A microscopic derivation of the Dirac composite fermion theory: aspects of non-commutativity and pairing instabilities
Dragoljub Go?anin, Sonja Predin, Marija Dimitrijevi? ?iri?, Voja Radovanovi?, Milica Milovanovi?
AA microscopic derivation of the Dirac composite fermion theory: aspects ofnon-commutativity and pairing instabilities
Dragoljub Goˇcanin, Sonja Predin, Marija Dimitrijevi´c ´Ciri´c, Voja Radovanovi´c, and Milica Milovanovi´c ∗ Faculty of Physics, University of Belgrade, Studentski Trg 12-16, 11000 Belgrade, Serbia Institute of Information Systems, Alfons-Goppel-Platz 1, 95030 Hof, Germany Scientific Computing Laboratory, Center for the Study of Complex Systems,Institute of Physics Belgrade, University of Belgrade, Pregrevica 118, 11080 Belgrade, Serbia
Building on previous work [1 and 2] on the system of bosons at filling factor ν = 1 we derivethe Dirac composite fermion theory for a half-filled Landau level from first principles and applyingHartree-Fock approach in a preferred representation. On the basis of the microscopic formulation,in the long-wavelength limit, we propose a non-commutative field-theoretical description, which ina commutative limit reproduces the Son’s theory, with additional terms that may be expected onphysical grounds. The microscopic representation of the problem is also used to discuss pairinginstabilities of composite fermions. We find that a presence of a particle-hole symmetry breakingleads to a weak (BCS) coupling p -wave pairing in the lowest Landau level, and strong coupling p -wave pairing in the second Landau level that occurs in a band with nearly flat dispersion - a thirdpower function of momentum. I. INTRODUCTION
The fractional quantum Hall effect (FQHE) is astrongly correlated problem of particles (electrons) intwo dimensions when there is a commensuration betweenthe number of particles and the number of flux quantaof the applied, orthogonal to the two-dimensional plane,magnetic field. Many phenomenological questions can beanswered by assuming that the most important physicstakes place in a fixed Landau level (LL) with the precisecommensuration of the number of particles and the num-ber of orbitals in a fixed LL (and other LLs are inert).That is why a mathematical, idealized problem of an iso-lated LL is so useful and relevant for the understandingof the FQHE.Some of the most interesting experimental phenomenaoccur at filling factors (ratio of the number of electronsand the number of flux quanta) ν = 1 / ν = 5 / ν =1 / ν =5 / p -wave pairingof underlying quasiparticles in the second LL (sLL), asproposed in Ref. 4. To understand more closely thesesystems, one may start by focusing on an isolated half-filled LL, the lowest LL (LLL) at ν = 1 / ν = 5 / ∗ Corresponding author: [email protected] theory in two dimensions. This can be of general interest:a system of interacting fermions on a non-commutative(NC) space of an LL, in which they fill half of the al-lowed, countable states, can be described by an effectiveDirac theory.The proposed Dirac CF theory is a phenomenologicaltheory, based on the assumption that an effective the-ory of an isolated, half-filled LL must be manifestly in-variant under the particle (electron)-hole transformation.Certainly, there is a need for a microscopic derivationof the Dirac CF theory, which can serve as a base forfurther understanding of this strongly correlated system.In this paper we develop a microscopic support for theDirac CF theory, and provide a framework for a moredetailed investigations. On the basis of a microscopicformulation, in the long-wavelength limit, we propose anon-commutative field-theoretical description, which in acommutative limit reproduces the Son’s theory, with ad-ditional terms that may be expected on physical grounds.We also discuss pairing instabilities within the developedmicroscopic framework, and provide a physical under-standing of the p -wave pairing instability in the LLL, andin the sLL. The pairing in the LLL is of the BCS, weak-coupling kind, and this may explain the scarcity of thepairing phenomena in the LLL. On the other hand, thepairing in the sLL is of the strong coupling (weak pair-ing) kind as proposed and discussed in Ref. 6, thoughwe find that the CF band dispersion, (cid:15) ( k ), is flatter - itobeys a third-power law, i.e. (cid:15) ( k ) ∼ k .The section that follows is a review of the bosonic prob-lem at ν = 1, in which we also introduce a point of viewof the formalism developed in Refs. 1 and 7, that will beuseful for the half-filled problem of electrons. Sections IIIand IV consider a (simpler) system, closely related to theone of the half-filled LL - a special-bilayer system withtwo kinds of particles, parallel to the existence of elec-trons and holes in the half-filled LL. A transformationinto holes of just one kind of particles in the special-bilayer system enables a formulation of the half-filled LL a r X i v : . [ c ond - m a t . s t r- e l ] F e b problem in Section V, with all necessary constraints. Fol-lowing the usual approach [8] to a formulation with con-straints (that enables a Hartree-Fock (HF) treatment) wediscuss a “preferred” form of the Hamiltonian in SectionVI, and in Section VII a Dirac form of the Hamiltonianin the HF approximation. In Section VIII we describehow in the long-wavelength limit of the microscopic for-mulation we can reach a field-theoretical description withgauge fields next to the Dirac composite fermions. Thestructure of the proposed NC field theory and its commu-tative limit is described in the Appendix. In Sections IXand X we discuss the description of possible pairing in-stabilities, while in Section XI we comment on the choiceof constraints and the inclusion of the invariance underthe unitary change of basis in a fixed LL. Conclusions aresummarized in Section XII. II. REVIEW OF THE ν = 1 BOSON SYSTEM
A CF is a composite object - a bound state of an un-derlying elementary particle with a whole number of vor-tices; a vortex represents an excitation of the FQHE sys-tem due to an insertion of one flux quantum that inducesa depletion of charge. At filling factors ν = 1 /q , where q is an integer, a composite fermion is a neutral object;a composite of an electron (fermion) and a hole (moreprecisely a depletion of charge) associated with q fluxquanta, when q is even, and a composite of a boson and ahole associated with q flux quanta, when q is odd. Theseintroductory remarks serve just to remind the reader ofthe physical picture of the CF, and for an elaborate in-troduction to the CF formulation the reader may consultRef. 