Electromagnetic response of composite Dirac fermions in the half-filled Landau level
EElectromagnetic response of composite Dirac fermions in the half-filled Landau level
Johannes Hofmann ∗ Department of Physics, Gothenburg University, 41296 Gothenburg, Sweden (Dated: February 25, 2021)An effective field theory of composite Dirac fermions was proposed by D. T. Son as a theory ofthe half-filled Landau level with explicit particle-hole symmetry. We compute the electromagneticresponse of this Son-Dirac theory on the level of the random phase approximation (RPA), wherewe pay particular attention to the effect of an additional composite-fermion dipole term that isneeded to restore Galilean invariance. We find that once this dipole correction is taken into ac-count, unphysical interband transitions and collective modes that are present in the response ofthe unmodified theory either cancel or are strongly suppressed. We demonstrate that this givesrise to a consistent theory of the half-filled Landau level valid at all frequencies, at least to leadingorder in the momentum. In addition, the dipole contribution modifies the Fermi-liquid response atsmall frequency and momentum, which is a prediction of the Son-Dirac theory within RPA thatdistinguishes it from a separate description of the half-filled Landau level by Halperin, Lee, andRead within RPA.
I. INTRODUCTION
The fractional quantum Hall effect (FQHE) in thelowest Landau level (LLL) is a prototypical exampleof a strong-interaction phenomenon, where due to thequenching of the kinetic electron energy in a magneticfield only a single (interaction) scale remains . Despitethe absence of a small parameter, significant progress hasbeen made by describing FQH states in terms of compos-ite fermions, which are quasiparticles formed of electronsand an even number of vortices . The field-theoreticaldescription of composite fermions is based on a Chern-Simons theory that is obtained from the Hamiltonian ofinteracting electrons in a magnetic field by a formallyexact singular gauge transformation, which attaches anumber of flux quanta to each electron . The advantageof this formulation is that standard many-body approx-imations — such as a mean-field approximation for theground state and a random phase approximation (RPA)for the fluctuations — provide an accurate descriptionof the FQHE. In particular, in the special case of the half-filled Landau level, mean field theory predicts that theAharonov-Bohm flux attached to each electron preciselycancels the external magnetic field and the compositefermions form a Fermi liquid, a field-theoretical analy-sis of which including RPA excitations was first given byHalperin, Lee, and Read (HLR) . There is considerableexperimental evidence for the existence of a compressiblestate of this form .In its original formulation, however, HLR theory in-cludes all electron Landau levels, and must be modifiedin the LLL limit to account for an effective mass renor-malization (which sets the interaction scale) and to en-sure Galilean invariance . A drawback of this mod-ified HLR description is that a particle-hole symmetry— which is an exact symmetry in the LLL and links theresponse at filling fractions ν and 1 − ν as well as con-strains the properties of the half-filled Landau level —is not apparent . This could either mean that calcu-lations within HLR theory have to be carefully revisited to establish consistency with particle-hole symmetry ,but it could also point to a breakdown of this frame-work . The latter point was addressed by Son, whoproposed an alternative effective Chern-Simons field the-ory of the half-filled Landau level in terms of compositeDirac fermions . The main difference between the Son-Dirac theory and HLR theory lies in the nature of low-energy excitations, with the Dirac composite fermionshaving an additional Berry phase . Possible discrep-ancies between the HLR and Son-Dirac theories are ofsignificant current interest, especially since recent exper-iments have been able to probe the FQHE while varyingthe electron density independently of a large magneticfield and thus probe effects of particle-hole symmetry .Even though the Son-Dirac theory is formulated interms of composite Dirac fermions, what is not expectedin the excitation spectrum is a Dirac cone, especiallyhigh-energy features associated with transitions betweena Dirac valence and conduction band — after all, thetheory is an effective approximation to the exact the-ory of non-relativistic composite fermions, where this isclearly absent . The response of the Son-Dirac the-ory should therefore not resemble the response of typicalDirac materials, such as graphene. However, what oughtto be true in principle is not always obvious in directcalculations. In this paper, therefore, we compute thedensity and current response function of the Son-Diractheory using the RPA and demonstrate how this givesrise to a consistent theory of the half-filled Landau level,at least at long wavelengths. This is not directly apparentsince the RPA links the response of the interacting sys-tem to the response of non-interacting 2D Dirac fermions,which naturally includes interband excitations andcollective modes that violate Kohn’s theorem (forgraphene, for example, these are well-established exper-imental features ). Indeed, as we show in this pa-per, such spurious contributions are suppressed only ifwe consider a Galilean-invariant modification of the Son-Dirac theory that includes an additional dipole term forthe composite Dirac fermion. The remaining response at a r X i v : . [ c ond - m a t . s t r- e l ] F e b q B B / v F S ( , q )/( q B ) Son Dirac = 1/2RPA0 1 2 3 4 5 6 q B B / v F S ( , q ) / ( q B ) intraband collective mode interband1/ MHLR
LLL
FIG. 1. Dynamic structure factor of the unmodified Son-Dirac theory at half-filling as predicted by the RPA. Thereis a Fermi-liquid intraband contribution at low frequencies ω < v F q , but also an additional high-frequency part caused byinter-band transition of the composite Dirac fermions. A col-lective mode at high frequencies and long wavelengths is alsoapparent. Bottom figure: Dimensionless dynamical struc-ture factor as a function of frequency in the long-wavelengthlimit (marked by a dashed line in the full figure). The high-frequency response (red shaded area) is unphysical. Inset:Dynamic structure factor as predicted by the modified HLRtheory, which shows no interband transitions. low-energy and small momenta is then consistent withthe predictions of HLR theory, but there are differencesin the excitation spectrum within RPA between the twotheories.This paper is structured as follows: We begin in Sec. IIby introducing the Son-Dirac theory and its symme-tries, and discuss aspects of the non-Galilean invariantresponse that motivate the present study. Section IIIcontains a field-theoretical derivation of the response inthe half-filled Landau level using the RPA. The results ofthis calculation are presented in Sec. IV, which discussesin particular the response at long wavelengths. The pa-per is concluded by a summary in Sec. V. There are twoappendices: The first App. A collects results for the re-sponse within a modified HLR theory projected to theLLL for reference. The second App. B computes the non-interacting response functions of two-dimensional Dirac q B B / v F S ( , q )/( q B ) Modified Son Dirac = 1/2indirect contribution
FIG. 2. Dynamic structure factor within RPA of the modifiedGalilean-invariant Son-Dirac theory that includes the dipolecorrection. Compared to Fig. 1, interband excitations and thecollective mode are now strongly suppressed. The same colorcoding is used in both figures for the intensity of the response. fermions used in the main text, which are linked to theelectron response via the RPA.
II. DEFINITIONS AND MOTIVATION
The theory proposed by Son is a Chern-Simons theoryfor Dirac fermions with a Son-Dirac Lagrangian density L SD = ψ † (cid:2) ( i (cid:126) ∂ t + ea ) + v F σ i (cid:0) i (cid:126) ∂ i + ea i (cid:1) (cid:3) ψ − e φ ε µνρ A µ ∂ ν a ρ + e φ ε µνρ A µ ∂ ν A ρ , (1)where φ = 2 π (cid:126) /e with e the electron charge, ψ is thetwo-component Dirac fermion field, a µ the Chern-Simonsgauge field, A µ the external vector potential, and theFermi velocity v F is an effective parameter that setsthe strength of the Coulomb interaction. A summationconvention is implied in this paper, where greek indicesrun over µ = 0 , ,
