International Workshop on Quantum Field Theory under the Influence of External Conditions, QFEXT07, Leipzig, Germany Quantum Hall effect in graphene: A functional determinant approach 8 0 C G Beneventano and E M Santangelo ‡ § 0 Departamento deF´ısica,UniversidadNacionaldeLaPlata 2 InstitutodeF´ısicadeLaPlata,UNLP-CONICET n C.C.67,1900LaPlata,Argentina a E-mail: [email protected], J [email protected] 0 2 Abstract. We start the paper with a brief presentation of the main characteristics ] of graphene, and of the Dirac theory of massless fermions in 2+1 dimensions h obtained as the associated low-momentum effective theory, in the absence of t - external fields. We then summarize the main steps needed to obtain the Hall p conductivity in the effective theory at finite temperature and density, with e emphasisonitsdependenceonthephaseoftheDiracdeterminantselectedduring h the evaluation of the effective action. Finally, we discuss the behavior, under [ gauge transformations, of the contribution due to the lowest Landau level, and interpretgaugetransformationsasrotationsofthecorrespondingspinorsaround 2 themagneticfield. v 8 2 9 PACSnumbers: 11.10.Wx,02.30.Sa,73.43.-f 4 . 0 1. Introduction 1 7 Graphene is a bidimensional array of carbon atoms, packed in a honeycomb crystal 0 structure. Actually, each layer of a graphene sample can be viewed as either an : v individual plane extracted from graphite, or else as an array of unrolled carbon i X nanotubes. Quite unexpectedly, in 2004, stable monolayer samples of such material were obtained [1] and, in 2005, the Hall conductivity was measured in such samples, r a independently, by two groups [2]. More recently, a different behavior of the Hall conductivity was reported [3] for bilayer samples. The main difference between the behavior of the Hall conductivity of mono- and bilayer samples is in the height of the jump around zero carrier density (or, equivalently, chemical potential). Fromatheoreticalpointofview,themostremarkablefeatureofgrapheneisthat, in a smallmomentum approximation,the chargecarriersorquasi–particlesbehaveas two “flavors” (to account for the spin of the elementary constituents) of massless relativistic Dirac particles in the two non–equivalent representations of the Clifford algebra (corresponding to the two non–equivalentverticesin the first Brillouin zone), with an effective “speed of light” about two orders of magnitude smaller than c [4]. ‡ MemberofCONICET § MemberofCONICET Quantum Hall effect in graphene 2 In[5],weshowedthatafieldtheorycalculationatfinitetemperatureanddensity, based upon ζ function regularization of the Dirac determinant leads, in the zero − temperature limit, to a sequence of plateaux in the Hall conductivity consistent with the measured ones, each time the chemical potential goes through a nonzero Landau level. Moreover, it was shown in [6] that two of the three possible combinations of phasesoftheDiracdeterminantinbothnonequivalentCliffordrepresentationspredict abehavioraroundzerochemicalpotentialconsistentwiththeonesmeasuredinmono- and bilayer graphene. For a complete presentation of other approaches to the study of graphene see, for instance, [7], and references therein. Thispaperpresents,insection2,abriefintroductiontothestructureofgraphene, andtothederivationofthecontinuousDiraceffectivetheory,intheabsenceofexternal fields. Section3containsareviewofourpreviousresultsonthesubject,withemphasis ontheroleofthe phaseofthe determinantingivingrisetodifferentbehaviorsaround zero chemical potential. In section 4, entirely new results are presented. In that section,weallowforcomplexchemicalpotentials,andconcentrateonthecontribution due to the lowestLandaulevel, inorderto study the invarianceofthe effective action under large gauge transformations. By relating gauge transformations to rotations in the plane, we analyze the effect of a 2π rotation for each of the three possible combinations of phases in both representations, and identify, in the zero temperature limit, the resulting geometrical or Berry’s phases. Our conclusions are presented in section 5. 