Quantum mechanics of composite fermions

We establish the quantum mechanics of composite fermions based on the dipole picture initially proposed by Read. It comprises three complimentary components: a wave equation for determining the wave functions of a composite fermion in ideal fractional quantum Hall states and when subjected to external perturbations, a wave function ansatz for mapping a many-body wave function of composite fermions to a physical wave function of electrons, and a microscopic approach for determining the effective Hamiltonian of the composite fermion. The wave equation resembles the ordinary Schr\"{o}dinger equation but has drift velocity corrections which are not present in the Halperin-Lee-Read theory. The wave-function ansatz constructs a physical wave function of electrons by projecting a state of composite fermions onto a half-filled bosonic Laughlin state of vortices. Remarkably, Jain's wave function ansatz can be reinterpreted as the new ansatz in an alternative wave-function representation of composite fermions. The dipole model and the effective Hamiltonian can be derived from the microscopic model of interacting electrons confined in a Landau level, with parameters fully determined. In this framework, we can construct the physical wave function of a fractional quantum Hall state deductively by solving the wave equation and applying the wave function ansatz, based on the effective Hamiltonian derived from first principles, rather than relying on intuitions or educated guesses. For ideal fractional quantum Hall states in the lowest Landau level, the approach yields physical wave functions identical to those prescribed by the standard theory of composite fermions. We further demonstrate that the reformulated theory of composite fermions can be easily generalized for flat Chern bands.


I. INTRODUCTION
Exotic correlated states of electrons emerge in fractional quantum Hall systems, where strong magnetic fields completely quench the kinetic energy of electrons, rendering conventional many-body techniques inadequate in addressing the effects of correlations between electrons [1].The theory of composite fermions, proposed by Jain in 1989, offers a comprehensive framework for understanding these exotic states [2].It introduces a new paradigm, interpreting correlated states of electrons as non-correlated or weakly-correlated states of fictitious particles called composite fermions, which are assumed to be the bound states of electrons and quantum vortices.Based on the insight, the theory prescribes an ansatz for constructing many-body wave functions that achieve nearly perfect overlaps with those determined by exact diagonalizations for various fractional quantum Hall states in the lowest Landau level [3].On the other hand, for predicting the responses of these states to external perturbations, one usually employs the effective theory proposed by Halperin, Lee and Read (HLR) [4], which has been shown to make predictions that align well with experimental observations [5].The two components of the theory, namely the wave function ansatz and the effective theory, complement each other, forming a versatile framework for understanding the rich physics of the fractional quantum Hall systems.
Despite the remarkable success, the theory still lacks a concrete foundation.On the one hand, one usually re- * junrenshi@pku.edu.cnlies on intuition or educated guesses when constructing composite fermion wave functions.This is in sharp contrast to the deductive approach typically employed for ordinary particles like electrons, for which one can confidently write down a Hamiltonian for a given physical circumstance, and obtain wave functions by solving the Schrödinger equation.On the other hand, the conjecture of the HLR theory -a composite fermion also obeys the ordinary Schrödinger equation -is only justified heuristically.Lopez-Fradkin's theory [6] is often cited as the rationale behind both the wave function ansatz and the effective theory [2].However, the theory can only be viewed as a tentative argument rather than a rigorous foundation for the theory of composite fermions due to two obvious issues.
Firstly, Jain's ansatz prescribes wave-functions of electrons in the form [3] which differs from the form suggested by Lopez-Fradkin's theory based on the singular Chern-Simons (CS) transformation [6] Ψ ( where Ψ and ΨCF represent the wave function of electrons and composite fermions, respectively, is the Bijl-Jastrow factor which presumably attaches a vortex with two flux quanta to each electron, PLLL is the projection operator to the lowest-Landau-level, and {z i ≡ (x i , y i )} and {z i ≡ x i + iy i } denote the coordinates of electrons in the vector and complex forms, respectively.Equation ( 2) is formulated in the full Hilbert space of free electrons while Eq. ( 1) is defined in the restricted Hilbert space of a single Landau level.Reconciling the two is non-trivial [7].
Secondly, the picture of composite fermions implied by Lopez-Fradkin's theory, which is also inherited by the HLR theory, differs from that obtained by directly inspecting the ansatz wave function Eq. (1).For the latter, Read's analysis indicates that the electron and the vortex in a composite fermion are spatially separated [8].The finding contradicts the picture implied by Eq. ( 2), which suggests that a composite fermion is a point particle, consisting of an electron and δ-function flux tubes.More recently, Son points out that the HLR theory lacks the particle-hole symmetry [9], whereas the ansatz wave function is shown to preserve the symmetry well [10].These observations raise doubts about whether the HLR theory accurately describes the composite fermions implied by the ansatz wave function.It prompts the need for an alternative effective theory, or ideally, an alternative foundation from which both the wave-function ansatz and the effective theory can be inferred.
The dipole picture of composite fermions, initially proposed by Read, offers an alternative picture for the fictitious particle [8,11].The picture differs from the HLR view in two fundamental ways.Firstly, instead of being a point particle, the composite fermion has a dipole structure with the electron and vortex spatially separated.Secondly, the electron and the vortex are confined in two separated Landau levels created by the physical magnetic field and an emergent CS magnetic field, respectively, as opposed to moving in a free space [12,13].The dipole picture is shown to yield low-energy and long-wavelength electromagnetic responses that are identical to those predicted by the Dirac theory of composite fermions [14], indicating that it satisfies the general requirements of particle-hole symmetry.The feature, along with the fact that the picture is inferred directly from microscopic wave functions, sets it apart from other alternatives.
Pasquier and Haldane investigate the dipole picture of a system of bosons in an isolated Landau level at filling factor one [15].Their work, along with subsequent developments by other researchers [16][17][18], sheds light on the construction of physical wave functions.They propose interpreting vortices as auxiliary degrees of freedom that extend the physical Hilbert space to a larger Hilbert space of composite fermions.To obtain a physical state from a state of composite fermions in the enlarged Hilbert space, one must eliminate the auxiliary degrees of freedom by projecting the state into a physical subspace defined by a pure state of the vortices.In this context, the Bijl-Jastrow factor is interpreted as the complex conjugate of the wave function of the vortex state, rather than the numerator of the singular gauge factor in Eq. ( 2).The interpretation naturally leads to a wave function ansatz alternative to Eq. ( 1), and avoids the difficulties associating with the singular CS transformation [16].
In this paper, we present a theory of quantum mechanics for composite fermions based on the dipole picture and Pasquier-Haldane's interpretation.The theory comprises three complimentary components: a wave equation for determining the wave functions of a composite fermion in ideal fractional quantum Hall states and when subjected to external perturbations, a wave function ansatz for mapping a many-body wave function of composite fermions to a physical wave function of electrons, and a microscopic approach for determining the effective Hamiltonian of the composite fermion.In our theory, the state of a composite fermion is represented by a bivariate wave function that is holomorphic (antiholomorphic) in the coordinate of its constituent electron (vortex), and is defined in a Bergman space with a weight determined by the spatial profiles of the physical and the emergent CS magnetic fields.The wave equation is derived by applying the rules of quantization in the Bergman space to the phenomenological dipole model proposed in Ref. 14.It resembles the ordinary Schrödinger equation but has drift velocity corrections which are not present in the HLR theory.The wavefunction ansatz constructs a physical wave function of electrons by projecting a state of composite fermions onto a half-filled bosonic Laughlin state of vortices.Remarkably, Jain's wave function ansatz, which underlies the success of the theory of composite fermions, can be recast to the form of the new ansatz using an alternative wave-function representation for composite fermions.The phenomenological dipole model can be derived from the microscopic model of interacting electrons confined in a Landau level by applying a Hartree-like approximation, with its parameters determined from first principles.In this framework, we can construct the physical wave function of a fractional quantum Hall state deductively by solving the wave equation and applying the wave function ansatz, rather than relying on intuition or educated guesses.We further demonstrate that the reformulated theory of composite fermions can be easily generalized for flat Chern bands, which are also predicted to host the fractional quantum Hall states [19,20].
The remainder of the paper is organized as follows.In Sec.II, we introduce the dipole model which is the basis of our discussions, and give an overview of the main results of this work.In Sec.III, we develop the theory for ideal fractional quantum Hall systems which are subjected only to uniform magnetic fields.In Sec.IV, the theory is established for general systems which could be subjected to spatially and temporarily fluctuating external perturbations and have inhomogeneous densities.In Sec.V, we provide a microscopic underpinning for our theory by deriving the phenomenological dipole model from the microscopic Hamiltonian of interacting electrons confined in a Landau level.In Sec.VI, as an application of our reformulation, we generalize the theory of composite fermions for flat Chern bands.In Sec.VII, we b = 2φ 0 ρ v n B Figure 1.Dipole model of composite fermions.A composite fermion consists of an electron (black) and a vortex (gray).The electron is confined in the Landau level induced by the physical magnetic field B, while the vortex belongs to a bosonic liquid of vortices in the ν = 1/2 Laughlin state.Under the mean-field approximation, the vortex is considered as an independent particle confined in the Landau level induced by an emergent CS magnetic field b.The electron and the vortex are bound together by a binding potential which can be modeled as the harmonic potential Eq. ( 23) in a lowest Landau level.
summarize and discuss our results.Certain details of derivations are presented in Appendices.