1. We conclude that it may be expected and nat-ural that an operator describing annihilation or creationof CF will carry two indices, one for the state of the ele-mentary particle and the other for the state of the hole inan orthonormal basis. In the following we will introducethe two-index formalism that was firstly proposed in Ref.7 and further elaborated in Refs. 1 and 2.We start from an enlarged space with (composite)fermion c † mn with two indecies, each corresponding to anorbital in the LLL (fixed LL): n = 1 , . . . , N φ ≡ N suchthat { c nm , c † m (cid:48) n (cid:48) } = δ n,n (cid:48) δ m,m (cid:48) . (1)Each c nm fermion represents a composite object. Wedefine physical subspace of bosonic states in the LLL by | n , . . . , n N (cid:105) = N φ (cid:88) m ,...,m N (cid:15) m ··· m N c † m n · · · c † m N n N | (cid:105) , (2)where (cid:15) m ··· m N is the Levi-Civita symbol. In this way,bosonic physical states have a property, defined by ρ Rmm (cid:48) = (cid:88) n c † mn c nm (cid:48) , (3) that ρ Rmm | n , . . . , n N (cid:105) = 1 · | n , . . . , n N (cid:105) , (4)expressing a single occupancy of each unphysical orbital m . The physical states are defined by this propertyso that unphysical orbitals make an uniformly occupiedbackground. They furnish a spin-singlet representationof SU ( N ) group, m (cid:54) = m (cid:48) , ρ Rmm (cid:48) | n , . . . , n N (cid:105) = 0 , (5)which is in agreement with the requirement that thephysics should not depend on the choice of basis in theunphysical sector R .But we may reformulate this requirement by demand-ing that (1) ρ Rmm = 1 (Eq. 4) in the physical sector (ofthe enlarged theory) and that (2) “unphysical” particles(objects), which are uniformly distributed in the LLL,are fermions. This will automatically lead to (5).Thus, in principle, we can discuss a possible phys-ical sector (system) in the enlarged theory for which ρ Rmm = 1, but with the unphysical particles being (“hard-core”) bosons, and the physical sector being a ν = 1fermionic system. The requirement (5) would be ful-filled, but as we know, this would not lead to a plausibleHartree-Fock description, i.e. a good representation inwhich the Hartree-Fock approach to the system of com-posite fermions c nm would make a good starting pointfor more refined descriptions. In this case | n , . . . , n N (cid:105) f = N φ (cid:88) m ,...,m N s m ··· m N c † m n · · · c † m N n N | (cid:105) , (6)where s m ··· m N = 1 only if m (cid:54) = m (cid:54) = · · · (cid:54) = m N , butotherwise zero. III. TOWARDS ν = 1 / FERMIONS - A SPECIALBILAYER SYSTEM
Now we discuss an enlarged space set-up for ν = 1 / c † mn and d † mn , and discuss an en-larged space with states c † m n · · · c † m N/ n N/ d † m (cid:48) n (cid:48) · · · d † m (cid:48) N/ n (cid:48) N/ | (cid:105) , (7)i.e. always there are N/ c nm fermions and N/ d nm fermions, where N , as before, is the number of availableorbitals in the LL. As the first step of the formulation ofthe problem in the enlarged space, we consider c nm and d nm fermions such that they react in the same way toexternal probes, and thus may be considered as a partof a description of a bilayer problem - each LL (layer)half-filled.Following the discussion in the previous part, we re-quire ρ R ( c ) mm + ρ R ( d ) mm = 1 , (8)i.e. we uniformly distribute particles in the unphysi-cal sector. Furthermore, we choose them to be bosonsand mutual bosons. The requirement (8) may be asso-ciated with special (identical for both c and d fermions)transformations of the LL basis in the unphysical sec-tor, which we denote by SU Rc ( N ) ( c stands for charge),a transformations that are realized identically on both c and d fermions by affecting their unphysical index.These transformations should leave the physical statesunchanged, i.e. m (cid:54) = m (cid:48) , ( ρ R ( c ) mm (cid:48) + ρ R ( d ) mm (cid:48) ) | n , . . . , n N/ n (cid:48) , . . . , n (cid:48) N/ (cid:105) = 0 , (9)and by assuming (8), we have that | n , . . . , n N/ , n (cid:48) , . . . , n (cid:48) N/ (cid:105) = N (cid:88) m ,...,m N/ ,m (cid:48) ,...,m (cid:48) N/ s m ··· m N/ m (cid:48) ,...,m (cid:48) N/ × c † m n · · · c † m N/ n N/ d † m (cid:48) n (cid:48) · · · d † m (cid:48) N/ n (cid:48) N/ | (cid:105) , (10)where s m ··· m N/ m (cid:48) ,...,m (cid:48) N/ is non-zero, equal to one onlyif no index is equal to any other index.In (10) we have not only required that the unphysical -bosonic degrees of freedom are uniformly distributed (8),but that they correlate mutually in a symmetric way.Thus we chose a sector of definite statistics in the en-larged space.If N = 2, we have the following candidates for physicalstates: | n, n (cid:48) (cid:105) = ( c † n d † n (cid:48) + c † n d † n (cid:48) ) | (cid:105) , (11)where n, n (cid:48) = 1 ,
2. Additionally, as a part of the def-inition, we require that in the special bilayer system,i.e. two half-filled LL system, LL orbitals in the physicalstates cannot be doubly occupied. Thus what is neededis to suppress the unwanted states (( n, n (cid:48) ) = (1 , , (2 , N = 2 example) of double occupancy (with an eyeon the half-filled problem). Therefore, we need also ρ L ( c ) nn + ρ L ( d ) nn = 1 , (12)This leads to | phy (cid:105) - physical states for which n (cid:54) = n (cid:48) , ( ρ L ( c ) nn (cid:48) ± ρ L ( d ) nn (cid:48) ) | phy (cid:105) = 0 . (13)We may associate the plus combination with SU Lc ( N ),and the minus combination with SU Ls ( N ) - “spin” trans-formations - which are inverse in the d sector with re-spect to the ones in the c sector. Together, (12) and (13)with the plus sign lead to conclusion that | phy (cid:105) statesare spin-singlet(s) under SU Lc ( N ). On the other hand,in the physical states, the generators of SU Ls ( N ) trans-formations, ρ L ( c ) nn − ρ L ( d ) nn , (14)may have expectation values from the interval [ − , { ρ L ( c ) nn (cid:48) } and { ρ L ( d ) nn (cid:48) } , with the constraints (cid:80) ρ L ( c ) nn = (cid:80) ρ L ( d ) nn = N/
2, furnished two adjoint representationsof SU ( N ) group. However, with the hard-core con-straint (12), we have only one non-trivial representationof SU ( N ) group, SU Ls ( N ). The physical states are in-variant under global U s (1) transformation because (cid:88) n ρ L ( c ) nn = (cid:88) n ρ L ( d ) nn = N/ , (15)and so are the unphysical ( R sector) states, (cid:88) n ρ R ( c ) nn = (cid:88) n ρ R ( d ) nn = N/ , (16)by definition.This completes a constraint ((8), (12), (15), (16)) andstatistics set-up for the description in an enlarged spaceof the problem that concerns a special bilayer at ν tot = 1with two kinds of electrons, i.e. composite fermions c and d , which cannot occupy the same orbital in the restrictedspace of a fixed LL. IV. A PARTICLE-HOLE TRANSFORMATIONAND ITS FORMULATION FOR THEHALF-FILLED PROBLEM
