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On the origin of multi-component anyon wave functions 3 9 Ansar Fayyazuddin 9 1 n Institute of Theoretical Physics a University of Stockholm J Vanadisvägen 9 2 1 S-113 46 Stockholm, Sweden 1 December 1992 v 1 4 0 Abstract 1 0 3 In this paper I discuss how the component structure of anyon wave functions arises in theories 9 / with non-relativistic matter coupled to a Chern-Simons gauge field on the torus. It is shown h -t that there exists a singular gauge transformationwhich brings the Hamiltonianto free form. The p e gauge transformationremoves a degree of freedom from the Hamiltonian. This degree of freedom h generates only a finite dimensional Hilbert space and is responsible for the component structure : v i of free anyon wave functions. This gives an understanding of the need for multiple component X anyon wave functions from the point of view of Chern-Simons theory. r a USITP-92-15 1 1 Introduction Anyons, particles which obey fractional statistics [1, 2] (for a review see e.g. [3]), are now a well established phenomenon in theoretical physics. A controversy over whether arbitrary fractional statistics could be defined on compact surfaces has now been resolved by the work of several independentgroups[4,5,6]. Theyfindthatanyonwavefunctionswitharbitraryrationalstatistics parameter θ = πq/p can be defined on the torus if the wave functions have p components and N /κ is an integer (where N is the number of anyons). In general one cannot simultaneously A A diagonalize the operators which translate the anyons along the different cycles of the surface. However,one can pick a basis of wave functions so that the component index indicates the phase it picks up under translation along one cycle while translation along the other cycle shifts the component index by one unit. In this paper I reconsider the problem of non-relativistic particles coupled to a U(1) Chern- Simons gauge field on the torus. I will consider a non-relativistic quantum field theory of bosons minimally coupled to the gauge field. The structure of the argument is as follows. Section two deals with the gauss constraint imposed by the Chern-Simons term and its solution on the torus. In section three the physical degrees of freedom of the gauge field are quantized followed by a discussion of the particle vacuum which is essentially a summary of an argument due to Poly- chronakos [11]. In section four the first quantized Hamiltonian is derived and the center of mass Hamiltonian is explicitly diagonalized. It is demonstrated in section five that only single component wave functions are needed in the full theory. Finally in section six it is shown that when the Hamiltonianisbroughttofreeformbyasingulargaugetransformationthesinglecomponentwave functions in the originaltheorybecome multiple componentwavefunctions. It is arguedthat this is a consequence of removinga dynamicaldegree of freedomfrom the center of mass Hamiltonian which only generates a p dimensional Hilbert space. Anyons on tori have been the subject of a number of recent investigations. In particular, discussions addressing similar issues to the ones presented in sections two and the first part of sectionfourcanbefoundin[6,8,17,16]Asmentionedearlierthediscussionoftheparticlevacuum in section four is essentially a summary of arguments originally presented in [11]. Throughoutthispaperthetoruswillbetakentobe theL ×L rectangleinthexy planewith 1 2 opposite sides identified and the Chern-Simons coupling will be given by κ=p/q, where p and q 2 are relatively prime integers. The modular parameter τ appearing in the Jacobi theta functions is given by τ =iL /L . 