2, with 0 (or t ) a temporal index andthe latin index i = 1 , x, y ) a space index, such that ε µνρ is the total antisymmetric tensor with ε = 1.Particle-hole symmetry is realized as a combination oftime-reversal and charge conjugation, with a transfor-mation ( A (cid:48) ( t, x ) , A (cid:48) i ( t, x )) = ( − A ( − t, x ) , A i ( − t, x )),( a (cid:48) ( t, x ) , a (cid:48) i ( t, x )) = ( a ( − t, x ) , − a i ( − t, x )), and ψ (cid:48) ( t, x ) = − iσ ψ ( − t, x ) . Different from HLRtheory, the composite Dirac fermions do not coupledirectly to the external gauge field but only indirectlythrough the mixed Chern-Simons term [the first termin the second line of Eq. (1)]. In the mean field ap-proximation at half-filling, where the electron density is j = 1 / π(cid:96) B with (cid:96) B = (cid:112) (cid:126) /eB the magnetic length,the composite Dirac fermions experience no effectiveChern-Simons magnetic field (cid:104) b (cid:105) = (cid:104) ε ij ∂ i a j (cid:105) = 0. Onthat level, composite fermions have a valence and aconduction band with linear dispersion ± (cid:126) v F q , and(barring spontaneous symmetry breaking ) theyform a Fermi sea with Fermi momentum k F = 1 /(cid:96) B anda Fermi energy that is detuned from the Dirac pointby E F = (cid:126) v F /(cid:96) B . Note that Eq. (1) is an effectivefield theory, which may contain additional terms that,for example, involve higher derivatives or powers of thefields.The Son-Dirac theory (1) is not invariant under Galileitransformation. A modified version of the Son-Dirac the-ory (which we shall refer to as modified Son-Dirac the-ory) is brought to Galilean-invariant form by coupling thecomposite Dirac fermions directly to the external electricfield E i by adding a dipole term of the form L D = d · E (2)to the action (1) with E i = ∂ i A − ∂ A i and a composite-fermion dipole moment d i = ε ji B (cid:2) ψ † (cid:0) i (cid:126) ∂ j + ea j (cid:1) ψ + (cid:0) − i (cid:126) ∂ j + ea j (cid:1) ψ † ψ (cid:3) . (3)A Galilei transformation to a moving inertial framewith coordinates x (cid:48) = x − V t is then implemented by A (cid:48) ( t (cid:48) , x (cid:48) ) = A ( t, x ) + V i A i ( t, x ), A (cid:48) i ( t (cid:48) , x (cid:48) ) = A i ( t, x )[such that ε ij E (cid:48) i = ε ij E i + V j B and B (cid:48) = B ] and ψ (cid:48) ( t (cid:48) , x (cid:48) ) = ψ ( t, x ) (the Chern-Simons field a µ transformsin the same way as the field A µ ).To motivate the current investigation, consider thedensity response of the theory (1) without including thedipole term that ensures Galilean invariance. In thiscase, the frequency- and momentum-dependent densityresponse Π within the RPA is given by Π ( ω, q ) = (cid:18) q π (cid:126) (cid:19) K xx ( ω, x )] ∗ , (4)where K xx is the transverse non-interacting current re-sponse function of Dirac fermions [a discussion andderivation of this result is given in the remainder of thepaper]. Figure 1 shows a density plot of the correspond-ing dynamic structure factor S ( ω, q ) = 1 π Im Π ( ω, q ) , (5)where the long-wavelength region q → ω < v F q due to intraband excita-tions [i.e., particle-hole excitations within the conductionband], and further weight at larger frequencies due tointerband transitions between the valence and the con-duction band of the Dirac fermions. The weight of boththe intra- and interband excitation at long wavelengthsis of order O ( q ), and the inter-band contribution athigh frequencies decays as O ( q /ω ). There is no con-tinuous spectral weight in a wedge | ω − ω c | < v F q due to phase-space restrictions. In this region, the RPA pre-dicts a well-defined collective mode at long wavelengths q(cid:96) B / (cid:126) (cid:46) . ≈ . ω c with aresidue of order O ( q ). For comparison, we show as aninset in Fig. 1 the corresponding result for the dynamicstructure factor of a modified version of HLR theory thatis projected to the lowest Landau level [cf. App. A fora detailed discussion]. The HLR theory shows a (pre-sumably spurious) collective mode that decouples fromthe Fermi-liquid continuum at finite wave vectors, butcrucially, the response contains no interband transitions.While the high-frequency response as apparent inFig. 1 is typical for Dirac fermions, as discussed it is unex-pected for a theory of electrons in the half-filled Landaulevel, for which a Dirac cone does not exist. Indeed, theonly large-frequency response of electrons in a magneticfield is associated with transitions between electron Lan-dau levels, with a typical energy scale of the order ofthe cyclotron frequency , which is much larger than thescales considered here and projected out in a theory re-stricted to the LLL. In addition, the only collective finite-frequency mode is the magnetoplasmon, the frequency ofwhich is fixed by Kohn’s theorem again at the cyclotronfrequency . This magnetoplasmon exhausts the f -sumrule f ( q ) = (cid:90) ∞ dωπ ω Im Π ( ω, q ) (6)at long wavelengths and is thus the only mode that con-tributes at O ( q ) to the dynamic structure factor. Whenrestricted to the lowest Landau level, the f -sum rule is oforder O ( q ) , which is in contrast to the weight of theDirac inter-band spectrum in Fig. 1. Even if this con-tribution had the correct weight, the f -sum rule woulddiverge linearly on account of the 1 /ω high-frequency tail[cf. Fig. 1]. Note that spurious spurious interband excita-tions and a collective mode also appear in the transversecurrent response, the spectral function of which takes asimilar structure as the density response in Fig. 1.The aim of this paper is to demonstrate that the com-posite Dirac fermion theory (1) can be extended to giverise to a RPA response valid at all frequencies, at leastto leading order in the wave number. We show that thisis the case for the modified Son-Dirac theory with thedipole term (2) that restores Galilean invariance. As anillustration and preempting a main result derived in theremainder of this paper, Fig. 2 shows the response forthe Galilean-invariant theory, where the spurious high-frequency response is indeed strongly suppressed. Pre-vious literature focusses on the non-Galilean invarianttheory or on the semiclassical limit in the vicinity ofthe half-filled state ν = 1 / ± / N , for which theresponse is accurate in a 1 /N -expansion for small wavenumbers O (1 /(cid:96) B N ) and energies O ( (cid:126) v F /(cid:96) B N ). Such arigorous power-counting argument does not apply at half-filling, and a full effective theory of the half-filled Landaulevel will include additional terms beyond (1) and (2).Here, a consistent theory that eliminates high-frequencymodes will constrain the effective theory beyond the lead-ing order. III. RPA FOR THE HALF-FILLED LANDAULEVEL
In this section, we consider the functional integral ofthe Son-Dirac partition function and derive the effectiveaction to second order in fluctuations of the gauge fieldsaround the mean-field result, which gives the RPA re-sponse functions. To this end, we first summarize inSec. III A the various constraints as well as the conjugatedensity and current corresponding to the Lagrangian (1).In Sec. III B, we then consider the Euclidean path inte-gral and derive the effective action.
A. Constraints and conjugate fields
The action (1) is linear in the Chern-Simons field a ,which is a Lagrange multiplier that enforces a constrainton the composite fermion density J = ψ † ψ = B φ . (7)The external magnetic field thus sets the density of com-posite fermions. This is different from HLR theory,where the electron density (which in HLR is equal tothe composite fermion density) constrains the Chern-Simons magnetic field b . Likewise, the spatial compo-nents a i enforce a constraint on the composite fermioncurrent J i = v F ψ † σ i ψ = − φ ε ij E j + 12 φ ε ij E j = 0 . (8)The second term, which cancels the first contribution,arises from the dipole correction (2) in conjunction withthe constraint (7). Galilean invariance thus inducesa backflow correction such that the composite Diracfermion current does not respond to the electric field.Unlike in HLR theory, the composite particle density J and current J i in the Son-Dirac theory are not equalto the conserved electron density and current. The elec-tron density is conjugate to the field eA and given by j = 12 φ ( B − b ) − ∂ i d i , (9)where b = ε ij ∂ i a j and d i is the dipole moment definedin Eq. (3). The last term is just the polarization chargeof dipoles with dipole moment d i . Using the definitionof the covariant derivative D i in Eq. (3), this expressioncan be cast in a different form, j = B φ − ε ij ∂ i (cid:16) − i (cid:126) B ( ψ †↔ ∂ j ψ ) (cid:17) . (10) Identifying the external magnetic field B with the densityof composite fermions, Eq. (7), density fluctuations arethus linked to the vorticity of the Dirac field. Likewise,the current conjugate to the external field eA i is j i = ε ij φ ( E j − e j ) + ∂ d i + ε ji ∂ j m (11)with e j = ∂ j a − ∂ a j and m = ε jk E k B i (cid:126) ( ψ †↔ ∂ j ψ ). Thelast two terms again follow from the dipole term (2) thatrestores Galilean invariance. The second-to-last terms isthe contribution induced by electric dipoles with dipolemoment d i , and the last term — which arises from vary-ing with respect to the magnetic field in the denominatorof Eq. (3) — is characteristic for the current induced bymagnetic dipoles with magnetization M = m ˆ e z . On amean field level, the definition of the density (9) alongwith ν = φ B (cid:104) j (cid:105) = 12 (12)implies that the expectation value of the Chern-Simonsmagnetic field (cid:104) b (cid:105) is zero, such that the composite Diracfermions do not experience an effective magnetic field.As we consider an isotropic system, the corrections tothe particle density and current arising from the dipoleterm (2) do not contribute to the mean field result. Theywill, however, change the fluctuations (which now coupledirectly to the external gauge field through the dipoleterm) and thus the response functions. B. Random phase approximation
