dynamical theory captures the feature of theincompressible fluid. Indeed one can derive Chern-Simons action by integrating out fermionmodes[2]. This situation is similar to topological string theory omitting a string fluctuation,but mainly treating its topological structure.Then we remark that the relation between incompressibility and noncommutativity. Themost primitive example of the noncommutativity is the canonical commutation relation[ x, p ] = i ~ . According to this noncommutativity, quantum mechanical phase space becomesfuzzy, and a quantum state covers an area ∼ ~ , which is called a quantum droplet, and ispreserved while its shape is transformed. On the other hand, a classical mechanical state isindicated by a point on classical phase space. In the case of the magnetic system, a momen-tum ~p = m ˙ ~x − ~A includes a coordinate component via the vector potential ~A . Therefore thecoordinate space, which is the phase space itself, becomes noncommutative and a state ofparticles is interpreted as the incompressible fluid. In general, the noncommutativity of thespace-time is induced by the effect of background fields[3].To manifest the noncommutativity of the quantum Hall state, the noncommutative ana-logue of Chern-Simons theory was proposed as the effective theory of the incompressiblefluid[4]. Since the canonical commutation relation can be realized by infinite dimensionalmatrices, the corresponding system is infinitely extended without the boundary. Then theregularized finite model, the Chern-Simons matrix model was also presented[5, 6], in whichthe commutation relation is modified by the boundary effect. On the other hand, the samerelation was discovered in the context of vortex theory, in particular the usual quantum Hallstate corresponds to the Abelian vortex state[7]. As a result, the phase space of the quantumHall state turns out to be identified with the moduli space of the vortex theory[8].In this paper, we investigate non-Abelian generalizations of the correspondence betweenthe quantum Hall state and the vortex theory. When two dimensional layers are stacked,we can consider the internal degree of freedom labeling the layers. It is well known as thepseudo spin. The quantum Hall state with the spin or pseudo spin degree of freedom iscalled a quantum Hall ferromagnet. The enhanced SU (2) symmetry is decomposed to theelectron part U (1) and the spin part SU (2) /U (1) = C P , and this is interpreted as thespin-charge separation. Hence a spin wave as the Nambu-Goldstone mode is induced bythe spontaneous symmetry breaking, and we can observe a skyrmion which is a topologicalexcitation characterized by a non-trivial homotopy class π [ SU (2) /U (1)] = Z .The C P space is also obtained as the internal space of the non-Abelian vortex. Thus,identifying these internal spaces, we can investigate the quantum Hall ferromagnet by thevortex theory and apply the C P valued field theory, especially N = (2 ,
2) supersymmetric C P model[9, 10] to vortex world-sheet theory[11]. This model has been studied in thecontext of the mirror symmetry[12] and also applied to superconductivity[13].The structure of this paper is the following. In section 2, we review the relationship2etween the noncommutativity and the incompressibility of the quantum Hall state and theeffective theory of the incompressible fluid for the finite system. In section 3, it will beshown that the regularized commutation relation of the incompressible fluid is also obtainedfrom vortex theory and the moduli space of vortices is identified with the phase space ofthe incompressible fluid. Due to the internal symmetry of the vortex, we introduce N =(2 ,