2 1 2 Solving the Chern-Simons constraint This section will deal with the problem of solving the Gauss Law constraint imposed by the Chern-SimonstermintheLagrangianandisolatingtheremainingdegreesoffreedomofthegauge field which are to be quantized. On the torus the Gauss Law constraint does not completely determine the gauge potential, unlike the situation on the plane. This is because the Gauss Law only constrains the curvature associated with the C-S connection and does not completely specify the connection (up to gauge transformations). In fact, it is well known that to specify the connection completely on topologically non-trivial spaces one has to specify the Wilson lines expi a dxµ along all non-trivial loops [7]. In mathematical language this amounts to specifying µ the cHohomology class of the connection. On the plane the de Rham cohomology group is trivial so that the curvature does determine the connection up to gauge transformation. This is not the case on the torus. There is no entirely natural way of dividing the gauge field into real and constrained degrees of freedom. In particular the Wilson loops around the non-trivial cycles of the torus are affected by the amount of flux coming out of the torus and therefore depend implicitly on the constraint. To see that this is the case one can convince oneself that the flux flowing out of an area bounded by two loops along, say, the x-axis is given by the product of the Wilson loops evaluated along these two (oppositely oriented) loops. Therefore the Wilson loops must know about not only the flux flowing through the holes of the torus but also the flux flowing out of the torus. Nonetheless it is possible to divide the gauge field into real and constrained degrees of freedom by specifying a canonical solution to the constraint so that the connection is unambiguously solved for. Any terms which one can add to the connection while preserving the constraint will be real degrees of freedomandshouldbequantized. Theremainingpartofthissectionwillbeaconcreteelaboration of this point. Consider the following Lagrangian: κ 1 L = ǫµνρa ∂ a +ψ†iD ψ− (D ψ)†(D ψ) µ ν ρ 0 j j 4π 2m D = ∂ −ia 0 0 0 3 D = ∂ −ia j j j (1) Where a is the Chern-Simons gauge field and the fields ψ are bosonic matter fields. From the µ above Lagrangianwe get the constraint: δL κ 0= = f +J (2) 12 0 δa 2π 0 where J =ψ†ψ (3) 0 It is important to realize that if the gauge field supports a non-zero total flux then there will be no globally well defined gauge potential on the torus. The mathematical reason for this is that the U(1) bundle over the torus is twisted. This can be seen by trying to calculate the total flux flowing out of the torus: φ = dxdy(∂ a −∂ a ) x y y x Z Z y0+L2 = dy(a (x +L ,y)−a (x ,y)) (4) y 0 1 y 0 Zy0 x0+L1 − dx(a (x,y +L )−a (x,y )) (5) x 0 2 x 0 Zx0 If the gauge field were periodic along the two cycles of the torus the flux would vanish. If the gauge field does support flux and satisfies the following ”quasi-periodic” boundary conditions: a (~r+eˆL )=a (~r)+∂ Λ (6) j i i j j i where the Λ are suchthat expiΛ (x,y) arewelldefined(i.e. single-valued)gaugetransformation i i on the torus 1. and if, in addition, the matter fields satisfy: ψ(~r+eˆL )=exp(iΛ )ψ(~r) (7) i i i then all gauge invariant observables such as the J are well defined on the torus and there is no µ problem with consistency. I will work in the ∇·a = 0 gauge, in which the Gauss law constraint 1When there is an external electromagnetic field present the condition that expiΛi(x,y) be single valued can be relaxed to requiring that expi Λcis(x,y)+Λeixt(x,y) be single valued on the torus where Λeixt(x,y) is the correspondinggaugefunctionoftheexternalfield. (cid:0) (cid:1) 4 can be written as: 2π ∇2a = ǫij∂ J (8) i j 0 κ Itremainstogiveacanonicalsolutiontotheaboveequationandtoidentifytheremainingdegrees of freedom. I pick the following solution: 2π yQ aˆ = − ∂ G r~′−~r J d2r′ 1 2 0 κ L L (cid:18) 1 2 Z (cid:19) 2π (cid:0) (cid:1) aˆ = ∂ G r~′−~r J d2r′ (9) 2 1 0 κ Z (cid:0) (cid:1) where Q= d2rJ (10) 0 Z is the particle