A. Dipole model
Our theory is based on the dipole picture which was originally proposed by Read for the half-filled Landau level [8].The picture can be generalized to a dipole model of composite fermions which can be applied to arbitrary filling factors [13,14,21].
The model is illustrated in Fig. 1.According to the model, a composite fermion consists of an electron and a vortex confined in two separate Landau levels, one for the electron is the Landau level induced by the physical magnetic field B = −Bn, another for the vortex is the fictitious Landau level induced by an emergent CS magnetic field b = bn with its strength determined by the CS selfconsistent condition b = (2h/e)ρ v , where ρ v is the density of vortices, and n denotes the normal vector of the twodimensional plane of the system.In general, both the physical magnetic field and the CS magnetic field can be non-uniform.The two particles are bounded by a binding potential, which can be shown to be well approximated by a harmonic potential for the lowest-Landau-level (see Sec. V B).
The non-interacting dipole model is actually a meanfield approximation for an underlying correlated system of composite fermions.First of all, electrons confined in the physical Landau level are interacting.On the other hand, in Pasquier-Haldane's interpretation [15,16], vortices are auxiliary degrees of freedom introduced to extend the physical Hilbert space of electrons to a larger Hilbert space of composite fermions.They are assumed to form a collective half-filled bosonic Laughlin state (see Sec. III B).In Sec.V, we will show how the non-interacting dipole model emerges after applying a Hartree-like approximation in the enlarged Hilbert space.We note that the standard interpretation of the vor-tex, namely, an entity consisting of two quantized microscopic vortices, is actually a property derived from the particular collective state assumed for the vortices (see Sec. IV B).
It is also possible to interpret the composite fermion as a point-particle by defining its momentum p and coordinate x.A definition of the momentum, as pointed out by Read [8], could be where z and η are the coordinates of the electron and the vortex, respectively, and l B ≡ /eB is the magnetic length of the B-field.We can define x = η, as suggested in Refs.13 and 14.The composite fermion can then be interpreted as a particle that is subjected to a uniform momentum-space Berry-curvature [13,14,22] and obeys the Sundaram-Niu dynamics [23].It is notable that such a particle has a modified phase space measure [24].Consequently, for a Landau level with the particle-hole symmetry, the kinetic energy of the composite fermion should be modeled as [14] where m * denotes the effective mass of the composite fermion, and D = b/B is the density-of-states correction factor due to the modified phase space measure [14].In Sec.V B, we will show that the microscopic derivation of the dipole model does give rise to the peculiar form of the kinetic energy.Finally, we note that one is actually free to choose the definition of (x, p) and have a different interpretation.The physical results do not depend on the interpretation.This is demonstrated in Ref. 14 where two different choices of the definition and their interpretations are compared.