2. Structure of graphene. Effective continuous model In this section, we sketch the main steps leading to the effective Dirac model for graphene, in the absence of external fields. For more detailed presentations see, for instance, [4]. The structure of the direct lattice for graphene is shown in figure 1. The direct lattice is a superpositionoftwotriangularlattices,A andB.The generatorsoflattice A are a = √3a(1, √3), and a = √3a(1,√3), where a is the lattice spacing. The 1 2 −2 2 2 2 vectors s = a(0, 1), s = a(√3,1) and s = a( √3,1) connect each site in the 1 − 2 2 2 3 −2 2 lattice A to its nearest neighbor sites in the lattice B. The tight binding Hamiltonian can then be written as 3 H = t a (r)b(r+s )+b (r+s )a(r) , 0 † i † i − rX∈ΛAXi=1(cid:2) (cid:3) where t is the uniform hopping constant. a(k) a(r) In momentum space, with = e ikr , it reads b(k) − · b(r) (cid:26) (cid:27) rǫΛA (cid:26) (cid:27) P H = Φ(k)a (k)b(k)+Φ (k)b (k)a(k) 0 † ∗ † k X (cid:0) (cid:1) 3 with Φ(k)= t eik·si. After defining two-component spinors as − i=1 ψ(k) (a(k), b(kP))T ,ψ (k) a (k), b (k) , one gets † † † ≡ ≡ 0 (cid:0) Φ(k) (cid:1) H = . 0 Φ (k) 0 ∗ (cid:18) (cid:19) Quantum Hall effect in graphene 3 Figure 1. Directlatticeforgraphene H vanishes at the six corners of the first Brillouin zone. Among these, only two 0 are inequivalent, and can be chosen as 4π k=K = ,0 ; Φ(K )=0. ± ± 3√3a ± (cid:18) (cid:19) When H is linearized around these two points , k=K +p, one obtains 0 ± 0 p ip Hk=K±+p =c p ip x∓0 y . (1) x y (cid:18) ± (cid:19) This last expression shows that each Fermi point gives rise to an effective Dirac theory, with an effective “velocity of light” c = 3ta, in one of the two inequivalent 2 representations of the gamma matrices. Thus, the total Hamiltonian can be taken as the direct sum of both (equivalently, as the Dirac Hamiltonian in the reducible 4x4 representationoftheCliffordalgebra). Moreover,anoverallfactoroftwo(twofermion species or “flavors”) must be included to take the spin of the original particles into account. 3. The Hall conductivity from the interacting quantum field theory at finite temperature and density As shown in our previous work on the subject [5, 6], the Hall conductivity can be determined by first evaluating the partition function (equivalently, the effective action) for massless Dirac fermions at finite temperature and density, in two spacial dimensions, in the presence of an external magnetic field perpendicular to the plane, andthenperformingaboosttoareferenceframewithorthogonalelectricandmagnetic fields. Inthissection,wesketchourmainresultsinthosereferences,withemphasison the role played by the phase of the Dirac determinant, which appears when treating the infinite tower of states associated to the lowest Landau level. We first consider a single flavor,andoneofthe twononequivalentrepresentationsofthe Cliffordalgebra. Inordertoconsidertheeffectsduetofinitetemperatureanddensity,westudythe theory in Euclidean three-dimensional space, with a compact Euclidean “time” 0 ≤ x β,whereβ = 1 (here,k istheBoltzmannconstantandT isthetemperature). 0 ≤ kBT B We introduce the (real) chemical potential and the magnetic field through a minimal coupling of the theory to an electromagnetic potential A = ( iµ,0,Bx ). Natural units (c=~=1) will be used, unless otherwise stated. µ − e 1 In this scenario, the Euclidean effective action is given by log = log det(i/∂ Z − eA/) , where the subindex AP indicates that antiperiodic boundary conditions must AP Quantum Hall effect in graphene 4 be imposed in the x direction, in order to ensure Fermi statistics. Now, this is a 0 formalexpression,whichwe willdefine throughazeta-functionregularization[8], i.e., d (i/∂ eA/) d ω s S =log ζ(s, − AP)= − , (2) eff Z ≡−ds α −ds α (cid:23)s=0 (cid:23)s=0Xω (cid:16) (cid:17) where ω represents the eigenvalues of the Dirac operator acting on antiperiodic, square-integrable functions, and α is a parameter introduced to render the zeta function dimensionless (as expected on physical grounds, our final predictions will be α-independent). So, in order order to evaluate the partition function, we first determine the eigenfunctions, andthe correspondingeigenvalues,ofthe Dirac operator. We propose eiλlx0eikx2 ϕ (x ) π Ψ (x ,x ,x )= k,l 1 λ =(2l+1) . k,l 0 1 2 √2πβ χk,l(x1) l β (cid:18) (cid:19) Notethat,inthelastexpression,λ , l= ,..., aretheMatsubarafrequencies l −∞ ∞ adequate to the required antiperiodic conditions, while the continuous index k represents an infinite degeneracy in the x direction. 