PHYSICAL REVIEW X 5, 031027 (2015)

Is the Composite a Dirac Particle?

Dam Thanh Son Kadanoff Center for Theoretical Physics, University of Chicago, Chicago, Illinois 60637, USA (Received 19 February 2015; published 2 September 2015) We propose a particle-hole symmetric theory of the Fermi-liquid ground state of a half-filled Landau level. This theory should be applicable for a Dirac fermion in the magnetic field at charge neutrality, as 1 well as for the ν ¼ 2 quantum Hall ground state of nonrelativistic in the limit of negligible inter- Landau-level mixing. We argue that when particle-hole symmetry is exact, the composite fermion is a massless Dirac fermion, characterized by a Berry phase of π around the Fermi circle. We write down a tentative effective field theory of such a fermion and discuss the discrete symmetries, in particular, CP. The Dirac composite fermions interact through a gauge, but non-Chern-Simons, interaction. The particle-hole conjugate pair of Jain-sequence states at filling factors n=ð2n þ 1Þ and ðn þ 1Þ=ð2n þ 1Þ, which in the conventional composite fermion picture corresponds to integer quantum Hall states with different filling 1 factors, n and n þ 1, is now mapped to the same half-integer filling factor n þ 2 of the Dirac composite fermion. The Pfaffian and anti-Pfaffian states are interpreted as d-wave Bardeen-Cooper-Schrieffer paired states of the Dirac fermion with orbital angular momentum of opposite signs, while s-wave pairing would give rise to a particle-hole symmetric non-Abelian gapped phase. When particle-hole symmetry is not exact, the Dirac fermion has a CP-breaking mass. The conventional fermionic Chern-Simons theory is shown to emerge in the nonrelativistic limit of the massive theory.

DOI: 10.1103/PhysRevX.5.031027 Subject Areas: Condensed Matter Physics

I. INTRODUCTION particle-hole symmetry [12]—is not explicit within the CF paradigm. The CF is constructed by attaching a magnetic The theory of the fractional quantum Hall (FQH) effect flux to the electron before projecting to the lowest Landau [1,2] is based on the paradigm of the composite fermion level (LLL); the CS field theory formalism, strictly speak- [3–5], which provides a unified explanation of a large ing, does not allow one to attach fluxes to holes in the LLL. amount of observed phenomena, among which the most In one manifestation of the particle-hole asymmetric nature early ones are the Jain sequences—series of quantum of the formalism, the two Jain-sequence states with Hall plateaux at filling factors ν near 1=2, 1=4, etc. The ν ¼ n=ð2n þ 1Þ and ν ¼ðn þ 1Þ=ð2n þ 1Þ, which form composite fermion picture gives rise to extremely accurate a particle-hole conjugate pair, receive slightly different wave functions of FQH ground states [6]. interpretations in the CF language: The former fraction is At half filling, the composite fermion (CF) picture, an integer quantum Hall (IQH) state of CFs with n filled developed into a mathematical framework of the Chern- Landau levels, while in the latter, n þ 1 Landau levels Simons (CS) field theory by Halperin, Lee, and Read are filled. A related issue of the CF picture is its failure to (HLR) [5], predicts that the ground state is a Fermi liquid, account for the anti-Pfaffian state [13,14]—the particle- providing an explanation for the results of acoustic-wave- hole conjugate of the Moore-Read state—in a simple propagation experiments [7]. Near, but not exactly at half manner. filling, the CF is predicted to feel a small residual magnetic The zero bare mass limit, where particle-hole symmetry field, and the semiclassical motion of the CF in such a field is exact, is particularly difficult to analyze within the HLR has been observed experimentally [8,9]. The composite theory. Kivelson et al. [15] analyzed the HLR theory at fermion theory also provides an elegant interpretation 1 ν ¼ and found that particle-hole symmetry requires the of the Pfaffian (or Moore-Read) state [10] as a p þ ip 2 x y liquid of CFs to have a Hall conductivity σCF ¼ − 1.Ina Bardeen-Cooper-Schrieffer (BCS) paired state [11]. xy 2 Despite its success, one of the symmetries of FQH zero net magnetic field, this means that the CF liquid has an systems in the limit of zero Landau-level (LL) mixing—the anomalous Hall coefficient and seems to contradict the Fermi liquid nature of the CFs. Kivelson et al. did not find any set of Feynman diagrams that could lead to a nonzero σCF. As one possible solution, they proposed that Published by the American Physical Society under the terms of xy the Creative Commons Attribution 3.0 License. Further distri- the problem lies in the noncommutativity of the limit of bution of this work must maintain attribution to the author(s) and m → 0 (the LLL limit) and the limit of taking the density of the published article’s title, journal citation, and DOI. impurities to zero (the clean limit). This proposal leaves

2160-3308=15=5(3)=031027(14) 031027-1 Published by the American Physical Society DAM THANH SON PHYS. REV. X 5, 031027 (2015)