B. Summary of results
We summarize the main results of this work as follows: A) The state of the composite fermion can be represented by a bivariate wave-function that is holomorphic (anti-holomorphic) in the coordinate of its constituent electron (vortex) (see Sec. III A).The Hilbert space is the tensor product of two Bergman spaces, one for the electron and one for the vortex, with weights determined by the spatial profiles of the physical and CS magnetic fields, respectively (see Sec. IV A).
B) A new wave function ansatz can be logically inferred from the dipole model (see Sec. III B): where Pv denotes the projection onto the collective state assumed for vortices.Remarkably, the new ansatz and the standard ansatz Eq. ( 1) are equivalent, although they use two different wave-function representations for composite fermions.The two representations can be related by a transformation shown in Eq. ( 20) for ideal states and Eq. ( 48) generally.
C) A general wave equation can be established for composite fermions, valid not only for ideal systems, but also when external perturbations are present.The wave equation has a biorthogonal form, shown in Eqs.(73, 74), and its Hamiltonian in the long-wavelength limit has corrections from the drift velocities of the electron and the vortex, shown in Eq. (71).For ideal systems, the wave equation yields wave functions identical to those prescribed by the standard theory.However, the responses to external perturbations predicted by the wave equation will differ from those predicted by the HLR theory because of the drift-velocity corrections.It has been shown that the dipole model yields longwavelength responses identical to those predicted by the Dirac theory of composite fermions with a dipole correction [14].
D) The non-interacting dipole model can be derived from the underlying microscopic model of a set of interacting electrons confined in a Landau level by applying a Hartree-like approximation in the enlarged Hilbert space of composite fermions.The origin of the fictitious Chern-Simons fields is clarified.They are introduced to impose the requirement of consistency of orthonormalities between the physical Hilbert space and the enlarged Hilbert space (see Sec. V A).The derivation also confirms that the kinetic energy, which is basically the Coulomb attraction energy between the electron and the charge void induced by the vortex, is indeed proportional to D = 2ν, where ν is the filling factor of the system, and can be well approximated by the parabolic form Eq. ( 5) for a lowest-Landaulevel (see Sec. V B).
E) The reformulated theory of composite fermions can be generalized for a flat Chern band with a Chern number |C| = 1 by substituting the Chern band in place of the physical Landau level in the dipole model shown in Fig. 3.We find that the effective band dispersion experienced by a composite fermion is renormalized by the combination of the quantum metric and the Berry-curvature of the band that gives rise to the heuristic "trace condition" [25].The observation rationalizes the "trace condition" for the stability of a fractional Chern insulator state.It also suggests the possibility of stabilizing a fractional Chern insulator state in a non-flat Chern band when the renormalization cancels the dispersion of the Chern band (see Sec. VI).

III. THEORY FOR IDEAL SYSTEMS
In this section, we develop the quantum mechanics of composite fermions in systems that are subjected only to uniform external magnetic fields and have homogeneous densities.We will establish a new wave function ansatz and a set of wave equations for composite fermions.Remarkably, our approach can be shown to reproduce the well-established results of the standard theory.The principles established in this section will serve as the foundation for developing a general theory.

A. Hilbert space
The Hilbert space of a composite fermion shown in Fig. 1 is the tensor product of two Hilbert spaces with respect to the two Landau levels.It differs from that of an ordinary quantum particle in a free space as assumed in the HLR theory.
The Hilbert space spanned by a Landau level is a weighted Bergman space [26,27].For a disc geometry, the space includes all holomorphic polynomials in the complex electron coordinate z = x + iy.The inner product between two states ψ 1 (z) and ψ 2 (z) in the space is defined by A Bergman space with the Gaussian weight is also known as the Segal-Bargmann space [28].
The Hilbert space of a vortex is also a Segal-Bargmann space consisting of all anti-holomorphic polynomials in the complex conjugated vortex coordinate η = η x − iη y , where η x and η y are the Cartesian components of the vortex coordinate η ≡ (η x , η y ).Note that because the direction of the b-field is opposite to that of the B-field, wave functions for the vortex are anti-holomorphic functions.The corresponding integral measure is where l b = eb/ is magnetic length of the b-field.
The Hilbert space of a composite fermion is the tensor product of the two Segal-Bargmann spaces for the electron and the vortex, respectively.The state of a composite fermion can thus be naturally represented by a bivariate function: which is holomorphic (anti-holomorphic) in the complex coordinate z (η) of the electron (vortex).Unlike the wave function ψ(z) ≡ ψ(z, z) for an ordinary particle, the two coordinates of the wave function Eq. ( 9) belong to different particles.
For a Bergman space, we can define a reproducing kernel, which is basically the coordinate representation of the identity operator of the space [28].For the spaces of the electron and the vortex, their reproducing kernels are respectively.The kernels transform wave functions in the respective Bergman spaces back to themselves: The kernels can also be used to project nonholomorphic functions into the Segal-Bergmann spaces [28].Actually, PLLL in Eq. ( 1), the projection operator to the lowest Landau level, can be written as an integral transform using the reproducing kernel: where f (z) is shorthand notation of a non-holomorphic function f (z, z).We will use the notations interchangeably in this paper.The projection into the η-space can be defined similarly using the reproducing kernel

B. Wave function ansatz
The wave function ansatz Eq. ( 1) maps a manybody wave function in the fictitious world of composite fermions to a physical wave function of interacting electrons in the real world.Although the ansatz is customarily expressed in a form that suggests its connection with the singular CS transformation Eq. ( 2), it can actually be more naturally inferred from the dipole model, as we will demonstrate in this subsection.
Pasquier and Haldane presented an alternative approach of constructing the many-body wave functions of fractional quantum Hall states [15].The approach was further developed by Read [16] and Dong and Senthil [17].They investigate a system of bosons at filling factor one.Vortices of one flux quantum, which are fermions, are introduced as auxiliary degrees of freedom for extending the physical Hilbert space of bosons to a Hilbert space of composite fermions.It is envisioned that in the enlarged Hilbert space, it may become feasible to apply mean-field approximations for the composite fermions.To obtain physical wave functions, on the other hand, one needs to eliminate auxiliary degrees of freedom by projecting states of composite fermions into a physical subspace.This leads to a relation between a wave function of composite fermions Ψ CF ({z i , ηi }) and its physical counterpart Ψ({z i }) [16]: where Ψ v ({η i }) is the wave function of a vortex state that defines the physical subspace in the enlarged Hilbert space, and Pv denotes the projection into the subspace.
For a system of bosons, the vortex state is assumed to be a ν = 1 incompressible state of fermions with The corresponding physical wave function describes a Fermi-liquid like state of bosons.
The general idea of Pasquier-Haldane-Read's approach can be adapted for a system of electrons.We can introduce bosonic vortices as the auxiliary degrees of freedom.We assume that the vortices form a ν = 1/2 bosonic Laughlin state with the wave function By substituting the vortex wave function into Eq.( 16), we obtain an ansatz for constructing physical wave functions of electrons: Remarkably, the ansatz can be shown to be equivalent to the standard (Jain's) ansatz Eq. (1).To see this, we express Eq. ( 1) in an integral form by using Eq. ( 14): where we insert the reproducing kernel Eq. ( 11) for each of the composite fermions.Comparing it with the new ansatz Eq. ( 18), we have We see that the two ansatze are equivalent but use different wave-function representations for a state of composite fermions.In the following, we will refer to the two representations as the dipole representation (Ψ CF ) and the standard representation ( ΨCF ), respectively.

