# Quantum mechanics of composite fermions

Junren Shi<EMAIL_ADDRESS>International Center for Quantum Materials,
Peking University, Beijing 100871, China  Collaborative Innovation Center of
Quantum Matter, Beijing 100871, China

###### Abstract

The theory of composite fermions consists of two complementary parts: a
standard ansatz for constructing many-body wave-functions of various
fractional quantum Hall states, and an effective theory for predicting
responses of these states to external perturbations. Conventionally, both the
ansatz and the de facto standard effective theory, the Halperin-Lee-Read (HLR)
theory, are justified by Lopez-Fradkin’s theory based on the singular Chern-
Simons transformation. In this work, we aim to provide an alternative basis
and unify the two parts into a coherent theory by developing quantum mechanics
of composite fermions based on the dipole picture first proposed by Read. We
argue that states of a composite fermion in the dipole picture are naturally
described by bivariate wave functions which are holomorphic (anti-holomorphic)
in the coordinate of its constituent electron (vortex), defined in a Bergman
space with its weight determined by the spatial profiles of the physical and
the emergent Chern-Simons magnetic fields. Based on a semi-classical
phenomenological model and the rules of quantization in the Bergman space, we
establish wave equations for composite fermions subject to external
perturbations. The wave equations resemble the ordinary Schrödinger equation
but have drift velocity corrections not present in the HLR theory. Using
Pasquier-Haldane’s interpretation of the dipole picture, which considers
vortices as auxiliary degrees of freedom, we develop a general wave-function
ansatz for constructing many-body wave functions of electrons by projecting
states of composite fermions solved from the wave equation into a half-filled
bosonic Laughlin state of vortices. It turns out that for ideal fractional
quantum Hall states the general ansatz and the standard ansatz are equivalent,
albeit using different wave-function representations for composite fermions.
To justify the phenomenological model, we derive it from the microscopic
Hamiltonian and the general variational principle of quantum mechanics. The
microscopic derivation yields an effective mass of composite fermion four
times larger than the commonly adopted value, and a kinetic energy with a
density-of-states correction which was first introduced in the
phenomenological model.

##  I Introduction

Exotic correlated states of electrons emerge in fractional quantum Hall
systems, in which strong magnetic fields completely quench the kinetic energy
of electrons, rendering conventional many-body techniques inadequate in
addressing the effect of correlations between electrons [1]. The theory of
composite fermions, proposed by Jain in 1989, provides a comprehensive
framework for understanding these states [2]. It suggests a new paradigm in
which correlated states of electrons are interpreted as non-correlated or
weakly-correlated states of fictitious particles called composite fermions,
which are contemplated 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 numerically determined ground-state wave functions for various fractional
quantum Hall states [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 agree well with experimental observations [5]. The two
parts of the theory, i.e., the wave function ansatz and the effective theory,
complement each other and form a versatile framework for understanding the
rich physics of the fractional quantum Hall systems.

Despite remarkable success of the theory, it still lacks a concrete
foundation. Lopez-Fradkin’s theory [6] is often cited as justification for
both the wave function ansatz and the effective theory. However, the theory
can only be regarded as a tentative argument for the theory of composite
fermions instead of a rigorous basis due to two obvious issues.

Firstly, the ansatz prescribes wave-functions of electrons in the form [3]

$\Psi\left(\left\\{z_{i}\right\\}\right)=\hat{P}_{\mathrm{LLL}}J\left(\left\\{z_{i}\right\\}\right)\tilde{\Psi}_{\mathrm{CF}}\left(\left\\{\bm{z}_{i}\right\\}\right),$
(1)

which differs from the wave function suggested by Lopez-Fradkin’s theory based
on the singular Chern-Simons (CS) transformation [6]

$\Psi\left(\left\\{\bm{z}_{i}\right\\}\right)=\frac{J\left(\left\\{z_{i}\right\\}\right)}{\left|J\left(\left\\{z_{i}\right\\}\right)\right|}\tilde{\Psi}_{\mathrm{CF}}\left(\left\\{\bm{z}_{i}\right\\}\right),$
(2)

where $\Psi$ and $\tilde{\Psi}_{\mathrm{CF}}$ represent the wave function of
electrons and composite fermions, respectively,

$J\left(\left\\{z_{i}\right\\}\right)\equiv\prod_{i<j}\left(z_{i}-z_{j}\right)^{2}$
(3)

is the Bijl-Jastrow factor that presumably attaches a vortex of two flux
quanta to each electron, $\hat{P}_{\mathrm{LLL}}$ is the projection operator
to the lowest-Landau-level, and $\\{\bm{z}_{i}\equiv(x_{i},y_{i})\\}$ and
$\\{z_{i}\equiv x_{i}+\mathrm{i}y_{i}\\}$ denote the coordinates of electrons
in the vector and complex forms, respectively. Equation (2) is formulated in
the full Hilbert space of electrons while Eq. (1) is defined in a truncated
Hilbert space in a Landau level. Reconciling the two forms is non-trivial [7].

