Dirac Composite Fermion Theory of General Jain's Sequences

We reconsider the composite fermion theory of general Jain's sequences with filling factor $\nu=N/(4N\pm1)$. We show that Goldman and Fradkin's proposal of a Dirac composite fermion leads to a violation of the Haldane bound on the coefficient of the static structure factor. To resolve this apparent contradiction, we add to the effective theory a gapped chiral mode (or modes) which already exists in the Fermi liquid state at $\nu=1/4$. We interpret the additional mode as an internal degree of freedom of the composite fermion, related to area-preserving deformations of the elementary droplet built up from electrons and correlation holes. In addition to providing a suitable static structure factor, our model also gives the expected Wen-Zee shift and a Hall conductivity that manifests Galilean invariance. We show that the charge density in the model satisfies the long-wavelength version of the Girvin-MacDonald-Platzman algebra.

Since the discovery of the fractional quantum Hall effect (FQHE) in 1982 [1,2], many theoretical models have been invented to explain different aspects of this fascinating phenomenon. However, with no proposed theory capable of explaining all the richness of the FQHE, the latter remains one of the most non-trivial questions of condensed matter physics. One breakthrough idea was proposed by Jain [3], who suggested that the low energy degree of freedom of FQHE is the composite fermion (CF), which can be thought of as an an electron moving together with an even number of magnetic flux quanta, a picture inspired by previous ideas [4][5][6]. The The composite fermion model has achieved significant success. At the mean-field level, the composite fermion picture explains the presence (though not the magnitude) of the FQH gap, mapping the latter to an integer quantum Hall (IQH) gap of the composite fermions. Inspired by Jain's CF, Halperin, Lee, and Read introduced [7] a field theory-the HLR theory-that predicts a Fermiliquid-like behavior of composite fermions near half-filling (ν = 1 2 ), which been confirmed experimentally [8]. Despite its enormous success, the HLR theory leaves many issues unresolved. One issue is the scale of the energy gap, which is incorrectly predicted to be the cyclotron frequency in the simplest version of the flux attachment procedure. This issue can be resolved by treating the effective mass of the composite fermion as a phenomenological parameter, and by adding new terms in the effective Lagrangian, which, in the lowest Landau level limit, amounts to assigning an electric dipole moment to the composite fermion. A more serious problem is the apparent breaking of particle-hole symmetry (PH), which is the emergent symmetry of the lowest Landau level (LLL) [9]. Flux attachment equates the composite fermion density with the electron density, breaking the symmetry between electrons and holes, which holds in experiment [10]. The Dirac composite fermion theory [11] has been proposed as an effective field theory for FQH states near half-filling, which explicitly incorporates PH symmetry by assigning a Berry phase of π for the CF around the Fermi surface. The theory gives reasonable predictions for the electromagnetic response functions [12].
In contrast to the case of ν = 1/2, at ν = 1/4 there is no symmetry that can be used to constrain the Berry phase of the CF. Recently Goldman and Fradkin, motivated by the experimentally observed reflection symmetry relating the I − V curves of states with filling fraction ν < 1/4 and ν > 1/4 [8,10,13], proposed a Dirac composite fermion model for the ν = N/(4N + 1) Jain sequences (and for the general Jain's sequences with ν = N 2nN ±1 ). The Berry phase of the CF in this model is equal to π. The similar effective theory for FQH states near ν = 1/4 was proposed by Wang in [14] motivated by the numerical study of the many-body Berry phase.
We will show in Goldman and Fradkin's model the static structure factor of the ν = N/(4N + 1) [and, more generally, ν = N/(2nN + 1)] states violates the Haldane bound, which places a lower bound on the coefficient of the leading q 4 behavior of the projected static structure factor [15] where s 4 is the coeffient of q 4 ℓ 4 B of the projected static structure factor and S is the Wen-Zee shift. This paper aims to construct an effective theory that adequately describes the physics of the ν = N/(4N ± 1) Jain's sequences in the lowest Landau level limit. We combine the ideas of preceding works [11,[16][17][18], including Haldane's idea of a new geometric degree of freedom of the FQH system [17] and that of the bimetric theory of this degree of freedom [18]. Our model is a hybrid model that includes a Dirac composite fermion sector and an additional spin-2 mode with the effective action of the form of the bimetric theory. In this model, in addition to the low-energy GMP mode at the energy scale of the effective cyclotron energy, which tends to zero as N → ∞, there is an extra high-energy mode with energy which remains finite in the limit N → ∞, and hence must already exist in the ν = 1/2n Fermi-liquid state. With the extra mode, which we call the Haldane mode, the issue with the Haldane bound of the projected static structure factor in the previous Dirac composite fermion model [16] can be fixed. We derive the topological Wen-Zee shift of Jain's sequences that matches the expected results in the literature [19,20]. We also evaluate the wave-number dependence of the Hall conductivity and verify its connection with the Hall viscosity [21]. We then rederive the long-wavelength limit of GMP algebra as evidence that the theory is appropriate for the LLL limit.
The plan of the paper is as follows. In Sec. II we propose our effective theory, introducing its ingredients and briefly discussing the motivations. Section III is devoted to calculation of the Wen-Zee shift, the Hall viscosity, and the Hall conductivity. In Sec. IV, we derive the long-wavelength version of GMP algebra. In Sec. V we calculate the projected static structure factor using a semiclassical approach developed in Refs. [12,22] and comment on the importance of the extra mode. Lastly, we conclude our paper and discuss the open questions in Sec. VI. Some technical details of the calculations are left to the Appendices.