2) supersymmetric C P theory describing internal particle state of the quantum Hallferromagnet. In section 4, particle states of the quantum Hall ferromagnet are investigatedby the vortex theory. In the C P theory, there exist two phases, which are strong andweak coupling phase, and they are separated by the curve of marginal stability. The strongcoupling phase is considered as the quantum Hall ferromagnet state, but the weak couplingphase is not. Subsequently, we propose that only a composite particle state appears as acharged particle but a pure electron state is absence from the quantum Hall ferromagnet. In this section we review the derivation of the noncommutative Chern-Simons theory as theeffective theory of the incompressible Quantum Hall fluid, based on [4, 5, 6].In a two dimensional system with perpendicular magnetic field, cyclotron motion of anelectron is quantized as the harmonic oscillator, and discretized energy levels are calledLandau levels. The density of states in the lowest Landau level (LLL) is uniform and inproportion to the strength of the magnetic field, ρ = 12 πl (2.1)where l = 1 / √ B is the magnetic length characterizing the scale of the wave function, andthus almost all electrons fall into the LLL in strong magnetic limit. Since the density isspatially constant, occupied area is exactly determined by fixing the number of particles.While the area is preserved, positions of particles can be changed by gauge transformation.Therefore, the electron state in the strong magnetic field behaves as incompressible fluid.Although any dynamical degrees of freedom do not exist because we neglect excitations tohigher Landau levels, we should consider residual degrees of freedom for the fluid, geometricalconfigurations of particles, related to area preserving transformation.The effective theory of the LLL state is often derived by integrating out fermion modes[2].In this paper, we show another way to obtain the effective theory. We firstly introduceintegration constants of the cyclotron motion describing the residual degrees of freedomcalled a guiding center: X = x + l Π y , Y = y − l Π x (2.2)where ~ Π = ~p + ~A is the magnetic momentum. These operators satisfy the following commu-tation relations [ X, Y ] = il , [Π x , Π y ] = − il . (2.3)This spatial noncommutativity is considered as an example of correspondences betweencommutative theory with background field and noncommutative theory which is well knownas Seiberg-Witten map[3]. 3hen the magnetic field becomes so strong, contributions of the magnetic momentumto the guiding center and the canonical momentum can be neglected as ~X ≃ ~x and ~p ≃ − ~A .Thus the Lagrangian can be written in terms of the guiding center coordinates L = ~p · ~ ˙ x − H = B (cid:16) X ˙ Y − ˙ XY (cid:17) , H = 12 m (cid:12)(cid:12)(cid:12) ~ Π (cid:12)(cid:12)(cid:12) . (2.4)This Lagrangian induces only the Lorentz force and mechanical work can be zero. Since ourtheory is not dynamical, we can generalize (2.4) to the n -body state action S = B Z dt n X α =1 ǫ ab X aα ˙ X bα (2.5)where X = X , X = Y and the subscript α = 1 , · · · , n , is an index for particles. In thelarge n limit, a fluid dynamical description becomes available n X α =1 → Z d x ρ ( ~x ) , ~X α ( t ) → ~X ( ~x, t ) , ~X ( ~x,
0) = ~x. (2.6)The initial state is a reference configuration of the fluid. We will consider fluctuation modesfrom the reference state as the residual degree of freedom.The constraint for the incompressibility is the constant density condition, ρ ( ~x ) = ρ e .Since the density of particles is the Jacobian of the fluid dynamical field, the constraint canbe written with Poisson bracket form ρ e = ρ ( ~x ) = ρ e (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ ~X∂~x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 12 ρ e ǫ ab (cid:8) X a , X b (cid:9) . (2.7)Adding this Jacobian preservation constraint to (2.5) with temporal gauge field A as theLagrange multiplier, the action is modified as S = B ρ e Z dt d x h ǫ ab X a (cid:16) ˙ X b − θ (cid:8) X b , A (cid:9)(cid:17) + 2 θA i (2.8)where θ = 1 / (2 πρ e ) will become the noncommutative parameter. Then, satisfying theconstraint, we can decompose X a as X a = x a + θ ab A b , θ ab = θǫ ab . (2.9)Here we can regard gauge fields as the fluctuation mode from the reference state, and thegauge transformation corresponds to area preserving transformation