Secondly, the picture of composite fermions implied by Lopez-Fradkin’s theory,
which is inherited by the HLR theory, is different from that obtained by
directly inspecting the ansatz wave function Eq. (1). For the latter, Read
shows that the electron and the vortex in a composite fermion are spatially
separated, contradicting the picture implied by Eq. (2) that a composite
fermion is a point particle of an electron attached with $\delta$-function
flux tubes [8]. More recently, Son notes 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 suggest that the HLR
theory may not accurately describe the composite fermion implied by the ansatz
wave function. We need an alternative effective theory, or ideally, an
alternative foundation from which both the wave-function ansatz and effective
theory can be derived.

The dipole picture of composite fermions, originally proposed by Read,
provides an alternative description of the fictitious particle [8, 11]. The
picture differs from the HLR picture in two significant 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 external
magnetic field and an emergent CS magnetic field, respectively, instead of
moving in a free space [12, 13]. The picture is shown to yield a low-energy
and long-wavelength electromagnetic response that is identical to that of the
Dirac theory of composite fermions [14], indicating that it satisfies the
general requirements imposed by the particle-hole symmetry. The feature, as
well as the fact that the picture is inferred directly from microscopic wave
functions, distinguish it 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 and generalizations by other researchers [16, 17, 18],
offers an insight into the construction of physical wave functions. They
interpret 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 has to 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, instead of the numerator of the
singular gauge factor in Eq. (2). The interpretation naturally leads to a wave
function ansatz alternative to Eq. (1), and is free of the difficulties
associating with the singular CS transformation [16].

In this paper, we develop a theory of quantum mechanics for composite fermions
based on the dipole picture and Pasquier-Haldane’s interpretation. We argue
that the state of a composite fermion can be naturally described by a
bivariate wave function which is holomorphic (anti-holomorphic) in the complex
coordinate of its constituent electron (vortex), defined in a Bergman space
with its weight determined by the spatial profiles of the physical and CS
magnetic fields. By applying the rules of quantization of the Bergman space to
the semi-classical phenomenological model proposed in Ref. 14, we are able to
establish wave equations for composite fermions. The wave equations resemble
the ordinary Schrödinger equation but have drift velocity corrections which
are not present in the HLR theory. Using Pasquier-Haldane’s interpretation of
the dipole picture, we develop a general wave-function ansatz for constructing
many-body wave functions of electrons by projecting states of composite
fermions solved from the wave equations into a half-filled bosonic Laughlin
state of vortices. We show that for ideal fractional quantum Hall states the
standard wave function ansatz Eq. (1) and the general ansatz are equivalent,
although they use different wave-function representations for composite
fermions. To establish a microscopic underpinning for the theory, we derive
the phenomenological model from the microscopic Hamiltonian and the general
variational principle of quantum mechanics. As a whole, our formalism provides
an alternative basis for the theory of composite fermions, and a comprehensive
framework of predicting behaviors of composite fermions under general physical
circumstances.

The remainder of the paper is organized as follows. In Sec. II, we develop the
quantum mechanics of composite fermions for ideal fractional quantum Hall
systems which are subject only to uniform magnetic fields. In Sec. III, the
quantum mechanics of composite fermions is established for general physical
circumstances when systems are subject to non-uniform external magnetic and
electric fields and have inhomogeneous densities, based on generalizations of
the concepts and methods developed in Sec. II. In Sec. IV, we provide a
microscopic underpinning for our theory by deriving the phenomenological
dipole model from the microscopic Hamiltonian and the general variational
principle of quantum mechanics. In Sec. V, we summarize and discuss our
results. Certain details of derivations are presented in Appendices.

##  II Theory for ideal systems

In this section, we develop the quantum mechanics of composite fermions for
ideal fractional quantum Hall systems that are subject only to uniform
external magnetic fields and have homogeneous densities. We establish a new
wave function ansatz and a set of wave equations for composite fermions. The
standard wave function ansatz Eq. (1), which underlies the success of the
composite fermion theory, is shown to be equivalent to the new ansatz.
Principles established in this section will be the basis of developing a
theory for general systems.

