Fractional Quantum Hall States in Graphene
aa r X i v : . [ h e p - t h ] A p r UCDTPG 08-01arXiv:0805.2388
Fractional Quantum Hall States in Graphene
Ahmed Jellal a,b, ∗ and Bellati Malika b, † a The Abdus Salam International Centre for Theoretical Physics , Strada Costiera 11, 34014 Trieste, Italy b Theoretical Physics Group, Faculty of Sciences, Choua¨ıb Doukkali University , Ibn Maˆachou Road, PO Box 20, 24000 El Jadida, Morocco
Abstract
We quantum mechanically analyze the fractional quantum Hall effect in graphene. This willbe done by building the corresponding states in terms of a potential governing the interactionsand discussing other issues. More precisely, we consider a system of particles in the presence of anexternal magnetic field and take into account of a specific interaction that captures the basic featuresof the Laughlin series ν = l +1 . We show that how its Laughlin potential can be generalized todeal with the composite fermions in graphene. To give a concrete example, we consider the SU ( N )wavefunctions and give a realization of the composite fermion filling factor. All these results willbe obtained by generalizing the mapping between the Pauli–Schr¨odinger and Dirac Hamiltonian’sto the interacting particle case. Meantime by making use of a gauge transformation, we establish arelation between the free and interacting Dirac operators. This shows that the involved interactioncan actually be generated from a singular gauge transformation. ∗ [email protected], [email protected] † [email protected] Introduction
Nowadays the observation of the famous quantum Hall effect (QHE) [1] does not remain at thestage of the semiconductors, but it can be seen in other materials. This is for example the caseof graphene, which is a projected graphite of the group symmetry C into two-dimensional spaces.When such system is submitted to a perpendicular magnetic field, it appears an anomalous integerHall conductivity [2, 3] that is a manifestation of the relativistic particle motions. It is originated fromdifferent sources, which are the four-fold spin/valley and the Berry phase due to the pseudospin (orvalley) precision when a massless (chiral) Dirac particle exercises cyclotron motion. This result wastheoretically suggested by two groups [4, 5] independently.The emergence of QHE in graphene during 2005 offered a laboratory for different investigationsand applications of some mathematical formalism. Many of them brought from the QHE studies insemiconductors and used the Dirac operator as the starting point instead of the Landau Hamiltonian.Indeed, different questions have been partially or completely solved but still some attentions to bepaid to others. This concerns an eventual fractional quantum Hall effect (FQHE) [6] in graphene andrelated issues. Such interest can be linked to the fact that FQHE is a fascinating subject becausestarting from its appearance is still capturing a great attention. This is due to its relations to differentareas of physics and mathematics.There are many interesting theories have been appeared dealing with some problems related tothe anomalous FQHE, for instance one may refer to the papers [7, 8, 9, 10]. In particular, basedon different arguments some wavefunctions have been suggested as good candidates to describe thesubject [11, 12, 13]. In fact, some of them used the composite fermion approach [14] and other madea straightforward generalization of the Halperin theory [15] for non-polarized spin. However, somewavefunctions can be linked to each other and recovered from a general proposal as has been seenin [13].Another theory was proposed by the first author [13] who studied the effect exhibited by therelativistic particles living on two-sphere S and submitted to a magnetic monopole. In fact, hestarted by establishing a direct connection between the Dirac and Landau operators through thePauli–Schr¨odinger Hamiltonian H SPs . This was helpful in the sense that the Dirac eigenvalues andeigenfunctions were easily derived. In analyzing H SPs spectrum, he showed that there is a compositefermion nature supported by the presence of two effective magnetic fields. For the lowest Landau level(LLL), he argued that the basic physics of graphene is similar to that of two-dimensional electrongas, which is in agreement with the planar limit. For the higher Landau levels, he proposed a SU ( N )wavefunction for different filling factors that captures all symmetries. Focusing on the graphene case,i.e. N = 4, he gave different configurations those allowed to recover some