To see the relevance of the special bilayer problem forthe problem of a half-filled LL of electrons, we can imag-ine a description in which the latter problem is mapped toa problem where both particles (electrons) and holes are(fermionic) degrees of freedom, but with the constraintthat particle and hole cannot simultaneously occupy thesame orbital in the LL. Formally, we can apply a particle-hole transformation on holes and define a problem withtwo kinds of electrons and composite fermions, c and d ,that cannot occupy the same orbital, i.e. the problem ofthe special bilayer that we discussed in the previous sec-tion. The formulation of the special bilayer problem inthe enlarged space of composite fermions, c nm and d nm ,was previously given, and we can use this formulation andapply the particle-hole transformation on the d nm com-posite fermions (i.e. apply the transformation in reverse)to get a formulation of the half-filled LL of electrons interms of two composite fermions with constraints.Note that, here, by a particle-hole transformation, ageneral transformation that relates particles and holes isunderstood. In this section we will first introduce theusual particle-hole transformation in the bilayer system(in the d sector) as a change of variable(s). It will turnout, one may say naturally, that what we need in order totransform the special bilayer problem into the half-filledLL problem, is the particle-hole conjugation (similar tothe transformation from the ν = 1 / ν = 1 / e c ≡ e and e d .We have to perform a particle-hole transformation on e d ( e d → h ) and exclude the double occupancy of e and h to reach the formulation of the problem of our interest- the half-filled LL. To introduce the particle-hole trans-formation in the enlarged-space description of the specialbilayer we will discuss in the following the representationof c and d composite fermions in the inverse (momentum)space.Following the previous studies on the ν = 1 bosonicproblem we introduce the decompositions, c nm = (cid:90) d k (2 π ) (cid:104) n | τ k | m (cid:105) c k , (17)and d nm = (cid:90) d k (2 π ) (cid:104) n | τ k | m (cid:105) d k , (18)with τ k = exp ( i k · R ), where R is the guiding-centercoordinate of a single particle in the external magneticfield, and {| n (cid:105)} are single-particle states (orbitals) in afixed LL.With these decompositions we find that ρ L ( c ) nn (cid:48) = (cid:88) m c † mn c n (cid:48) m = (cid:90) d q π (cid:104) n (cid:48) | τ q | n (cid:105) ρ L ( c ) q , (19)where ρ L ( c ) q = (cid:90) d k (2 π ) c † k − q c k exp (cid:18) i k × q (cid:19) , (20)and similarly for ρ L ( d ) nn (cid:48) . Note the inverse order of indices, n and n (cid:48) , on the left and right side of (19). Similarly, ρ R ( c ) mm (cid:48) = (cid:88) n c † mn c nm (cid:48) = (cid:90) d q π (cid:104) m | τ q | m (cid:48) (cid:105) ρ R ( c ) q , (21)where ρ R ( c ) q = (cid:90) d k (2 π ) c † k − q c k exp (cid:18) − i k × q (cid:19) , (22)and analogously for ρ R ( d ) nn (cid:48) . Note the positions of indices m and m (cid:48) on the left and right side of (21).We have [ ρ L q , ρ L q (cid:48) ] = 2 i sin (cid:18) q × q (cid:48) (cid:19) ρ L q + q (cid:48) , (23)and [ ρ R q , ρ R q (cid:48) ] = − i sin (cid:18) q × q (cid:48) (cid:19) ρ R q + q (cid:48) , (24)i.e. GMP algebra for two kinds of particles - particleswith opposite electric charge.We introduce a particle-hole transformation in the d sector by taking d k → d †− k ( d † k → d − k ) . (25) This implies d mn → d † mn ( d † mn → d mn ) , (26)and also for ρ L ( d ) nn (cid:48) = (cid:88) m d † mn d n (cid:48) m , n (cid:54) = n (cid:48) (27)we have ρ L ( d ) nn (cid:48) → (cid:88) m d mn d † n (cid:48) m = − (cid:88) m d † n (cid:48) m d mn = − ρ R ( d ) n (cid:48) n . (28)In the inverse space this implies ρ L ( d ) ( q ) → − ρ R ( d ) ( q ) , (29)which is consistent with our expectation of what aparticle-hole transformation will imply on the physicaldensity in the d sector; it will induce a density of par-ticles of opposite charge in the magnetic field ( L → R )(and a minus sign that is always accompanied with sucha transformation). We can reach the same result by con-sidering ρ L ( d ) ( q ) for q (cid:54) = 0, ρ L ( d ) ( q ) = (cid:90) d k (2 π ) d † k − q d q exp (cid:18) i k × q (cid:19) , (30)and applying the transformation d k → d †− k ( d † k → d − k ).Thus this transformation may be identified to be the onethat corresponds (in the enlarged space) to the particle-hole transformation on the elementary (fundamental) de-grees of freedom, e d , the second kind of electrons in thespecial bilayer: e d → h † d , e † d → h d (where e d , e † d , h d , h † d are annihilation and creation operators).Above we introduced the effect of the particle-holetransformation (on electrons e d ) in the d sector on com-posite fermion operators, while putting aside the ques-tion of the diagonal terms, ρ L ( d ) nn , and the necessaryexistence of a constant term, equal to N , due to theanti-commutation relation of d mn ’s. This would implyan additional delta function contribution ( ∼ δ ( q )) inthe inverse space for the particle-hole transformation of ρ L ( d ) ( q ).To comply with the restrictions of physical spaces inthe enlarged spaces of composite fermion operators, d nm and c nm , of the half-filled and special bilayer problem,we expect ρ L ( d ) nn → − ρ R ( d ) nn , (31)because the summation on n on both sides, and the re-strictions (cid:88) n ρ L ( d ) nn = (cid:88) n ρ R ( d ) nn = N , (32)would be consistent with the particle-hole (single layer)symmetry of the physical system and restrictions on thespecial-bilayer system.Thus, in order to project the transformation d mn → d † mn on the physical spaces of half-filled and special bi-layer problems we demand ρ L ( d ) nn → − ρ R ( d ) nn , (33)which requires an additional subtraction of a constantterm ( N −
1) after the d mn → d † mn ( d † mn → d mn ) trans-formation, in order to project out the unphysical de-grees of freedom. In the inverse space this affects thedelta function contribution; thus for q (cid:54) = 0 we still have ρ L ( d ) ( q ) → − ρ R ( d ) ( q ).In the following we would like to examine how thisparticle-hole transformation affects the constraints im-posed on the bilayer system in order to see how they looklike in the (enlarged) space of the special bilayer system.The two “hard-core” constraints, ρ R ( c ) nn + ρ R ( d ) nn = 1 in (8)and ρ L ( c ) nn + ρ L ( d ) nn = 1 in (12) become ρ R ( c ) nn = ρ L ( d ) nn , (34)and ρ L ( c ) nn = ρ R ( d ) nn , (35)respectively, which is consistent with the view that now d sector is described by hole degrees of freedom. Onthe other hand the operator ρ L ( c ) nn − ρ L ( d ) nn transforms into ρ L ( c ) nn + ρ R ( d ) nn −