of the fluid. Writingthe action (2.8) in terms of the gauge fields, we obtain S = 14 πν Z dt d x ǫ µνλ (cid:18) ∂ µ A ν A λ + θ { A µ , A ν } A λ (cid:19) . (2.10)The constant 1 /ν = 1 / ( Bθ ) is an integer, which is the level of the Chern-Simons theory, and ν = ρ e /ρ is a filling fraction for the LLL states. Furthermore, this action can be regardedas a leading contribution of noncommutative Chern-Simons action[4] S NCCS = 14 πν Z dt d x ǫ µνλ (cid:18) ∂ µ A ν ⋆ A λ − iA µ ⋆ A ν ⋆ A λ (cid:19) (2.11)4here ⋆ -product is the Moyal product defined as f ( x ) ⋆ g ( x ) = f ( x ) exp (cid:18) i ← ∂ µ θ µν → ∂ ν (cid:19) g ( x ) . (2.12)Because this product is noncommutative, the commutation relation is naively modified[ x , x ] ⋆ = x ⋆ x − x ⋆ x = iθ. (2.13)This noncommutativity is analogous to (2.3). This means that the noncommutative relationfor one particle state is generalized to multi-particle fluid state.The noncommutative relation can be also represented by regarding X a as an infinitedimensional matrix acting on Hilbert space. The corresponding matrix model becomesChern-Simons matrix model S MCS = B Z dt Tr h ǫ ab X a (cid:16) ˙ X b − i (cid:2) A , X b (cid:3)(cid:17) + 2 θA i . (2.14)The spatial integration is replaced with taking the matrix trace. Then we immediately obtainthe equation of motion for the non-dynamical variable A as (cid:2) X , X (cid:3) = iθ ∞ . (2.15)Here the right hand side of (2.15) is in proportional to the infinite dimensional identitymatrix. That means this action is well defined only when the number of particles is infinite.However natural quantum Hall states are realized with the finite system where the boundarystate plays an essential role on the transport phenomena. To regularize the infiniteness ofthe Hilbert space, one should introduce a boundary field which seems to correspond to theedge state. Thus we obtain a regularized finite matrix model proposed in [5, 6], S MCS = B Z dt Tr h ǫ ab X a (cid:16) ˙ X b − i (cid:2) A , X b (cid:3)(cid:17) + 2 θA − ω ( X a ) i + Z dt Ψ † (cid:16) i ˙Ψ − A Ψ (cid:17) . (2.16)The quadratic term ω ( X a ) is the confinement potential and Ψ is n component bosonic fieldabsorbing boundary anomaly. Thus the equation of motion for the Lagrange multiplier A is obtained as (cid:2) X , X (cid:3) = iθ n − iB ΨΨ † (2.17)with the normalization condition, Ψ † Ψ = nBθ = n/ν . In this case, the modified commu-tation relation (2.17) is realized with n × n matrices X a . Introducing a complex matrix X = ( X + iX ) / √ X † = ( X − iX ) / √
2, the noncommutative relation (2.17) isrewritten as 1 B ΨΨ † + (cid:2) X, X † (cid:3) − θ n = 0 . (2.18)The number of parameters for the physical phase space satisfying this constraint, the di-mension of the phase space, is 2 n + 2 n − n = 2 n . Thus these parameters can be regardedas two dimensional coordinates of particles. This relation is of quantum Hall state withoutinternal degrees of freedom. In the following section we will see this relation also appears inthe vortex theory, and thus its non-Abelian generalization is considered.5 Vortex theory and quantum Hall ferromagnets
Topological excitations, e.g. vortices, instantons, play an important role on non-perturbativeaspects of quantum field theory. Although solutions of k -instanton with arbitrary k wasconstructed in [14], an explicit vortex solution is not yet found. However, the structure ofthe vortex moduli space was recently conjectured by the stringy method[7]. In this section,we start with a review of the vortex moduli space based on [7]. We then discuss a relationshipto the incompressible fluid, and show that quantum Hall state is considered as vortex fluidstate. We want to investigate vortices in 2 + 1 dimensional N = 4 supersymmetric gauge theory.The U ( N ) G vector multiplet consists of a gauge field A µ , a triplet of adjoint scalar fields φ r ,and fermionic partners. The N fundamental hypermultiplets are complex scalars q , ˜ q andfermions. Furthermore, considering SU ( N ) F flavor symmetry, fundamental