In this section, we derive the linear response functionfor the density and current of the Son-Dirac on the levelof the RPA. To this end, we consider the Euclidean pathintegral and expand around the mean-field saddle pointup to second order in the external gauge fields. The ker-nel of this expansion is related to the response functionsby analytical continuation. As discussed in the introduc-tion, the advantage of the RPA approximation is that theresponse functions of the interacting theory are expressedin terms of the free non-interacting response functions of2D Dirac fermions, which can be computed in closed an-alytical form (cf. App. B).The starting point is the Euclidean partition function Z [ A ] = (cid:90) D [ ψ † , ψ, a µ ] e − S E [ ψ † ,ψ,a µ ,A µ ] , (13)where S E is the Euclidean action corresponding to theLagrangian (1) [i.e., obtained after a Wick rotation toimaginary time t = − iτ (cid:126) ], L ESD = ψ † (cid:2) ( ∂ τ − ea ) + v F σ i (cid:0) − i (cid:126) ∂ i − ea i (cid:1) (cid:3) ψ − d i E i + e φ ε µνρ A µ ∂ Eν a ρ − e φ ε µνρ A µ ∂ Eν A ρ , (14)where the derivative is now ∂ Eµ = ( i (cid:126) ∂ τ , ∂ x ). In the fol-lowing, we denote by ¯ A µ a background-field configurationwith constant magnetic field ¯ B such that B = ¯ B + δB and E i = δE i , i.e., we split off the gauge field fluctuationsas A µ = ¯ A µ + δA µ . (15)In addition, we have ¯ a i = 0 in the half-filled Landau level,and split off fluctuations in the Chern-Simons field as a = ¯ a + δa and a i = δa i . Within linear response, (cid:104) j µ ( x ) (cid:105) = (cid:90) dy Π µν ( x, y ) eδA ν ( y ) , (16)where the density with µ = 0 is given in Eq. (9) and thecurrent with µ = i in Eq. (11) and we use a three-vectornotation x = ( τ, x ) for the coordinates. The responsekernel is given byΠ µν ( x, y ) = − δ ln Z [ δA ] δ ( eδA µ ( x )) δ ( eδA ν ( y )) (cid:12)(cid:12)(cid:12)(cid:12) A µ = ¯ A µ . (17)We shall work in Coulomb gauge where ∇ · A = 0. InFourier space with an external momentum oriented alongthe y -axis, q = (0 , | q | ), this implies A y = 0 [note that wewill continue to use greek indices for the summation overthe 0 and x component]. In this convention, Π denotesthe density response function, Π xx the transverse currentresponse functions, and Π x the mixed density-currentresponse.The Son-Dirac action is quadratic in the compositefermion fields, such that the Grassmann integral can beperformed directly. This gives an effective Euclidean ac-tion S eff = − Tr ln[ − G − ]+ eφ (cid:90) q (cid:110) A T ( − q ) C ( q )2 a ( q ) − A T ( − q ) C ( q )4 A ( q ) (cid:111) , (18)where G − is the inverse Green’s function, and we define A = ( A , A ) and C ( q ) = (cid:18) − iq y iq y (cid:19) . (19)In position space, the Green’s function can be written ina form that separates out the fluctuations, G − ( x ) = G − ( x ) − V ( x ) , (20)where G is the bare propagator with Fourier transform G − ( q ) = i (cid:126) ω − ( v F (cid:126) q · σ − ¯ a ) . (21)This is the free propagator of two-dimensional Dirac elec-trons, where the mean-field contribution ¯ a acts as achemical potential. The Fourier transform of the fluc-tuation terms reads V (1) ( k | q ) = − eδa ( q ) − v F σ i ( eδa i ( q )) G ( k ) k − φ C µν ( q ) q φ C µν ( q ) q (a) eδa ( q ) k eδa i ( q ) kk + q ℓ B ( k j + q j ) T jν ( q ) eδA ν ( q ) − v F σ i − − ℓ B k j e ¯ B T jν ( q )( iq y ) kkkk − e ¯ B T ν ( q ) eδA ν ( q ) eδA ( − q ) eδA ν ( q ) eδa ( − q ) kk + qk + q (b) FIG. 3. (a) Feynman rules for the Dirac propagator and theChern-Simons terms. The Dirac field is indicated by a contin-uous line, the Chern-Simons gauge field by a dashed line, andthe external vector potential by a wavy lines. (b) Feynmanrules for the vertex terms that couple the composite Diracfield to the Chern-Simons gauge field and the external vectorpotential. Terms on the left-hand side contribute at leadinglinear order in the gauge field fluctuations, and terms on theright-hand side at second order. For the latter, we only showmomentum configurations that will contribute a diamagneticterm to the response. + ( (cid:126) k + (cid:126) q / × δ E ( q )¯ B (22)to leading order in the fluctuations and V (2) ( k | q, q (cid:48) ) = δ E ( q ) × eδ a ( q (cid:48) )¯ B − ( (cid:126) k + (cid:126) ( q + q (cid:48) ) / × δ E ( q )¯ B δB ( q (cid:48) ) (23)to second order, where the electric field is in the plane[we neglect fluctuations beyond quadratic order, whichwill not contribute to the RPA response]. Pictograph-ically, the fluctuations V (1) and V (2) describe vertexterms that couple the Dirac electrons to the external andChern-Simons gauge fields, respectively. These verticesare shown in Fig. 3, where it is convenient to introducethe conversion matrix T iν ( q ) = (cid:18) iq y ω (cid:19) , (24)which maps the external vector potential to the elec-tric field fluctuation, ε ij δE j ( q ) = [ T ( q )] iν δA ν ( q ). InFig. 3, continuous lines denote the Dirac field, the wavyline the external field, and the dashed line the Chern-Simons field. Only the second vertex contains Pauli ma-trices, while the remaining terms are diagonal in spinor (a) eδa µ ( − q ) eδa ν ( q ) eδA ν ( q ) eδA µ ( − q ) eδA ν ( q ) eδA µ ( − q ) eδa µ ( − q ) eδA ν ( q ) eδa µ ( − q ) eδA ν ( q ) eδA ν ( q ) eδA µ ( − q ) eδa µ ( − q ) eδA ν ( q ) (b) FIG. 4. Diagrams contributing to the effective action up tosecond order in the gauge field fluctuations. (a) Contributionsat linear order. The first two diagrams represent tadpole dia-grams, some of which are zero. (b) Contributions at quadraticorder in the gauge fields. The first line shows paramagneticresponse functions, the second line the Chern-Simons terms,and the third line diamagnetic terms. space. The last term in Eq. (22) and both terms in (23)are due to the dipole term. The remaining Feynmanrules are then as usual, where one imposes momentumand energy conservation at each vertex and integratesover each undetermined loop-momentum with measure (cid:82) k = (cid:82) d ( (cid:126) ω ) d k / (2 π ) .In the following, we use the decomposition (20) to ex-pand the trace of the logarithm (18) to second order inthe field fluctuations δa µ and δA µ . At the leading lin-ear order, the effective action reads [omitting terms thatevaluate to zero] S (1)eff = − (cid:90) k tr[ G ( k )] δa (0) + ¯ B φ δa (0) − ¯ B φ δA (0) , (25)where the trace runs over the spinor indices. The corre-sponding Feynman diagrams are shown in Fig. 4(a). The first term in Eq. (25) follows from the leading-order ex-pansion of the logarithm in Eq. (18) [diagrammatically,this is the first tadpole diagram in Fig. 4(a), with thesecond evaluating to zero], the second line is the mixedChern-Simons term, and the last line is the Chern-Simonsterm for the external gauge field. Evaluated at the sad-dle point, this contribution has to vanish. Indeed, vary-ing with respect to δa and δa i , we reproduce the con-straints (7) and (8) to leading order, (cid:104) J (cid:105) = (cid:90) k tr[ G ( k )] = ¯ B φ , (26) (cid:104) J i (cid:105) = v F (cid:90) k tr[ G ( k ) σ i ] = 0 . (27)The value of the mean field contribution ¯ a is adjustedto set the density (26). Relation (27) is satisfied by dintof the symmetry properties of free Dirac fermions [thecorresponding term is omitted in Eq. (25)]. In addition,the first variation with respect to δA µ gives a mean-fielddensity of n = ¯ B/ φ and a vanishing mean field current,as expected for the half-filled Landau level.To second order in the field fluctuation, the effectiveaction is [again avoiding terms that evaluate to zero] S (2)eff = 12 (cid:90) q (cid:110) e δa T ( − q ) K ( q ) δa ( q ) − e φ δa T ( − q ) (cid:2) R ( q ) T ( q ) (cid:3) δA ( q ) − e φ δA T ( − q ) (cid:2) T T ( − q ) R T ( − q ) (cid:3) δa ( q )+ e δA µ ( − q ) (cid:2) T T ( − q )∆( q ) T ( q ) (cid:3) δA ( q ) (cid:111) − e φ (cid:90) q δa T ( − q ) C ( q ) δA ( q )+ e φ (cid:90) q δA T ( − q ) C ( q ) δA ( q )+ 1 e ¯ B (cid:90) k tr[ G ( k )] (cid:90) q eδa ( − q ) [ T ( q )] ν eA ν ( q ) . (28)Diagrammatically, the various contributions are shownin Fig. 4(b). The first four terms in Eq. (28) in curlybrackets [first line in Fig. 4(b)] are contributions to aparamagnetic response, where the Dirac fermions are in-tegrated out at the one-loop level. They are defined as K µν ( q ) = (cid:90) k tr (cid:18) G ( k ) G ( k + q ) v F G ( k ) σ G ( k + q ) v F σ G ( k ) G ( k + q ) v F σ G ( k ) σ G ( k + q ) (cid:19) (29) R νi ( q ) = 4 π(cid:96) B (cid:90) k tr (cid:32) − (cid:126) k x G ( k ) G ( k + q ) ( (cid:126) k y + (cid:126) q y / G ( k ) G ( k + q ) v