1, i.e. ρ L ( c ) nn + ρ R ( d ) nn acquires expectationvalues in the physical states ranging from 0 to 2. Thus weintroduced the hole view in the d sector and this increasedthe allowed occupancy of c particles and d holes of a singlesite to 2. But we want to introduce a description in termsof holes not in the way of change of variables but in theway of a real change in the d sector: where are particlesthere should become holes and vice versa. FIG. 1. An illustration of the particle-hole transformationin the special bilayer problem. The transformation is donein the layer with particles (electrons) 2 and thus also on theassociated composite fermion d . FIG. 2. An illustration of the “active” particle-hole transfor-mation, i.e. the particle-hole conjugation on particles (elec-trons) 2 which become holes. In this way the special bilayerproblem is transformed into the half-filled LL problem. Thus ρ L ( d ) → ρ R ( d ) and ρ R ( d ) → ρ L ( d ) . In this way theoperator ρ L ( c ) nn − ρ L ( d ) nn becomes ρ L ( c ) nn − ρ R ( d ) nn and describefluctuating charge of the half-filled LL. Also the followingaction on operators d mn and d † mn is implied: d mn → d nm ( d † mn → d † nm ) , (36)(compare the definitions of the density operators in (19)and (21), with d instead of c operators, in L and R sec-tors). V. THE FORMULATION OF THEHALF-FILLED PROBLEM
On the basis of the discussion in the previous sec-tion, we can conclude that the charge fluctuations aroundmean density ( πl B ) are given by ( ρ L ( c ) nn − ρ R ( d ) nn ) / ρ R ( c ) nn + ρ L ( d ) nn = 1 and ρ L ( c ) nn + ρ R ( d ) nn = 1 is H = 12 (cid:90) d q V ( | q | ) × (37)( ρ L ( c ) ( q ) − ρ R ( d ) ( q ))2 ( ρ L ( c ) ( − q ) − ρ R ( d ) ( − q ))2 . The charge operator ( ρ L ( c ) ( q ) − ρ R ( d ) ( q )) / ρ L ( c ) ( q ) − ρ R ( d ) ( q ))does) because of the doubling of the degrees of free-dom (extra 2 in the GMP algebra). But together withthe constraint ρ L ( c ) ( q ) + ρ R ( d ) ( q ) = 0 it does, because( ρ L ( c ) ( q ) − ρ R ( d ) ( q )) / ρ L ( c ) ( q ) (or − ρ R ( d ) ( q )due to the particle-hole symmetry) with the constraint,and represents the physical charge that satisfies the GMPalgebra.In the Hamiltonian formulation we used ρ L ( c ) ( q ) + ρ R ( d ) ( q ) = 0 , (38)the constraint that eliminates the hole degrees of free-dom as additional degrees of freedom, by precluding thedouble occupancy as in (12) in the special bilayer prob-lem. In the following paragraph we will recapitulate thenecessary constraints in the formulation of the half-filledLL problem.We summarize that ρ R ( c ) nn + ρ L ( d ) nn = 1 , (39)together with the definite statistics requirement in theunphysical sector (i.e. a unique spin-singlet realizationof the SU ( N ) symmetry in the unphysical sector or ofunphysical degrees of freedom), and ρ L ( c ) nn + ρ R ( d ) nn = 1 , (40)with global constraints, (cid:88) n ρ L ( c ) nn = (cid:88) n ρ R ( d ) nn = N/ , (41)and (cid:88) n ρ R ( c ) nn = (cid:88) n ρ L ( d ) nn = N/ , (42)form a set of constraints that define the half-filled LLproblem. VI. PREFERRED FORM OF THEHAMILTONIAN
The most natural binding in H is the Cooper pair bind-ing (cid:104) c k d − k (cid:105) (cid:54) = 0 in the s -wave channel. In a Hartree-Focktreatment, the mean-field description would have kineticterms with quadratic dispersions, for c and d degrees offreedom, that do not conform to our expectation thatthey are dipoles - distinct dipole objects that pair, andthat their dispersion comes from the polarization energydue to their dipole moments in a Hartree contribution asemphasized in [2].As in the ν = 1 bosonic case we may wonder whetherexists a “preferred” form of the Hamiltonian, i.e. theHamiltonian with some of constraints included in its for-mulation but with the same description (and action) asthe original one in the physical space. The “preferred”form should capture the basic physics in the most effi-cient way, enabling the description of the basic physicsin a Hartree-Fock treatment.It is not hard to see that a unique low-momentum pos-sibility for a kinetic (non-pairing) term can be reached byan addition of the following term, H→ H + 12 (cid:90) d q V ( | q | ) × (43)( ρ R ( c ) ( q ) + ρ L ( d ) ( q ))2 ( ρ R ( c ) ( − q ) + ρ L ( d ) ( − q ))2 , which uses the following constraint, ρ R ( c ) ( q ) + ρ L ( d ) ( q ) = 0 , (44) in the physical sector for the unphysical degrees of free-dom that directly follows from the requirement (39).In this way we removed the cause for the s -waveCooper pairing and modified the relevant term from ∼ (cid:90) d q (cid:104) − ρ L ( c ) ( q ) ρ R ( d ) ( − q ) (cid:105) V ( | q | ) (45)to ∼ (cid:90) d q (cid:104) − ρ L ( c ) ( q ) ρ R ( d ) ( − q ) + ρ R ( c ) ( q ) ρ L ( d ) ( − q ) (cid:105) V ( | q | ) ∼ (cid:90) d q (cid:90) d k (cid:90) d k c † k − q c k d † k + q d k (46) × [ i ( k + k ) × q ] V ( | q | ) . Clearly this term in the Hartree-Fock treatment can leadonly to an excitonic, (cid:104) c † k d k (cid:105) (cid:54) = 0, instability and a Dirac-like description of the low-momentum physics. VII. THE DIRAC THEORY FROM THE MEANFIELD
Thus we apply the Hartree-Fock approach to the rel-evant part of the Hamiltonian (we neglect the quadraticcontributions from the other terms): H D = (cid:90) d q V ( | q | ) (cid:104) − ρ L ( c ) ( q ) ρ R ( d ) ( − q ) + ρ R ( c ) ( q ) ρ L ( d ) ( − q ) (cid:105) ≈ (cid:90) d q (cid:90) d k (cid:90) d k V ( | q | )4(2 π ) [ i ( k + k ) × q ] × (cid:104) (cid:104) c † k − q d k (cid:105) d † k + q c k + c † k − q d k (cid:104) d † k + q c k (cid:105) − (cid:104) c † k − q d k (cid:105)(cid:104) d † k + q c k (cid:105) (cid:105) = (cid:90) d k (2 π ) (cid:16) ∆ ∗ k d † k c k + ∆ k c † k d k (cid:17) + C , (47)where C is a constant and∆ k = | k | (cid:90) d q V ( | q | )2(2 π ) ( i ˆ k × q ) (cid:104) d † k + q c k + q (cid:105) . (48)We diagonalize H D by introducing α k and β k operators, (cid:20) c k d k (cid:21) = 1 √ (cid:20) − exp {− iδ k } exp { iδ k } (cid:21) (cid:20) α k β k (cid:21) , (49)where δ k is defined by ∆ k = | ∆ k | exp {− iδ k } .The very important question is how we choose the oc-cupation of the momentum k states in the ground statethat is subjected to the constraints, ρ R ( c ) nn (cid:48) + ρ L ( d ) n (cid:48) n = δ nn (cid:48) (50)and ρ L ( c ) nn (cid:48) + ρ R ( d ) n (cid:48) n = δ nn (cid:48) . (51)In a mean field treatment we expect that at least theglobal constraints, (cid:90) d k (2 π ) c † k c k = (cid:90) d k (2 π ) d † k d k = ¯ ρ e = 12 12 πl B , (52)will be satisfied.In the α k , β k language this implies (cid:90) d k (cid:16) e − iδ k α † k β k + e iδ k β † k α k (cid:17) = 0 . (53)This is a complex constraint and we may try to satisfythe requirement on α ’s and β ’s, by demanding that alsothe number of α ’s and β ’s is conserved. We might expect, (cid:90) d k (2 π ) α † k α k = (cid:90) d k (2 π ) β † k β k = ¯ ρ e . (54)This seems very crude “translation” of (53), but it in-corporates the basic idea of our approach: to treat theparticles and holes in an equal way, with their dynamicsnot independent but constrained, and in this way dupli-cated in a theory. The constraint implies two sectors, α and β , in the ground state configuration: half-filled α sector and half-empty β sector. FIG. 3. A schematic illustration of the implementation ofthe global constraint in (52) via (54), i.e. half-filled positiveenergy sector and half-empty negative energy sector.