number operator. G is the periodic Green’s function on the torus: 1 ∇2G(r)=δ(~r)− (11) L L 1 2 and is given by [8, 9]: 1 |θ (z |τ)|2 y2 1 G(x,y)= ln + (12) 4π |θ1′(0|τ)|2 2L2L1 where z = x+iy and θ is the odd Jacobi theta function. The above solution fixes Λ = 0 and 1 1 Λ = 2πQx/κL . It is easy to see that the only terms consistent with the constraint and the 2 1 choice of transition functions (the Λ s) that one can add to a are position independent terms: i i θ i a = +aˆ (13) i i L i In fact, only θ mod 2π is observable (the rest being gauge equivalent to 0)2. The single valued i transition functions respect the seperation of the θ from the aˆ . i i The flux flowingout ofthe torus (2πQ/κ)mustbe quantizedaccordingto the Dirac quantization conditionwhich follows fromrequiring that the holonomyof any homotopically trivialclosed path should be well defined. This imposes the condition that Q/κ be an integer. So the theory restricts the number of particles to be an integer multiple of p. This condition also makes the transition functions single valued on the torus. 2Itshouldbeclearfromthecontextwhethertheθi standfortheJacobithetafunctionsorthegaugedegreesof freedom. 5 3 Quantization and the Structure of the Vacuum I turn now to the quantization of the θ . The relevant term in the Lagrangianis: i κ κ π ǫij a˙ a d(2)r = θ˙ θ −θ˙ θ − θ Q˙ −θ˙ Q + aˆ˙ aˆ −aˆ˙ aˆ (14) i j 1 2 2 1 2 2 1 2 2 1 4π 4π κ Z (cid:18) (cid:16) (cid:17) Z (cid:16) (cid:17)(cid:19) By partial integration of the action one can remove all the time derivatives from θ up to a total 2 derivative term which contributes a surface term irrelevant to quantization. The variation of the action with respect to θ˙ gives the corresponding conjugate momentum 3: 1 δS κ Π≡ = θ (15) δθ˙ 2π 2 1 Imposing the canonical commutation relations gives: κ [θ ,Π]=i= [θ ,θ ] (16) 1 1 2 2π Finally, it is necessary to construct a vacuum on which physical states can be built. The first quantizedHamiltonianwill dependonthe particlecoordinates,momenta andthe phasesθ which i are really global degrees of freedom (they are to be interpreted as the amount of flux flowing through the holes of the torus). A basis for the Hilbert space is provided by the set of states {|θ i|r ...r i}. 1 1 n Now try to construct a complete set of states in the θ sector. A natural first guess is to constructstates inthe ’θ representation’: {|θ i},θˆ |θ i=θ |θ i, θˆ |θ i= −i2π ∂ |θ i with 1 1 1 1 1 1 2 1 κ ∂θ1 1 the completeness relation: dθ | θ ihθ |= 1. Since the θ are not observables the summation 1 1 1 i overcountsbyincludingphyRsicallyequivalentstates,statesrelatedbygaugetransformations. The onlyobservableswhichcanbeconstructedfromtheθ areU =expiθ andthereforeitisreasonable i i i to construct a complete set of states with respect to these observables. The U satisfy i 2π U U = U U expi (17) 1 2 2 1 κ U U p = U pU (18) 1 2 2 1 Let |θ ,α i represent a state on which 1 2 U |θ ,α i = eiθ1 |θ ,α i (19) 1 1 2 1 2 U p |θ ,α i = eipα2 |θ ,α i (20) 2 1 2 1 2 3Onemayofcoursetreatbothθ1 andθ2ascoordinatesbutthenonehastouseDiracbracketstoquantizesince themomentum conjugate toθ2 vanishes identically. Ofcoursebothprocedures givethesamefinalresult. Ithank T.H.Hanssonforpointingoutanerrorinanearlierversionofthemanuscriptconcerningthispointandbringing ref[10]tomyattention wherethisquestionisdiscussedinitsgenerality 6 2π 2π The new completeness relation reads dα dα | α ,α ihα ,α |= 1. Even this relation 0 0 1 2 1 2 1 2 is not quite correct since the state | θR1 +R2π,α2i does not lie on the same ray as | θ1,α2i [11] even though they represent the same physical state. This is easily seen from the fact that the operators which perform these gauge transformations, T =exp −2π ∂ , can be represented as i ∂θi T =expiκθ and T =exp−iκθ and satisfy T T =T T e−i2π(cid:16)κ. The ir(cid:17)reducible representaions 2 1 1 2 1 2 2 1 of the T