fields q and ˜ q obey ( N, ¯ N ) and ( ¯ N , N ) representation respectively. Thus we write the bosonic part of theLagrangian of this theory L = − Tr h e F µν F µν + 12 e D µ φ r D µ φ r + D µ q † D µ q + D µ ˜ qD µ ˜ q † + e | q ˜ q | + 12 e | [ φ r , φ s ] | + (cid:0) ˜ q † ˜ q − qq † (cid:1) φ r φ r + e (cid:0) qq † − ˜ q † ˜ q − ζ N (cid:1) i (3.1)where ζ is the Fayet-Iliopoulos (FI) parameter which ensures the symmetry broken vacuum.The ground state of this model is gapped, and then vortices appear with the broken symmetry U ( N ) G × SU ( N ) F −→ SU ( N ) diag . (3.2)Then we construct this model by D-brane configuration with N D3-branes and k D1-branes which are regarded as the space-time and vortices respectively. In the decouplinglimit of the string fluctuation, dynamics of D1-branes can be described by N = (2 ,
2) super-symmetric quantum mechanics . In this model, U ( k ) vector multiplet consists of a gaugefield and adjoint scalars φ r corresponding to vortex fluctuations of perpendicular directions.Thus two dimensional positions of vortices are described as a complex scalar Z of the ad-joint chiral multiplet. The fundamental chiral multiplets, complex scalars ψ , come fromexcitations of D1-D3 strings. Then the bosonic Lagrangian on D1-branes becomes L vortex = Tr h g D t φ r D t φ r + D t Z † D t Z + D t ψ i D t ψ † i − g [ φ r , φ s ] − | [ Z, φ r ] | − ψ i ψ † i φ r φ r − g (cid:16) ψ i ψ † i − (cid:2) Z, Z † (cid:3) − r k (cid:17) i . (3.3)The FI parameter of this model is identified with the original gauge coupling as r = 2 π/e .For finite r = 0, we should consider Higgs branch in the decoupling limit g → ∞ , and thus When we consider 4 dimensional N = 2 supersymmetric theory, vortex theory becomes 1+1 dimensional N = (2 ,
2) supersymmetric theory. × k D -term condition reads ψ i ψ † i − (cid:2) Z, Z † (cid:3) − r k = 0 (3.4)where i is the index of the gauge group U ( N ) G running as i = 1 , · · · , N . The number ofparameters of the Higgs branch is 2 kN + 2 k − k = 2 kN since Z and ψ are k × k and k × N matrices. These matrix valued fields parametrize positions of vortices. Such kinds ofparameters for the solution space are called moduli, and the corresponding parameter spaceis called moduli space. Therefore we can regard this Higgs branch as the moduli space M k,N for k -vortex solution with U ( N ) G gauge symmetry.Adjusting some normalizations, the noncommutative relation for the incompressible fluid(2.18) is equivalent to the Abelian ( N = 1) case of the vortex relation (3.4) when we identifythe number of particles n with the vortex number k and the noncommutative parameter θ with the FI parameter r . This means that we can regard particles of the incompressible fluidas Abelian vortices. In this aspect, the geometry of the Abelian vortex moduli space wasdiscussed in [8]. In fact, since the vortex width l v is evaluated as l v ∼ √ r , the particle densitybecomes ρ e ∼ / (2 πl v ) ∼ / (2 πr ). This estimation is consistent with our identification r ∼ θ .Due to this relation, we want to consider incompressible fluid consisting of non-Abelianvortices. Indeed quantum Hall state with internal symmetry is known as a quantum Hallferromagnet and its internal degree corresponds to not only spin of a particle but an indexof multilayer systems, which is called a pseudo spin.To discuss the relationship between quantum Hall state and vortex theory, we investigatethe moduli space of vortices. From the vortex relation (3.4), the moduli space of 1-vortexstate is determined, M ,N ∼ = C × C P N − . (3.5)This means that the 1-vortex moduli is decomposed to a position of vortex center C andinternal C P N − space. On the other hand, the moduli space for k -Abelian vortex is obtainedby [15] as M k, ∼ = C k / S k (3.6)where S k is symmetric group. Then the higher k moduli space is also represented as sym-metric product[16] M k,N ∼ = (cid:0) C × C P N − (cid:1) k / S k , (3.7)and it is suggested that the orbifold singularity of vortex collision is smoothed out[16, 17, 18].We