In this we implicitly assumed the finiteness of the avail-able volume of k : the number of available k ’s is N - thenumber of orbitals in the fixed LL. We expect that thedescription is duplicated by treating particles and holesin an equal way, and, in the first (mean-field) approxi-mation, the dynamics of α and β are separate and inde-pendent, and we may consider one or the other sector asa description of the problem.Thus for ∆ k we get, by self-consistency, in the α sector,with a cut-off q F = l B ,∆ k = | k | e − iφ k (cid:90) d q V ( | q | )4(2 π ) ( i ˆ k × q ) e − iφ k + q + iφ k (55)In this expression for ∆ k , because of the Gaussian in V ( | q | ), the contribution of the α sector is dominant and we neglected the contribution from the β sector. Wechoose δ k to describe a definite momentum state, δ k = φ k , where φ k is the phase of the complex variable, k = k x + ik y . It follows that | ∆ k | = | k | π π ) (cid:90) q F dq q V ( q ) . (56)The strength of the excitonic amplitude is zero for highermomenta.Thus by applying the Hartree-Fock approach to thepreferred form of the Hamiltonian, (44), we reached alow-energy description of the problem in terms of H D = (cid:90) d k (2 π ) (cid:16) ∆ ∗ k d † k c k + ∆ k c † k d k (cid:17) , (57)where ∆ k = ( k x − ik y )∆ with ∆ = π π ) (cid:82) q F dq q V ( q ),at the finite density of the Dirac system. VIII. THE INCLUSION OF U (1) INVARIANCEIN THE EFFECTIVE DIRAC THEORY
The original SU ( N ) gauge invariance (i.e. invarianceunder a change of basis in the fixed LL) is broken down[2] to U (1) in the mean-field (Hartree-Fock, averaged)description in (57). The residual U (1) gauge symmetry,in terms of gauge fields, should be present in the effectivedescription. As we already detailed, the SU ( N ) gaugeinvariance is realized by the following two constraints, ρ R ( c ) nn + ρ L ( d ) nn = 1 , (58)i.e. equal charge distribution of unphysical degrees offreedom, and ρ L ( c ) nn + ρ R ( d ) nn = 1 , (59)i.e. the exclusion of the double occupancy between par-ticles and holes (extra unphysical degrees of freedom).This defines and implies, in the effective description,simultaneous U (1) transformations in R and L sectorsto which we may associate, in a continuum description,gauge fields a µ and a µ . We may also consider a (back-ground) field A µ that is associated with physical degreesof freedom (particles and holes), ρ L ( c ) nn − ρ R ( d ) nn . Thus inthe continuum we expect, D µ c = ∂ µ c − ic (cid:63) a µ − i ( A µ + a µ ) (cid:63) c, (60)and D µ d = ∂ µ d − ia µ (cid:63) d − id (cid:63) ( − A µ + a µ ) , (61)in the non-commutative description of the low-energyphysics, following the considerations in [2] for the ν = 1system of bosons.But there is a problem with this proposal for a non-commutative description: the implied theory of the Diractype is not gauge invariant. The gauge invariance is re-stored only if c and d fields couple in the same way togauge fields, i.e. ( L and R ) gauge fields should be thesame in D µ c and D µ d with the same sign.We have to step back to understand why this problemoccurs. The formulation of the half-filled LL with con-straints (58) and (59) is distinct from the case of bosonsat ν = 1 filling, in which there is a clear distinction be-tween L and R , physical and unphysical sector, with noadditional requirements (on the statistics) in the unphys-ical sector. The formulation of the half-filled LL systemneeds an additional requirement next to constraints (58)and (59), and it is more complicated since it includesexchange of L and R sectors, physical and unphysicaldegrees of freedom. Also we may point out that the two(sets of) constraints are related by exchange between c and d , which mimics a particle-hole transformation, and,in a way, we may consider only one as independent. (Weincorporated only one in the preferred form of the Hamil-tonian.)Having this in mind, we may reconsider the questionof constraints and gauge invariance in an effective, long-wavelength theory that we are looking for, a theory thatwill nevertheless include some non-commutative aspectsof the physical system. In order to get a gauge-invariantdescription we also have to apply the long-wavelengthlimit on the constraints (not just in the derivation ofthe Hamiltonian). This boils down to making a rightchoice for the form of covariant derivatives D µ c and D µ d in the long-wavelength limit. We may expect that A µ (background field) couples symmetrically, with oppositesign, in the L and R sectors due to the requirement forthe gauge invariance. Thus in this long-wavelength de-scription there is no distinction between physical (par-ticles or holes) and unphysical (quantum of flux excita-tion) degrees of freedom. And the previous constraint(59), “where electrons cannot be holes”, or (58) “wheretwo Laughlin (quantum flux) quasielectrons cannot betwo Laughlin (quantum flux) quasiholes”, may simplify(because of the size of composite fermions ( (cid:46) l B )) into“where electrons cannot be two Laughlin (quantum flux)quasielectrons”, or “where holes cannot be two Laughlin(quantum flux) quasiholes”. In other words, constraints(58) and (59) become one in the long-wavelength descrip- tion (compare (20) and (22) in the small q limit, etc.). Wejust described in physical terms what the gauge invari-ance requires as a type of a single constraint in the long-wavelength limit. Thus we may define covariant deriva-tives D µ c and D µ d in a gauge invariant way, D µ c = ∂ µ c − iA µ (cid:63) c − ic (cid:63) ( a µ − A µ ) , (62) D µ d = ∂ µ d − iA µ (cid:63) d − id (cid:63) ( a µ − A µ ) . (63)This can define an NC description, and it should bechecked whether in