known results.Motivated by different analysis concerning the anomalous FQHE and in particular [13], we developa real approach in order to describe the subject. Taking advantage of our knowledge about the phe-nomena in semiconductors, we introduce some effective interactions as relevant ingredient in formingthe Hamiltonian system. For this, we consider M -particles in the presence of a perpendicular magneticfield B where the interaction is taken to be two and three-body types. These have been successfullyapplied to the conventional FQHE, for instance one may see [16, 17]. Based on the connection be-tween the Pauli–Schr¨odinger and Dirac Hamiltonian’s, we construct an appropriate Dirac operator1hat captures the basic feature of the interacting system and where the Laughlin wavefunction [18] isits groundstate.Because of the basic physics is the same in both systems: semiconductors and graphene forthe Laughlin series, we start by establishing the corresponding effective potential. It will be gen-eralized to describe the composite fermions in graphene [19] of filling factor ν cf = 4 (cid:0) n + (cid:1) , with n = 0 , ± , ± , · · · . This of course captures the physics behind the fractional filling factors beyondLaughlin series l +1 , with l is integer, for FQHE in graphene. Moreover, we return to the SU ( N )wavefunctions [13] to firstly give a concrete example showing a particular type of potential. Secondly,we show how the corresponding filling factor can be linked to that generated from the compositefermions pictures [19]. Different discusions will be reported elsewhere about the matter.The present paper is organized as follows. In section 2, in order to show the difference betweenthe interacting and free particles cases, we start by studying a system without interaction. We makeuse an approach based on the mapping between different Hamiltonian’s to get the correspondingspectrum and eigenfunctions. In section 3, for the later convenience, we give some discussions about theLaughlin potential that captures the interaction effect. We introduce two and three-body interactionsand analyze the behaviour the new system in section 4. In particular, we show that the Laughlinwavefunction is a groundstate of the present system and derive the spectrum’s for the excited states.We note that the interactions can be generated from a singular transformation and allow us to establisha relation between both Dirac systems: free and interacting. In section 5, we consider the compositeDirac fermions in terms of an appropriate Jain potential, which can be obtained by making useof straightforward generalization of the Laughlin one. In this framework, we analyze the SU ( N )wavefunctions and show its basic features in section 6. Finally, we conclude and give some perspectivesof the present work. We start by developing our approach that will be subsequently generalized to the case of the interactingsystem. This concerns a mapping between different spectrum’s: Landau, Pauli–Schr¨odinger and Dirac.Indeed, let us consider one-relativistic particle living on the plane ( x, y ) in presence of a perpendicularmagnetic field B . This can be described by H PS = 12 m h ~σ · (cid:16) ~P − ec ~A (cid:17)i (1)where ~σ are the Pauli matrices and verify the usual relations, namely { σ i , σ j } = 2 δ ij , [ σ i , σ i ] = 2 ǫ ijk σ k . (2)We will see how the Pauli–Schr¨odinger Hamiltonian H PS can be used to get the Dirac operator for thepresent system. Specifically, it will be derived from the square root of H PS . This mapping is useful insense that the Dirac eigenvalues and eigenfunctions will be easily obtained.One way to establish the mentioned mapping is to express H PS in terms of the Landau Hamiltonian,which describe free particle. This connection has an important advantage because it leads to exactlyderive different quantities related to the subject. Thus, by choosing the symmetric gauge ~A = B − y, x ) (3)2e show that H PS can be mapped in the form H PS = H L H L ! − B − ! (4)where H L is simply the Landau Hamiltonian and reads as H L = 12 m (cid:16) ~P − ec ~A (cid:17) (5)which is extremely used in different contexts and in particular in analyzing QHE in the semiconductorsystems. To easiest way to resort different spectrum is to write H L in terms of the annihilation andcreation operators as H L = 14 ( a † a + aa † ) (6)where, in the complex notation z = x + iy , a and a † are given byˆ a = 2 ∂∂z + B z, ˆ a † = − ∂∂ ¯ z + B | z | . (7)They verify the commutation relation [ˆ a, ˆ a † ] = 2 B. (8)Since we are wondering