now consider non-Abelian vortex fluid state as quantum Hall state with internalsymmetry. In fact, for N -layered quantum Hall state, each particle has SU ( N ) symmetry,but its U (1) part is decoupled as electromagnetic part. Thus the residual C P N − partis interpreted as internal symmetry of a particle. This phenomenon is called spin-chargeseparation. Although explicit derivation of non-Abelian generalization of (2.18) has notbeen found, relying on the coincidence of the internal symmetry, we identify quantum Hallferromagnets with non-Abelian vortex fluid.Therefore, according to the supersymmetry of the original field theory, we choose 1 + 1dimensional N = (2 ,
2) supersymmetric C P N − model for the vortex world sheet theory[11].Although the physical meaning of superpartners in quantum Hall ferromagnets is not clear,since (3.4) is derived from D -term condition of the supersymmetric theory, we apply su-persymmetric effective theory to the vortex fluids. However, some properties of solitons in7upersymmetric theory are actually observed in the incompressible fluid. We then explainthem as follows.The well known vortical model describing the superconductor, which is called the Ginzburg-Landau model, has two independent coupling constants, the electromagnetic constant e andthe condensate coupling λ , L GL = − e F µν F µν + D µ φD µ φ † − λ (cid:0) φφ † − ζ (cid:1) . (3.8)Each parameter corresponds to characteristic lengths of the superconductor, coherence lengthand penetration length. Thus the type of the superconductor is determined by the ratio ofthese lengths called Ginzburg-Landau parameter, κ = p λ/ (2 e ). For κ ≪
1, the supercon-ductor is of type I in which the interaction between vortices are attractive. For κ ≫
1, itbecomes the type II superconductor where vortices are repulsive. In the case of (3.1), sinceour superconductor κ = 1 / √ Then we consider the field theory describing the dynamics of the non-Abelian vortex. Todiscuss the quantum Hall ferromagnets, we now give a brief review of the supersymmetric C P N − model based on [9, 10, 12].To consider the supersymmetric generalization of the bosonic model, it is convenienceto introduce the superfield formulation . Chiral and anti-chiral superfield are defined asΦ j ( x µ + i ¯ θγ µ θ ) and Φ † ¯ j ( x µ − i ¯ θγ µ θ ). Thus D -term Lagrangian of the supersymmetric C P N − model is written as L = Z d θ K (Φ , Φ † )= G i ¯ j (cid:20) ∂ µ φ † ¯ j ∂ µ φ i + i ¯ ψ ¯ j γ µ D µ ψ i − R i ¯ jk ¯ l (cid:16) ¯ ψ ¯ j ψ i (cid:17) (cid:16) ¯ ψ ¯ l ψ k (cid:17)(cid:21) (3.9)with K¨ahler metric G i ¯ j = ∂ i ∂ ¯ j K , Riemann tensor R i ¯ jk ¯ l and covariant derivative D µ . In thecase of the C P N − model, the K¨ahler potential is defined as K (Φ , Φ † ) = 2 g log (cid:0) † Φ (cid:1) . (3.10)Although the fields in this model are different from those in the previous section, the couplingconstant g is same as that of the corresponding gauge theory. Actually, at the low energyregion, it goes to infinity g → ∞ , and is consistent with the decoupling limit of stringfluctuation. Not to be confused with the fermionic parameter θ and the noncommutative parameter θ = 1 / (2 πρ e ). m a which are the classical vacuumexpectation values of the twisted chiral superfield, Σ = σ + √ θ ˜ χ + θ S , is obtained bymodifying the usual K¨ahler potential (3.10) for C P N − manifold, K (Φ , Φ † , V ) = 2 g log (cid:0) † e V a T a Ψ (cid:1) (3.11)where ( T a ) ij = δ ia δ aj ( a = 1 , · · · , N −