the commutative limit via Seiberg-Witten map we can recover the Son’s theory to the linearorder in the small parameter θ = − l B . Because of thesimultaneous presence of small θ and long-wavelength ex-pansions, we will seek an effective description by consid-ering only lowest order terms. To find the first correctionin the commutative limit we start with the (Euclidean)action S NC of the form: S NC = (cid:90) dτ d r (cid:16) c † (cid:63) D τ c + d † (cid:63) D τ d (64)+ c † (cid:63) ( iD x + D y ) d + d † (cid:63) ( iD x − D y ) c + i ( a − A )¯ ρ e (cid:17) . In S NC we have a linear term in a (non-commutative)field ( a − A ) that fixes the total number of c ’s and d ’s inthis description. We have to recall that our descriptionis for the following k , 0 (cid:46) | k | (cid:46) /l B = k F in the upperhalf of Fig. 3. Thus (cid:90) d k (2 π ) (cid:16) c † k c k + d † k d k (cid:17) | upper half = ¯ ρ e . (65)In the Appendix we detail the small θ expansion withcommutative fields ˆ c, ˆ d, ˆ a µ , ˆ A µ . We find S NC = S (0) + S (1) + . . . , where the classical limit ( θ = 0) is simply S (0) = (cid:90) dτ d r (cid:16) ˆ c † D τ ˆ c + ˆ d † D τ ˆ d (66)+ ˆ c † ( iD x + D y ) ˆ d + ˆ d † ( iD x − D y )ˆ c + i (ˆ a − ˆ A )¯ ρ e (cid:17) , and the linear NC correction reads (see the Appendix fordetails) S (1) = i ¯ ρ e θ (cid:15) αβγ (cid:90) dτ d r (ˆ a α − ˆ A α ) ∂ β (ˆ a γ − ˆ A γ )+ θ (cid:90) d τ d r (cid:20) ˆ c † (cid:18)
12 ˆ f − ˆ F (cid:19) D τ ˆ c − ˆ c † (cid:18)
12 ˆ f − ˆ F (cid:19) D y ˆ c + ˆ c † (cid:18)
12 ˆ f − ˆ F (cid:19) D x ˆ c (cid:21) + θ (cid:90) d τ d r (cid:20) ˆ d † (cid:18)
12 ˆ f − ˆ F (cid:19) D τ ˆ d − ˆ d † (cid:18)
12 ˆ f − ˆ F (cid:19) D y ˆ d + ˆ d † (cid:18)
12 ˆ f − ˆ F (cid:19) D x ˆ d (cid:21) , (67)where we introduced classical (commutative) gauge field strengths, ˆ F µν = ∂ µ ˆ A ν − ∂ ν ˆ A µ , (68)ˆ f µν = ∂ µ ˆ a ν − ∂ ν ˆ a µ . (69)The classical action S (0) and the Chern-Simons term in S (1) give the description of a version of the Son’s theory[5] of the Dirac composite fermion [9, 10] that assumesthe Pauli-Villars type of regularization [11, 12] becauseof the presence of the Chern-Simons term for field a µ .Note that CS term has the correct coefficient, π . Inthe linear NC correction S (1) we also have that the Diracmomentum density couples to the external electric fieldas expected from the Galilean invariance [7, 13]. On theother hand we also find the presence of a coupling tothe internal (ˆ a µ ) electric field, which is quite natural andexpected given the influence (ˆ a µ ) of other particles on aselected one. Also we find that the presence of internaland external (i.e. departure from the uniform) magneticfield induces a change in the coefficient of the kineticterms ˆ c † ∂ τ ˆ c and ˆ d † ∂ τ ˆ d . Thus, we can conclude that theNC formulation up to the first order in θ recovers knownresults but also systematically adds terms that we mayexpect on physical grounds. IX. INCLUSION OF PAIRING
We eliminated unphysical degrees of freedom in a wayof constraint (39), although in the case of the half-filledLandau level we also had to impose additional, bosoniccorrelations of the unphysical degrees of freedom to fixa unique subspace of physical states - we called it spin-singlet sector of the SU ( N ) gauge symmetry of unphys-ical degrees of freedom. (Subsequently, we also had toimpose (40) - to eliminate hole degrees of freedom.)We may search for another such a state for unphysicaldegrees of freedom by imposing other constraint(s), suchas ρ R ( c ) nn (cid:48) = ρ L ( d ) n (cid:48) n . (70)In the inverse space this corresponds ρ R ( c ) ( q ) − ρ L ( d ) ( q ) = 0 . (71)This seems choice seems natural as a requirement thatwill equalize and uniformly distribute the electric chargeof the unphysical degrees of freedom (similarly to thespecial bilayer system).The Hamiltonian that we can consider now is H p = H − (cid:90) d q V ( | q | ) × (72)( ρ R ( c ) ( q ) − ρ L ( d ) ( q ))2 ( ρ R ( c ) ( − q ) − ρ L ( d ) ( − q ))2 , which contains the same relevant two-body part for theDirac physics as in the previous inclusion of constraints(44), but now diagonal terms in the c and d sector, like ∼ (cid:90) d q (cid:104) ρ L ( c ) ( q ) ρ L ( c ) ( − q ) − ρ R ( c ) ( q ) ρ R ( c ) ( − q ) (cid:105) V ( | q | ) ∼ (cid:90) d q (cid:90) d k (cid:90) d k c † k − q c k c † k + q c k (73) × i [( k − k ) × q ] V ( | q | ) , in the c sector, that can lead to p -wave (Pfaffian in the c sector and anti-Pfaffian in the d sector) instabilities.We will assume a particle-hole symmetry breaking andan effective Hartree-Fock-BCS Hamiltonian of the follow-ing form, H BCS = (cid:90) d k (2 π ) (cid:16) ∆ ∗ k d † k c k + ∆ k c † k d k (cid:17) + (cid:90) d k (2 π ) (cid:16) ˜∆ ∗ k c − k c k + ˜∆ k c † k c †− k (cid:17) . (74)Here ∆ k is defined in (48) and˜∆ ∗ k = | k | (cid:90) d q V ( | q | )4(2 π ) ( i ˆ k × q ) (cid:104) c † k − q c †− k + q (cid:105) . (75)We project to the α sector by taking c k → √ α k , (76)and d k → √ e iφ k α k . (77)Thus, H αBCS = (cid:90) d k (2 π ) (cid:34) | ∆ k | α † k α k + (cid:32) ˜∆ ∗ k α − k α k + h.c. (cid:33)(cid:35) . (78)Equations [6] that follow and need to be solved self-consistently are | ∆ k | = | k | (cid:90) d q V ( | q | )8(2 π ) | q | sin ( φ q ) (cid:18) − | ∆ k + q | − | ∆ q F | E k + q (cid:19) , (79)and | ˜∆ k | = | k | (cid:90) d q V ( | q | )16(2 π ) | q | sin ( φ q ) | ˜∆ k + q | E k + q , (80)where E q = ( | ∆ q | − | ∆ q F | ) + | ˜∆ q | . In this way, byspecifying V ( | q | ) that will include factors due to the pro-jection to a fixed Landau level, we can find amplitudesin ˜∆ k = | ˜∆ k | exp {− iφ k } and ∆ k = | ∆ k | exp {− iφ k } . X. PAIRING SOLUTIONS