C. Wave equation: the dipole representation
The theory of composite fermions often relies on intuition or educated guesses when selecting wave functions for composite fermions.The resulting ansatz wave functions are then justified a posteriori by showing high overlaps with wave functions obtained from exact diagonalizations [3].Is it possible to determine appropriate wave functions for composite fermions a priori, as we do for ordinary electrons?In this subsection, we take the first step towards demonstrating the possibility by developing a wave equation for composite fermions in ideal systems.
The wave equation can be derived from the variational principle δL = 0, with the Lagrangian L defined by where ǫ is the Lagrange multiplier for the normalization constraint of the wave function and T is the binding energy of a composite fermion modeled as a harmonic potential It becomes the kinetic energy Eq. ( 5) in the point-particle interpretation discussed in Sec.II A.
Differentiating the Lagrangian with respect to ψ * (z, η) gives rise to the wave equation ǫψ = Ĥψ, with the Hamiltonian defined by Ĥψ (z, η) ≡ P T (z, η) ψ (z, η) , (24) where P denoting the projection into the Hilbert space of the composite fermion defined in Sec.III A. Applying the rule of the projection into Landau levels, we map z and η to the operators ẑ ≡ 2l 2 B ∂ z and η ≡ 2l 2 b ∂ η , respectively [2].The stationary-state wave equation of the composite fermion can then be written as: where an unimportant constant term in the Hamiltonian due to the ordering of operators is ignored.
We can transform the wave equation to an ordinary Schrödinger equation for a charged particle subjected to a uniform magnetic field by applying the transformation For ϕ(ξ) ≡ ϕ(z, η)| z→ξ,η→ ξ , we have with l ≡ /e|B| being the magnetic length of the effective magnetic field B = B − b, and σ ≡ sgn(B) indicating its direction.

D. Wave equation: the standard representation
We can also have a wave equation for the standard representation.In the case of non-interacting composite fermions, both Ψ CF and ΨCF are Slater determinants of single-particle wave functions.The single-particle counterpart of the transformation Eq. ( 20) is where ψ(ξ) denotes a single-particle wave function in the standard representation.Substituting it into Eq.( 25), we obtain the wave equation (see Appendix E 1) and We see that the wave equation for φ(ξ) is just the ordinary Schrödinger equation for a charge particle in the uniform effective magnetic field.
Our theory reproduces the well-established results of the standard theory for ideal states.The eigen-solutions of the wave equation (see Appendix A) are exactly the Λ orbits of the standard theory of composite fermions [2].The interpretation of the fractional series is also the same: a fractional state of electrons corresponds to an integer filling state of composite fermions.The filling factor ν = n/(2n + 1) < 1/2, n ∈ Z corresponds to n filled Λ-levels, and ν = (n + 1)/(2n + 1) > 1/2 corresponds to n + 1 filled Λ-levels with σ = −1.For the special case of ν = 1/2, the effective magnetic field vanishes.The eigen-solutions of the wave equation become the plane-wave states, and composite fermions will form a Fermi sea.The resulting physical wave function is exactly the Rezayi-Read wave function of the composite Fermi-liquid [11].
We note that the states {ϕ i }and { φi } are dual to each other, and form a biorthogonal system.This can be seen by applying Eq. ( 28) to rewrite the orthonormal condition dµ IV. GENERAL THEORY In this section, we generalize our theory to systems that are subjected not only to strong uniform magnetic fields, but also to spatial and temporal fluctuations of electromagnetic fields, and generally have inhomogeneous densities.In the HLR theory, this can be done trivially by assuming that the composite fermion obeys the ordinary Schrödinger equation.In the dipole model, however, the composite fermion is far from being an ordinary particle, as we see in Sec.II A. We need to derive the general quantum theory of composite fermions in a logical way, as we demonstrate in the last section for the ideal systems.