2 The resulting spectrum has two pieces: An asymmetric piece, associated to the lowest Landau level of the Hamiltonian: π ω =λ˜ , with λ˜ =(2l+1) +iµ and l = ,..., , l l l β −∞ ∞ with corresponding eigenfunctions ψk,l(x1)= eπB 14 e−e2B(x1−ekB)2 , (3) 0 ! (cid:0) (cid:1) and a symmetric piece ω = λ˜2+2neB with n=1,..., l = ,..., , l,n ± l ∞ −∞ ∞ correspondingtoeigenfunqctionswithbothcomponentsdifferentfromzero. Inallcases, the degeneracy per unit area is given by the well known Landau factor, ∆ = eB. L 2π Theasymmetricpartofthespectrumisquiteparticular. Inthefirstplace,asseen from (3), the corresponding eigenfunction is an eigenfunction of the Pauli matrix σ , 3 with eigenvalue +1. The eigenfunction with the opposite “chirality” was eliminated by the square integrability condition in x . Moreover, in the other nonequivalent 1 representation, this part of the spectrum appears with the opposite sign. Such transformation is equivalent to µ µ. This is nothing but charge conjugation → − (which,forrealµ,isalsoequivalenttoaparitytransformation[9]). Aswewilldiscuss in what follows, this part of the spectrum is the one which requires the consideration of a phase of the determinant when evaluating the effective action. Before going to such evaluation, it is interesting to note the invariance of the whole spectrum under µ µ+ 2ikπ. This invariance is a natural one, since such → β transformationspreservetheantiperiodicityoftheeigenfunctionsand,thus,theDirac statistics. They are nothing but the so-called large gauge transformations. The invariance of the effective action under such transformations is also required for topological reasons. We will discuss this point in more detail in section 4. As is clear from (2), in evaluating the effective action, one must perform the analytic extension of the contributions to the zeta function coming from both pieces of the spectrum, s ∞ π µ − ζ (s,µ)=∆ (2l+1) +i , 1 L αβ α l= (cid:20) (cid:21) X−∞ Quantum Hall effect in graphene 5 and s ∞ 2neB π µ 2 −2 ζ (s,µ,eB)=(1+( 1) s)∆ + (2l+1) +i . 2 − − L nX=1 " α2 (cid:18) αβ α(cid:19) # l= −∞ The analytic extension of ζ (s,µ,eB) is quite standard, and it relies mainly on 2 performingaMellintransformandmakinguseoftheinversionpropertiesoftheJacobi theta functions. A detailed presentation can be found in [5]. The final result for the contribution to the effective action coming from this piece is (always consideringonly one representation of the gamma matrices and one fermion species) SII =∆ β√2eBζ 1 +∆ ∞ log 1+e (√2neB µ)β 1+e (√2neB+µ)β . (4) eff L R −2 L − − − (cid:18) (cid:19) nX=1 h(cid:16) (cid:17)(cid:16) (cid:17)i The other nonequivalent representation of the Clifford algebra gives rise to an identical contribution, since this part of the spectrum is the same in both irreducible representations. Assaidbefore,theextensionofζ (s,µ,eB)requiresacarefulconsiderationofthe 1 phase of the determinant. In fact, ζ can be written as 1 s s s 2π − ∞ 1 µβ − ∞ 1 µβ − ζ (s,µ)=∆ (l+ )+i + (l+ ) i , (5) 1 L (cid:18)αβ(cid:19) "l=0(cid:20) 2 2π(cid:21) l=0(cid:20)−(cid:18) 2 − 2π(cid:19)(cid:21) # X X andthedefinitionoftheoverallminussigninthesecondsumdependsontheselection of the cut in the complex plane of eigenvalues. As discussed in detail in [6], the usual prescriptionistochoosethecutsuchthatonedoesnotgothroughvanishingarguments when continuously transforming eigenvalues with positive real part into eigenvalues with negative real part [10]. This prescription then gives rise to what will be called inthe followingthe standardphase ofthe determinant(characterizedfromnowonby κ= 1). One could certainly choose the opposite prescription, which we will call the − nonstandard phase (κ = +1). Once one of the phases is selected, the contribution of ζ tothe effectiveactioncanbeevaluatedbymakinguseofthe well-knownproperties 1 of the Hurwitz zeta function, to obtain µβ µβ SI (κ)=∆ log 2cosh( ) +κ| | . eff L 2 2 (cid:26) (cid:20) (cid:21) (cid:27) When this last contribution is added to the one in (4), one gets for the effective action µβ µβ 1 S (κ)=∆ log 2cosh( ) +κ| | +β√2eBζ eff L R 2 2 −2 (cid:26) (cid:20) (cid:21) (cid:18) (cid:19) + ∞ log 1+e−(√2neB−µ)β 1+e−(√2neB+µ)β . ) nX=1 h(cid:16) (cid:17)(cid:16) (cid:17)i From