CF 1 unanswered the question of why σxy is exactly − 2 at all (i) The fermion is, by nature, a Dirac fermion. frequencies, independent of the physics of impurities. Few (ii) The fermion is its own particle-hole conjugate. attempts have been made to resolve the apparent inability to (iii) The fermion has no Chern-Simons interactions. This fit particle-hole symmetry to the CF picture; examples is an important point as the Chern-Simons term is include Ref. [16]. not consistent with particle-hole symmetry [26]. One logical possibility is that particle-hole symmetry is As per point (i), one may wonder if there is any difference spontaneously broken and the HLR theory describes only between the Fermi liquid of Dirac fermions and that of 1 nonrelativistic fermions: Both are characterized by a linear one of the two ν ¼ 2 ground states, which become degenerate at zero Landau level mixing. In this case, the anti-Pfaffian dispersion relation of quasiparticles near the Fermi surface. state would be inaccessible from within the HLR theory. The difference is in the Berry phase when the quasiparticle Numerical simulations, however, seem to be consistent with is moved around the Fermi surface (a circle in 2D), which is 1 π in the case of a Dirac fermion. The importance of the a particle-hole symmetric ν ¼ ground state [17].Away 2 Berry phase as a property of the Landau fermion quasi- from half filling, despite the apparent asymmetry in the particle was emphasized by Haldane [28]. treatments of the ν ¼ n=ð2n þ 1Þ and ν ¼ðnþ1Þ=ð2nþ1Þ The Berry phase of π offers a resolution to the puzzle of states, particle-hole symmetry maps one CF trial wave the anomalous Hall conductivity of the CFs. As shown by function to another with high accuracy [18]. In addition, a Haldane [28], the unquantized part of the anomalous Hall recent experiment, set up to measure the Fermi momentum conductivity of the CF Fermi liquid is equal to the global using commensurability effects in a periodic potential, Berry phase γ, which the quasiparticle receives when it implies that the Fermi momentum is determined by the moves around the Fermi disk, density of particles at ν < 1 and of holes at ν > 1 [19].One 2 2 ν 1 2 interpretation of this experiment is that the ¼ 2 state allows γ e σCF ∈ Z two alternative, but equivalent, descriptions as a Fermi liquid xy ¼ 2π þ n ;n: ð1Þ 1 h of either particles or holes. This would mean that the ν ¼ 2 CF 1 ground state coincides with its particle-hole conjugate. When γ ¼π, this equation is consistent with σxy ¼ − 2. The problem of particle-hole symmetry is more acute for The resolution to the puzzle is not the noncommutativity systems with Dirac fermions. The Dirac fermion is realized of the m → 0 limit and the clean limit; rather, it is in an in graphene and on the surface of topological insulators ingredient of the missing in all treat- (TIs) and is a relatively new venue for studying the ments of the CFs so far: the global Berry phase. quantum Hall (QH) effect. For Dirac fermions, the IQH Beside these differences, there are also many similarities plateaux occur at half-integer values of the Hall conduc- between the picture proposed here and the standard CF σ 1 2 tivity: xy ¼ðn þ 2Þðe =hÞ, which (after accounting for the picture. In particular, the fermion quasiparticle is electri- fourfold degeneracy of the Dirac fermion) has been seen in cally neutral. The Jain-sequence states near half filling are graphene [20,21]. FQH plateaux have also been observed mapped to IQH states of the CFs; however, in our theory, in graphene [22,23]; there is intriguing evidence that the particle-hole conjugate states map to IQH states with the FQH effect may exist on the surface of TIs [24]. In contrast same half-integer filling factors. to the nonrelativistic case, for Dirac fermions, particle-hole We show that the theory suggests the existence of a 1 symmetry is a good symmetry even with Landau level particle-hole symmetric gapped state at ν ¼ 2, distinct from mixing. It is not obvious that the flux attachment procedure the Pfaffian and anti-Pfaffian states. can be carried out for Dirac fermions; the usual workaround The structure of the paper is as follows. In Sec. II,we is to work in the limit of zero Landau-level mixing where start by writing down a model consisting of a single two- the projected Hamiltonian is identical to the nonrelativistic component Dirac fermion, localized on a ð2 þ 1ÞD “brane,” one [25]. This method explicitly breaks particle-hole coupled to electromagnetism in four dimensions. This symmetry and, furthermore, does not work at finite model does not contain complications specific for graphene Landau level mixing. or the surface state of TIs (the fourfold degeneracy in In this paper, we propose an explicitly particle-hole graphene, or the Zeeman coupling in the case of TIs) and symmetric effective theory describing the low-energy should realize a Fermi-liquid state in a finite magnetic field. dynamics of the Fermi liquid state of a half-filled We show that the electromagnetic response in this model is Landau level. This theory is constructed to respect all related directly to that of the nonrelativistic electrons on the discrete symmetries and to satisfy phenomenological con- LLL. We use the model to discuss discrete symmetries, straints, in particular, the existence of the two Jain which are also shared by the nonrelativistic model in the 1 sequences below and above ν ¼ 2. Our proposal is similar LLL limit, emphasizing that there are two independent to the fermionic Chern-Simons (HLR) theory in that the discrete symmetries in the finite magnetic field at charge elementary degree of freedom is a fermion; however, it neutrality: CP and PT . We then put forward, in Sec. III, differs from it in several ways: our proposal for the low-energy effective field theory and

031027-2 IS THE COMPOSITE FERMION A DIRAC PARTICLE? PHYS. REV. X 5, 031027 (2015) discuss physical implications. The possible connection A. Equivalence between Dirac fermions and to the conventional fermionic Chern-Simons theory is nonrelativistic fermions on the LLL discussed in Sec. IV. Finally, Sec. V contains concluding 1. Universality of the LLL limit remarks. We have argued that the problem of finding the ground state of a Dirac fermion in a magnetic field, in the weak II. A RELATIVISTIC MODEL REALIZING coupling regime e2 ≪ 1, is equivalent to finding the ground ν 1 THE ¼ 2 FQH STATE state of a nonrelativistic fermion in the LLL limit m → 0. To be more specific, we discuss a theory of a massless In this subsection, we show that the equivalence can be fermion localized on a (2 þ 1)-dimensional brane placed at extended further: The full electromagnetic response of one z ¼ 0, interacting through a U(1) gauge field in the (3 þ 1)- theory can be obtained from that of another theory. It is dimensional bulk, obvious that the density response to an external potential is the same in the two theories. The relationship between the Z Z 1 currents in the two theories is, however, slightly more 3 Ψ¯ γμ ∂ − Ψ − 4 2 S ¼ d xi ð μ iAμÞ 2 d xFμν; ð2Þ complicated. 4e To establish the connection, we recall a recent procedure → 0 where Ψ is a two-component spinor. We choose the to derive the expression for the current in the m limit following representation for the gamma matrices: of a nonrelativistic theory [29]. We start from the Lagrangian γ0 ¼ σ3; γ1 ¼ iσ2; γ2 ¼ −iσ1: ð3Þ 1 L ¼ iψ †D ψ − D ψ †D ψ þ L : ð4Þ t 2m i i int One can think of Ψ as the fermion zero mode localized on a L domain wall. In the condensed-matter language, Ψ is the In later formulas, we drop the interaction term int, always surface mode of a 3D TI. The theory has one dimensionless implicitly implying it. We now add to the Lagrangian a coupling constant e. In contrast to the usual QED, e does magnetic-moment term, giving the particle a gyromagnetic not run since it determines the strength of the electromag- factor g ¼ 2 (and assuming the system is fully spin netic interactions infinitely far away from the brane, where polarized), there is no matter that would renormalize it. We are mostly 1 2 † † B † interested in the weak coupling regime where e ≪ 1,but L 2 ¼ iψ D ψ − D ψ D ψ þ ψ ψ: ð5Þ g¼ t 2m i i 2m most of our statements should be valid up to some finite value of e2. For a constant magnetic field, the added term is propor- Our task is to understand the ground state of the system tional to the total number of particles, which commutes in the finite magnetic field B ¼ Fxy and its excitations. This with the Hamiltonian and does not alter the dynamics. problem is nonperturbative even at e2 ≪ 1 since it maps to However, the expression for the electromagnetic current 2 δ δ a FQH problem. To see this, assume at first that e ¼ 0, and ji ¼ S= Ai has changed. Denoting the current corre- i i recall that the Landau levels of the Dirac Hamiltonian are sponding to Eq. (4) by j 0 and to Eq. (5) by j 2, we find pffiffiffiffiffiffiffiffiffi g¼ g¼ 2 E ¼ nB. In the ground state, the states with negative 1 energy are filled and those with positive energy are empty, ji ji − ϵij∂ ρ: 6 g¼0 ¼ g¼2 2 j ð Þ but the noninteracting Hamiltonian gives us no prescription m for the n ¼ 0 LL (the zeroth LL), whose states have exact The two currents differ only by a solenoidal term, and if we zero energy. The true ground state is only singled out when know the density response, we can get one current from the i interaction is turned on. It is worth notingffiffiffiffi that the energy other. As we will see later, j 2 is finite in the limit m → 0, 2p g¼ scale induced by the interaction is e B, while the spacing i pffiffiffiffi but jg¼0 is, in general, divergent. In the subsequent between LLs is B. 1 discussion, by the electromagnetic current we have in To see that the problem maps to the ν ¼ 2 QH problem, i mind jg¼2 by default. Note that for linear response at we notice that, because of particle-hole symmetry, at zero q ¼ 0 (but any frequency), the two currents coincide. chemical potential, the zeroth LL must be half full (or half 2 Ignoring a total derivative, the Lagrangian (5) can be empty). At small e , all essential physics occurs on the rewritten as zeroth LL, and the Hamiltonian projected to this LL is the same as the usual projected QH Hamiltonian since the 1 L ¼ iψ †D ψ − ðD − iD Þψ †ðD þ iD Þψ: ð7Þ Dirac orbitals on the zeroth LL are the same (neglecting one t 2m x y x y component of the Dirac spinor which vanishes for these states) as the Landau orbitals of a nonrelativistic particle Introducing auxiliary fields χ and χ†, we can recast the [see Eq. (27) below]. Lagrangian in the form