A. Bergman space
In this subsection, we show that the Hilbert space of a particle confined in a Landau level by a non-uniform magnetic field is generally a Bergman space with its weight determined by the spatial profile of the magnetic field.Consequently, the Hilbert space of a composite fermion is the tensor product of two Bergman spaces with their weights determined by the spatial profiles of the physical and the CS magnetic fields, respectively.
We consider a non-relativistic electron confined in the lowest Landau-level by a non-uniform magnetic field B(z) = −B(z)n, and assume B(z The Hamiltonian of the system, in complex coordinates, is given by [ with A ≡ A x (z) + iA y (z) and Ā ≡ A * being the complex components of the vector potential of the magnetic field.
The first term of the Hamiltonian yields zero-energy for a state with the wave function ϕ(z) satisfying the constraint All such states form the lowest Landau level in the non-uniform magnetic field [29], and define the physical Hilbert space of the electron in the zero-electron-mass limit m e → 0. The second term of the Hamiltonian, on the other hand, can be interpreted as the orbital magnetization energy of the electron, and will become a part of the scalar potential experienced by composite fermions [30].We note that for a two-dimensional massless Dirac particle, Equation ( 33) is an exact constraint for the zero-energy Landau level, and there is no orbital magnetization energy.
To fulfill the constraint, a wave function in the Hilbert space must have the form where ψ(z) is a holomorphic function in z, and f B (z, z) is determined by the equation Fixing the vector potential in the Coulomb gauge, we have We can then choose f B (z, z) to be a real solution of the equation.
The Hilbert space of the electron is therefore a weighted Bergman space consisting all holomorphic polynomials that are normalized by the condition dµ B (z)|ψ(z)| 2 = 1, where the integral measure is modified to with w B (z) being the weight of the Bergman space, and l B ≡ /eB 0 .We can choose the constant of integration for f B to normalize the measure: dµ B (z) = 1.
Similarly, for a vortex in a non-uniform CS magnetic field b(η) = b(η)n, b(η) > 0, its Hilbert space is a Bergman space consisting all anti-holomorphic polynomials in η with the modified integral measure where f b (η, η) is a real solution of the equation The counterpart of Eq. ( 35) for the vortex is where ā ≡ a x − ia y denotes the complex-conjugated component of the vector potential (a x , a y ) of the CS magnetic field.
As in the ideal case, the state of a composite fermion is represented by a bi-variate wave function that is holomorphic in the coordinate of the electron and antiholomorphic in the coordinate of the vortex, defined in the Hilbert space that is the tensor product of the two Bergman spaces.
We can also define the reproducing kernels K B (z, z′ ) and K b (η, η ′ ) for the weighted Bergman spaces of electrons and vortices, respectively [28].K B transforms a wave function defined in the B-Bergman spaces back to itself: It also defines the projection into the space: In general, we do not have a closed form of the reproducing kernel like Eq. (10).We formally express the reproducing kernels as by introducing the function F B ( ξ, z).In the longwavelength limit, F B and f B can be related approximately, see Appendix C. The reproducing kernel of the b-space has a similar set of properties.

B. Wave function ansatz
Using the modified integral measure Eq. ( 38), we can generalize the wave function ansatz Eq. ( 16) straightforwardly: (44) where we only change the integral measures for {η i }, and assume that the wave function of the vortices defining the physical subspace remains the same as in the ideal case.
Due to the change of the weight of the η-Bergman space, vortices are actually in a deformed bosonic Laughlin state with an inhomogeneous density.The joint density distribution of vortices is proportional to Using Laughlin's plasma analogy [31], we can interpret it as the distribution function of a set of classical particles, each of which carries two unit "charges", on a non-uniform neutralizing background with the "charge" density ∂ η ∂ η f b (η, η)/π.Such a system is expected to be nearly "charge-neutral" everywhere.It implies that the single-particle density of vortices should be where we make use of Eq. ( 39).We see that the CS selfconsistent condition, which relates the vortex density to the strength of the CS magnetic field, arises as a result of the constraint of the physical subspace.
The standard ansatz can be generalized and shown to be equivalent to the new ansatz.By using the reproducing kernel of the electron Bergman space, the standard ansatz Eq. ( 1) can be written as The general relation between the dipole representation and the standard representation reads

C. General wave equation
In this subsection, we generalize the wave equation Eq. (25) for systems that are subjected to external perturbations.We assume that the external magnetic field has a strong uniform component B 0 and a small fluctuating component B 1 (z) which varies slowly over space with |B 1 (z)|/B 0 ≪ 1, |∇B(z)|l B /B 0 ≪ 1, and the strength of the external electric field is weak and does not induce inter-Landau-level transitions.The resulting theory will be adequate for predicting long-wavelength responses to electromagnetic fields [14].In this limit, we can establish a general wave equation while not obscured by excessive microscopic details.A more general theory would require taking into account microscopic details which will be elucidated in Sec.V.
The Lagrangian of the dipole model for a set of composite fermions, in terms of the single-particle wavefunctions {ψ i (z, η)}, can be generally written as: where the summation is over the occupied states of composite fermions, and ǫ i is the Lagrange multiplier for the normalization constraint of the wave functions.The second term in the braces is the kinetic energy, which is basically the harmonic binding potential Eq. ( 23) written in a form that implies anti-normal ordering when quantizing η (see Appendix B), with a space-dependent coefficient parametrized in the local magnetic lengths of the external field l B (z) ≡ /eB(z) and the CS field l b (z) ≡ /eb(z).The third term is the energy due to the single-body scalar potential Φ(z) experienced by electrons, which includes the scalar potential of the external electromagnetic field as well as the orbital magnetization energy discussed in Sec.IV A, and is the local density of electrons.The next two terms are the Coulomb energy and an exchange-correlation energy functional E xc [ρ e ] which accounts for the exchange and correlation effects of composite fermions.The last term imposes the CS constraint which relates the local density of vortices to the local strength of the CS magnetic field b(η), with φ(η) serving as the Lagrange multiplier.The effects of the non-uniform physical and CS magnetic fields are included implicitly in the integral measures dµ B and dµ b , respectively.In Sec.V, we will derive the Lagrangian from first principles.
Differentiating the Lagrangian with respect to ψ * i , we obtain a generalized wave equation for the stationary state of a composite fermion: where we drop the state index subscripts for brevity, Φ eff is the effective scalar potential experienced by electrons, defined by which is obtained by differentiating the kinetic energy with respect to ρ e (z) ≈ ρ v (z) = 1/4πl 2 b (z) and applying the quantization (see below), and φ(η) can be interpreted as the scalar potential experienced by vortices.The orthonormal condition between two eigen-states is: Applying the rules of quantization defined in Appendix B, we can write the wave equation as with the effective Hamiltonian operator where ẑ and η are defined by (see Appendix B 1) [ηψ] (z, η) and N + [• • • ] denotes the normal ordering that places ẑ and η on the left of all z's and η's.We apply the approximations l 2 B (z) ≈ l 2 B (η, z) and l 2 b (z) ≈ l 2 b (η, z) for the coefficient of the kinetic energy.
The wave equation is complemented by a set of CS self-consistent conditions, which are obtained by differentiating the Lagrangian Eq. ( 49) with respect to a and φ.We have where E v and b are the CS electric and magnetic fields, respectively, j v (η) denotes the current density of vortices, which can be written as (see Appendix F)