F (cid:126) k x G ( k ) σ G ( k + q ) − v F ( (cid:126) k y + (cid:126) q y / G ( k ) σ G ( k + q ) (cid:33) (30)∆ ij ( x, y ) = (cid:96) B (cid:90) k tr (cid:32) k x G ( k ) G ( k + q ) k x ( k y + q y / G ( k ) G ( k + q ) k x ( k y + q y / G ( k ) G ( k + q ) ( k y + q y / G ( k ) G ( k + q ) (cid:33) . (31)These “Dirac Lindhard functions” can be evaluated inclosed analytical form, which is done in App. B. Themixed response function R ( q ) as well as the direct dipoleresponse to the external field ∆( q ) are due to the dipoleterms in Eqs. (9) and (11). Returning to Eq. (28), thenext two terms in the second and third line [second linein Fig. 4(b)] arise from the Chern-Simons terms. Finally,the last term [third line in Fig. 4(b)] is a diamagnetic con-tribution. This term cancels with the x C ( q ) = (cid:18) − iq y /ω (cid:19) (32)such that the mixed Chern-Simons term reads − e φ (cid:82) q δa T ( − q ) ˜ C ( q ) T δA ( q ). A similar cancellationbetween the dipole correction and the Chern-Simonsterm was noted in Ref. .The RPA approximation consists of bringing the effec-tive action to quadratic form in the Chern-Simons fluctu-ation δa µ , in which case the Gaussian path integral overthat field decouples. This is accomplished by shifting eδa ( q ) → eδa ( q ) − φ [ K − ]( R + ˜ C ) T δA ( q ) , (33)which gives the full effective action at second order in theexternal field: S eff = − (cid:90) q [ T eδA ( − q )] T (cid:8) ∆ + (cid:16) π (cid:126) (cid:17) × ( R + ˜ C ( − q )) T K − ( − q )( R + ˜ C ( q )) (cid:9) [ T eδA ( q )]+ e φ (cid:90) q δA T ( − q ) C δA ( q ) . (34)Using Eq. (17), this effective action determines the linearresponse functions. The retarded response functions arethen obtained by analytic continuation from Euclideanimaginary frequencies to real frequencies iω → ω + i IV. RESULTS
In this section, we discuss the results for density, trans-verse current, and Hall response function. Using the theresult (34) to compute the response function (17) givesΠ µν ( q ) = 14 π (cid:126) C µν − [ T ( − q )∆ T ( q )] µν + (cid:16) π (cid:126) (cid:17) (cid:2) T ( − q )( R + ˜ C ) T K − ( R + ˜ C ) T ( q ) (cid:3) µν . (35)There are three separate contributions to this function:The first term arises from the AdA term; the second oneis a direct response of non-interacting dipoles that couple directly to the external field; and the third term is the in-direct response, where composite fermions couple to theexternal probe through the Chern-Simons field. Differ-ent from the non-Galilean invariant theory, this indirectcoupling is no longer merely mediated by the mixed
Ada
Chern-Simons terms but may occur through the current-dipole response R of composite fermions.As derived in Appendix B, the free Dirac responsefunctions without magnetic field have the followingnonzero components: K µν = (cid:18) K K xx (cid:19) (36) R νi = (cid:18) R R x (cid:19) (37)∆ ij = (cid:18) ∆
00 ∆ (cid:19) . (38)These response functions and their analytic continuationto real frequency that gives the retarded Dirac responseare calculated in App. B. In terms of these components,we obtain the following results for the retarded responsefunctions:Π x ( ω, q ) = − iq y eφ (39)Π ( ω, q ) = (cid:18) q π (cid:126) (cid:19) Z L ( ω, q )[ K xx ( ω, q )] ∗ + q ∆ ( ω, q ) (40)Π xx ( ω, q ) = (cid:18) q π (cid:126) (cid:19) Z T ( ω, q )[ K ( ω, q )] ∗ + ω ∆ ( ω, q ) (41)with the dimensionless residue function Z L ( ω, q ) = |R x ( ω, q )) | (42) Z T ( ω, q ) = | ωq R ( ω, q ) | . (43)Note that the absence of a term of order O (1) in theresidue (42) is due to the cancellation of the Chern-Simons term with the tadpole correction as discussedfollowing Eq. (28).For reference, we also note the result for the densityand transverse current response in the non-Galilean in-variant theory [i.e., without the dipole term (2)] first de-rived by Son . The density response is stated in Eq. (4)and transverse current response isΠ xx ( ω, q ) = (cid:18) q π (cid:126) (cid:19) K ( ω, q )] ∗ . (44)There is no direct dipole response ∆, and the residueterms Z L and Z T are equal to unity. As discussed in theintroduction, these response functions show an unphysi-cal behavior at finite frequencies, which is rectified whenincluding the dipole corrections. In the following sec-tions, we will discuss the properties of the full responsefunctions (40) and (41) in detail.Before we proceed, note that it is straightforward toextend our calculation to a more general case that in-cludes an additional interaction potential . On the levelof the RPA, this term is taken into account as a Hartreecorrection that changes the effective vector potential by∆ A ( x ) = (cid:90) d x (cid:48) V ( x − x (cid:48) ) (cid:104) δj ( x (cid:48) ) (cid:105) . (45)Electrons are then assumed to respond to the Hartreepotential in addition to the external vector potential δA µ .In Fourier space, we have (cid:104) δj (cid:105) = Π[ δA + V (cid:104) δj (cid:105) ], whereΠ µν is the RPA response derived previously without theinteraction potential and V = (cid:0) V ( q ) 00 0 (cid:1) . The full RPAresponse ˜Π is then linked to the response Π without thepotential by ˜Π − = Π − − V. (46)In particular, for the density response, we have ˜Π − =Π − − V ( q ). A. Hall response function
The first response (39) is the Hall response function,which is completely fixed by the
AdA
Chern-Simons termand receives no corrections on the RPA level for the ver-sions of the Son-Dirac theory discussed here. It is con-nected to the Hall conductivity by σ H = lim ω → lim q y → ie q y Π x ( ω, q y ) = e h . (47)This result agrees with the exact long-wavelength limitpredicted by particle-hole symmetry . Particle-holesymmetry also predicts a subleading correction of order O ( q ), σ H = e h (1 − q (cid:96) B /
4) for v F q (cid:28) ω , which isnot reproduced by the RPA calculation. However, thisproblem is shared between HLR and the Son-Dirac the-ory in the form considered in this paper, and the lattertheory incorporates this correction if half the action ofa fully filled Landau level is added to the Son-Dirac ac-tion , S = − eφ (cid:90) q A ( − q ) 1 − q − q (cid:96) B / − q (cid:96) B / q(cid:96) B A ( q ) , (48)with further modifications if an explicit Coulomb inter-action term is taken into account. Note that this termdoes not affect the density and current response functiondiscussed in the next sections. B. Density response function
Consider first the density response (4) of the unmodi-fied Son-Dirac theory, which is proportional to the inverseDirac current response K xx . In the long-wavelength limit, the Dirac current response takes the form (in di-mensionless notation where ν = ω(cid:96) B /v F , x = q(cid:96) B , andˆ K xx = 2 π (cid:126) (cid:96) B K xx /v F ):ˆ K xx ( ν, x ) = −
12 + ν (cid:12)(cid:12)(cid:12)(cid:12) ν − ν (cid:12)(cid:12)(cid:12)(cid:12) + i πν ν − . (49)The full expression valid at all momenta is given inApp. B. The real part of this response crosses zero ata finite frequency ν = 1 . O ( x /ν ) that leads to aUV-divergence of the f -sum rule (6). In the static limit,the Dirac current response readsˆ K xx ( ν, x ) = (cid:18) − πx − √ x − x + x x (cid:19) Θ( x − . (50)This function vanishes for momenta q(cid:96) B <
2, which islinked to a divergent orbital susceptibility of Dirac elec-trons at finite detuning . For the Son-Dirac theory,this implies a divergent static response function, which ischanged to a linear function of momentum if a Coulombinteraction is included, the latter result being consistentwith HLR theory . Finally, in the Fermi-liquid scal-ing regime at small momentum and frequency, we have(defining the scaling variable s = ω/v F q )lim ω,q → S SD ( ω, q ) = q (cid:96) B π (cid:126) v F √ − s s Θ( s < . (51)Indeed, neglecting the high-frequency response present inthe Son-Dirac theory, the density response (51) at smallfrequencies is equal to the HLR result [App. A] if weidentify the Fermi velocity in the Son-Dirac theory andthe effective mass parameter in HLR theory in the natu-ral way v F = (cid:126) k F /m ∗ . The real parts of the density re-sponse are equal as well, such that both theories predict a(sub)diffusive mode with frequency ν = − ix ( α + x ) ,where α = e / (cid:126) v F the dimensionless Coulomb interac-tion strength . Moreover, evaluating the f -sum rule us-ing the low-frequency response (51), we obtain the stan-dard scaling ¯ f ( q ) = π q(cid:96) B ) , (52)as required in the LLL from the Girvin-Mac Donald-Platzman algebra . Evaluating the static structure fac-tor in this limit gives an asymptotic IR-divergent scaling S ( q ) = O (( q(cid:96) B ) log( q(cid:96) B )), consistent with a compress-ible phase at half-filling and again the same as for HLRtheory [cf. App. A]. Of course, these sum rule calcu-lations ignore the spurious interband excitations and col-lective modes discussed previously, which give a divergentcontribution to the sum rules. v F q Z T /( ql B ) Z L ( v F q / ) FIG. 5. Small-momentum and small-energy scaling limit ofthe