1) are the generators with a diagonal form, and corre-sponding external U (1) components are written as a complex form, V a = − m a ¯ θ (1 + σ ) θ − ¯ m a ¯ θ (1 − σ ) θ, (3.12) m a = A ay + iA ax , ¯ m a = m ∗ a = A ay − iA ax . (3.13)Here σ is Pauli matrix and we can set P Na =1 m a = 0 by shifting the twisted chiral field.Then let us discuss the vacuum structure of this model. The low energy effective actionis obtained by integrating out the chiral superfield and written in terms of the twisted chiralfield. To consider the additional contribution of F -term corresponding to FI-term and ϑ -term, we introduce the twisted superpotential at classical level, W = i τ Σ (3.14)where τ is a complex coupling constant obtained by introducing a theta angle ϑ , τ = 2 ig + ϑ π . (3.15)Although there exists only one classical vacuum at Σ = 0, we will show the quantum vacuumpossesses more rich structure. The dynamically generated mass is exactly evaluated by therenormalization group equation at one loop order with a reference point µ ,Λ = µe − πNg . (3.16)The twisted superpotential with the twisted masses is also corrected by the renormalizationeffect. Thus the effective potential is given by˜ W = i " τ Σ − πi N X a =1 (Σ − m a ) log (cid:18) µ (Σ − m a ) (cid:19) . (3.17)In this case, vacua of this potential can be determined by differentiating with the twistedchiral field ∂ ˜ W ∂ Σ = 0 −→ N Y a =1 ( σ − m a ) − ˜Λ N = 0 (3.18)where ˜Λ = ( µ/
2) exp (2 πiτ /N −
1) is a complexified dynamical mass. This condition ensuresthat there exist N vacua in the quantum level, and then we can consider a topological kinksolution. 9n the case of N = 2 theory, which corresponds to C P model, the renormalization point µ can be replaced with the twisted mass m , with m = − m = m/
2. The mass of the kinksolution becomes m D = ˜ W ( σ + ) − ˜ W ( σ − ) = − i π " m log m − p m + 4 ˜Λ m + p m + 4 ˜Λ ! + 2 p m + 4 ˜Λ (3.19)where σ ± = ± q m / are solutions of σ − m − ˜Λ = 0 . (3.20)From the mass of the topological excitation m D which is exactly evaluated in (3.19) andthe elementary mass m , we obtain the central charge of 1 + 1 dimensional superalgebra Z = n e m + n D m D which characterizes the BPS mass M = | Z | .To discuss the BPS states in the strong coupling region (cid:12)(cid:12)(cid:12) m / (cid:12)(cid:12)(cid:12) ≪ (cid:12)(cid:12)(cid:12) m / (cid:12)(cid:12)(cid:12) ≫
1, we expand the topological mass in terms of the mass parameter, m D = imπ iπ + log (cid:18) m ˜Λ (cid:19) + ∞ X k =1 c k ˜Λ m ! k (3.21)where c k = ( − k (2 k − / ( k !) . The first term is the tree level contribution, and the secondis the one loop correction. The infinite series of the last term comes from the instantoneffect.In the weak coupling limit (cid:12)(cid:12)(cid:12) m / (cid:12)(cid:12)(cid:12) ≫
1, since the ratio of two masses increases loga-rithmically, m D /m ∼ log m , the topological excitation is restricted, and thus surviving BPSstates are n e = ± n D = 0 and n D = ± n e . On the other hand, the situationat strong coupling (cid:12)(cid:12)(cid:12) m / (cid:12)(cid:12)(cid:12) ≪ C P model in the absence of thetwisted mass. When we shift the theta angle ϑ → ϑ + 2 π , a sign of the mass is inverted m → − m . This means that a relevant parameter is the squared mass m and we have acut singularity along negative part of the real axis of the complex m plane. The range ofthe cut is [ − , m plane is obtainedas ( m, m D ) → ( − m, − m D − m ), which is equivalently ( n e , n D ) → ( − n e + n D , − n D ). As aresult, the BPS states in the strong coupling region are only ( n e , n D ) = (0 , , −
1) andtheir anti-excitations.Since the structures of the BPS states in the strong and weak coupling regions are quitedifferent, they must be separated by a curve of marginal stability (CMS) on the complex m plane. The CMS is obtained as a coincidence condition of phases of the elementary massand the topological mass, simply written asIm (cid:16) m D m (cid:17) = Re log − q /m q /m + 2 q /m = 0 . (3.22)10he BPS masses are satisfying M (1 , = M (1 , − + M (0 , on this curve, and thus ( n e , n D ) =(1 ,
0) as the bound state of (1 , −
1) and (0 ,
1) becomes unstable and decays into fundamentalstates when the mass parameter is inside of the CMS.We have discussed the correspondence between quantum Hall state as the incompressiblefluid and vortex state. Thus elementary excitations on a non-Abelian vortex is investigatedin this section 3.2. Then we will consider a physical interpretation of these elementaryexcitations in terms of quantum Hall ferromagnets in the following section.