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† † † L ¼ iψ Dtψ þ iψ ðDx − iDyÞχ þ iχ ðDx þ iDyÞψ energies. The states in the Dirac sea contribute a uniform charge density −B=4π; thus, þ 2mχ†χ: ð8Þ

→ 0 ρ ρ − B Now, taking the m limit, the Lagrangian becomes D ¼ NR 4π ; ð13Þ L iψ †D ψ iψ † D − iD χ iχ† D iD ψ: ¼ t þ ð x yÞ þ ð x þ yÞ where the subscripts “D” and “NR” refer to “Dirac” and ð9Þ “nonrelativistic,” respectively. To find the contribution of the sea to the current, we notice that the electrons in the The last form of the Lagrangian makes it clear that the Dirac sea are subject to an additional scalar potential limit m → 0 is finite. In fact, Eq. (9) can be thought of created by the electrons on the zeroth Landau level, as a Lagrangian formulation of the problem on the lowest Z χ χ† Landau level. The variables and are Lagrange multi- x y x − y ρ x ≡ ρ x pliers enforcing the constraints Veffð Þ¼ d Vð Þ ð Þ ðV · Þð Þ; ð14Þ

ψ 0 − ψ † 0 ðDx þ iDyÞ ¼ ; ðDx iDyÞ ¼ ; ð10Þ where Vðx − yÞ is the interaction potential between elec- trons (i.e., the Coulomb potential). The response of the which is nothing but the LLL constraint. For example, in Dirac sea to this scalar potential gives an extra contribution − 1 1 the symmetric gauge Ax ¼ 2 By, Ay ¼ 2 Bx, the con- to the current, leading to the following relationship between straints imply that ψ is a linear combination of Landau’s the currents: 2 orbitals zne−Bjzj =4, n ¼ 0; 1; 2…. Equation (9) implies that the physics on the LLL is universal, i.e., independent of 1 ji ji ϵij∂ V · ρ : 15 how the LLL limit is approached. In Ref. [29], it was shown D ¼ NR þ 4π jð Þ ð Þ that the current computed from Eq. (9) coincides with that found in Ref. [30]. This current differs from the one The two formulas can be summarized as the following obtained in Ref. [31] by a solenoidal term, which is not relationship between the logarithms of the partition func- surprising since Ref. [31] assumes a zero geomagnetic tions (W ¼ −i ln Z) in the two theories, valid for any i i factor and should gives jg¼0 instead of jg¼2. external A0 and to linear order in perturbations of Ai Let us now consider the Dirac field theory. Denoting the around the background, two components of the Dirac spinor as   − A0B V · B ψ WD½A0;Ai¼ 4π þ WNR A0 þ 4π ;Ai : ð16Þ ψ ¼ ; ð11Þ χ Differentiating with respect to A0 and Ai we recover the Dirac action can be written as Eqs. (13) and (15). The formulas are written in the long-wavelength limit † † † L ¼ iψ Dtψ þ iχ Dtχ þ iψ ðDx − iDyÞχ of external probes. This is sufficient for the purpose of the † rest of the paper. For completeness, we mention that there þ iχ ðD þ iD Þψ: ð12Þ x y are corrections to the formulas from two sources: (i) the wave-number dependence of the Hall conductivity of If one concentrates on states on the zeroth Landau level, the Dirac spinor will have χ much smaller than ψ. In the the filled levels, and (ii) the exchange interactions between † the electrons in the Dirac sea and those on the zeroth Lagrangian, the term χ ∂ χ is quadratic in χ and can t Landau level [only the Hartree interaction has been taken be neglected compared to other terms. Terms linear in χ into account in Eq. (14)]. Both effects are small in the long- will still have to be kept since the time derivative D is of t wavelength limit for the potential V sufficiently long the same order of smallness as χ. Thus, the Lagrangian ranged so that its Fourier transform V q diverges at becomes exactly Eq. (9). ð Þ q → 0 (as the Coulomb potential). 2. Relationship between currents in relativistic and nonrelativistic theories 3. Relativistic convention Although the expressions of the current coming from the The discussion above shows that there is a direct modes on the LLL in the relativistic and nonrelativistic relationship between physical observables in the cases theories are the same, the physics of the relativistic model is of Dirac and nonrelativistic fermions in the LLL limit. slightly different from that of the nonrelativistic one by the Here, we collect formulas that relate quantities in the two presence of a Dirac sea of Landau levels with negative theories, which will be useful in our further discussion.

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We define the filling factor as the ratio ν1 þ ν2 ¼ κ1 þ κ2 ¼ 0: ð23Þ ρ ν ¼ ; ð17Þ Another quantity, which will not be discussed in this B=2π paper, is the chiral central charge c. The relationship between the relativistic and nonrelativistic convention for where ρ is the charge density. According to Eq. (13), the − 1 ν κ this charge is c ¼ cNR 2. As for and , particle-hole relationship between the relativistic and nonrelativistic conjugation flips the sign of c. filling factors is 1 ν ν − B. Discrete symmetries ¼ NR ; ð18Þ 2 We recall the theory (2) has the following sym- metries [33]: where we have dropped the index “D” on the left-hand side. (i) Charge conjugation, In the relativistic theory, ν ¼ 0 occurs at zero chemical potential (the charge neutrality point), where the n ¼ 0 −1 CAμC ¼ −Aμ; ð24aÞ Landau level is half filled. The latter corresponds to ν ¼ 1=2. NR CΨC−1 σ1Ψ: Similarly, one can derive a relationship between the ¼ ð24bÞ conductivities in the relativistic and nonrelativistic theories from Eq. (15). At zero wave number (but any frequency), (ii) Spatial parity x x; y → x0 x; −y , the spatial derivative term on the right-hand side of Eq. (15) ¼ð Þ ¼ð Þ vanishes, giving −1 0 PA0ðt; xÞP ¼ A0ðt; x Þ; ð25aÞ σ ω σNR ω ; 19 xxð Þ¼ xx ð Þ ð Þ −1 0 PA1ðt; xÞP ¼ A1ðt; x Þ; ð25bÞ 1 σ ω σNR ω − : 20 −1 0 xyð Þ¼ xy ð Þ 2 ð Þ PA2ðt; xÞP ¼ −A2ðt; x Þ; ð25cÞ