###  II.1 Hilbert space

The Hilbert space of a composite fermion in the dipole picture is not that of
an ordinary quantum particle in a free space as assumed in the HLR theory.
This is obvious from the dipole picture [8, 14], which conjectures that a
composite fermion is comprised of an electron and a vortex confined in two
separated Landau levels, one for the electron is the physical Landau level
induced by the uniform physical magnetic field $\bm{B}=-B\bm{n}$, another for
the vortex is a fictitious Landau level created by the emergent CS magnetic
field $\bm{b}=b\bm{n}$ with its strength determined by the CS self-consistent
condition $b=(2h/e)\rho_{0}$, where $\rho_{0}$ is the density of electrons,
and $\bm{n}$ denotes the normal vector of the two-dimensional plane of the
system considered. The Hilbert space of such a composite fermion should be the
tensor product of the Hilbert spaces with respect to the two Landau levels.

The Hilbert space spanned by a Landau level is a weighted Bergman space [19,
20]. For a disc geometry, the space includes all polynomials of the complex
electron coordinate $z=x+\mathrm{i}y$. The inner product between two states
$\psi_{1}(z)$ and $\psi_{2}(z)$ in the space is defined by
$\braket{\psi_{1}}{\psi_{2}}=\int\mathrm{d}\mu_{B}^{(0)}(\bm{z})\psi_{1}^{\ast}(z)\psi_{2}(z)$
with the integral measure

$\mathrm{d}\mu_{B}^{(0)}(\bm{z})\equiv\frac{\mathrm{d}\bm{z}}{2\pi
l_{B}^{2}}e^{-\left|z\right|^{2}/2l_{B}^{2}},$ (4)

where $l_{B}=\sqrt{eB/\hbar}$ is the magnetic length of $B$. A Bergman space
with the Gaussian weight is also known as the Segal-Bargmann space [21].

The Hilbert space of a vortex is also a Segal-Bargmann space consisting of all
polynomials of the conjugated complex vortex coordinate
$\bar{\eta}=\eta_{x}-\mathrm{i}\eta_{y}$, where $\eta_{x}$ and $\eta_{y}$ are
the components of the vortex coordinate $\bm{\eta}\equiv(\eta_{x},\eta_{y})$.
Because the direction of the $b$-field is opposite to that of the $B$-field,
wave functions in the Landau level for the vortex are anti-holomorphic
functions in $\bar{\eta}$. The corresponding integral measure is

$\mathrm{d}\mu_{b}^{(0)}(\bm{\eta})\equiv\frac{\mathrm{d}\bm{\eta}}{2\pi
l_{b}^{2}}e^{-\left|\eta\right|^{2}/2l_{b}^{2}},$ (5)

where $l_{b}=\sqrt{eb/\hbar}$ is magnetic length of $b$.

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 in the dipole picture is thus naturally described by a
bivariate function:

$\psi(z,\bar{\eta}),$ (6)

which is holomorphic (anti-holomorphic) in the complex coordinate $z$
($\bar{\eta}$) of the electron (vortex). Unlike a wave function
$\psi(\bm{z})\equiv\psi(\bar{z},z)$ for an ordinary particle, the two
coordinates of the wave function Eq. (6) belong to different particles.

For a Bergman space, one can define a reproducing kernel, which is basically
the coordinate representation of the identity operator of the space [21]. For
the spaces of the electron and the vortex, their reproducing kernels
$K_{B}^{(0)}(z,\bar{z}^{\prime})$ and $K_{b}^{(0)}(\bar{\eta},\eta^{\prime})$
are

$\displaystyle K_{B}^{(0)}(z,\bar{z}^{\prime})$
$\displaystyle=e^{z\bar{z}^{\prime}/2l_{B}^{2}},$ (7) $\displaystyle
K_{b}^{(0)}(\bar{\eta},\eta^{\prime})$
$\displaystyle=e^{\bar{\eta}\eta^{\prime}/2l_{b}^{2}},$ (8)

respectively. As identity operators of the respective Segal-Bargmann spaces,
the kernels have the properties

$\displaystyle\psi(z)$
$\displaystyle=\int\mathrm{d}\mu_{B}^{(0)}(\bm{z}^{\prime})K_{B}^{(0)}(z,\bar{z}^{\prime})\psi(z^{\prime}),$
(9) $\displaystyle\varphi(\bar{\eta})$
$\displaystyle=\int\mathrm{d}\mu_{b}^{(0)}(\bm{\eta}^{\prime})K_{b}^{(0)}(\bar{\eta},\eta^{\prime})\varphi(\bar{\eta}^{\prime}),$
(10)

for arbitrary holomorphic functions $\psi(z)$ and anti-holomorphic functions
$\varphi(\bar{\eta})$.