The effective interaction in a fixed LL, m = 0 , , , . . . is given by the following expression V ( | q | ) = V c ( | q | ) e − | q | (cid:20) L m (cid:18) | q | (cid:19)(cid:21) , (81)where V c ( | q | ) = V | q | , (82)0 FIG. 4. The red line and the blue line represent solutions ofequations (79) and (80), respectively. The black dot marksthe corresponding self-consistent solution. represents the Coulomb interaction, l B = 1, and L m de-notes the Laguerre polynomial associated with a fixed LLwith the quantum number m .In the LLL, the equations (79) and (80) lead to self-consistent solutions with the following amplitudes, in thekinetic part, | ∆ k | = ∆ | k | where ∆ ≈ . V , andin the pairing part, | ˜∆ k | = ˜∆ | k | where ˜∆ ≈ . V .The self-consistent solution is numerically obtained using Mathematica with error estimation to 5 × − . Obvi-ously this represents a weak coupling case in which thekinetic part dominates. In Figure (4) is illustrated so-lutions of equations (79) and (80) with correspondingself-consistent solution.In the sLL, we found that equations (79) and (80) donot support a coexistence of (non-zero) kinetic and pair-ing amplitudes, and thus if only pairing is present it leadsto a gapless (critical) p -wave state at this level of approx-imation. We considered the question of coexistence whena cubic term is generated in the expansion (47), in the ki-netic part. The resulting equations are slightly modifiedequations (79) and (80) ( V → V , etc.), and lead to a solu-tion that describes a coexistence of pairing, | ˜∆ k | = ˜∆ | k | where ˜∆ ≈ . V , and now | ∆ k | = ∆ | k | where∆ ≈ . V ( l B = 1). The numerically obtainedsolution using Mathematica is shown in Figure 5. Theerror is estimated to 1 × − . Thus this is a strong cou-pling, weak pairing case that can be identified with theusual Pfaffian-state case in which all composite fermionsare paired in the same way of a p -wave. FIG. 5. The red line and the blue line illustrate solutions ofmodified equations (79) (with V → V , etc.) and (80), respec-tively. The black dot shows the corresponding self-consistentsolution with pairing | ˜∆ k | = ˜∆ | k | , and | ∆ k | = ∆ | k | . XI. GAUGE INVARIANCE ANDCONSTRAINTS
The basic (gauge) invariance that we have as we ap-proach the problem of electrons in a fixed Landau levelis the SU ( N ) invariance of the theory (the form of theHamiltonian, Lagrangian). The SU ( N ) invariance is theinvariance under unitary change of basis in the fixedLandau level. We also consider an enlarged space withunphysical degrees of freedom which contribute an ad-ditional redundancy in our description. The physicalstates become a unique - “spin-singlet” realization of the SU ( N ) symmetry on the unphysical degrees of freedom,i.e. the generators of the SU ( N ) transformation on theunphysical degrees of freedom should annihilate physi-cal states. In other words, the change of basis in theunphysical sector should not affects the physics. In the ν = 1 bosonic state we have only one kind of redundantparticle, and thus a single set of requirements, (5) and(4).In the ν = 1 / c and d . We can consider constraints (50) and(70). The diagonal constraints on a single orbital are in-compatible, and we may fix those to be either all (50)with n = n (cid:48) (“charge”) constraints, or all (70) with n = n (cid:48) (“spin”) constraints. By fixing the diagonal con-straints we will not obtain a single “spin-singlet” state.For example, fixing (50) in the case with two orbitals,the states c † n d † n (cid:48) | (cid:105) , (83)and c † n d † n (cid:48) | (cid:105) , (84)1(note the inversion of indices in the d sector as a conse-quence of the particle-hole transformation), can be con-sidered as eigenstates of ρ R ( c ) nn − ρ L ( d ) nn , and thus each re-alize a SU (2) spin-singlet. Obviously, this will lead tosectors of the theory with inhomogeneous distribution ofunphysical degrees of freedom. Secondly, this will leadto a description where SU ( N ) transformations are inde-pendent for each, c and d sector, which is unphysical; wewould expect that the change of basis in the unphysicalsector would affect both c and d . Thus we have two nat-ural possibilities for a unique state unaffected with the SU ( N ) transformation in the unphysical sector,( c † n d † n (cid:48) + c † n d † n (cid:48) ) | (cid:105) , (85) FIG. 6. An illustration of the unphysical sector (quantum-fluxexcitations) in state (85). which is characterized by ρ R ( c ) nn + ρ L ( d ) nn = 1 and the ex-pectation value of ρ R ( c ) nn − ρ L ( d ) nn with respect to this stateis zero, associated with “synchronized”, “charge” trans-formation, or ( c † n d † n (cid:48) + c † n d † n (cid:48) ) | (cid:105) , (86) FIG. 7. An illustration of the unphysical sector (quantum-fluxexcitations) in state (86). which is characterized by ρ R ( c ) nn − ρ L ( d ) nn = 0 and the expec-tation value of ρ R ( c ) nn + ρ L ( d ) nn − XII. CONCLUSIONS