We have measured the conductivities in units of e2=h. PΨðt; xÞP−1 ¼ σ1Ψðt; x0Þ: ð25dÞ One can interpret Eq. (20) as the statement that the Hall conductivity of the Dirac fermion consists of a contribution NR → − from the zeroth Landau level, equal to σxy , and the (iii) Time reversal t t, contribution from the filled negative-energy Landau levels, 1 T x T −1 − x equal to − 2. Note that these formulas are valid also in the A0ðt; Þ ¼ A0ð t; Þ; ð26aÞ presence of impurities. T x T −1 − − x Next we consider the shift. In the relativistic context, Aiðt; Þ ¼ Aið t; Þ; ð26bÞ instead of the shift, it is convenient to parametrize a given gapped quantum Hall state by a parameter κ giving the T Ψðt; xÞT −1 ¼ −iσ2Ψð−t; xÞ: ð26cÞ offset between the total charge Ne and the total magnetic flux in the unit of flux quantum Nϕ, on a sphere [32]: T is an anti-unitary operator (T iT −1 ¼ −i). These indi- Ne ¼ νNϕ þ κ: ð21Þ vidual symmetries are broken by the magnetic field B which changes sign under each of C, P, and T . However, To compare κ with the shift of the corresponding QH state CP, CT , and PT leave the magnetic field unchanged and on the LLL, one has to take into account two facts: There hence remain the discrete symmetries of the Dirac fermion 1 is a 2 offset between the definitions of the relativistic and in a magnetic field [32]. We lose one symmetry because ν ν − 1 CT ∼ nonrelativistic filling factors, ¼ NR 2, and that Dirac only two out of these three are independent, e.g., particle has a direct coupling to the spin connection. As a ðCPÞðPT Þ. The chemical potential further breaks CP and result, the connection between κ and the shift S is [32] CT , leaving PT as the only symmetry of the Dirac fermion in a magnetic field at nonzero chemical potential. As we are κ ν S − 1 CP CT ¼ NRð Þ: ð22Þ interested mostly in the charge neutrality point, and will be exploited as symmetries [34]. In particular, if two states, “1” and “2,” are particle-hole When e2 ≪ 1, all interesting physics occurs on the NR NR NR conjugates of each other, then ν1 þ ν2 ¼ ν1 S1þ n ¼ 0 Landau level, and the CP, CT , and PT operators NR ν2 S2 ¼ 1, which in the relativistic notations become map to symmetries of the LLL Hamiltonian. Let us choose simply the symmetric gauge, where the single-particle orbitals on

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m −jzj2=4 σ Π Π the LLL are z e . The field Ψ projected to the LLL can The Hall conductivity xy is related to H by H ¼ be written as ði=2πÞωσxy. At nonzero chemical potential, PT is the Π −ω only symmetry, and we get the constraints L;Tð ;qÞ¼ X − 2 4 A zme Bjzj = Π ðω;qÞ and σ ð−ω;qÞ¼σ ðω;qÞ, but at charge Ψ ¼ m c ; ð27Þ L;T xy xy m neutrality, CP implies that the Hall conductivity vanishes, m 0 σ ðω;qÞ¼0; ð34Þ where Am is a normalization coefficient and cm is the xy operator annihilating a fermion on the orbital m. Then, ω from Eqs. (24b), (25d), and (26c)), we find the action of the at any value of and q. This is true not only for clean systems but also in the presence of impurities, provided symmetries on cm, that the latter do not statistically break the particle-hole −1 † symmetry. CPc ðCPÞ ¼ cm; ð28Þ m For nonrelativistic fermions on the LLL, we can use −1 Eq. (20) to find, for q ¼ 0, PT cmðPT Þ ¼ cm; ð29Þ 1 CT CT −1 † σNRðωÞ¼ ; ð35Þ cmð Þ ¼ cm: ð30Þ xy 2

Note that CT is different from CP: The former is an anti- generalizing a result derived in Ref. [15] to nonzero unitary transformation, while the latter is unitary. In the frequencies. For q ≠ 0, this relationship is replaced by a σ Π quantum Hall literature, by particle-hole symmetry, one linear constraint on xy and L [36]. normally has in mind CT . Likewise, PT is not an identity An explicitly particle-hole symmetric theory of the half- operator: It replaces the wave function of a given state, in filled state should imply Eq. (34) automatically. We now † the basis obtained by acting cm’s on the vacuum, with its construct such a theory. complex conjugate. More precisely, if jfi is the following quantum N-body state on the LLL, III. PROPOSAL FOR THE LOW-ENERGY X EFFECTIVE FIELD THEORY † † † 0 jfi¼ fn1n2…n cn1 cn2 cn þj i; ð31Þ N N Lacking a better way, we are going to simply guess the fnig form of the low-energy effective theory. We start by stating then a few requirements that our theory should satisfy: CP PT X (i) The theory should be invariant under and . PT † † † 0 (ii) At charge neutrality and nonzero magnetic field, the jfi¼ fn1n2…nN cn1 cn2 cnN þj i: ð32Þ fn g ground state should be a Fermi liquid. i (iii) The theory should explain the Jain sequences. ’ Note that the Laughlin states [2] and the Moore-Read states The requirement (ii) is particularly nontrivial, as Luttinger s [10] are invariant under PT : In both cases, the coefficients theorem requires the volume of the Fermi sphere to be appearing in the holomorphic polynomial in the wave proportional to the density of some charge, but at charge function are all real. We are unaware of any trial wave neutrality, the electromagnetic charge density is equal function that is not invariant under PT . Assuming PT to zero. symmetry, CP is equivalent to CT ; thus, we will sometimes It turns out that the three requirements above are satisfied call CP the particle-hole symmetry, although in the by the following action, which we propose as the low- literature the latter usually corresponds to CT . energy effective theory: Z 1 3 ψγ¯ μ ∂ 2 ψ ϵμνλ ∂ C. Consequences of discrete symmetries Seff ¼ d x i ð μ þ iaμÞ þ 2π Aμ νaλ for linear response Z 1 4 2 Consider the linear response of a QH system to a − d xFμν þ: ð36Þ 4e2 small perturbation of the electromagnetic potential: μ Πμν Πμν 0 j ðqÞ¼ ðqÞAνðqÞ, qμ ¼ . Assuming rotational Here, ψ is a Dirac field describing the fermionic invariance, there are three independent components of quasiparticle, aμ is an emergent gauge field, and … stands Πμν the polarization tensor , for other terms, including a possible Maxwell kinetic term for aμ and interaction terms. The quasiparticle ψ, which qiqj qiqj Πij Π δij − Π ϵijΠ will also be called the Dirac CF, has quantum numbers ¼ 2 L þ 2 T þ H: ð33Þ q q different from the electron: It is electrically neutral and