II. EFFECTIVE FIELD THEORY
Inspired by previous works [16][17][18]23], we propose an effective theory for the Jain sequences around ν = 1/(2n), ν ± = N 2nN ±1 , where n and N are integers. In practice, only Jain sequences have been seen only around ν = 1/2 and ν = 1/4, so in practice for n = 2, though for generality we will keep n arbitrary in our formulas.
As we will see in Section V, the projected static structure factor for the ν = N/(4N + 1) calculated in the Dirac composite fermion model by Goldman and Fradkin [16] violates the Haldane bound. In order to fix this issue, we add an extra chiral mode to the model. Though the microscopic nature of this mode is not important for our effective theory, one can, following Haldane [17,23], think about this mode as corresponding to the area-preserving deformation of the composite object ("elementary droplet") built up from electrons and correlation holes. In this paper, we show that with the proper choice of the coupling of the extra mode with external background electromagnetic field, we can reproduce the known physical results for the general Jain's sequence.

A. Symmetry of the Lowest Landau Level
The system of electrons with Landé factor g = 2 [24] in a constant magnetic field B has a well defined lowest Landau level (LLL) limit where the electron mass goes to zero [25,26]. Inheriting the symmetry of non-relativistic massless fermion with g = 2 [25,27] [28], the external gauge field appears in the effective action as the combinatioñ where v i is the drift velocity, given by v i = ǫ ij Ej B at the leading order, and ω µ is the spin connection of the Newton-Cartan space that was defined through derivative of vielbein e a i [25] In particular, in flat space and uniform magnetic field Effective theories on the LLL, including composite fermion models, should couple with the external gauge field throughÃ µ .

B. Composite fermion sector
In this paper, we propose the following effective action of Jain's sequences where the Dirac composite fermion action is given by where σ a is the Pauli matrix and a µ is the emergent gauge field. As in Ref. [16], the Dirac composite fermion is massless. Here we chose the Fermi velocity of composite fermion v F = 1 to simplify the formulae [12]; the main results of this paper do not depend on v F . The covariant derivative of the composite fermion is The CF is charged neutral without direct coupling to the external electromagnetic field. The third term of (7) implies that composite fermion has a dipole moment perpendicular to its momentum. This term is new compared to previous work on Dirac composite fermion model with the filling fraction is near 1/2n [14,16]. The neutrality of the composite fermions , as well as its dipole moment, are features of the Dirac composite fermion near half-filling [11,12].
Differentiating the action over a 0 gives us a constraint (we work in flat space) while differentiating the action with respect to A 0 and taking average over space gives the average physical charge density,ρ .
The Haldane mode will not change the above equation; physically, the area-preserving deformations of the elementary droplets do not change the average density. The specific values of emergent magnetic field correspond to FQH filling fractions of the Jain states where the filling fraction of composite fermion is which are the filling factors of IQH states of a massless Dirac fermion.