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Here we give a short review of the basic structure ofNC gauge field theory based on the Seiberg-Witten (SW)construction, first presented in the context of string the-ory [14]. We review the case of Moyal-Weyl star-product.Generalization of the SW map for other types of star-product can be found in [15]. A more comprehensiveaccount on various aspects of NC geometry and NC fieldtheory can be found in [16].Let T A be a set of generators of a gauge group G . In-finitesimal SW gauge variation of an NC field Φ trans-forming in the fundamental (vector) representation of thegauge group (e.g. a matter field) is defined by δ SW Λ Φ = i Λ (cid:63) Φ , (87)where Λ stands for an NC gauge parameter. This trans-formation rule is analogous to the familiar one in ordinarycommutative gauge field theory, except for the noncom-mutative Moyal-Weyl star-product.There is a notorious problem concerning the closureaxiom for SW gauge transformations. Namely, assum-ing that NC gauge parameter is Lie algebra valued,Λ = Λ A T A , the commutator of two SW transformationsis given by[ δ SW (cid:63) , δ SW ]Φ = (Λ (cid:63) Λ − Λ (cid:63) Λ ) (cid:63) Φ (88)= 12 (cid:0) [Λ A (cid:63) , Λ B ] { T A , T B } + { Λ A (cid:63) , Λ B } [ T A , T B ] (cid:1) (cid:63) Φ . Due to the appearance of anticommutator { T A , T B } , in-finitesimal SW transformations do not in general close inthe Lie algebra of a gauge group. A way to surmount thisdifficulty is to assume that NC gauge parameter Λ be-longs to the universal enveloping algebra (UEA), which isalways infinite dimensional [17]. This, however, leads toan infinite tower of new degrees of freedom (new fields),which is not a preferable property. To see this, considera SW variation of the covariant derivative δ SW Λ D µ Φ = i Λ (cid:63) D µ Φ , (89)with D µ Φ = ∂ µ Φ − iV µ (cid:63) Φ. This implies the followingtransformation rule for the NC gauge potential, δ SW Λ V µ = ∂ µ Λ + i [Λ (cid:63) , V µ ] , (90) meaning that it is also an UEA-valued object, whichleaves us with an infinite number of new fields in thetheory (one for each basis element of UEA).Seiberg-Witten map resolves this issue by demandingthat NC fields can be expressed in terms of the NC pa-rameter θ , commutative gauge parameter λ = λ A T A ,commutative gauge potential v µ = v Aµ T A , and theirderivatives, Λ = Λ( θ, λ, v µ ; ∂λ µ , ∂v µ , . . . ) , (91) V µ = V µ ( θ, v µ , ∂v µ , . . . ) , (92)where dots stand for higher derivatives. In this way, NCtheory is defined by the corresponding commutative one.There are no new degrees of freedom, just new interactionterms in the NC action.NC gauge transformations are now induced by the cor-responding commutative ones, δ SWα
Λ = Λ( λ, v µ + δ α v µ ) − Λ( λ, v µ ) , (93) δ SWα V µ = V µ ( v µ + δ α v µ ) − V µ ( v µ ) , (94)with δ α v µ = ∂ µ α + i [ α, v µ ].From (88) and (91) follows a consistency condition forNC gauge parameter,Λ α (cid:63) Λ β − Λ β (cid:63) Λ α + i ( δ SWα Λ β − δ SWβ Λ α ) = i Λ − i [ α,β ] , (95)which can be solved perturbatively (this makes sense be-cause the star-product is also defined perturbatively). Tothis end we represent NC gauge parameter as an expan-sion in powers of NC parameter θ , with coefficients builtout of fields from the commutative theory,Λ = λ + θ Λ (1) + θ Λ (2) + . . . . (96)The first term in the expansion (zeroth order in θ ) is thecommutative gauge parameter λ = λ A T A .In the case of U (1) (cid:63) NC gauge theory, the NC gaugeparameter, up to first order in θ , is given byΛ = λ − θ (cid:15) αβ v α ∂ β λ + O ( θ ) , (97)where (cid:15) αβ ( α, β = 0 , ,
2) is an antisymmetric matrixdefined by (cid:15) i = 0 ( i = 1 ,
2) and (cid:15) = − (cid:15) = 1. Notethat we work in 2 + 1 dimensions; generalization to anarbitrary number of dimensions is straightforward.Using the expansion (97) and the transformation rule(87) we readily obtainΦ = φ + θ (cid:15) αβ v α ∂ β φ + iθ (cid:15) αβ v α v β φ + O ( θ ) . (98)Also, from (90) we get V µ = v µ − θ (cid:15) αβ v α ( ∂ β v µ + F βµ ) . (99)In connection to our model of NC gauge field the-ory, with left and right NC U (1) gauge transformations,3we combine NC fields c and d (describing compositefermions) into a doubletΨ = (cid:18) cd (cid:19) (100)Under NC gauge transformations (simultaneous left andright U (1) NC gauge transformation)Ψ → Ψ (cid:48) = U L (cid:63) Ψ (cid:63) U R (101)or infinitesimally δ Ψ = i (cid:0) Λ L (cid:63) Ψ + Ψ (cid:63) Λ R (cid:1) . (102)Left and right NC gauge parameters are labeled Λ L andΛ R , respectively. Covariant derivative of Ψ is given by D µ Ψ = ∂ µ Ψ − iV Lµ (cid:63) Ψ − i Ψ (cid:63) V Rµ = ∂ µ Ψ − iA µ (cid:63) Ψ − i Ψ (cid:63) ( a µ − A µ ) , (103)where we introduced left, V Lµ := A µ , and right, V Rµ := a µ − A µ , gauge potential. In terms of c and d we recover(62) and (63) D µ c = ∂ µ c − iA µ (cid:63) c − ic (cid:63) ( a µ − A µ ) ,D µ d = ∂ µ d − iA µ (cid:63) d − id (cid:63) ( a µ − A µ ) . (104)One can show that D µ Ψ transforms covariantly( D µ Ψ) (cid:48) = U L (cid:63) ( D µ Ψ) (cid:63) U R (105)provided that the NC gauge potentials V Lµ and V Rµ trans-form in the following way( V Lµ ) (cid:48) = U L (cid:63) V Lµ (cid:63) ( U L ) ∗ − i ( ∂ µ U L ) (cid:63) ( U L ) ∗ , ( V Rµ ) (cid:48) = ( U R ) ∗ (cid:63) V Rµ (cid:63) U R − i ( U R ) ∗ (cid:63) ( ∂ µ U R ) . (106) Left and right NC gauge parameter (up to first order)are given by (note the sign difference)Λ L = ˆΛ L − θ (cid:15) αβ ˆ A α ∂ β ˆΛ L , Λ R = ˆΛ R + θ (cid:15) αβ (ˆ a α − ˆ A α ) ∂ β ˆΛ R . (107)Likewise, left and right NC gauge potential are V Lµ = ˆ A µ − θ (cid:15) αβ ˆ A α (cid:16) ∂ β ˆ A µ + ˆ F Lβµ (cid:17) , (108) V Rµ = (ˆ a α − ˆ A α ) + θ (cid:15) αβ (ˆ a α − ˆ A α )( ∂ β (ˆ a µ − ˆ A µ ) + ˆ F Rβµ ) , where we introduced relevant gauge field strengthsˆ f αβ = ∂ α ˆ a β − ∂ β ˆ a α , ˆ F αβ = ∂ α ˆ A β − ∂ β ˆ A α , ˆ F Lαβ = ∂ α ˆ V Lβ − ∂ β ˆ V Lα = ˆ F αβ , ˆ F Rαβ = ∂ α ˆ V Rβ − ∂ β ˆ V Rα = ˆ f αβ − ˆ F αβ . (109)SW expansion of the NC matter field doublet isΨ = ˆΨ + θ (cid:15) αβ (cid:16) (ˆ a α − A α ) ∂ β ˆΨ − i ˆ a α ˆ A β ˆΨ (cid:17) , (110)and for its covariant derivative, D µ Ψ = ∂ µ Ψ − iV Lµ (cid:63) Ψ − i Ψ (cid:63) V Rµ = D µ ˆΨ + θ (cid:15) αβ (cid:104) (ˆ a α − A α ) ∂ β D µ ˆΨ − i ˆ a α ˆ A β D µ ˆΨ − ( ˆ f αµ − F αµ ) D β ˆΨ (cid:105) , (111)where D µ ˆΨ = ∂ µ ˆΨ − i ˆ a µµ