are q dimensional. The states | θ +2πl,α i, l = 0,...,q−1 are eigenstates of T but i 1 2 2 transform into each other under the action of T and do not lie on the same ray. Thus the full 1 Hilbert space consists of q copies of the physical Hilbert space. Since the θ are being treated as i phases, each physical sector of the Hilbert space just specifies a way of picking a branch for the phases and one is free to remainwithin one such sector. Indeed all physicalobservables commute with these transformations and thus restriction to one sector is equivalent to fixing a gauge. Having constructed a basis of states in the θ representation in the gauge field sector of the 1 theory any eigenvector of θ can be expressed as a linear superposition of these states. Which 2 states does one need in order to construct the eigenstate of expiθ with eigenvalue expiδ? To 2 specify the state unambiguously one has to give the value of expipθ as well. Consider the action 1 ofU onthestate|θ ,βi. Sinceθ =−i2π ∂ ,thestate|θ ,βiismappedonto|θ −2π,βiupto 2 1 2 κ ∂θ1 1 1 κ a phase factor. After p actions of U the state returns to itself up to a phase (recalling that the 2 states|θ +2πq,βiand|θ ,βilie onthesameray). Thereforeexpiθ canalwaysbe diagonalized 1 1 2 by the p states {| α +2πl/κ,βi}, l = 0...p−1, with possible eigenvalues β +2πn/κ where 1 expipα is the eigenvalue of Up [11]. If δ belongs to the set {β+2πn/κ} then the corresponding 1 1 eigenstate is a linear combination of the p states {|α +2πl/κ,βi}. Thus the gauge field Hilbert 1 space is divided up into sectors labeled by the eigenvalues Up = eipα1 and U p = eipα2 and each 1 2 sector is p dimensional [11]. With respect to the operators U and U the physical Hilbert space 1 2 has the form H=α ,α H where each H is p-dimensional. The particle vacuum state is 1 2 α1,α2 α1,α2 L then p-dimensional. This direct sum structure of the Hilbert space will turn out to be intimately related to the multi-component structure of anyon wave functions. 4 The Schrödinger Equation InthissectionIwillwritedowntheHamiltonianinfirstquantizedformanddefinetheSchrödinger wavefunctions. IwillpointoutsomenewqualitativefeaturesintheHamiltonianwhichdistinguish 7 the torus from the plane. In particular I will concentrate on the center of mass Hamiltonian and arguethatitcontainsessentiallyallthequalitativelynewfeaturesoftheHamiltonianonthetorus. Following Jackiw and Pi [12], I take the Hamiltonian to be: 1 H=− (D ψ)†D ψ (21) i i 2m where the covariant derivatives have already been defined above. The wave functions are defined by: φ(θ,x ,y )=hθ |ψ(r )...ψ(r )|φi (22) i i 1 n TomakethenotationlesscumbersomeIhaveadoptedthefollwingabbreviatednotation: θ stands for θ , the eigenvalue of U p issuppressed, and | θi is an abbreviation of | θi | 0i, | 0i being the 1 2 particle vacuum. The Schrödinger equation is then given by: ∂ i φ(θ,x ,y ) = hθ |[ψ(r )...ψ(r ),H]|φi (23) i i 1 n ∂t This determines the first quantized Hamiltonian, in the N particle sector, to be: A 1 NA H = − D~ ·D~ (24) α α 2m α=1 X D~ = ∇~ −i~a (25) α α α θ 2πN 2π ∂ A a = + y + G(x −x ,y −y ) (26) αx α α β α β L κL L κ ∂y 1 1 2 β6=α(cid:18) α (cid:19) X 2π ∂ 2π ∂ a = −i − G(x −x ,y −y ) (27) αy α β α β κ ∂θ κ ∂x β6=α(cid:18) α (cid:19) X whereGistheperiodicGreen’sfunctiongivenabove. Theexpressionsforthe~a canbewritten α in the simpler form: θ 2πN i ∂ θ∗(z −z |τ) a = − A Y + ln 1 α β (28) αx L κL L 2κ∂x θ (z −z |τ) 1 1 2 α 1 α β β6=α X 2π ∂ i ∂ θ∗(z −z |τ) a = −i + ln 1 α β (29) αy κ ∂θ 2κ∂y θ (z −z |τ) α 1 α β β6=α X 8 where X and Y are the center of mass coordinates defined by X ≡ 1 NA x and Y ≡ NA α=1 α N1A Nα=A1yα. P TPhe Hamiltonian conveniently splits up into a center of mass plus a relative piece which commute with each other. The wave functions will then be of the product form ψcm ⊗ψrel where each factor will satisfy the Schrödinger equation with respect