C. Haldane sector
The Haldane mode (named so because it embodies Haldane's idea of an emergent degree of freedom of the FQHE [17,23]) is a gapped dynamical spin-2 degree of freedom described by the extra metricĝ ij . We define the emergent vielbein as the square root of the ambient metricĝ and we also denote the matrix inverse ofê α i byÊ i α . The compatibility conditions, are satisfied bŷ We further introduce the definition of the emergent Ricci curvature ofĝ ij aŝ Restricting the dynamical part of the the emergent metric to fluctuations that preserve the area, settingĝ = g, we can rewrite the emergent Ricci curvature aŝ We propose to use as the action of the spin-2 mode the leading (in derivatives) terms of bimetric theory introduced in Refs. [18,29].
with ν is the FQH filling fraction, and the magnetic field in curved space is defined as B = ∂1A2−∂2A1 √ g . The mass m > 0 in the potential term sets the energy gap for the extra spin-2 mode. We require γ < 1 so that the ground state is isotropic [18,29]. The last term comes from the symmetry of LLL, where the temporal component of the emergent spin connection enters the Lagrangian through the combinationω 0 + 1 2 ǫ ij ∂ i v j , hence its coefficient is fixed by that of the first term. This last term contributes to the finite-wave-number Hall conductivity at the k 2 order. Besidesm there is only one additional parameter, ζ [18]. In the subsequent section, we will show that one can fix the coefficient ζ for general Jain's sequences using the Wen-Zee shift. We will find that ζ only depends on n but not N , as expected.

III. WEN-ZEE SHIFT S AND GALILEAN INVARIANT
In this section, we find the value of ζ. We show that, with ζ = n − 1, one can obtain the expected Wen-Zee shift, the finite-wave-number Hall conductivity and the Hall viscosity for the general Jain's sequences ν ± . The results demonstrate that the topological Wen-Zee shift as well as the relationship between the finite-wave-number Hall conductivity and the shift [21] are reproduced successfully in our model. At the filling fraction ν + , the filling fraction of composite fermion is ν CF = N + 1/2. The Dirac composite fermion then forms an IQH state. One then integrate out the fermion fields ψ and obtain the Chern-Simons (CS) action The first term encodes the Hall conductivity of IQH state of the Dirac fermion [30]. The second term represents the effective coupling of Dirac composite fermions with the background curvature, which reproduces the known Wen-Zee shift of Dirac IQH states [27,31] [32]. We have the following leading terms in the effective Lagrangian of composite fermion sector [33] We then integrate out the emergent gauge field a µ to obtain the final effective action of external background fields From Eq. (26) we obtain the contribution to the charge density from the Dirac composite sector in a static background without electric field [34] with R ≡ 2 √ g (∂ 1 ω 2 − ∂ 2 ω 1 ) being the Ricci curvature of the background metric g ij . The first term of (27) comes from the electromagnetic CS termÃdÃ. The second term receives contributions from the mixed CS termÃdω and theÃdÃ tern due to the spin connection in the definition ofÃ [35]. Now let us consider the response of the Haldane sector to background fields. At zero frequency, in the isotropic phase, the potential term in (23) constraints the ambient metric to follow the background metricĝ ij = g ij [18]. As a consequence, one can read off the contribution to charge density of the Haldane sector in a static background without an electric field, Combining Eqs. (27) and (28), we obtain the charge density, and from it the total charge on a closed manifold where N φ is the total number of flux quanta N φ = 1 2π √ gB, and χ is the Euler character of the Riemann surface which means that the Wen-Zee shift [36] is where we dropped the term of subsub leading in 1/N expansion.