031027-6 IS THE COMPOSITE FERMION A DIRAC PARTICLE? PHYS. REV. X 5, 031027 (2015) carries charge with respect to the emergent gauge field aμ. −1 T a ðt; xÞT ¼ a ð−t; xÞ; ð39bÞ Note that in the HLR theory, the CF is also electrically i i neutral, a point which has been emphasized by T ψðt; xÞT−1 ¼ σ3ψ ð−t; xÞ: ð39cÞ Read [37,38]. Going from the microscopic action (2) to Eq. (36),it looks as if the electron Ψ has been stripped off of its charge, [We omit the transformation laws for Aμ which are the same whose dynamics is now governed by a dual gauge field as in Eqs. (24), (25), and (26)]. In particular, our effective aμ. Interestingly, a similar phenomenon has recently been theory is invariant under CP and PT . We note that the suggested to occur in the “composite Dirac liquid” state parity anomaly is avoided in this theory by the charge 2 of [39]—a possible parent to several gapped topological the CF. Note that the fermion mass term mψψ¯ and the phases of the strongly interacting surface of 3D TI Chern-Simons term for aμ are allowed by PT but for- [40–43]. As in our proposed effective theory, the only bidden by CP or CT . But we expect the terms not forbidden low-energy mode of the composite Dirac liquid is a neutral “ ” Dirac fermion. While it is certain that the composite- by symmetries to appear in the in Eq. (36). fermion liquid phase cannot occur in the theory (2) in It is instructive to write down the transformation law for 2 ψ under CP, CT , the zero magnetic field in the limit e ≪ 1, what we are suggesting is that as soon as one turns on a magnetic field, CPψ t; x CP −1 σ1ψ t; −x ; 40a the most convenient representation of the low-energy ð Þð Þ ¼ ð Þ ð Þ dynamics is not in terms of the original fermion but in CT ψ x CT −1 − σ2ψ − x terms of a composite Dirac fermion of the type considered ðt; Þð Þ ¼ i ð t; Þ; ð40bÞ in Ref. [39]. PT ψ x PT −1 σ3ψ − −x In Eq. (36), we normalize the field aμ so that a 2π ðt; Þð Þ ¼ ð t; Þ: ð40cÞ magnetic flux of a carries a unit electric charge [see CP CT ψ Eq. (41) below]. With this normalization, the charge of Note that under particle-hole symmetries ( and ), ψ the ψ field is fixed to 2. As we will see below, this value of does not transform into the complex conjugated field but ψ ψ CP the charge is required for the theory to be consistent with remains . In fact, the transformation laws of under CT Ψ Jain’s sequences, the Fermi momentum of the half-filled and are the same as those of the original electrons P T state, and the e=4 charged excitation in the Pfaffian state. under and [Eqs. (25d) and (26c)]. Thus, particle-hole symmetry does not transform the composite fermion ψ into its antiparticle but leaves it as a particle. This result can also A. Discrete symmetries of the effective field theory be seen from the fact that a0, which is proportional to the It is easy to check that the Lagrangian (36) exhibits the chemical potential of the CFs, does not change sign under full set of discrete symmetries of the original theory (2), CP and CT . including (i) charge conjugation, B. Fermi liquid and Jain sequences −1 Caμðt; xÞC ¼ −aμðt; xÞ; ð37aÞ To establish the nature of the QH state at charge neutrality, we first note that the electromagnetic current Cψðt; xÞC−1 ¼ σ1ψ ðt; xÞ; ð37bÞ defined by Eq. (36) is

μ 1 μνλ j ¼ ϵ ∂νaλ: ð41Þ (ii) spatial parity, 2π

−1 0 In particular, charge density is related to the emergent Pa0ðt; xÞP ¼ −a0ðt; x Þ; ð38aÞ magnetic field: b ¼ 2πρ. At charge neutrality, ρ ¼ 0, −1 0 and therefore the fermionic quasiparticles feel a zero Pa1ðt; xÞP ¼ −a1ðt; x Þ; ð38bÞ magnetic field. −1 0 On the other hand, differentiating the action with respect Pa2ðt; xÞP ¼ a2ðt; x Þ; ð38cÞ to a0, one finds a relationship between the density of the Dirac CF and the magnetic field, Pψðt; xÞP−1 ¼ ψ ðt; x0Þ; ð38dÞ ρ~ ≡ ψγ¯ 0ψ B h i¼4π : ð42Þ (iii) time reversal, Therefore, in a nonzero magnetic field, the CFs have a finite T x −1 − − x a0ðt; ÞT ¼ a0ð t; Þ; ð39aÞ density and live in a zero magnetic field.