D. Biorthogonal quantum mechanics
As in the theory for ideal systems, we can determine the wave equation for the standard representation, and define a biorthogonal system of wave functions.In general, the single-particle wave functions of the dipole representation and the standard representation are related by the transformation The operators in the dipole representation can be mapped to their counterparts in the standard representation accordingly, see Appendix E 2.
We further introduce the transformations which are the counterparts of the transformations Eq. ( 26) and Eq. ( 30), respectively.The orthonormal condition Eq. ( 56) can then be rewritten as We see that as in ideal systems, {ϕ i } and { φi } are dual to each other and form a biorthogonal system.The wave equations for ϕ and φ have the form of the biorthogonal quantum mechanics [32].In general, we can show that the Hamiltonian for φ(ξ) is the complex conjugate of that for ϕ(ξ)(see Appendix E 3).Therefore, we have where the first equation is transformed from Eq. ( 57) with Note that Ĥ is non-Hermitian in general.In the long-wavelength limit, the effective Hamiltonian of a composite fermion can be written as (see Appendix C): where ( Ā, A) denotes the effective vector potential experienced by composite fermions: and V (η, z) ≡ 2i∂ η Φ eff (η, z)/eB(η, z) and v(η, z) ≡ 2i∂ z φ(η, z)/eb(η, z) are the complex components of the drift velocities the presence of the electric fields E ≡ e −1 ∇Φ eff and E v ≡ e −1 ∇φ for electrons and vortices, respectively.The set of wave equations can be further generalized for time-dependent systems.We have (see Appendix D): where Ĥ is formally identical to the stationary state Hamiltonian Eq. ( 71) in the long-wavelength limit, but the electric fields E and E v , which determine the drift velocities, are replaced by their gauge-invariant forms The wave equations (74, 73), together with the CS selfconsistent conditions Eqs. (61, 62) and the self-consistent equation for the effective potential Eq. ( 54), define the effective theory of composite fermions in the presence of long-wavelength external perturbations.It is evident that the effective theory differs from the heuristic HLR theory because of the corrections from the drift velocities in Eq. ( 71).The corrections have previously been identified as either anomalous velocity corrections [14] or sidejump corrections [33] in the context of the semi-classical theory of composite fermions.Equation (71) shows how these corrections are manifested in the quantum mechanics of composite fermions.

V. MICROSCOPIC UNDERPINNING
In this section, we derive the phenomenological dipole model, which underlies our derivation of the quantum mechanics of composite fermions, from the microscopic model of interacting electrons confined in a Landau level in the zero-electron-mass limit.The microscopic Lagrangian of such a system can be written as where Ψ denotes the many-body wave function of electrons, E is a Lagrange multiplier for the normalization constraint Ψ|Ψ = 1, V ee = (e 2 /4πε) i<j |z i − z j | −1 + V B denotes the Coulomb interaction between electrons with V B being the potential from a uniform neutralizing positive charge background, and Φ ≡ i Φ(z i ) denotes the energy of an externally applied scalar potential.The kinetic energy of electrons is ignored since it is completely quenched in a Landau level.
Our derivation is based on the general variational principle of quantum mechanics.By introducing the fictitious degrees of freedom of the vortices, we basically embed the physical Hilbert space into a larger Hilbert space of composite fermions, in the hope that the strongly correlated state of electrons can be viewed as the projection of a non-correlated state of composite fermions onto a lower-dimensional subspace.Therefore, we choose the trial electron wave functions for |Ψ to be the ansatz form Eq. ( 44) with Ψ CF being the Slater determinant of a set of single-body trial wave-functions {ψ i } of composite fermions.We will show that the Lagrangian Eq. ( 49) in terms of {ψ i } can be derived from the microscopic Lagrangian Eq. ( 75).The set of the single-body trial wavefunctions should be determined by applying the variational principle which gives rise to the wave-equations and the CS selfconsistent conditions.

A. Chern-Simons constraints
A notable feature of the theory of composite fermions is the presence of the fictitious CS fields which are determined self-consistently by Eqs.(61, 62).In this subsection, we show how the CS fields and the self-consistent conditions emerge in a microscopic theory.
It is easy to see that for the Slater determinant wave function two sets of single particle trial wave functions that are related by a non-singular linear transformation yield the same physical wave function after applying Eq. ( 15) or Eq. ( 44).To eliminate the redundancy, it is necessary to impose the orthonormal condition: We note that the orthonormality depends on the weight in dµ b , which is not yet defined at this point.
To proceed, we adopt an approximation analogue to the Hartree approximation.Basically, we determine the state of a composite fermion in an effective medium formed by other composite fermions.In the spirit of the Hartree approximation [34], we introduce a test particle that is distinguishable from other composite fermions but interacts and correlates just like them.The physical wave function of a system with N composite fermions plus such a test particle can be written as where the test particle has the wave function ψ(z, η), and correlates with other composite fermions via the Bijl-Jastrow factor.Because the test particle has no exchange symmetry with other composite fermions, it can occupy any state, including those already occupied in Ψ CF .Our approximation is to assume that the set of single particle trial wave functions for constructing Ψ CF can be chosen from eigen-wave-functions of the test particle.
With the approximation, we can determine the weight of dµ b , self-consistently, by requiring that the orthonormality Eq. (78) in the Hilbert space of composite fermions is consistent with that of the physical Hilbert space.This is to require where |Ψ t i and |Ψ t j denote two physical states obtained by setting ψ = ψ i and ψ = ψ j in Eq. (79), respectively, and ψ i and ψ j satisfy the orthonormal condition Eq. (78).Equation (81) can be rewritten as with To make Eq.( 82) consistent with Eq. ( 78), we can adjust the weight of dµ b so that K b (η, η ′ ) is the corresponding reproducing kernel.Equation (82) can then be reduced to Eq. ( 78) by integrating out η ′ .The requirement that K b (η, η ′ ) is the reproducing kernel of the η-space gives rise to the CS constraint Eq. (61) in the long-wavelength limit.To see this, we rewrite Eq. (83) as K b (η, η ′ ) = e F , with and To the lowest order, we ignore the fluctuation and higher order corrections, and have To evaluate the i-th term of the summation, we expand the Slater determinant Eq. ( 77) along its i-th row, substitute the expansion into Eq.( 84), and ignore contributions involving particle exchanges.We obtain where is defined by Eq. ( 83) but with one composite fermion in the state ψ a being removed from Eq. (77).We assume that the effect of removing a composite fermion from the effective medium of N composite fermions is negligible, thus have After integrating out η ′ i , we obtain where ρ v (η 1 ) is the vortex density defined in Eq. (51).
Applying the identity In the long-wavelength limit, we have Substituting the relation into Eq.(89), ignoring the spatial gradient of the magnetic length, and applying Eq. ( 39), we obtain the CS constraint Eq. (61).
We can then replace the normalization constraint Ψ|Ψ = 1 in Eq. ( 75) with normalization constraints of the single-body wave-functions as well as the CS constraint Eq. (61), and introduce ǫ i and φ(η) as the respective Lagrange multipliers.The Lagrangian becomes