dipole residue terms Z L and Z T as a function of the scal-ing parameter s = ω/v F q . Both residues decay rapidly forfrequencies larger than ω > v F q , which leads to a strong sup-pression of the response at large frequencies ω ∼ (cid:126) v F /(cid:96) B . Consider now the density response (40) of the modifiedSon-Dirac theory, which involves the density responseof the original Son-Dirac theory modified by a dipoleresidue factor Z L as well as a direct response term thatdescribes the direct coupling of dipoles to the externalprobe. Crucially, the dipole residue Z L [Eq. (42)] leavesthe low-frequency response of the Son-Dirac theory dis-cussed above unchanged but strongly suppresses the spu-rious large-frequency response. To see this, consider thelow-momentum scaling form of (42):lim ω,q → Z L ( ω, q )= s < − s + 8 s − s (2 s − √ s − s > . (53)This result is shown as a continuous orange line in Fig. 5,and the full result for this part of the dynamics struc-ture factor is shown in Fig. 2. For ω < v F q the residueterm in Eq. (53) is unity [such that the dynamic struc-ture factor in the scaling regime is unchanged from theSon-Dirac results (51)] and then decays very quickly[on a scale of O ( q )] to zero with an asymptotic form(1 / s ) = v F q / ω . The spurious interband transi-tions and the collective mode (which set in at a muchlarger frequency) are then suppressed by four further or-ders of magnitude as O ( q ) at small momentum. Indeed,as is apparent from Fig. 2, this suppression of spectralweight at ω > v F q holds for all momenta. In particular,if one restricts the response to the Chern-Simons contri-bution, the f -sum rule is finite for all momenta, whichis shown in Fig. 6. At small momentum, the f -sum ruletakes the form (52), shows a kink at the point where thespurious collective mode joins the particle-hole contin-uum, and then crosses over to an asymptotic power-lawscaling O (( q(cid:96) B ) ).The full RPA response (40) contains an additional di-rect dipole-dipole response term q ∆ . In the small-momentum limit, this response reads (in dimensionless q B f ( q ) / ( q l B ) FIG. 6. f -sum rule of the indirect density response functionto the modified Son-Dirac response. form ˆ∆ = 2 π (cid:126) ∆ /v F (cid:96) B )ˆ∆ ( ν, x )= − x (4 + 3 ν )8 ν + 3 x ν
32 ln (cid:12)(cid:12)(cid:12)(cid:12) ν − ν (cid:12)(cid:12)(cid:12)(cid:12) + i iπνx
32 Θ( ν − O ( q ) in the low-momentum limit and thus sub-leading compared to theoriginal Son-Dirac theory, where they are of order O ( q ).However, beyond this order, due to the linear frequencybehavior (54), sum rules at arbitrary momentum are nolonger finite, and higher-order corrections to the Son-Dirac theory will have to be taken into account to ob-tain a consistent theory of the half-filled Landau level.At quadratic order, where the high-frequency response isabsent, the dipole response will affect the Fermi liquidscaling regime, where it contributes a termlim ω,q → S MSD ,d ( ω, q ) = q (cid:96) B π (cid:126) v F s (cid:112) − s Θ( s <
1) (55)to the dynamic structure factor in addition to Eq. (51).The static response of the full result is unchanged, butthere is added spectral weight near threshold as ω (cid:39) v F q . Note that this direct dipole contribution to theRPA response of the Son-Dirac theory is not containedin the RPA response of the modified HLR theory.To illustrate the difference between the RPA responseof the Son-Dirac theory, the modified Son-Dirac theoryas well as the modified HLR theory, we show in Fig. 8the dynamic structure factor of the modified HLR theory,the Son-Dirac theory, and the modified Son-Dirac theory(top to bottom row) including a dimensionless Coulombinteraction strength α = 0 . r s = 2 α when setting v F = (cid:126) k F /m ∗ ]. The panels on the left-hand side showa density plot of the dynamic structure factor and plotson the right-hand side show it as a function of frequencyfor five fixed momenta q(cid:96) B = 0 . , . , . , . , and 0 . q B B / v F S ( , q )/( q B ) Modified Son Dirac = 1/2direct contribution
FIG. 7. Spectral function of the direct dipole response func-tions q ∆ as a function of momentum and frequency. For small momenta, the Coulomb interaction is sublead-ing and the dominant feature is the low-frequency diver-gence (51) that is also sketched at the bottom plot ofFig. 1. This divergence is cut off at finite momenta bythe Coulomb interaction, such that the response is lin-ear at small frequencies. At larger frequency, there isadditional incoherent spectral weight that slowly decaysup to the phase-space boundary ω = v F q . As apparentfrom the figure and discussed in this section, the RPAresults of Son-Dirac theory and the modified HLR the-ory in this regime are very similar. There is, however, aclearly visible difference compared to the modified Son-Dirac theory, which has increased incoherent weight nearthreshold. This distinguishes the RPA response of themodified Son-Dirac theory from the RPA response of themodified HLR theory. C. Transverse current response function
A similar discussion as for the density response func-tion applies to the transverse current response func-tion (41). Figure 9 shows from left to right the dynamicstructure factor as computed using the non-Galilean re-sponse (44), the indirect Galilean-invariant response thatarises from the coupling to the Chern-Simons field [firstterm in Eq. (41)] as well as the direct dipole response[second term in Eq. (41)]. In the small-momentum limit,the Dirac density response is given by (in dimensionlessform ˆ K = 2 π (cid:126) v F (cid:96) B K )ˆ K ( ν, x ) = − x ν + x ν ln (cid:12)(cid:12)(cid:12)(cid:12) ν − ν (cid:12)(cid:12)(cid:12)(cid:12) + iπx ν Θ( ν − , (56)which as for the density response gives rise to a pole at ν = 1 . O ( ωq ). The small- q B B / v F MHLR
LLL B / v F S ( , q )/ q = 0.50.00 0.25 0.50 q B B / v F Son-Dirac 0.00 0.25 0.50 B / v F q B =0.1 q B =0.2 q B =0.3 q B =0.4 q B =0.5 q B B / v F Modified Son-Dirac 0.00 0.25 0.50 B / v F FIG. 8. Dynamic structure factor of the modified HLR the-ory (top row), Son-Dirac theory (middle row) and modifiedSon-Dirac theory (bottom row) for a fixed Coulomb interac-tion strength α = 0 .
5. While HLR and Son-Dirac theoriesmake similar predictions, there is a difference compared tothe modified Son-Dirac theory, which is marked by a shiftin the incoherent spectral weight closer to the particle-holethreshold. momentum limit of the dipole residue Z T is given bylim ω,q → Z T ( ω, q ) = s − s s ≤ (cid:16) s − s √ s − (cid:17) s > . (57)which at large frequencies decays as 9 q / ω , thus sup-pressing the collective mode and interband contributions.This result is shown as a continuous green line in Fig. 5.Different from the density response function, the dipoleresidue is not equal to unity in the Fermi-liquid scaling re-gion and thus the response differs from the non-Galileanresult (44). Indeed, the corresponding spectral functiontakes the formlim ω,q → Im Π xx,a ( ω, q ) = v F q (cid:96) B π (cid:126) s (1 + 3 s )4 √ − s Θ( s < . (58)The small-momentum limit of the additional di-rect dipole response ω ∆ is (in dimensionless form1 q B B / v F Im xx ( , q )/ Son Dirac = 1/2 q B B / v F Modified Son Dirac = 1/2indirect contribution q B B / v F Modified Son Dirac = 1/2direct contribution
FIG. 9. Spectral function Im Π xx of the transverse current response as a function of momentum and frequency for (a) thenon-Galilean invariant theory [Eq. (44)]; (b) the full indirect response [Eq. (41)] including the dipole residue; and (c) the directdipole response functions ω ∆ . ˆ∆ = 2 π (cid:126) ∆ /v F (cid:96) B )ˆ∆ ( ν, x )= − x (12 + ν )8 + x ν
32 ln (cid:12)(cid:12)(cid:12)(cid:12) ν − ν (cid:12)(cid:12)(cid:12)(cid:12) + i iπν x
32 Θ( ν − ω,q → Im Π xx,b ( ω, q ) = v F q (cid:96) B π (cid:126) s √ − s Θ( s <
1) (60)to the spectral function. As for the density function, thiscontribution is negligible in the static limit but changesthe response near threshold. The enhancement of spec-tral weight is both due to the dipole residue, which di-verges as Z T ∼ / | − s | near s (cid:39) . V. SUMMARY
In summary, we have discussed the response of theSon-Dirac theory of the half-filled Landau level using therandom phase approximation. If a dipole correction isincluded that renders the Son-Dirac theory Galilean in-variant, we find that the response is free of spurious high-frequency excitations, which are natural features in theresponse of Dirac materials but not expected for a the-ory of the LLL. Furthermore, while the response of theSon-Dirac theory reproduces many features of HLR the-ory at small frequencies and momenta, the dipole termincreases the response near the particle-hole boundary.This is a prediction of the Son-Dirac theory within RPAthat differs from the modified HLR theory within RPA.
ACKNOWLEDGMENTS
I thank G. M¨oller and C. Turner for discussions, and N.Cooper for discussions and comments on the manuscript.This work is supported by Vetenskapsr˚adet (grant num-ber 2020-04239).