to generalize different Hamiltonian’s entering in the game, let us convert H L into its analytical form. It is not hard to obtain H L = − (cid:26) ∂ ∂z i ∂ ¯ z i − B (cid:18) z ∂∂z − ¯ z ∂∂ ¯ z (cid:19) − B z (cid:27) . (9)Hereafter, we set the fundamental constants ( e, c, ~ , m ) to one. Clearly, the spectrum and eigenfunc-tions of H PS can be derived from that of H L .The above tools can be applied to analysis the anomalous QHE in graphene. Indeed in suchsystems, the two-Fermi points, each with a two-fold band degeneracy, can be described by a low-energy continuum approximation with a four-component envelope wavefunction whose componentsare labelled by a Fermi-point pseudospin = ± H D = i √ v F a † − ˆ a ! . (10)where v F ≈ c is the Fermi velocity, which will be set to one, and the many-body effects are neglected.Its spectrum can be determined in a simple way if we introduce its square. This is H D = 12 ˆ a † ˆ a
00 ˆ a ˆ a † ! . (11)It is related to the Pauli-Schr¨odinger Hamiltonian (4) up to some multiplicative constants. It is clearthat, H D is written in terms of the diagonal form of the Landau Hamiltonian (9). Therefore, itspectrum can easily be obtained.We start by solving the eigenvalue equation H D Ψ = E D Ψ (12)3ince H D has to do with the Landau Hamiltonian (9), the wavefunctions Ψ should be written in anappropriate form. This is Ψ m,n = ψ m,n ψ m − ,n ! (13)where the eigenfunctions ψ m,n are given by ψ m,n ( z, ¯ z ) = ( − m √ B m m ! p n +1 π ( m + n )! z n L nm ( z ¯ z e − B z ¯ z , m, n = 0 , , · · · . (14)The corresponding Landau levels are given by E D ( m ) = Bm. (15)We can show that the normalized eigenfunctions of H D take the formΨ m =0 ,n = 1 √ ψ | m | ,n − i ψ | m |− ,n ! (16)with the convention ψ − ,n = 0. Note that, the zero mode wavefunction isΨ (0 ,n ) = ψ ,n ! · (17)Their energy levels read as E D ( m ) = ± p B | m | . (18)This makes a difference with respect to the Landau spectrum for the present system. In fact, it has nozero energy and is discrete as well, unlike (18). These what make the integer QHE is an unconventionaleffect in graphene.The above analysis can easily be generalized to many-body system without interaction. This leadsto a spectrum as sum over all single particle ones and eigenfunctions as tensor products. However, wewill give a generalization of the system by introducing a kind of interaction that is behind FQHE insemiconductors and see what is its influence on graphene. To develop our main idea we need first to start from what we know so far about the Laughlin wave-function. This latter is involving a Jastrow factor that is originated from a specific interaction betweenparticles. In fact, it corresponds to an artificial model that captures the basic physics of the fillingfactor ν = l +1 . This is an typical example to get more general results, which concern the compositefermions as well as the SU (4) wavefunctions for graphene. For our task it is necessary to start fromthe Laughlin theory for FQHE in graphene.To talk about the Laughlin series for FQHE, it is convenient to introduce some discussions aboutLLL because it is a rich level and has many interesting properties. Indeed, in such level the Landausystem behaves like a non-commutative one, which is governed by the commutation relation[ z, ¯ z ] LLL = 2
B . (19)4his effect can be interpreted as follows. In the present level, the potential energy is strong enoughthan kinetic energy and therefore the particles are glue in the fundamental level. Using the sameanalysis as for the case of H L , is not hard to get a basis as set of the eigenstates. They are [20] | m i LLL = 1 √ m ! (ˆ a † | LLL ) m | i LLL . (20)where the annihilation and creation operators reduce now toˆ a | LLL = B z, ˆ a † | LLL = B z. (21)The projection into the complex plane leads to the eigenfunctionsΨ m ( z, ¯ z ) = r B m +1 m +1 πm ! z m e − B | z | . (22)As we will see very soon, this can be generalized to get more interesting results. In particular, thesewill offer a way to get in contact with the Laughlin theory [18] for FQHE.Returning now to discuss FQHE in graphene. In doing so, Let us consider M -particles in LLL,which of course means that all m i = 0 with i = 1 , · · · , M and each m i corresponds to the spectrum (16–18). The total wavefunction of zero-energy Landau level (17) can be written in terms of the Slaterdeterminant. This is ψ ( z, ¯ z ) = ǫ i ··· i M z m i · · · z m M i M exp − B M X i | z i | ! (23)where ǫ i ··· i M is the fully antisymmetric tensor and m i are integers. It is relevant to write