B. Energy
In this subsection, we determine the expectation value Ψ | V ee + Φ | Ψ .We shall show how the kinetic energy of a composite fermion, i.e., the electron-vortex binding potential, as well as its peculiar density-of-states correction factor (see Sec. III C), would emerge.We first determine the expectation value of the scalar potential Ψ|Φ|Ψ .Similar to Eq. (86), we have Applying the approximation Eq. (87), we obtain Next, we determine the expectation value of the Coulomb interaction energy.It can be written as where |Ψ is the two-particle reduced density matrix of electrons.We decompose V ee into two parts.The first part is the meanfield contribution of the Coulomb interaction which gives rise to the Coulomb energy term of the Lagrangian Eq. ( 49).The second part is the correlation contribution which gives rise to the binding energy between electrons and vortices.We determine the two-particle reduced density matrix by applying the Hartree-like approximation introduced in the last subsection.We have 2 with z 1 = z and z 2 = z ′ .We treat the first particle (z 1 ) as a test particle, and the ensemble of other N − 1 particles as an effective medium.By expanding the Slater determinant Eq. (77) along its first row, ignoring exchange terms in |Ψ({z i })| 2 , and replacing the N − 1 particle effective medium with the N -particle one as in Eq. (79), we can approximate ρ 2 as and ρ c (z 1 ; η, η ′ ) ≡ ρ c (z 1 , η)| η→η ′ .ρ c (z 1 ; η) is the density profile of electrons surrounding a vortex at η, which suppresses the electron density in its vicinity, resulting in a void of electrons.The Coulomb attraction between the test (first) electron and the void gives rise to the binding energy of a composite fermion.Substituting Eq. ( 96) and (50) into Eq.(95), we obtain and ∆ρ e (z 1 ; η) ≡ ρ c (z 1 ; η) − ρ e (z 1 ).
The form of ∆ρ e (z 1 ; η) is constrained [35].The electron density is suppressed near the center of the vortex, and recovers in a length scale ∼ l B (see the inset of Fig. 2).Thus we have ∆ρ e (z 1 ; η) < 0 for z 1 → η and ∆ρ e (z 1 ; η) → 0 for |z 1 − η| ≫ l B .Moreover, because the insertion of a 2h/e vortex should induce a charge void with a total charge of 2νe, where ν is the filling factor, we have the sum rule where ν(η) denotes the local value of the filling factor, and we assume that the electron density is nearly homogeneous.
The fact that the binding energy of a composite fermion originates from the Coulomb attraction between an electron and a void with a total charge of 2νe suggests that it should be proportional to 2ν, which is exactly the density-of-states correction factor D ≡ b/B = 2ν appeared in Eq. ( 5).The peculiar factor in the kinetic energy turns out to be a natural consequence of the interaction origin of the binding energy.
In the long-wavelength limit, we can approximate ∆ρ e (z 1 ; η) as h(z 1 ; η) ≈ ρ e (η)h 0 (|z 1 − η|/l B (η)), where h 0 (r) ≡ ∆ρ e (r)/ρ 0 is the electron-vortex correlation function of a homogeneous system, with ∆ρ e (r) being the change of electron density relative to the average density ρ 0 in the vicinity of a vortex at the origin of space.We expect that the binding energy, apart from the densityof-states correction factor, should depend only weakly on the density since h 0 (r) is constrained by the densityindependent sum rule ∞ 0 drh 0 (r)r = −2 and the overall form.We thus estimate the binding energy using the Laughlin state at ν = 1/3, for which we can complete the integrals with respect to {η i } in Eq. ( 80), and obtain The density profile of the electrons near the origin can be determined numerically using the Monte-Carlo method.The result is shown in Fig. 2. We find that the binding energy can be well fitted by a quadratic function for |z − η| 2l B [36], and approximated as with g 0 ≈ 0.5 and 2 m * ≈ 0.08 The estimated effective mass is about four times larger than the one usually assumed in the literature ( 2 /m * ≈ 0.3e 2 l B /4πε) [4,37].On the other hand, the effective masses determined in experiments vary with the measurement methods [2].Our estimate is actually close to the cyclotron effective mass measured by Kukushkin et al. [38].There is also a theoretical proposal that the effective mass should be four times larger [39].We can define an exchange-correlation functional which includes the contribution of the first term of Eq. (102), as well as contributions that are ignored in our derivation, in particular the effect of particle exchanges.
In the spirit of the Kohn-Sham approach of the density functional theory, we could define E xc [ρ e ] as the difference between the exact ground state energy of a system with a uniform density ρ e and the total kinetic energy of non-interacting composite fermions at the same density [37,40].Combining all, we obtain the Lagrangian Eq. (49).