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Assuming no Cooper instability (a possibility that we sphere, the degeneracy of the nth Landau level is will consider later), the ground state is then a Fermi liquid Nϕ þ 2jnj. Therefore, of the CFs, and Luttinger’s theorem fixes the Fermi Z Z momentum to the inverse of the magnetic length. The 1 2b dx ψ †ψ n dx n n 1 : 45 value of the Fermi momentum is the same as the value in h i¼ þ 2 2π þ ð þ Þ ð Þ the standard HLR theory [5]. Note that the Fermi velocity † vF of the CF is, in general, different from 1; at weak But ρ ¼ b=2π and hψ ψi¼B=4π; therefore, we have coupling, vF should scale with the interaction energy: 2 v ∼ e . The renormalization of the Fermi velocity is Nϕ F ¼ð2n þ 1ÞN þ nðn þ 1Þ: ð46Þ related to Landau’s Fermi liquid parameters [44]. The 2 e effective theory can be trusted to describe the dynamics of the fermions near the Fermi surface but not far away This means [see Eq. (21)] from it. In principle, one should be able to reformulate the 1 nðn þ 1Þ theory completely in terms of the degree of freedom near ν ¼ ; κ ¼ − ; ð47Þ the Fermi surface [45,46], but since preserving the infor- 2ð2n þ 1Þ 2n þ 1 mation about the Berry phase (which will be important ν 1 2 1 S when we discuss the Jain sequences) is quite nontrivial, we so NR ¼ðn þ Þ=ð n þ Þ and, from Eq. (22), ¼ − 1 ν 2 1 will not try to do so. n þ . The shift of the NR ¼ n=ð n þ Þ can be com- S 2 The fermion quasiparticle is the CP conjugate of itself. puted similarly to be ¼ n þ . These values of the shift To see it, we note that CPψðCPÞ¼σ1ψ. Consider a coincide with what is known about these states [48]. particle moving with momentum p ¼ðp; 0Þ, choosing the p to be invariant under y → −y. Its wave function is D. Pfaffian, anti-Pfaffian, and a particle-hole an eigenvector of the Dirac Hamiltonian σ · p ¼ σ1p, symmetric non-Abelian state which is ð1; 0ÞT. This wave function is invariant under CP. We can construct various gapped states by letting ψ form Away from charge neutrality, the fermion quasiparticles a BCS pair. Because the Dirac fermion has an additional feel an effective magnetic field equal to Berry phase around the origin in momentum space, pairing occurs in channels with even angular momentum. B~ ¼ −2∇ × a ¼ −4πρ: ð43Þ Consider first s-wave pairing, with the order parameter T being ψ σ2ψ. It is easy to see that the order parameter is Equations (42) and (43) tell us that from the point of view invariant under CP, CT , and PT symmetries (combined of the composite fermions, the notions of density and with phase rotations of ψ when required). magnetic field get swapped, similarly to what happens Since this is a particle-hole symmetric state, it cannot under particle-vortex duality for bosons [47]. As a result, be the Pfaffian or anti-Pfaffian state. In the conventional the filling factor of the original electrons ν ¼ 2πρ=B and composite fermion picture, this state can be understood as a the effective filling factor of the fermion quasiparticles state where the composite fermions form Cooper pairs in ~ − ν~ ¼ 2πρ~=B are inversely proportional to each other, the px ipy channel with the orbital angular momentum opposite to the Moore-Read state. While the conventional 1 picture does not immediately tell us so, we find here that 2ν ¼ − : ð44Þ 2ν~ this state coincides with its particle-hole conjugate. In particular, for this state, κ ¼ 0, corresponding to the shift This means the Jain-sequence state with filling factor S ¼ 1. Because of the condensation of Cooper pairs that ν 2 1 − 1 −1 2 2 1 ¼ n=ð n þ Þ 2 ¼ =½ ð n þ Þ maps to an IQH carry charge 4 under aμ, the vortices in the condensate ν~ 1 state of the fermion quasiparticle with ¼ n þ 2. Note correspond to quasiparticles with charge 1=4. The vortex 1 that the filling factor ν ¼ðn þ 1Þ=ð2n þ 1Þ − 2 maps to has fermionic zero mode and should have non-Abelian 1 ν~ ¼ −ðn þ 2Þ, making explicit the fact that the two states statistics. This state is a particle-hole symmetric Pfaffian- are particle-hole conjugates of each other. like state, which we will term the PH-Pfaffian. In fact, its construction parallels that of the time-reversal-invariant T-Pfaffian state on the surface of TIs [41,42]. C. Shift of states on the Jain sequences The Pfaffian and anti-Pfaffian states correspond to A nontrivial check for the proposed effective field pairing of ψ in the l ¼2 channels. The vortices in these theory is the computation of the shift [48] of the states states also have charges 1=4 as the s-wave paired state. ν on the Jain sequences on a sphere. Consider the NR ¼ Further confirmations are obtained when one computes the ðn þ 1Þ=ð2n þ 1Þ state. The fermions ψ are effectively shift. If one puts a d-wave bosonic condensate on a sphere, in the magnetic field of 2b in the IQH state with without any magnetic field we would have four vortices or ν − 1 1 4 eff ¼ ðn þ 2Þ. From solving the on a antivortices. Since each vortex has charge = , this means

031027-8 IS THE COMPOSITE FERMION A DIRAC PARTICLE? PHYS. REV. X 5, 031027 (2015)

κ ¼1, which corresponds to S ¼ 3 and S ¼ −1 [see In the limit q → 0, the formulas simplify considerably. Eq. (22)]. These are the values of the shift for the Pfaffian Substituting and anti-Pfaffian states, respectively [49]. ν 5 iω In real systems, for example, in the ¼ 2 QH state, Π Π σ T ¼ L ¼ xx; ð54Þ whether the l ¼ 0 or l ¼2 is favored is the question 2π about the energy of the ground state and cannot be iω determined from general principles. It is possible that there Π σ H ¼ 2π xy; ð55Þ is no pairing instability at l ¼ 0, while there are insta- bilities at finite l (Ref. [50]). where σxx and σxy are the longitudinal and Hall conduc- tivities for the electrons, in units of e2=h ¼ 1=2π (and the E. Electromagnetic response version with a tilde for the composite fermions), we find In the random phase approximation (RPA), one can [assuming VðqÞ diverges slower than q−2 at q → 0] relate the response functions of electrons Πμν to the Π~ 1 σ~ response functions of the Dirac composite fermion, μν. σ xx ; xx ¼ 4 σ~ 2 σ~ 2 ð56Þ We use the notation of Eq. (33) and assume the Fermi xx þ xy velocity of the quasiparticles to be much smaller than the   1 σ~ 1 speed of light, so the interaction is an instantaneous σ − xy : xy ¼ 4 σ~ 2 σ~ 2 þ 2 ð57Þ interaction. The RPA calculation can be done in a straight- xx þ xy NR forward way, and one finds the response functions of the electrons to be The term 1=2 on the right-hand side of Eq. (57) should be included in the nonrelativistic case. We concentrate on this ω2 Π − Π~ case from now on. From Eqs. (56) and (57), we can express L ¼ 2 L; ð48Þ 16π Δ σ~ xy in terms of the measurable ρxx and ρxy as follows: ω2 q2VðqÞ 1 ρ − 2 Π ¼ − Π~ − ; ð49Þ σ~ xy T 16π2Δ T 16π2 xy ¼ 2 þ 2 2 : ð58Þ ρxx þðρxy − 2Þ ω2 ρ ρ − 2 ∼ ρ2 Π Π~ Typically, xx is small, and in the regime xy xx, the H ¼ H; ð50Þ 16π2Δ above equation can be written as where ρ − 2 1 xy σ~ − 2 ¼ xy 2 : ð59Þ 2 ρxx Δ Π~ − q VðqÞ Π~ − Π 2 ¼ T 2 L j Hj : ð51Þ 16π Equation (59) opens up a possibility to experimentally distinguish the Dirac CFs and the standard HLR theory. ν 0 At ¼ , for massless Dirac composite fermions, we have The HLR theory, as we will argue in Sec. IV, corresponds to Π~ 0 H ¼ , and we find a vanishing Hall conductivity of the σ~ xy ¼ 1=2; thus, the right-hand side of Eq. (59) is zero at Dirac electrons, Π ¼ 0, as required by CP. The formulas H exact half filling (e.g., ρxy ¼ 2 at exact half filling). On the for the other response functions simplify to other hand, for a system near particle-hole symmetry, σ~ xy is −1 2 ω2 small and the right-hand side should be close to = .One Π ¼ − ; ð52Þ can thus plot the left-hand side of Eq. (59) as a function of L 16π2Π~ − 2 T q VðqÞ the magnetic field B in a range of temperature where ρxx depends nontrivially on the temperature. Both HLR and our ω2 Π ¼ − : ð53Þ theory then predict that all curves go through one point, T 16π2Π~ L whose position on the horizontal axis is the value of the magnetic field at exact half filling B1=2, and the position on The first equation coincides with the formula in the theory the vertical axis is 0 in the HLR theory and −1=2 in the proposed in Ref. [16], but the second does not. The density- theory of the massless Dirac composite fermion. When Π density correlation function L shows the same behavior in particle-hole symmetry is not exact, the position of the the infrared as in the HLR theory. For example, because of point on the vertical axis is −γ=2π, where γ is the Berry Π~ the Landau damping in T (note that the CFs always have a phase of the composite fermions around the Fermi disk Π ω ∼ − 2 Fermi surface), L has a pole at iq for the Coulomb (assuming Hall conductivity is dominated by the Berry Π ∼ ω ω ≫ 2 γ π interaction and behaves as L q for q , reproduc- phase). Furthermore, we expect the deviation of from to ing two key features of the HLR theory [5]. be proportional to the amount of Landau-level mixing, i.e.,