At last, we now discuss a relationship between the supersymmetric C P model and thequantum Hall ferromagnets. Let us start with the meaning of the kink solution of the C P model in the context of the quantum Hall state. According to the internal degree of freedom,two isolated vacua appear in the vortex theory. Since this internal space C P = SU (2) /U (1)corresponds to the spin degree of the electron, the C P coordinate is interpreted as the spinor pseudo spin direction. Thus the kink excitation interpolates two polarized states. In termsof the bilayer quantum Hall system, these two vacua correspond to the top and bottom layers.Therefore a kinked vortex ( n e , n D ) = (0 ,
1) is a magnetic flux penetrating two layers and( − ,
1) excitation has also an electric charge. This is an electron attached with a flux whichis namely a composite particle state[19, 20]. We propose that a pure electron state is absencefrom the strong coupling region but only a composite particle state is present.Then the kink excitation is obtained by the dimensional reduction of the monopole inthe four dimensional theory[11]. The singularity of the monopole, which is the Dirac string,can transmute the statistics of the particle. As a result of the C P model, permitted vortexstates are only the penetrating magnetic flux and the composite particle state, and the pureelectron state is forbidden in the small m region[Fig.4.1(a)]. (a) (b) Figure 4.1: Particle states as elementary excitations in the vortex theory (a) Charged particle(electron) must accompany a magnetic flux in the small m region. (b) Pure electron statecan be observed in the large m region.In the large m region, however, the restriction of elementary excitations is soften. Thatmeans the stability of the composite particle becomes ambiguous and a pure electric excita-tion n e = 1, n D = 0 can be observed[Fig.4.1(b)]. This excitation is considered as the boundstate of the kink n e = 0, n D = 1 and the opposite composite particle n e = 1, n D = − m region is the quantum Hall phasebut the quantum Hall ferromagnet state as the non-Abelian vortex fluid state is broken asthe twisted mass becomes larger. Furthermore, by the analysis of the C P model, the CMS11iving the phase transition point is exactly evaluated. However, we have applied the 1 + 1dimensional model to the vortex theory although the vortex is not elongated infinitely. Thusthe kink solution spectrum can be corrected by finite size effect.To confirm the above discussion, we should discuss a meaning of the twisted mass pa-rameter in terms of the quantum Hall ferromagnets. In the classical limit where the effect ofthe dynamical mass can be neglected, the two vacua are obtained as σ ± ≃ ± m/
2, and thespectrum of the kink excitation becomes M ≃ m . Because the kink spectrum is consideredas difference of energy levels between the top and bottom layers, we now identify the twistedmass parameter m with bias gate voltage between layers V bias , which corresponds to Zeemanenergy in the context of the spin system.Furthermore the twisted mass includes the imaginary part induced by the theta angle ϑ .When the mass is pure imaginary at ϑ = π , the topological mass m D vanishes, and thus thevortex worldsheet obtains superconformal symmetry. In general, this superconformal pointof the C P N − model at m k = − exp(2 πik/N ) ˜Λ for k = 1 , · · · , N is related to the A N − seriesof the ADE classification[21]. At the critical point, the real part of the twisted mass vanishes,and this fact suggests that the bias gate also vanishes. Since ϑ is originally considered asthe mixing angle of the electric and magnetic field, the electric field is fully converted tomagnetic at ϑ = π . Another considerable interpretation is that ϑ/ ϑ = 0, and is parallel at ϑ = π . However this interpretation isa little confusing because the incompressible fluid can be hardly constructed with the parallelexternal field. On the other hand, in the case of the n -vector model, the twisted mass isassociated with the coupling constant, J/kT = ˜Λ /m [13]. According to the imaginary partof the coupling, the corresponding Hamiltonian becomes non-hermitian operator. In thissense, we might apply superconformal field theory to non-hermitian quantum mechanics.Finally we mention the filling fraction of the quantum Hall state. When the numbersof the kink excitations ( n e , n D ) = (0 ,
1) and the composite states ( − ,