Appendix A: Halperin-Lee-Read response in theLLL
This appendix collects for reference results for the elec-tron response within HLR theory at half-filling restrictedto the lowest Landau level, which are compared in themain text with the findings of this paper.The starting point is a modified version of HLR theorythat contains as a parameter an interaction-renormalizedmass m ∗ instead of the bare mass m (for a review seeRef. ). In order to restore Galilean invariance, thistheory must include a Fermi-liquid back-flow term, thestrength of which is set by the p -wave Fermi liquid pa-rameter F = mm ∗ −
1. Within this framework, responsefunctions can be computed in closed analytic form onthe level of the RPA and linked to the response func-tions of the non-relativistic two-dimensional electron gas(2DEG). The LLL limit corresponds to the limit of fixedfilling fraction ν ∼ O ( m ) and diverging cyclotron fre-quency ω c ∼ O ( m − ). This is the limit of vanishingbare mass, m → F → ∞ , respectively. Units arenow set by the renormalized mass, such that energies aremeasured in units of (cid:126) ˜ ω c = (cid:126) eB/m ∗ with a dimensionlessfrequency ν = (cid:126) ω/ (cid:126) ˜ ω c and wave number x = q(cid:96) B [notethat this unit of energy differs from the standard defini-tion of a 2DEG E F = (cid:126) k F / m ∗ with Fermi momentum k F = 1 /(cid:96) B by a factor of 2].2At half-filling, the dimensionless density response is [ ˆΠ ] − ( ν, x ) = m ∗ π (cid:126) [Π ] − = [ ˆ K ] − − r s x − ν x − x (cid:16) [ ˆ K xx ] − − (cid:17) − , (A1)where ˆ r s = r / ˜ a is the dimensionless Coulomb inter-action strength, defined as the ratio of average elec-tron spacing r = 1 / √ πn and interacting Bohr radiusˆ a = (cid:126) /m ∗ e . The expression contains the standard2DEG response functions Re ˆ K αα ( ν, x ) = − h α ( ν, x ) − x g α> (cid:0) νx + x (cid:1) B1 g α> (cid:0) νx + x (cid:1) − g α> (cid:0) νx − x (cid:1) C1 g α> (cid:0) νx + x (cid:1) + g α> (cid:0) νx − x (cid:1) D1 (A2)Im ˆ K αα ( ν, x ) = − x g α< (cid:0) νx + x (cid:1) − g α< (cid:0) νx − x (cid:1) A1 − g α< (cid:0) νx − x (cid:1) B10 C10 D1 , (A3)where the four regions are sketched in Fig. 10(a), with h α ( ν, x ) = − α = 0 x
12 + ν x α = x (A4) g α> ( ν, x ) = √ z − α = 01 − z √ z − α = x (A5) g α< ( ν, x ) = √ − z α = 01 − z √ − z α = x . (A6)The full results is shown in the inset of Fig. 1. It isinteresting to note that there is a collective mode thatemerges at larger momentum q(cid:96) B (cid:38) . r s = 0)ˆΠ ( ν, x ) = (cid:18) −
14 + i √ − s s (cid:19) x s < (cid:18) −
14 + √ s − s (cid:19) x s > x = q B = B / v F C1 B1 D1A1 q B f ( q ) / ( q l B ) q B S ( q ) FIG. 10. Left panel: Phase-space regions used to parameter-ize the free-particle response functions of 2D electrons. Rightpanels: f -sum rule (top) and static structure factor (bottom)of HLR theory projected onto the lowest Landau level. Thecontinuous blue lines indicate the full result, and the dashedred line excludes a collective mode that decouples from thecontinuum response at finite momentum. The results for the f -sum rule and the static struc-ture factor within HLR theory are shown in Figs. 10(b)and 10(c), where the blue continuous lines denote thefull contribution of both the continuum and the collec-tive mode that emerges at larger momenta [cf. the insetin Fig. 1], and the red dashed line excludes the collectivemode. The projected f -sum rule takes the value (52) forall momenta. Moreover, the static structure factor van-ishes as S ( x ) = O ( x log x ), the same as discussed in themain text after Eq. (52). Appendix B: Linear response of composite Diracfermions
In the main text, the response of electrons in the half-filled lowest Landau level is expressed via the RPA interms of six linearly independent response functions ofnon-interacting 2D Dirac fermions. In this appendix,we compute these response functions in closed analyticalform. Two of these Dirac response functions — the den-sity and current response — are discussed in the grapheneliterature as well , while to the best of our knowl-edge the four independent response functions involvingthe dipole term are new. Here, we derive all responsefunctions for completeness.Evaluating the frequency integral in Eqs. (29)-(31)gives the standard Lindhard form of the response func-tion M µν ( iω, q )= (cid:90) d k (2 π ) (cid:88) ss (cid:48) f ( E s (cid:48) ( k (cid:48) )) − f ( E s ( k )) E s (cid:48) ( k (cid:48) ) − E s ( k ) − i (cid:126) ω F µνss (cid:48) ( k , k (cid:48) ) , (B1)3 I m K q B =0.5 0 1 2 3 4 5 60.00.51.01.52.0 q B =1 0 1 2 3 4 5 60.00.51.01.52.0 q B =2 0 1 2 3 4 5 60.00.51.01.52.0 q B =30 1 2 3 4 5 6 B / v F R e K B / v F B / v F B / v F FIG. 11. Dimensionless density response function of Dirac electrons for four different momenta (left to right panel) q(cid:96) B =0 . , ,
2, and 3. The top panels show the imaginary part and the bottom panels the real part. I m K xx q B =0.5 0 1 2 3 4 5 60.00.51.01.52.02.53.0 q B =1 0 1 2 3 4 5 60.00.51.01.52.02.5 q B =2 0 1 2 3 4 5 60.00.51.01.52.02.5 q B =30 1 2 3 4 5 6 B / v F R e K xx B / v F B / v F B / v F FIG. 12. Dimensionless transverse current response function of Dirac electrons for four different momenta (left to right panel) q(cid:96) B = 0 . , ,
2, and 3. The top panels show the imaginary part and the bottom panels the real part. where M is either one of the response functions K , R , and ∆ discussed in the main text, f ( E s ( k )) = n F ( E s ( k ) − µ ) is the Fermi-Dirac distribution for a sys-tem with chemical potential µ ( µ = E F = (cid:126) v F /(cid:96) B at zerotemperature), E s ( k ) = ± (cid:126) v F k is the single-particle en-ergy of a Dirac state with wave vector k and chiral bandindex s = ±
1, and F µνss (cid:48) ( k , k (cid:48) ) is the matrix element of thevertex terms between two single-particle eigenstates | k s (cid:105) and | k (cid:48) s (cid:48) (cid:105) . Here, F µνss (cid:48) ( k , k (cid:48) ) is formed by an appropri-ate product of two of the following single-particle matrixelements: (cid:104) k (cid:48) s (cid:48) | I | k s (cid:105) = 12 (1 + ss (cid:48) e i ( φ − φ (cid:48) ) ) (B2) (cid:104) k (cid:48) s (cid:48) | σ x | k s (cid:105) = 12 ( se iφ + s (cid:48) e − iφ (cid:48) ) (B3) (cid:104) k (cid:48) s (cid:48) | − i ↔ ∂ x | k s (cid:105) = k cos φ + k (cid:48) cos φ (cid:48) ss (cid:48) e i ( φ − φ (cid:48) ) )(B4) (cid:104) k (cid:48) s (cid:48) | − i ↔ ∂ y | k s (cid:105) = k sin φ + k (cid:48) sin φ (cid:48) ss (cid:48) e i ( φ − φ (cid:48) ) ) , (B5)where φ and φ (cid:48) are the angles of the vectors k and k inthe complex plane.It is convenient to split the response into an intrinsicpart M µν − that describes the response of a system with µ = 0 (i.e., where the Fermi level is precisely at the Diracpoint) as well as an extrinsic part M µν + that contains thecorrection of finite detuning:4 I m q B =0.5 0 1 2 3 4 5 60.00.51.01.52.0 q B =1 0 1 2 3 4 5 60123456 q B =2 0 1 2 3 4 5 6024681012 q B =30 1 2 3 4 5 6 B / v F R e B / v F B / v F B / v F FIG. 13. Dimensionless longitudinal dipole response function of Dirac electrons for four different momenta (left to right panel) q(cid:96) B = 0 . , ,
2, and 3. The top panels show the imaginary part and the bottom panels the real part. I m q B =0.5 0 1 2 3 4 5 60.00.20.40.60.81.0 q B =1 0 1 2 3 4 5 601234 q B =2 0 1 2 3 4 5 6024681012 q B =30 1 2 3 4 5 6 B / v F R e B / v F B / v F B / v F FIG. 14. Dimensionless transverse dipole response function of Dirac electrons for four different momenta (left to right panel) q(cid:96) B = 0 . , ,
2, and 3. The top panels show the imaginary part and the bottom panels the real part. M µν ( iω, q ) = M µν − ( ω, q ) + M µν + ( ω, q ) (B6) M µν − ( iω, q ) = (cid:90) d k (2 π ) (cid:20) f ( E − ( k (cid:48) )) − f ( E − ( k )) E − ( k (cid:48) ) − E − ( k ) − i (cid:126) ω F µν −− ( k , k (cid:48) ) + f ( E − ( k (cid:48) )) F µν + − ( k , k (cid:48) ) E − ( k (cid:48) ) − E + ( k ) − i (cid:126) ω − f ( E − ( k )) F µν − + ( k , k (cid:48) ) E + ( k (cid:48) ) − E − ( k ) − i (cid:126) ω (cid:21) (B7) M µν + ( iω, q ) = (cid:90) d k (2 π ) (cid:20) f ( E + ( k (cid:48) )) − f ( E + ( k )) E + ( k (cid:48) ) − E + ( k ) − i (cid:126) ω F µν ++ ( k , k (cid:48) ) + f ( E + ( k (cid:48) )) F µν − + ( k , k (cid:48) ) E + ( k (cid:48) ) − E − ( k ) − i (cid:126) ω − f ( E + ( k )) F µν + − ( k , k (cid:48) ) E − ( k (cid:48) ) − E + ( k ) − i (cid:126) ω (cid:21) . (B8)In the following, we evaluate both contributions in turn,computing first the Euclidean response at imaginary fre-quency M µν ( iω, q ) and then performing the analytic con-tinuation iω → ω + i q = k F x = x /(cid:96) B (B9) (cid:126) ω = E F ν = (cid:126) v F ν/(cid:96) B (B10)In addition, we introduce dimensionless response func- tions, which we shall indicate by a hat, as follows: K ( iω, q ) = N ˆ K ( ν = ωv F k F , x = qk F ) (B11) K xx ( iω, q ) = v F N ˆ K xx ( ν = ωv F k F , x = qk F ) (B12)∆ ( iω, q ) = (cid:96) B N ˆ∆ ( ν = ωv F k F , x = qk F ) (B13)∆ ( iω, q ) = (cid:96) B N ˆ∆ ( ν = ωv F k F , x = qk F ) (B14) R ( iω, q ) = v − F ˆ R ( ν = ωv F k F , x = qk F ) (B15) R x ( iω, q ) = ˆ R x ( ν = ωv F k F , x = qk F ) , (B16)5 I m q B =0.5 0 1 2 3 4 5 60.00.51.01.52.0 q B =1 0 1 2 3 4 5 60.00.51.01.52.0 q B =2 0 1 2 3 4 5 601234 q B =30 1 2 3 4 5 6 B / v F R e B / v F B / v F B / v F FIG. 15. Dimensionless density-dipole response function of Dirac electrons for four different momenta (left to right panel) q(cid:96) B = 0 . , ,
2, and 3. The top panels show the imaginary part and the bottom panels the real part. I m x q B =0.5 0 1 2 3 4 5 61.00.50.00.51.01.52.0 q B =1 0 1 2 3 4 5 61.00.50.00.51.01.52.0 q B =2 0 1 2 3 4 5 61.00.50.00.51.01.52.0 q B =30 1 2 3 4 5 6 B / v F R e x B / v F B / v F B / v F FIG. 16. Dimensionless current-dipole response function of Dirac electrons for four different momenta (left to right panel) q(cid:96) B = 0 . , ,
2, and 3. The top panels show the imaginary part and the bottom panels the real part. where N is the density of states of a non-interactingsystem at the Fermi surface N = k F π (cid:126) v F = 12 π (cid:126) v F (cid:96) B . (B17)Note that if we define an effective mass m by v F = (cid:126) k F /m , this is equivalent to the non-interacting densityof states of the 2DEG, for which N = m/ π (cid:126) .