031027-9 DAM THANH SON PHYS. REV. X 5, 031027 (2015) the ratio of the Coulomb energy scale and the cyclotron † 1 2 L ¼ iψ ð∂t − iA0 þ ic0Þψ − jð∂i − iAi þ iciÞψj energy. Note that, because of the smallness of ρxx, 2m distinguishing Berry phases of 0 and π requires a suffi- 1 1 − AdA þ cdc þ; ð62Þ ciently accurate measurement of ρxy: For a relatively large 8π 8π 2 −3 2 ρxx ∼ 0.1h=e , an accuracy better than 5 × 10 h=e is needed. where cμ is the statistical gauge field. Now, changing The formulas (56) and (57) are valid at zero frequencies. notation, At finite frequencies, one needs to modify the theory for it to be consistent with Galilean symmetry and Kohn’s cμ ¼ Aμ þ 2aμ; ð63Þ theorem. The modification and the results for the con- ductivities are discussed in the Appendix. the Lagrangian becomes 1 L ψ † ∂ 2 ψ − ∂ 2 ψ 2 F. Infrared divergences ¼ i ð t þ ia0Þ jð i þ iaiÞ j 2m One can directly check that despite the fact that the 1 1 þ ada þ Ada þ: ð64Þ dynamical gauge field aμ does not have a Chern-Simons 2π 2π interaction, the propagator of the transverse component of ai has the same infrared behavior as in the HLR theory: The form of the action is very similar to Eq. (36), but there are two crucial differences: (i) The CF is nonrelativistic, 1 and (ii) there is a Chern-Simons interaction ada. ha⊥ð0; −qÞa⊥ð0; qÞi ∼ : ð60Þ We now suggest that theory (64) can be obtained from jqj Eq. (36) by adding to the latter a CP-breaking mass term and taking the large mass limit. The mass term for the Dirac Therefore, we expect that our theory would have the CF, −mψψ¯ , is generally expected once one relaxes the same logarithmic divergences as in the HLR theory with condition of particle-hole symmetry (the coefficient of the Coulomb interaction. Chern-Simons term ada cannot change continuously and thus has to remain zero). Now consider the regime where the mass m is very large, and the CF is in the nonrelativistic regime. Let us remind ourselves that if one has a massive IV. CONNECTION WITH THE FERMIONIC Dirac field, coupled to a U(1) gauge field CHERN-SIMONS THEORY L ψ¯ ∂ − ψ − ψψ¯ μψγ¯ 0ψ The QH physics of the Dirac fermion shares the same ¼ i ð μ iaμÞ m þ ; ð65Þ LLL limit with nonrelativistic electrons. The standard theory describing the FQH states for nonrelativistic electrons is the and if the chemical potential is inside the gap, i.e., μ ψ fermionic CS (HLR) theory. One can then ask the following j j < jmj, then one can integrate out to obtain an question: Can one connect the fermionic CS theory and the effective action for aμ [51,52], relativistic theory (36)? This may seem impossible since the 1 m former has a CS term in the Lagrangian, while the latter does − ada: ð66Þ not. However, we will argue that it is at least possible to go 8π jmj continuously from the relativistic theory to the conventional μ μ nonrelativistic fermionic CS theory, if one allows breaking Now imagine that is slightly above the mass, > jmj but μ − ≪ of particle-hole symmetry. jmj m. In this case, one has to take into account the The starting theory of the fermionic CS theory is the gapless fermions, which are nonrelativistic, but one should Lagrangian also not forget the induced Chern-Simons term (66), which can be thought of as coming from the filled Dirac sea and is largely unaffected by a small density of added fermions. † 1 2 1 L ¼ iψ ð∂ − iA0Þψ − jð∂ − iA Þψj − AdA þ: Hence, low-energy dynamics is described by t 2m i i 8π

ð61Þ † 1 2 2 L ¼ iψ ð∂ − ia0Þψ − jð∂ − ia Þ ψj t 2jmj i i We have included a Chern-Simons term −ð1=8πÞAdA ≡ 1 m μνλ − ada þ: ð67Þ −ð1=8πÞϵ Aμ∂νAλ to take into account the contribution of 8π jmj negative energy states to the Hall conductivity. We now use the standard flux attachment procedure and attach two flux By the substitution a → −2a, and assuming m<0, quanta to the fermion ψ. The resulting action is then one obtains the action (64) before coupling to

031027-10 IS THE COMPOSITE FERMION A DIRAC PARTICLE? PHYS. REV. X 5, 031027 (2015) electromagnetism. Thus, the usual fermionic CS theory is fermionic vortex which was interpreted as a bound state obtained in a large mass, nonrelativistic limit of the Dirac of a gaugino with a dual photon. It is tempting to speculate CF theory. In other words, one can deform the Dirac CF a connection between mirror symmetry and the physics of theory without a CS term into the conventional non- the half-filled Landau level. relativistic theory with a CS term, at the price of breaking We have suggested that careful measurement of transport particle-hole symmetry. near half filling can determine the Berry phase of the This does not mean that we have been able to derive composite fermion around the Fermi surface, and it dis- the effective field theory (36) using the familiar flux- tinguishes the standard HLR scenario and the scenario of attachment procedure, but now we can look at the old Dirac CF. Our theory implies the existence of a new gapped puzzle of Ref. [15] in a new light. If one performs the RPA phase, the PH-Pfaffian state, distinct from the Pfaffian and calculation using the action (64), we would find expres- anti-Pfaffian states by being its own particle-hole conju- sions identical to Eqs. (48)–(50) with the replacement gate. It would be interesting to see if there exist physical systems where such a phase is realized. Besides GaAs and iω 1 the surface of 3D TIs, systems with a tunable parameter like Π~ → σCF ; H 2π xy þ 2 ð68Þ bilayer graphene in a perpendicular magnetic field [55,56] seem to be promising venues to search for this new phase. CF where σxy is the Hall conductivity of the CFs (as a function 1 of frequency and wave number) and 2 comes from the ACKNOWLEDGMENTS Chern-Simons term ada. To be consistent with Eq. (34), The author thanks Alexander Abanov, Clay Córdova, σCF − 1 xy must be equal to 2 for all frequencies and wave Eduardo Fradkin, Siavash Golkar, Jainendra Jain, Woowon numbers. However, at least in the regime ω ≫ vFq, the Kang, Michael Levin, Eun-Gook Moon, Sergej Moroz, Hall conductivity of a Fermi liquid is expected to be zero. Sungjay Lee, Andrei Parnachev, Xiao-Liang Qi, Subir This is essentially the puzzle identified in Ref. [15], and this Sachdev, Steve Simon, Boris Spivak, and Paul puzzle would not exist if the low-energy dynamics is indeed Wiegmann for valuable discussions and comments. This described by the Dirac composite fermion. For the latter, work is supported, in part, by the U.S. DOE Grant No. DE- CF the nonzero value of σx can be interpreted as coming from FG02-13ER41958, MRSEC Grant No. DMR-1420709 by the Berry phase of the CF around the Fermi disk. It seems the NSF, ARO MURI Grant No. 63834-PH-MUR, and by a that the key to understanding the particle-hole symmetry Simons Investigator grant from the Simons Foundation. in the composite fermion theory is to understand the The author thanks Maissam Barkeshli and Michael appearance of this Berry phase. Mulligan for illuminating discussions.