Filling fraction
One can repeat the calculations in for ν + and obtain the effective Lagrangian of composite fermion sector for We again combine the charge density of CF sector and Haldane sector and obtain The first two terms of (34) come from the effective Lagrangian of composite fermion sector (33), the last term of (34) come from the Haldane sector (23). We then obtain the Wen-Zee shift, up to leading and first subleading orders in the 1/N expansion.
We now see that if we chose we can reproduce the expected Wen-Zee shift for both ν + and ν − sequences [19,20] S ν+ = N + 2n, For Jain's sequences near half-filling with n = 1, the coefficient ζ = 0. In this case, there is no need for a Haldane mode. We argue that, due to the particle-hole symmetry at ν = 1/2, the electrons and holes are uniformly distributed in the elementary droplets, so their areapreserving deformations do not change the local charge density and then decoupled from the external electromagnetic field.

B. Hall viscosity
To derive the zero frequency Hall viscosity, one needs to work out the coefficient of the mixed CS term Adω in the effective action. This coefficient is determined by the Wen-Zee shift as νSAdω [36]. One can also do the calculation explicitly using the effective Lagrangian of the CF sector and replace, in the Haldane sector, the emergent spin connectionω by the spin connection of the background metric ω at low energy. We found that for both sequences ν − and ν + .

C. Finite-wave-number Hall conductivity
In a flat background where g ij = δ ij , using the definitions of ω 0 andÃ, we can rewrite the CF action (26) as AdA.
The Lagrangian of the Haldane sector for Jain's sequence ν + state at low energy can be rewritten as One can derive the current in flat space, with ζ = n − 1, as a function of the applied electric field (we work up to leading and first subleading order in the 1/N expansion).
where we have assumed the Landé factor of electron is g = 2. We then obtain the Hall conductivity at finite momentum with the magnetic length is ℓ B = 1/ √ B. One can perform the same calculation to yield the finite momentum Hall conductivity of Jain sequence ν − σ ν− The wave-number dependence of the Hall conductivity of both sequences ν + and ν − satisfy the relationship with the Wen-Zee shift [21]. The results imply that our model satisfies the symmetries of the LLL, as designed.

IV. THE GMP ALGEBRA
In the classic paper [37], Girvin, MacDonald, and Platzman showed that the LLL projected density operator ρ e (k) obeys the following (GMP) algebra The GMP algebra was discovered independently in string theory with the name W ∞ [38,39]. In the FQHE, the GMP algebra is a consequence of charge conservation and the LLL constraint. In the long wave length approximation, the GMP algebra is reduced to the algebra of area-preserving diffeomorphisms [40] [ρ(k), Near half-filling, the long wavelength limit of GMP algebra has been shown to arise from Dirac composite fermion theory in Ref. [12]. In this section, we generalize the derivation of the GMP algebra for the general Jain sequences. From the action (6), one can read off the charge density operator in the flat background where g ij = δ ij ,R = 2ǫ ij ∂ iωj up to subleading order in the momentum expansion, where ρ (1) and ρ (2) come from the CF and the Haldane sectors, respectively. Explicitly, The second term of Eq. (47) implies that the composite fermion has a dipole moment perpendicular and proportional to its momentum. Employing the canonical equaltime anti-commutation relation of the CF field, and the commutation relation of emergent gauge field due to the Chern-Simon term of a µ in the action (7) [ one can derive the algebra up to lowest order in momentum of ρ (1) [ρ (1) Interested reader can find the detailed derivation of Eq. (51) in Appendix B. Equation (51) implies the GMP algebra of the CF sector. Following Ref. [18], one find the charge density in the Haldane sector also satisfies the GMP algebra, Combining equations (51) and (52), we arrive at the GMP algebra of the electron density operator at the lowest order in momentum.
The GMP algebra is the key feature of a LLL effective theory. For the algebra, it is crucial for the Dirac composite fermion to have an electric dipole moment [12]. The dipole moment term in Eq. (7) thus not only preserves the LLL invariance of the DCF action but also plays an essential role in the GMP algebra (53).

V. DYNAMICS OF THE CF SURFACE AND THE STATIC STRUCTURE FACTOR
In the theory of Dirac composite fermion near halffilling, the dynamics of the composite fermion sector at low energy can be interpreted as the deformation of the composite Fermi surface [12,22,29,41]. One can apply the semiclassical equation of motion to analyze the physics at the long-wavelength limit [42]. In this section, we will extend the semiclassical approach for Dirac composite fermion theory of general the Jain's sequences and derive the static structure factor (SSF).