this last expression, the finite-temperature charge density can be obtained as j (κ) = e d log . In the zero-temperature (β ), and recovering physical 0 −β dµ Z → ∞ units, it reduces to (n+ 1+κ)ce2B j0(2ec2~Bn<µ2 <2eBc2~(n+1))= − 2 sign(µ), h Quantum Hall effect in graphene 6 3 2 2 2 1 1 1 -3-2-1 1 2 3 -3-2-1 1 2 3 -3-2-1 1 2 3 -1 -1 -1 -2 -2 -3 -2 Figure 2. Hall conductivity for different selections of the phase of the determinant. Left to right: K = 1, K = 2, K = 0. In all cases, the horizontal axisrepresentsνC =sgn(µ)µ2/2eB~c2 andtheverticalone,σxyh/4e2. where n=[ µ2 ], and [x] is the integer part of x. 2eB~c In orderto obtainthe Hallconductivity,one must performaboost to a reference frame with crossed electric and magnetic fields. The final contribution to the Hall conductivity from each fermion species and one irreducible representation is given by (n+1+κ)e2 [6] σxy = − h2 sign(µ). Now, the phases of the determinant in both irreducible representations can be selectedwiththesameorwithoppositecriteria. Whenthis istakenintoaccount,and anoverallfactorof2isincluded,totakebothfermionspeciesintoaccount,onobtains for the total zero-temperature Hall conductivity 4(n+ K)e2 σ = − 2 sign(µ), xy h where K =0 corresponds to selecting the standard phase of the determinant in both irreducible representations, K = 1 corresponds to choosing opposite criteria for the phases,andK =2,tochoosingbothphasesinthenonstandardway. Thedependence of the Hall conductivity on the classical filling factor (ν ) is presented in figure 2, for C the three values of K. From that figure, it is clear that the behavior of monolayer graphene, as presented in [2], corresponds to K =1, i.e., to choosing opposite phases ofthedeterminantinbothrepresentations(or,equivalently,ignoringthephaseinboth representations, as done in [11]). In fact, in this case the (rescaled) Hall conductivity shows a jump of height 1 for ν = 0, and further jumps of the same magnitude for C ν = 1, 2,.... Inturn,thebehaviorofbilayergraphene,asreportedin[3]isexactly C ± ± reproduced by K =2 (nonstandard selection of the phase in both representations). 4. Invariance under large gauge transformations. Interpretation in terms of rotations To analyze the physical meaning of the invariance of the effective action under large gauge transformations in this context, we go back to the zeta function associated to the asymmetric part of the spectrum, for two fermion species and one representation, this time allowing for an imaginary part in the chemical potential, µ˜ =µ+iγ, while alwayskeepingµ=0. Inthiscase,onemustbecarefulwhensplittingtheinfinitesum 6 asin(5)(detailofthe calculationswillappearin[12]). Infact,suchsplitting mustbe different for different γ-ranges, to make sure that all the eigenvalues in each infinite sum have a real part with the same sign, which is crucial in defining the phase. For Quantum Hall effect in graphene 7 example, for 1 < γβ < 1, one has −2 2π 2 SeIff(−12 < γ2πβ < 12)= −2∆L dds(cid:23)s=0(l∞=0[(2l+1)π/β+iµ−γ]−s X + ∞ e−isθ (2l+1)π/β+i(µ+iγ)e−iθ −s . ) l=0 X (cid:2) (cid:3) Now,the values ofθ suchthatthe secondterminthe RHSdoes vanisharethoseones for which, simultaneously, (2l+1)π/β+µsinθ γcosθ = 0 =µcosθ+γsinθ. As − before,weconsiderheretwodifferentdefinitionsofthephaseofthedeterminant,which correspond to the standard definition for the phase κ = 1, and to the nonstandard − oneκ=+1. Witheachoneoftheseprescriptions,the contributionofthe asymmetric spectrum to the effective a action is given by 1 γβ 1 (κ+1)β SI ( < < )=2∆ sgnµ(µ+iγ) eff −2 2π 2 L 2 (cid:26) +log e−β2(µ+iγ)(1+sgnµ)+eβ2(µ+iγ)(1−sgnµ) . (6) (cid:16) (cid:17)o Things are entirely different for γβ = 1. In this case, one mode in the infinite 2π ±2 sum defining the zeta function has a vanishing real part. A careful treatment shows that, at such points, SI is discontinuous. For instance, SI (γβ = +1) coincides eff eff 2π 2 with limγ2πβ→21− of (6). An equally carefully treatment of the case γ2πβ = −21 shows that SI (γβ = 1)=SI (γβ = 1). This analysis can be extended to other ranges eff 2π −2 eff 2π 2 of variation of γβ, to obtain, with k = ,..., , 2π −∞ ∞ 1 γβ 1 (κ+1)β 2kπ SI ((k )< (k+ ))=2∆ sgnµ[µ+i(γ )] eff − 2 2π ≤ 2 L 2 − β (cid:26) + log e−β2(µ+i(γ−2kβπ))(1+sgnµ)+eβ2(µ+i(γ−2kβπ))(1−sgnµ) . (7) (cid:16) (cid:17)o This expression shows that the contribution to the effective action of the nonsymmetric part of the spectrum, in this representation of the gamma matrices, is invariantunderlargegaugetransformations,nomatterwhichphaseofthedeterminant is selected. As already said, such transformations must constitute an invariance. In fact,anincreaseofiγinthechemicalpotentialcorrespondstothemultiplicationofthe eigenfunctions (3) with a phase, i.e., ψk,l(x)→eiγx0ψk,l(x). So, an increase iγ = 2βiπ is a pure gauge transformation which, moreover,preserves the antiperiodicity in x . 