VI. GENERALIZATION FOR FLAT CHERN BANDS
The fractional quantum Hall effect is also predicted to emerge in flat Chern bands, i.e., Bloch bands which are nearly dispersion-less and have non-zero Chern numbers [19,20].A flat Chern band is considered as a generalized "Landau level" which possesses essential properties for hosting the fractional quantum Hall effect.Conversely, a Landau level could be interpreted as an ideal flat Chern band with a Chern number |C| = 1 [41].One expects that interacting electrons confined in a flat Chern band behave similarly as in an ordinary Landau level.The expectation is recently confirmed in experiments [42][43][44][45].
The generalization of our approach for flat Chern bands with |C| = 1 is straightforward.A dipole model is shown in Fig. 3, where we replace the electron Landau level in Fig. 1 with a flat Chern band.The general idea presented in Sec.III B for constructing many-body wavefunctions of electrons is still applicable.We introduce vortices as auxiliary degrees of freedom which should be projected out in the end, and require that electrons always reside in their original and physical Hilbert space.We thus have the wave function ansatz for flat bands with C = −1 [46]: where {r i } denotes the set of electron coordinates.For a flat Chern band, unlike a Landau level, the wave functions Ψ and Ψ CF are generally not holomorphic in the electron coordinates.Instead, they should be expanded in the Bloch states of the flat band which span the physical Hilbert space.Thus, the single-body wave function of a composite fermion can be written as where u k (r) denotes the periodic part of the Bloch wave function at the quasi-wave-vector k of the flat band, and the state of the composite fermion is represented by the wave function ϕ k (η).
We can introduce an effective Hamiltonian for composite fermions.In the enlarged Hilbert space of composite fermions, each electron in the flat band is bound to a vortex.While the binding potential could be derived microscopically as we have demonstrated for a Landau level in Sec.V B, it is reasonable to assume that the harmonic form Eq. ( 23) is a good first approximation.Therefore, the effective Hamiltonian of a composite fermion can be written as where Te is the electron kinetic energy, and we define the effective magnetic length with V p being the area of the primitive cell of the system.We can obtain the effective Hamiltonian for ϕ k (η) by determining the expectation value ψ| ĤCF |ψ for ψ defined by Eq. ( 106).It is easy to prove the identities: where A k and G k are the Berry connection and quantum metric tensor of the flat band, respectively, defined by [25] A Applying the identities, we obtain: and where ǫ k and Ω k are the dispersion and Berry curvature of the flat band, respectively.The form of the η operator depends on the spatial profile of the vortex density.As a first approximation, one could assume a homogeneous vortex density, thus η = 2l 2 b ∂ η .We could predict the stability of a fractional Chern insulator by determining the eigen-spectrum of the singlebody effective Hamiltonian Eq. (114).For an ideal flat band with a uniform Berry curvature, only the last term remains, and it is easy to show that the Hamiltonian gives rise to the ordinary Λ-levels (see Appendix A).For the more general cases, however, we expect that the first two terms, which could be interpreted as the renormalized band dispersion experienced by composite fermions, will make Λ-levels non-degenerate and suppress excitation gaps.When the gaps are closed, the fractional Chern insulator state will be destroyed.The application of the effective Hamiltonian to real materials is left for future investigation.
The form of the effective Hamiltonian seems to justify the heuristic trace condition which requires TrG k − |Ω k | ≈ 0 everywhere in the Brillouin zone for the emergence of a fractional Chern insulator [25,47].We see that the second term, which is proportional to , renormalizes the dispersion ǫ k of electrons.As the renormalization tends to make a flat electron band non-flat, it destabilizes a fractional Chern insulator.On the other hand, it could be possible to engineer the correction to compensate the electron dispersion of a non-flat electron band and make it flatter after the renormalization.The latter suggests a novel possibility that fractional Chern insulators could be stabilized in non-flat Chern bands.

VII. SUMMARY AND DISCUSSION
In summary, we present a reformulation of the theory of composite fermions based on the dipole model.Some new insights emerge.
A) The states of composite fermions can be determined by solving a wave equation, with an effective Hamiltonian that can be derived from first principles.Such a deductive approach can reproduce the wellestablished results of the standard theory of composite fermions, namely the wave functions of the ideal fractional quantum Hall states in the lowest Landau level.It may also provide an alternative to intuition and educated guesses for understanding more complex states such as those observed in higher Landau levels [49].
B) A wave-function ansatz Ψ = Pv Ψ CF , or equivalently Jain's wave function ansatz in an alternative wave-function representation of composite fermions, can be naturally inferred from the dipole model.The Bijl-Jastrow factor in Jain's ansatz can be interpreted the complex conjugate of the wave function of the collective state of vortices, rather than the numerator of the singular CS transformation.
C) The effective theory specified by Eqs.(73, 74) differs from the HLR theory due to the drift-velocity corrections in the effective Hamiltonian Eq. (71).
D) The wave function ansatz and the effective theory can be unified on the common basis of the dipole model, and logically connected.
E) The Hilbert space of composite fermions has a simple structure, i.e., the tensor product of two separate Hilbert spaces for the physical and fictitious degrees of freedom, respectively.The simple structure makes it much easier and less prone to arbitrariness to generalize the composite fermion theory, e.g., for the flat Chern bands.
where w(z) denotes the weight of the Bergman space.

Current densities of a composite fermion system
The result derived in the last subsection can be applied to composite fermions with a straightforward generalization.The electron and vortex current densities for a state ψ(z, η) can be written as  58), and we ignore the orbital magnetization contribution.

Dipole approximation
We can also obtain approximate expressions for the particle and current densities of electrons and vortices by differentiating the action Eq.(D3) with respect to (Φ eff , A) and (φ, a), respectively.The approximation corresponds to the multipole expansion discussed in Ref. 14 We can show that a system of composite fermions always has a vanishing dipole density in the longwavelength limit.To see this, we apply the self-consistent condition Eq. (61), and find that the first term of the current density Eq. ( 63) becomes an anomalous Hall current with a half-quantized Hall conductance σ where we ignore the slow spatial variation of 1/l 2 B (z).
The vanishing dipole density suggests that, on average, the coordinate of an electron, always coincides with the coordinate of the vortex to which it is bound.The same identity has also been found in the semi-classical theory [14].In Ref. 16, it was considered that this condition could replace the CS self-consistent conditions and serves as the basis for a composite fermion theory without the CS fields.

Figure 3 .
Figure 3. Dipole model of composite fermions for a flat Chern band.Compared to the model presented in Fig. 1 for a Landau level, the electron is now confined in a Bloch band characterized by a Chern number C and other parameters such as the Berry curvature Ω k and the quantum metric G k .A Landau level can actually be interpreted as an ideal flat Chern band with C = −1, a constant Berry curvature, and vanishing TrG k − |Ω k |.The Landau level can be continuously evolved to a flat Chern band with the same Chern number.One expects that the continuous evolution should not induce a topological phase transition to the state of vortices.The possibility that the vortices adopt other topological collective states, in particular for flat Chern bands with |C| = 1, is not yet considered in this work.