1) are N f and N c respectively, the filling fraction which is the ratio of the particles and the total fluxes isrepresented as ν = N c N f + N c . (4.1)A disappointing part of our analysis is that we mainly focus on each vortex independentlybecause we treat the interactionless sector of the vortex fluid state. It is 1-body problem.Thus we cannot discuss the ratio of the elementary excitations more. On the other hand,according to the correspondence between the noncommutative parameter θ = 1 / (2 πρ e ) =1 / ( Bν ) and the FI parameter or the coupling constant of the C P model r = 2 /g , wecan associate the filling fraction with the twisted mass parameter from the relation (3.16), ν ≈ π/ ( B log( m/ Λ)). Although this estimation is at the classical level and available in theweak coupling region, this means the filling fraction decreases as the mass becomes larger,and supports the breakdown of our description for the quantum Hall state as the previousdiscussion. In the large m region where the bias gate voltage becomes larger, it seems thebilayer system is decoupled, and hence the non-Abelian vortex fluid is not good descriptionfor the quantum Hall ferromagnets. 12 Discussions
In this paper, we have discussed the incompressible fluid as the LLL state and its effectivetheory. The background magnetic field induces the spatial noncommutativity, but it isrealized by the infinite dimensional Hilbert space where the number of particles is infinite.This infiniteness is regularized by the boundary field modifying the commutation relation.As a result, one obtains the matrix Chern-Simons theory as the effective theory of theregularized incompressible fluid[5, 6]. Thus the phase space of the incompressible fluid isspanned by finite dimensional matrix variables satisfying the modified commutation relation.On the other hand, the modified commutation relation is also observed in the vortextheory. It is the supersymmetric vacuum condition for the vortex theory, and characterizingthe moduli space of vortices[7]. Thus the incompressible fluid is considered as the vortexfluid state. This correspondence suggests that the phase space of the incompressible fluid isequivalent to the moduli space of the vortex.The non-Abelian vortex possesses the internal space C P N − . Indeed the symmetry ofthe quantum Hall ferromagnet SU ( N ) is decomposed to the electric charge part U (1) andthe spin part C P N − . This decomposition is called the spin-charge separation. Thus thenon-Abelian vortex fluid state is considered as the quantum Hall ferromagnet. According tothis correspondence, particle state of the quantum Hall ferromagnet is investigated by thevortex theory.To study the vortex state, we have applied the supersymmetric C P N − model to thevortex world-sheet theory[11]. In the case of the C P model, the mass spectrum of thetopological excitation, the mass of the kink solution and the CMS which is the marginalline of the strong and weak coupling region on the complex twisted mass space are exactlyevaluated. This twisted mass m characterizes the mass scale of the kink excitation, and thusit is considered as the bias gate voltage between the top and bottom layers in the case ofthe bilayer quantum Hall system.The elementary excitations in the strong coupling, equivalently the small m region, areonly ( n e , n D ) = (0 ,
1) and (1 , −
1) modes, and this result implies only the composite particleappears, and the pure electron state is forbidden in this region. On the other hand, thecomposite particle is decomposed to the pure electron and the magnetic flux in the large m region. Thus it proposes the phase transition between the strong coupling region whichis the non-Abelian vortex fluid phase and the weak coupling region where the non-Abelianvortex description is not available. Therefore the CMS separating the two phases gives thetransition line for the breakdown of the non-Abelian vortex description of the quantum Hallferromagnet.Then we now comment some issues of our approach in perspective. In the quantum Hallstate, the edge excitation plays an important role on the transport phenomena, and theedge state is well described by conformal field theory, induced on boundary of a manifoldon which Chern-Simons theory is defined. This is an example of the holographic relation ofthe bulk/edge duality. In the case of Chern-Simons matrix theory[5, 6], one obtain the onedimensional quantum many-body model, which is called Calogero model[22], or Sutherlandmodel[23]. Thus it is expected that Calogero model with internal degrees of freedom isobtained from the quantum Hall ferromagnets. Furthermore, it is well known that thequantized filling fraction possesses the hierarchical structure, which has been discussed in the13ontext of the matrix model[24]. The elementary excitation of the vortex in the hierarchicalstate should be understood. Acknowledgments
The author would like to thank S. Hikami for stimulating conversations and reading themanuscript. The author also thank H. Shimada and T. Yoshimoto for useful discussions, G.Marmorini and M. Nitta for valuable comments.
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