to the appropriate Hamiltonian. It is illuminating to see the explicit form of the Hamiltonians: 1 ∂ N θ 2πN 2 2 ∂ 2πN ∂ 2 Hcm = − −i A −i A Y + − A (30) 2mN ∂X L κL L ∂Y κL ∂θ A "(cid:18) 1 1 2 (cid:19) (cid:18) 2 (cid:19) # 1 ∂ ∂ 1 ∂ ∂ 2 Hrel = − − − − Λ 2mN ∂x ∂x 2κ ∂x ∂x A α,β,α6=β"(cid:18) α β (cid:18) α β(cid:19) (cid:19) X 2 ∂ ∂ 1 ∂ ∂ + − − − Λ (31) ∂y ∂y 2κ ∂y ∂y (cid:18) α β (cid:18) α β(cid:19) (cid:19) # where θ∗(z −z |τ) Λ= ln 1 µ ν (32) θ (z −z |τ) 1 µ ν µ<ν X Hrel canbeunderstoodasthegeneralizationoftheHamiltonianfortherelativecoordinateson the plane. Itis wellknownthatthe θ (z |τ) arethe torusanalogsofz =x+iy onthe plane [13], 1 and therefore the expression lnθ1∗(zk−zl|τ) correponds to lnz∗ on the plane. The Hamiltonian is θ1(zk−zl|τ) z mapped to the free Hamiltonian by an obvioustransformationanalogousto the one on the plane. As far as Hrel is concerned everything is analogous to the case on the plane. The center of mass Hamiltonian, on the other hand, is quite a different object and there is no simple analogy between it and the corresponding Hamiltonian on the plane. On the plane the center of mass Hamiltonian is explicitly free and knows nothing about the flux tubes attached to the particles. On the torus, however, the Hamiltonian is not free, but, as I will show, there is a transformation which takes it to a free form. For such a transformation to exist it will turn out to be necessary that the θ be quantized. This crucial difference will be responsible for the i component structure of the anyon wave functions on the torus. In fact, I will show that if the θ i are not quantized the picture of anyons as interacting Aharonov-Bohmtubes breaks down. Since I am only interested in revealing the component structure of anyon wave functions and not in finding exact solutions for the entire Hamiltonian, I will restrictmy attention to the center of mass Hamiltonian in the following. In the previous section I discussed the non-trivialbehavior 9 of the gauge field sector under the transformations θ → θ +2π. In particular it was shown i i thatassociatedwitheachphysicalvalueofexpiθ therewereq linearlyindependentHilbertspace 1 rays any of which could be reached from any other by the action of T an appropriate number 1 of times. The second quantized Hamiltonian is invariant under the combined transformations T :θ →θ +2π andψ →exp i2πxj ψ. Thisinvarianceisreflectedinthefirstquantizedtheory j j j Lj by the set of conditions: (cid:16) (cid:17) 2πX T φ(X,Y,θ)=eiγexp i φ(X,Y,θ) (33) 1 L (cid:18) 1 (cid:19) 2πY T φ(X,Y,θ)=eiβexp i φ(X,Y,θ) (34) 2 L (cid:18) 2 (cid:19) Due to the non-commutativity of the operators which translate the θ it is not possible to si- i multaneously impose the above conditions. Instead, the most general conditions one may impose consistent with the commutation relations of the T are: i 2πXN T φ (X,Y,θ) = eiγexp i A φ (X,Y,θ) (35) 1 l l−1 L (cid:18) 1 (cid:19) 2πYN T φ (X,Y,θ) = e−iβ−i2πκlexp i A φ (X,Y,θ) (36) 2 l l L (cid:18) 2 (cid:19) The second condition states that exp−i(κθ+2πN Y/L )φ (X,Y,θ)=exp−i(β+2πκl)φ (X,Y,θ) (37) A 2 l l The first condition requires that 2πXN φ (X,Y,θ+2π)=eiγexp i A φ (X,Y,θ) (38) l l−1 L (cid:18) 1 (cid:19) Turning now to the Hamiltonian, note that: ∂ 2πN ∂ N 2πN 2 A A A − , θ+ Y =0 (39) ∂Y κL ∂θ L κL L (cid:20)(cid:18) 2 (cid:19) (cid:18) 1 1 2 (cid:19)(cid:21) This tells us that the operators ∂ , NAθ+ 2πNA2Y , and ∂ − 2πNA ∂ can be diagonalized X L1 κL1L2 Y κL2 ∂θ simultaneously. The situation is com(cid:16)plicated by the f(cid:17)act tha(cid:16)t the Hamilton(cid:17)ian is not periodic on the torus (because of the presence of a non-zero flux) making it necessary to impose boundary conditionswhichdonotrespectthecommutativityoftheseoperators. Inparticular,onehastosum over eigenstates of the Hamiltonian which carry distinct eigenvalues of the operator P = −i∂ X X 10