1. Intrinsic response
In this section, we consider the intrinsic contributionto the response M µν − , which is the full response if theFermi energy is at the band-touching Dirac point. In thiscase, only interband transitions (which have ss (cid:48) = −
1) contribute, Eq. (B7), such that M µν − ( iω, q ) = − (cid:90) d k (2 π ) (cid:20) F µν + − ( k , k (cid:48) ) E − ( k (cid:48) ) − E + ( k ) − i (cid:126) ω − F µν − + ( k , k (cid:48) ) E + ( k (cid:48) ) − E − ( k ) − i (cid:126) ω (cid:21) . (B18)To evaluate this part, it is convenient to shift the integra-tion variable k → k − q / q = (0 , q ) and transformto an elliptic coordinate system with ± q / , k x = q µ sin ν (B19) k y = q µ cos ν, (B20)where µ > − π < ν < π . The Jacobian of thetransformation is (cid:12)(cid:12)(cid:12)(cid:12) ∂ ( k x , k y ) ∂ ( µ, ν ) (cid:12)(cid:12)(cid:12)(cid:12) = q µ + sin ν ) . (B21)6In these coordinates, the matrix elements are (cid:104) k + q s (cid:48) | I | k − q s (cid:105) = 1sinh µ − i sin ν × sinh µ ss (cid:48) = +1 − i sin ν ss (cid:48) = − (cid:104) k + q s (cid:48) | σ x | k − q s (cid:105) = s sinh µ − i sin ν × cosh µ sin ν ss (cid:48) = +1 i sinh µ cos ν ss (cid:48) = − (cid:104) k + q s (cid:48) | k x | k − q s (cid:105) = q µ sin ν sinh µ − i sin ν × sinh µ ss (cid:48) = +1 − i sin ν ss (cid:48) = − (cid:104) k + q s (cid:48) | k y | k − q s (cid:105) = q µ cos ν sinh µ − i sin ν × sinh µ ss (cid:48) = +1 − i sin ν ss (cid:48) = − . (B25)The square of the joint denominator in all of these ex-pressions cancels with the Jacobian. Furthermore, notethe distance to the focal points | k + q | = q µ + cos ν ) (B26) | k − q | = q µ − cos ν ) , (B27)which implies that the denominator in Eq. (B18) onlydepends on µ , and the ν -integration in Eq. (B18) canbe performed directly. The subsequent µ -integration iselementary but requires a cutoff Λ = ˆΛ /(cid:96) B in momen-tum space to regulate the expression. The result of thiscalculation isˆ K − ( iν, x ) = πx √ x + ν (B28)ˆ K xx − ( iν, x ) = ˆΛ2 − π √ x + ν − ( iν, x ) = 3 x ˆΛ32 − πx √ x + ν
128 (B30)ˆ∆ − ( iν, x ) = x ˆΛ32 − πx ν √ x + ν . (B31)Note that there is no intrinsic contribution to the mixedcurrent-dipole response function R .
2. Extrinsic response
Shifting the integration variables in Eq. (B8), the fullextrinsic response readsˆ K µν + ( iν, x ) = (cid:90) d y (2 π ) f ( E + ( y )) (cid:20) ˆ F ++ µν ( − y (cid:48) , − y ) E + ( y ) − E + ( y (cid:48) ) − iν − ˆ F ++ µν ( y , y (cid:48) ) E + ( y (cid:48) ) − E + ( y ) − iν + ˆ F − + µν ( − y (cid:48) , − y ) E + ( y ) − E − ( y (cid:48) ) − iν − ˆ F + − µν ( y , y (cid:48) ) E − ( y (cid:48) ) − E + ( y ) − iν (cid:21) . (B32)Introducing polar coordinates for the y -integration, theresponse function is expressed asˆ K µν + ( iν, x ) = (cid:90) dy y ˆ J µνE ( iν, y, x ) , (B33)where ˆ J µνE ( iν, y, x ) is the angular integral of Eq. (B32).This angle integral is evaluated by transforming to a com-plex contour z = e iθ around the unit circle. The numer-ator reads z [( y ∓ iν ) − | y + x | ] = − xyz ( z − z )( z − z ) , (B34)where the positions of the three simple poles are z = 0 (B35) z = − x + ν ± iyν xy + 12 xy (cid:112) ( x + ν )( x − (2 y ∓ iν ) ) (B36) z = − x + ν ± iyν xy − xy (cid:112) ( x + ν )( x − (2 y ∓ iν ) ) . (B37)We have z z = 1 with | z | < | z | >
1. Theintegral is evaluated by applying the residue theorem andpicking up the two poles at z and z − inside the contour.Performing the integration yieldsˆ K ( iν, x ) = − x √ x + ν (cid:20) z (cid:112) − z + arcsin z (cid:21) (2 − iν ) /x − iν/x + ( ν → − ν )(B38)ˆ K xx + ( iν, x ) = − ν x + √ x + ν (cid:20) z (cid:112) − z − arcsin z (cid:21) (2 − iν ) /x − iν/x + ( ν → − ν )(B39)ˆ∆ ( iν, x ) = x √ x + ν (cid:20) z (2 z − (cid:112) − z + 14 arcsin z (cid:21) (2 − iν ) /x − iν/x + ( ν → − ν )(B40)ˆ∆ ( iν, x )= x √ x + ν (cid:20)
23 ( z − (cid:112) − z (cid:21) (2 − iν ) /x − iν/x + ( ν → − ν )(B41)7 x = q B = B / v F C1 B1 B2 C2A1 A2
FIG. 17. Frequency-momentum regions for the non-interacting Dirac response function. ˆ R ( iν, x ) = x √ x + ν (cid:20) z (2 z − (cid:112) − z −
34 arcsin z (cid:21) (2 − iν ) /x − iν/x + ( ν → − ν )(B42)ˆ R x ( iν, x ) = ( x + ν ) ˆ R ( iν, x ) . (B43)
3. Results
In this section, we perform the analytic continuationto real frequencies iν → ν + i K , K xx , ∆ , and ∆ :Re ˆ M αα ( ν, x )= − h α ( ν, x ) − f α ( ν, x ) g α> (cid:0) νx (cid:1) − g α> (cid:0) − νx (cid:1) A1 g α> (cid:0) νx (cid:1) B1 g α> (cid:0) νx (cid:1) − g α> (cid:0) ν − x (cid:1) C1 π A2 π g α< (cid:0) − νx (cid:1) B2 g α< (cid:0) νx (cid:1) + g α< (cid:0) − νx (cid:1) C2(B44) Im ˆ M αα ( ν, x )= − f α ( ν, x ) π A1 π g α< (cid:0) − νx (cid:1) B10 C1 − g α> (cid:0) νx (cid:1) + g α> (cid:0) − νx (cid:1) A2 − g α> (cid:0) νx (cid:1) B20 C2 (B45)with h α ( ν, x ) = − α = 0 ν x α = xx + ν ( ν +4) x − ν + 1) α = 1 − − ν x − ν x α = 2 (B46) f α ( ν, x ) = 18 (cid:112) | ν − x | x α = 0 ν − x α = x x ( ν − x ) α = 1 x ν α = 2 (B47)(B48)and g α< ( z ) = arcsin z + z √ − z α = 0arcsin z − z √ − z α = x arcsin z − z (2 z − √ − z α = 1arcsin z + z (2 z − √ − z α = 2 (B49) g α> ( z ) = − arccosh z + z √ z − α = 0 − arccosh z − z √ z − α = x − arccosh z − z (2 z − √ z − α = 1 − arccosh z + z (2 z − √ z − α = 2 . (B50)Taking the limit of small momentum gives the resultsstated in Eqs. (56), (49), (54), and (59) in the maintext. Figures 11–14 show the four response functions K , K xx , ∆ , and ∆ as a function of frequency forfour separate momenta q(cid:96) B = 0 . , ,
2, and 3, where toppanels show the imaginary part and bottom panels thereal part. The Dirac density response K and transversecurrent response K xx function have been computed be-fore in the graphene literature . Our results agreewith these works (where we identify k F = 1 /(cid:96) B and E F = (cid:126) v F /(cid:96) B ).8The result for the remaining two response functions R and R x is:Re ˆ R β ( ν, x )= − h β ( ν, x ) − f β ( ν, x ) g β> (cid:0) νx (cid:1) − g β> (cid:0) − νx (cid:1) A1 g β> (cid:0) νx (cid:1) B1 g β> (cid:0) νx (cid:1) + g β> (cid:0) ν − x (cid:1) C10 A2 g β< (cid:0) − νx (cid:1) B2 g β< (cid:0) νx (cid:1) + g β< (cid:0) − νx (cid:1) C2(B51)Im ˆ R β ( ν, x )= − f β ( ν, x ) g β< (cid:0) − νx (cid:1) B10 C1 − g β> (cid:0) νx (cid:1) + g β> (cid:0) − νx (cid:1) A2 − g β> (cid:0) νx (cid:1) B20 C2 (B52)with h β ( ν, x ) = − νx − ν ( ν − x )6 x β = 02 − ν x + ( x − ν ) x β = x f β ( ν, x ) = 18 (cid:112) | ν − x | x ν β = 02 x ( x − ν ) β = x g β< ( ν, x ) = 23 ( z − (cid:112) − z (B55) g β> ( ν, x ) = 23 ( z − (cid:112) z − . (B56)Note the limiting value at small momentumˆ R ( ν, x ) = − xν − x ν ν − ν − R x ( ν, x ) = x ν ( ν − , (B58)from which the scaling of the residue term Z L , Z T = O ( q ) at long wavelengths is directly apparent. a. Small-argument scaling limit In the small-frequency region and small momentumregion (A1 and A2), introduce the scaling variable ν = sx : ˆ K ( sx, x ) = i s √ − s A11 − s √ s − K xx ( sx, x ) = − s + is √ − s A1 − s + s √ s − ( sx, x ) = − s + i s √ − s A12 − s + 4 s √ s − ( sx, x ) = s + i s √ − s A12 + 4 s − s √ s − R ( sx, x ) = s + i s √ − s A12 s − s √ s − R x ( sx, x ) = − s + i s √ − s A11 − s + 2 s √ s − . (B64)Taking the absolute value of the last two expressions givesthe results for the residue terms (53) and (57) stated inthe main text. We also note the expansion of the inversefunctions[ ˆ K xx ( sx, x )] − = − − i √ − s s A1 − √ s − s A2 (B65)[ ˆ K ( sx, x )] − = − s − is √ − s A11 − s − s √ s − . (B66) ∗ [email protected] J. K. Jain,