A. Semiclassical equation of motions
Near the filling fraction ν = 1 2n , the Dirac composite fermion forms the Fermi surface with a small effective background magnetic fieldb induced by the emergent gauge a µ . Following Refs. [12,22,29,41], we introduce the deformation of composite Fermi surface at a given position x where u n (x) is a space dependent functions and θ is the polar angle in the momentum space. The Fermi momentum is related with the composite fermion density by Luttinger's theorem [43] u 1 (x) and u −1 (x) can be regard as the complex components of a vector u i (x), where we defined the complex coordinate and the complex derivative The composite fermion density and momentum operator in this essentially semiclassical formalism is [12,22] The u n modes satisfies the following commutation relation [22,41] [ The same commutation relation for Fermi surface in a magnetic field was derived by Haldane using bosonization [44]. The Fermi-liquid kinetic equation can be interpreted as the equations of motion that followed from the commutation relation (60) and the Hamiltonian [12,45] with e i = ∂ i a 0 − ∂ 0 a i . For m = 0, Eq. (62) giveṡ which can be understood as the conservation law of CF number: ∂ t ρ CF + ∇ · j CF = 0, with CF number current We will use this semiclassical approach to derive the density-density correlation function, and specifically the static structure factor to leading order in the momentum expansion.

B. Projected Static Structure Factor
In this section, we will use the semiclassical approach to calculate the leading order in gradient expansion of the projected static structure factor. We will show that the extra Haldane mode is crucial to satisfy the Haldane bound, and the fact that the latter is satisfied provides a nontrivial check for our proposal.
We combine Eqs. (47) and (59) to obtain the charge density operator of the CF sector in the semiclassical formalism, We take the time derivative of above equation and use the equation of motion (62) for m = 0, ±1, ±2 to obtain, up to leading order in the gradient expansion.
(66) We leave the detail derivation of above equation to Appendix C. Using the explicit value of k F from Eq. (55), we see that only term including u ±2 in (66) survives and we are left with Note that by the construction of the Hamiltonian (61), at long wave length limit the ground state is annihilated by u m with m < 0 (see Ref. [22] for details): u −2 |0 = 0|u 2 = 0, which allows us to derive the equaltime density-density correlation function, (68) We then use the commutation relation of u ±2 and obtain From Eq. (69) we then read off the contribution to the coefficient of q 4 ℓ 4 B of the projected SSF from the CF sectors where we defined the projected SSF following Girvin, MacDonald, and Platzman [37] s = 1 ρ e δρ(−q)δρ(q) .
From the action of the Haldane sector (23), one can calculate the contribution to the equal time correlation function δρ (2) (−q)δρ (2) (q) as well as the contribution of the Haldane sector to the projected SSF. The calculation follows directly from Ref. [18], we will not repeat the calculation and quote the result only: We then have the total static structure factor [47] s 4 =s 4 +s We are now in position to compute the coefficients 4 for both filling fraction ν ± of the general Jain's sequences.
Combining Eqs. (73), (9), (10) and (11) one obtain, for filling fraction ν + = N 2nN +1 and for ν − = N 2nN −1 , The O(1/N ) corrections in Eqs. (74) and (75) are beyond the limit of accuracy of the semiclassical approximation and will be dropped in future formulas. If we fix the coefficient ζ = n − 1 as in Section III to match the Wen-Zee shift, we finally have the coefficients 4 , Remembering that the shift of ν + is S ν+ = N + 2n, we see thats ν+ 4 saturates the Haldane bound which suggests that ν + is a chiral state [17,48]. However, the Haldane bound for the ν − state, is saturated only for n = 1 [49]. For example, the ν = N/(4N − 1) states are not chiral: the chirality of the Haldane mode is opposite to the chirality of the lowenergy magnetoroton. We emphasize here that the contribution to projected SSF of the Haldane sector is crucial in our proposal. Without the additional terms (2) 4 , the projected SSF of the Jain sequence ν + violates the Haldane bound for n > 1 [17,23]. The coefficient ζ = n − 1 not only helps us to obtain the correct Wen-Zee shift in Sec. III but also makes it possible to satisfy the lower bound on the projected SSF.