0 Due to the fact that these eigenfunctions are eigenfunctions of σ such that 3 σ ψ (x) = ψ (x), one can equivalently write gauge transformations in the form 3 n=o n=o ψk,l(x) eiσ232γx0ψk,l(x). This last expression shows that, as x0 grows from 0 to β, spinors →are rotated by 2γβ, since σ3 is the generator of rotations in the plane x x . 2 1 2 In particular, γ = 2π corresponds to a 4π-rotation around the magnetic field. Note β that not only the partition function, but the Abelian Chern-Simons term (and, thus, the effective action) is invariant under large gauge transformations. On the other hand,γ = π correspondstoa2π-rotation. Atfinitetemperature,suchtransformation β changesthestatisticstoabosonicone. Forκ=+1,italsogivesrisetoanoverallphase ofπperunitdegeneracyinthepartitionfunction. Suchphaseisthecontributionwhich survives in the zero temperature limit. The three possible combinations of phases of the determinant then give a total phase per unit degeneracy in the partition function Quantum Hall effect in graphene 8 of π (K = 1, which reproduces the behavior of the Hall conductivity for monolayer graphene), or 0 (both for K =0 and K =2, this last reproducing the behavior of the Hall conductivity for bilayer graphene). At his point, it is interesting to note that, in ordertohaveazero-temperaturepartitionfunctioninvariantunderrotationsof2πfor monolayer graphene, the reduced flux through a unit cell of area Ω must be given by Φ =Ω∆ =N,withN apositiveinteger. Thisispreciselytheconditionforphysical Φ0 L states to transform as unidimensional ray representations of the magnetic translation group [9]. 5. Conclusions The first conclusion, as already stated in [6], is that two of the three possible combinationsofphasesgivebehaviorsoftheHallconductivitywhichcoincidewiththe ones measuredinmono- andbilayergraphene. Inthe caseofbilayergraphene[3],the (rescaled)Hallconductivitypresentsajumpofheight2forν =0,andfurtherjumps C of height 1. The main point here concerns the positions of these subsequent jumps. As a matter of fact, according to figure 1.b in [3], these subsequent jumps appear for ν = 1, 2,..., which is exactly the behavior predicted, in our calculation, for C ± ± K =+2. However,thesamereferenceinterpretstheHallbehaviorofbilayergraphene throughatheoreticalpredictionfirstmade in [13]which, asdiscussedin[14], predicts a plateaux of larger width. Our calculation, instead, completely coincides with the measured behavior of the plateaux, both in height and width. An entirely new conclusion is that, in each representation, the effective action per unit degeneracy is invariant under large gauge transformations, with any of the two possible selections of phase. As a result, the invariance persists no matter which of the three possible combinations of phases is selected. Moreover, each of the two selections of phase in each representation corresponds to a different geometric phase undertherotationofspinorsalongaclosedpatharoundthemagneticfield(κ= 1: no − geometricphase; κ=+1: geometricphase ofπ). So, differentvalues ofK correspond to different total geometric phases per unit degeneracy, to be compared with the Berry phases studied, for instance, in [14]. Finally, by taking γβ = π, we note that, 3 for Φ =1,thesethreevaluesofK alsocorrespondtothethreenonequivalentunitary Φ0 representationsofthegeneratorofthecyclicgroupC ,whichistherelevantsymmetry 3 in the case of free graphene. To the best of our knowledge, the relation between the phase of the fermionic determinant and Berry’s phase hadn’t been noticed before. This point, as well as the connection with the magnetic translationgroup will be studied in more detail in [12]. 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