Composite Fermions (Cambridge UniversityPress, 2007). J. K. Jain, “Composite-fermion approach for the fractionalquantum Hall effect,” Phys. Rev. Lett. , 199 (1989). A. Lopez and E. Fradkin, “Fermionic Chern-Simons FieldTheory for the Fractional Hall Effect,” in
Composite Fermions: A Unified View of the Quantum Hall Regime ,edited by O. Heinonen (World Scientific Publisching (Sin-gapore), 1998) Chap. IV. G. Giuliani and G. Vignale,
Quantum Theory of the Elec-tron Liquid (Cambridge University Press, 2005). S. H. Simon and B. I. Halperin, “Response function ofthe fractional quantized Hall state on a sphere. I. FermionChern-Simons theory,” Phys. Rev. B , 1807 (1994). S. He, S. H. Simon, and B. I. Halperin, “Response functionof the fractional quantized Hall state on a sphere. II. Exactdiagonalization,” Phys. Rev. B , 1823 (1994). S. H. Simon, “The Chern-Simons Fermi liquid descriptionof fractional quantum Hall states,” in
Composite Fermions:A Unified View of the Quantum Hall Regime , edited byO. Heinonen (World Scientific Publisching (Singapore),1998) Chap. II. D. Tong, “Lectures on the Quantum Hall Effect,”arXiv:1606.06687 (2016). B. I. Halperin, P. A. Lee, and N. Read, “Theory of thehalf-filled Landau level,” Phys. Rev. B , 7312 (1993). R. L. Willett, M. A. Paalanen, R. R. Ruel, K. W. West,L. N. Pfeiffer, and D. J. Bishop, “Anomalous sound prop-agation at ν =1/2 in a 2D electron gas: Observation ofa spontaneously broken translational symmetry?” Phys.Rev. Lett. , 112 (1990). W. Kang, H. L. Stormer, L. N. Pfeiffer, K. W. Baldwin,and K. W. West, “How real are composite fermions?” Phys.Rev. Lett. , 3850 (1993). V. J. Goldman, B. Su, and J. K. Jain, “Detection of com-posite fermions by magnetic focusing,” Phys. Rev. Lett. , 2065 (1994). S. H. Simon and B. I. Halperin, “Finite-wave-vector elec-tromagnetic response of fractional quantized Hall states,”Phys. Rev. B , 17368 (1993). S. H. Simon, A. Stern, and B. I. Halperin, “Compositefermions with orbital magnetization,” Phys. Rev. B ,R11114 (1996). S. M. Girvin, “Particle-hole symmetry in the anomalousquantum Hall effect,” Phys. Rev. B , 6012 (1984). S. A. Kivelson, D-H. Lee, Y. Krotov, and J. Gan,“Composite-fermion Hall conductance at ν =1/2,” Phys.Rev. B , 15552 (1997). C. Wang, N. R. Cooper, B. I. Halperin, and A. Stern,“Particle-Hole Symmetry in the Fermion-Chern-Simonsand Dirac Descriptions of a Half-Filled Landau Level,”Phys. Rev. X , 031029 (2017). P. Kumar, M. Mulligan, and S. Raghu, “Emergent reflec-tion symmetry from nonrelativistic composite fermions,”Phys. Rev. B , 205151 (2019). M. Levin and D. T. Son, “Particle-hole symmetry and elec-tromagnetic response of a half-filled Landau level,” Phys.Rev. B , 125120 (2017). D. X. Nguyen, S. Golkar, M. M. Roberts, and D. T.Son, “Particle-hole symmetry and composite fermions infractional quantum Hall states,” Phys. Rev. B , 195314(2018). S. T. Son, “The Dirac Composite Fermion of the FractionalQuantum Hall Effect,” Ann. Rev. Cond. Mat. Phys. , 397(2018). D. T. Son, “Is the Composite Fermion a Dirac Particle?”Phys. Rev. X , 031027 (2015). S. D. Geraedts, M. P. Zaletel, R. S. K. Mong, M. A. Metlit-ski, A. Vishwanath, and O. I. Motrunich, “The half-filledLandau level: The case for Dirac composite fermions,” Sci- ence , 197 (2016). W. Pan, W. Kang, M. P. Lilly, J. L. Reno, K. W. Baldwin,K. W. West, L. N. Pfeiffer, and D. C. Tsui, “Particle-Hole Symmetry and the Fractional Quantum Hall Effect inthe Lowest Landau Level,” Phys. Rev. Lett. , 156801(2020). A. C. Balram and J. K. Jain, “Nature of compositefermions and the role of particle-hole symmetry: A mi-croscopic account,” Phys. Rev. B , 235152 (2016). B. I. Halperin, “The Half-Full Landau Level,”arxiv:2012.14478 (2020). E. H. Hwang and S. Das Sarma, “Dielectric function,screening, and plasmons in two-dimensional graphene,”Phys. Rev. B , 205418 (2007). W. Wunsch, T. Stauber, F. Sols, and F. Guinea, “Dynam-ical polarization of graphene at finite doping,” N. J. Phys. , 318 (2006). R. E. Throckmorton and S. Das Sarma, “Failure of Kohn’stheorem and the apparent failure of the f -sum rule inintrinsic Dirac-Weyl materials in the presence of a filledFermi sea,” Phys. Rev. B , 155112 (2018). J. Hofmann, “Quantum oscillations in Dirac magnetoplas-mons,” Phys. Rev. B , 245140 (2019). K. F. Mak, M. Y. Sfeir, Y. Wu, C. H. Lui, J. A. Misewich,and T. F. Heinz, “Measurement of the Optical Conductiv-ity of Graphene,” Phys. Rev. Lett. , 196405 (2008). L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao,H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, andF. Wang, “Graphene plasmonics for tunable terahertzmetamaterials,” Nature Nanotechnology , 630 (2011). A. N. Grigorenko, M. Polini, and K. S. Novoselov,“Graphene plasmonics,” Nature Photonics , 749 (2012). D. Kamburov, Yang Liu, M. A. Mueed, M. Shayegan, L. N.Pfeiffer, K. W. West, and K. W. Baldwin, “What Deter-mines the Fermi Wave Vector of Composite Fermions?”Phys. Rev. Lett. , 196801 (2014). M. Barkeshli, M. Mulligan, and M. P. A. Fisher, “Particle-hole symmetry and the composite Fermi liquid,” Phys.Rev. B , 165125 (2015). A. Mitra and M. Mulligan, “Fluctuations and magne-toresistance oscillations near the half-filled Landau level,”Phys. Rev. B , 165122 (2019). S. M. Girvin, A. H. MacDonald, and P. M. Platzman,“Magneto-roton theory of collective excitations in the frac-tional quantum Hall effect,” Phys. Rev. B , 2481 (1986). W. Kohn, “Cyclotron Resonance and de Haas-van AlphenOscillations of an Interacting Electron Gas,” Phys. Rev. , 1242 (1961). K. Prabhu and M. M. Roberts, “Electrons and com-posite Dirac fermions in the lowest Landau level,”arXiv:1709.02814 (2017). N. Read and E. H. Rezayi, “Hall viscosity, orbital spin, andgeometry: Paired superfluids and quantum Hall systems,”Phys. Rev. B , 085316 (2011). A. Principi, M. Polini, and G. Vignale, “Linear responseof doped graphene sheets to vector potentials,” Phys. Rev.B , 075418 (2009). J. Hofmann, E. Barnes, and S. Das Sarma, “Why DoesGraphene Behave as a Weakly Interacting System?” Phys.Rev. Lett. , 105502 (2014). G. Murthy and R. Shankar, “Field Theory of the FractionalQuantum Hall Effect,” in
Composite Fermions: A UnifiedView of the Quantum Hall Regime , edited by O. Heinonen(World Scientific Publisching (Singapore), 1998) Chap. III. R. E. Throckmorton, J. Hofmann, E. Barnes, andS. Das Sarma, “Many-body effects and ultraviolet renor- malization in three-dimensional Dirac materials,” Phys.Rev. B92