VI. CONCLUSION
In this paper, we propose a Dirac composite fermion model of general Jain's sequences ν = N 2nN ±1 . The proposal incorporates a Dirac composite fermion and an extra spin-2 mode at ν = 1/4 and the Jain sequences around it. We show that our model reproduces the known results of Hall conductivity and Hall viscosity. The results imply that our model satisfies the Galilean invariant and reproduces the Wen-Zee shift. Using the semiclassical approach, we calculated the coefficients 4 of the projected static structure factor. We show that, without the extra mode, the projected SSF violates the Haldane bound. We also derive the long-wavelength limit of the GMP algebra that emerges from the model. In summary, the results calculated in this paper suggest that our Dirac composite fermion model is a proper effective theory of LLL that provides the expected physical results of general Jain's sequences.
In this paper, we proposed a single Haldane mode with spin-2. However, we can come up with the Haldane sector that includes multiple extra spin-2 modes. In particular, if we consider s spin-2 modes with different energy gaps determined bym i and different coupling ζ i with background electromagnetic field as described in Eq. (23), then we can reproduce the physical results of this paper with the condition We then arrive at the other main conclusion of this paper, which is the extra spin-2 mode(s). We expect that those modes have higher energy gaps than the Dirac composite fermion effective cyclotron frequency but still lower than the electron's cyclotron gap. We suggest that the appearance of the extra spin-2 mode(s) can be confirmed in the numerical calculation of the correlation function of the stress tensor operator in the LLL [50,51] at or near the filling fraction 1 4 . We expect that there would be an additional peak(s) in the spectral function of the stress tensor at high frequency. We can also verify the extra spin-2 mode(s) with their chiralities by circularly polarized Raman scattering [52].
In this paper, we assume that the Berry phase of the composite fermion is equal to π as for a massless Dirac fermion, a choice motivated by the symmetry of the I −V curve around ν = 1/4. Without this motivation, one can consider other possibilities; for example, one in which the CF has zero Berry phase as in the HLR theory. The Haldane mode will also be needed here to avoid violation of the Haldane bound within the HLR theory (which already occurs at near half filling [12] For completeness, we present the derivation of the Haldane bound in this Appendix. 2D electrons in a constant magnetic field satisfies the following Ward's identities which are nothing but the conservations of U (1) charge and momentum density, where the momentum density of non-relativistic electron is P i = mj i . In the lowest Landau level limit, where we take electron's mass to zero, the momentum density conservation equation becomes the force balance equation from which one can derive j i in terms of T ij . Combining with the charge conservation, one obtain the lowest Landau level Ward's identity [48] Using the Ward identity (A3), one can derive the first spectral sum rule for lowest Landau level [41,48] [53] with s 4 is the coefficient of q 4 ℓ 4 B of the projected SSF and the spectral functions ρ T andρ T are defined as [41] where N is the total number of electrons, |0 is the ground state, the sum is taken over all excited states |n in the lowest Landau level, and E n is the energy of the state |n . The relation of Hall viscosity in term of the retard Green's functions of stress tensor at zero spatial momentum with the definition of retard Green's function at frequency ω Using Eq (A7), one can derive the second lowest Landau level sum rule By definition, both ρ T (ω) andρ T (ω) are non-negative, therefore the combination of sum rules (A4), (A9) leads us to an inequality which is the Haldane bound.
which is Eq. (51) of the main text.
Appendix C: Density operator in the semiclassical formalism In this Appendix, we provide a detailed derivation of the charge density operator of the CF sector in terms of u m operators. From Eqs. (B7) and (59) we havė We rewrite the equation of motion for u ±1 aṡ We also need the equation of motion for u ±2 up to leading order in spatial derivativeṡ Combing the time derivative of Eq. (65) and Eqs. (C2), (C3), we obtaiṅ With the help of the Bianchi identity ∂ z ez − ∂ze z = − i 2ḃ , the equation of motion for u ±2 , and a combination of Eqs. (63) and (C1), we obtaiṅ which is Eq. (66) in the main text.