V. CONCLUSIONS Note added.—After this work was completed, the author became aware of Ref. [57], which has some overlap with We have presented evidence that the composite fermion the current paper. In particular, the phase CFL2 in Fig. 5 of of the half-filled Landau level is a Dirac fermion, whose Ref. [57] differs from the picture advocated here only by Dirac mass vanishes when particle-hole symmetry is exact. the presence of a second Fermi surface, coupled to aμ and Although we did not try to derive Eq. (36) microscopically, carrying a trivial Berry phase. In addition, Ashwin it is hoped that such a derivation is possible, following one Vishwanath informed the author that the topological order of the methods used in Ref. [39] for deriving the composite of the PH-Pfaffian phase has been previously proposed Dirac-liquid Hamiltonian. We also notice that the LLL within the context of topological superconductors [58]. Lagrangian (9) suggests that the Dirac spinor may involve, as one of its components, the Lagrange multiplier χ that APPENDIX: GALILEAN INVARIANCE enforces the LLL constraints. For now, it is also clear that AND KOHN’S THEOREM the conventional flux attachment procedure cannot be applied to the for a Dirac fermion In the main text, we have avoided the issue of Galilean in a particle-hole symmetric manner. invariance and the related Kohn’s theorem. Kohn’s theorem The proposed map between the electromagnetic states that in the absence of disorder, the q ¼ 0, finite- current in the original and low-energy effective theory, frequency electromagnetic response of a system of ¯ μ μνλ Ψγ Ψ ¼ð1=2πÞϵ ∂νaλ, is reminiscent of mirror sym- interacting nonrelativistic electrons is the same as of metry [53] in three dimensions. Recently, Hook et al. used noninteracting electrons. In particular, the only pole in mirror symmetry to argue that the ground state of ð2 þ 1Þ- the response is at the cyclotron frequency. In the LLL limit dimensional supersymmetric QED in the presence of m → 0, the cyclotron frequency is infinite, and the response σ ω 0 σ ω ν magnetic impurities is characterized by an emergent is especially simple: xxð Þ¼ , xyð Þ¼ (the conduc- Fermi surface which encloses a volume proportional to tivities are measured in units of e2=h ¼ 1=2π). On the other the magnetic field [54]. The emergent fermion is a hand, Eq. (56) gives nonzero σxx at finite ω. The problem

031027-11 DAM THANH SON PHYS. REV. X 5, 031027 (2015) stems from the fact that the theory we were using does not σ~ ij σ~ ij ji −2e~ −2e m v_ : have the Galilean invariance of the original electrons. CF ¼ 2π ð jÞ¼2π ð j þ jÞ ðA7Þ The full solution to the problem of Galilean invariance, capable of reproducing the more subtle effects such as the Furthermore, the physical electromagnetic current is related 2 q dependence of the Hall conductivity of gapped states in to the field tensor constructed from aμ [Eq. (41)]. From the the clean limit [59,60], will be presented elsewhere. Here, formulas listed above, one can get the conductivities of the we only sketch a simplified version of the theory, adequate electrons, for finding the q ¼ 0, finite-frequency response. What 1 σ~ follows is, in a sense, a version (streamlined and trivially σ xx m ω “ xx ¼ 4 2 2 þ 2 i ; ðA8aÞ adapted to Dirac CFs) of the phenomenological modified σ~ xx þ σ~ xy B RPA” (MRPA) proposal of Ref. [61].(Atq ¼ 0, one does   not need to be concerned with the issues that have led to the 1 σ~ 1 σ − xy : “MMRPA” scheme of Ref. [62].) xy ¼ 4 σ~ 2 σ~ 2 þ 2 ðA8bÞ xx þ xy NR Galilean invariance is implemented by the introduction of a dynamic field vi, which transforms as a velocity under One sees that the only modification compared to Eqs. (56) Galilean boosts. The effective Lagrangian is schematically and (57) is the appearance of an additional term linear in ω 1 on the right-hand side of Eq. (A8a). L L ψ ψ † i ϵμνλ ∂ ¼ ð ; ;aμ;vÞþ2π Aμ νaλ: ðA1Þ It is instructive to consider the Drude model for the composite fermions, parametrizing disorders by a single For modes near the Fermi surface, the coupling of vi to other relaxation time τ. In this approximation, the conductivities fields can be subsumed into a modified gauge potential, of the CFs are

1 2 −1 a~ 0 ¼ a0 − m v ; ðA2Þ m ð−iω þ τ Þ 4 σ~ ¼ 2πρ ; ðA9Þ xx CF m2ð−iω þ τ−1Þ2 þ 4b2 1 a~ ¼ a þ m v ; ðA3Þ i i 2 i 2b σ~ −2πρ : xy ¼ CF 2 − ω τ−1 2 4 2 ðA10Þ mð i þ Þ þ b where m is the effective mass of the composite fermions on ~ the Fermi surface. In terms of aμ, the Lagrangian is Substituting these expressions into Eqs. (A8), we find approximately remarkably simple expressions,

† 1 μνλ m L ¼ Lðψ; ψ ; a~ μÞþ ϵ Aμ∂νaλ; ðA4Þ σ ¼ ; ðA11Þ 2π xx 2Bτ where ψ has a gauge coupling with a~ μ with charge −2.(Note b 1 σ ¼ þ ¼ ν: ðA12Þ that the mixed Chern-Simons term still involves aμ,nota~ μ.) xy B 2 One can integrate out vi to obtain interactions between the composite fermions, the major part of which is a Landau- In the relaxation time approximation, the electron con- type interaction required to restore Galilean invariance. To ductivities do not depend on frequency, and the Hall perform calculations in the RPA, it is, however, more conductivity is at its classical value. The clean case can τ → ∞ σ 0 convenient to leave vi in the Lagrangian. be recovered by taking . In this case, xx ¼ and σ ν ’ 0 Differentiating the action with respect to aμ, we get the xy ¼ , as required by Kohn s theorem at m ¼ . relationship between the composite fermion density and the current with the external field,

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031027-12 IS THE COMPOSITE FERMION A DIRAC PARTICLE? PHYS. REV. X 5, 031027 (2015)

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