this wave-function as Vandermonde determinant. We have ψ ( z, ¯ z ) = const M Y i 1) + 1] ! . (39)For later convenience, let us project these states on the complex plane. Indeed, by requiring that m = i − 1, we end up with the wavefunction ψ l L ( z, ¯ z ) = M Y k 00 ˆ A i ˆ A † i ! . (51)Summing over all M -particles and if we forget about all constants entering in different Hamiltonian’s,it is not hard to end up with the relation (cid:0) H intD (cid:1) = H intPS . (52)This equivalence shows that how the applied approach is relevant as well to analyze the interactingparticles. Remember that δ ( z i − z j ) and δ (¯ z i − ¯ z j ) can be dropped because they do not affect theantisymmetric wavefunction. Therefore, we can carry out our previous study concerning free Diracparticles and its connection to FQHE on graphene to the present case and underline the difference.It is clear that, (cid:0) H intD (cid:1) is written in terms of the diagonal form of the Hamiltonian (32) andtherefore its spectrum can easily be obtained. Then, the eigenstates of H intD take the form |{ m } 6 = 0 , η i D = |{| m i |} , η i L − i |{| m i | − } , η i L ! (53)and the zero mode is given by |{ } , η i D = |{ } , η i L ! . (54)The corresponding eigenvalues are E intD ( m i , η ) = ± vuut B (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M X i =1 m i + 12 ( η − M ( M − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (55)It can be projected on LLL that is realized by requiring that m i = 0. This operation leads to thelowest energy for the excited Dirac states, such as E intD ( m i = 0 , η ) = ± r 12 ( η − M ( M − . (56)Clearly, for η = 1 we get a zero energy mode. This is an agreement with the fully occupied state ν = 1see before fore the relativistic particles.To close this part, one can also report some discussions about the symmetry group that leavesinvariance the Dirac Hamiltonian for the present case. These of course will offer a theory groupapproach to analyze the interacting relativistic particles.10 .4 Gauge transformation Sometimes it is relevant to make the appropriate transformations in order to get some informationabout a system under consideration. This has an interst in sense that we can derive the integrabilityof the system from its old partner. For this, we are going to establish a concrete relation between thefree and interacting Dirac operators. This can be done by considering an appropriate singular gaugetransformation that captures the basic features the interacting term involved in different Hamiltonian’ssee before. It can also be seen as a kind of bosonization of the present system, for this we may referto the reference [16]. The relation can be served as a good candidate to test the previous analysisconcerning the interacting particle case.We start with a system of M -relativistic fermions of mass m in the presence of a external magneticfield. Without interaction, this system is described by the total Dirac Hamiltonian in the complexcoordinates, such as H D = i √ P Mi (cid:16) − ∂∂ ¯ z i + B z i (cid:17) − P Mi (cid:16) ∂∂z i + B ¯ z i (cid:17) . (57)This of course an immediate generalization of the Hamiltonian for one free particle. Therefore, itsspectrum is nothing but a summation over that of one particle and the wavefunctions are tensorproduct of one single eigenfunctions.To explicitly determine the element that governs such transformation, we adopt the method usedby Karabali and Sakita [16]. In doing so, let us consider two wavefunctions and establish a linkbetween them. This can be done by considering a singular gauge transformation, such asΨ( z , · · · , z N ) = U Φ( z , · · · , z N ) (58)where Ψ and Φ are, respectively, totally antisymmetric and symmetric wavefunctions. This transfor-mation can be identified to an element of the unitarty group U ( N ) and should be written in terms ofa suitable function Θ. This is U = e − i Θ( z , ··· ,z N ) . (59)Clearly from (58) we have some information about Θ. Indeed, it should be defined such that aninterchange of a pair of variables gives the constraintexp [ − i Θ( z , · · · , z i , · · · , z j , · · · , z N )] = − exp [ − i Θ( z , · · · , z j , · · · , z i , · · · , z N )] . (60)Consequentely, Θ should be realized in such way that (60) must be fullfiled. Thus, we map Θ in termsof our language as Θ( z , · · · , z N ) = η X i This work was completed during a visit of AJ to the Abdus Salam Centre for Theoretical Physics(Trieste, Italy) in the framework of junior associate scheme. 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