The kernels can also serve as projection operators to project non-holomorphic
functions into the Segal-Bergmann spaces [21]. Actually, the projection
operator to the physical Hilbert space, i.e., $\hat{P}_{\mathrm{LLL}}$ in Eq.
(1), can be written as an integral form using the reproducing kernel:

$\hat{P}_{\mathrm{LLL}}f\left(\bm{z}\right)\equiv\int\mathrm{d}\mu_{B}^{(0)}(\bm{z}^{\prime})K_{B}^{(0)}(z,\bar{z}^{\prime})f\left(\bm{z}^{\prime}\right),$
(11)

where $f(\bm{z})$ is shorthand notation of a non-holomorphic function
$f(\bar{z},z)$. We will use the notations interchangeably in this paper. The
projection operator into the $\eta$-space can be defined similarly using the
reproducing kernel $K_{b}^{(0)}(\bar{\eta},\eta^{\prime})$.

###  II.2 Wave function ansatz

The wave function ansatz Eq. (1) maps a many-body 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 linked to the dipole
picture, as we will demonstrate in this subsection.

Pasquier and Haldane present an alternative approach of constructing many-body
states in a fractional quantum Hall system [15], which is 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 enlarging the physical Hilbert
space of bosons to a Hilbert space of composite fermions. To obtain physical
wave functions, 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
$\Psi_{\mathrm{CF}}(\\{z_{i},\bar{\eta}_{i}\\})$ and its physical counterpart
$\Psi(\\{z_{i}\\})$ [16]:

$\Psi(\\{z_{i}\\})=\int\prod_{i}\mathrm{d}\mu_{b}^{(0)}(\bm{\eta}_{i})\Psi_{\mathrm{v}}^{\ast}(\\{\bar{\eta}_{i}\\})\Psi_{\mathrm{CF}}(\\{z_{i},\bar{\eta}_{i}\\}),$
(12)

where $\Psi_{\mathrm{v}}(\\{\bar{\eta}_{i}\\})$ is the wave function of the
vortex state which defines the physical subspace in the enlarged Hilbert
space. For a system of bosons, the vortex state is the $\nu=1$ incompressible
state of fermions with
$\Psi_{\mathrm{v}}(\\{\bar{\eta}_{i}\\})=\prod_{i<j}(\bar{\eta}_{i}-\bar{\eta}_{j})$.
The corresponding physical wave function describes a Fermi-liquid like state
of bosons, and the complex conjugate of $\Psi_{\mathrm{v}}$ becomes the Bijl-
Jastrow factor of the wave function.

The general idea of Pasquier-Haldane-Read’s approach can be adapted for a
system of electrons. We can introduce vortices of two flux quanta, which are
bosons, as the auxiliary degrees of freedom. We assume that the vortices form
a $\nu=1/2$ incompressible bosonic Laughlin state with the wave function

$\Psi_{\mathrm{v}}\left(\left\\{\bar{\eta}_{i}\right\\}\right)=J^{\ast}\left(\left\\{\eta_{i}\right\\}\right).$
(13)

By substituting the vortex wave function into Eq. (12), we obtain an ansatz
for constructing physical wave functions of electrons:

$\Psi\left(\left\\{z_{i}\right\\}\right)=\int\prod_{i}\mathrm{d}\mu_{b}^{(0)}(\bm{\eta}_{i})J\left(\left\\{\eta_{i}\right\\}\right)\Psi_{\mathrm{CF}}\left(\left\\{z_{i},\bar{\eta}_{i}\right\\}\right).$
(14)

We see that the dipole picture and Pasquier-Haldane-Read’s interpretation
naturally lead to the ansatz, and the singular CS transformation is not
needed.

It is important to see that the standard ansatz Eq. (1) can be recast to the
form of the new ansatz. To achieve this, we express Eq. (1) in an integral
form by using Eq. (11):

$\displaystyle\Psi(\\{z_{i}\\})=$
$\displaystyle\int\prod_{i}\mathrm{d}\mu_{B}^{(0)}(\bm{\xi}_{i})e^{\sum_{i}z_{i}\bar{\xi}_{i}/2l_{B}^{2}}J(\\{\xi_{i}\\})\tilde{\Psi}_{\mathrm{CF}}(\\{\bm{\xi}_{i}\\})$
$\displaystyle=$
$\displaystyle\int\prod_{i}\mathrm{d}\mu_{B}^{(0)}(\bm{\xi}_{i})\mathrm{d}\mu_{b}^{(0)}(\bm{\eta}_{i})e^{\sum_{i}(z_{i}\bar{\xi}_{i}/2l_{B}^{2}+\bar{\eta}_{i}\xi_{i}/2l_{b}^{2})}$
$\displaystyle\times
J(\\{\eta_{i}\\})\tilde{\Psi}_{\mathrm{CF}}(\\{\bm{\xi}_{i}\\}).$ (15)

where we insert the reproducing kernel Eq. (8) for each of the composite
fermions. Comparing it with Eq. (14), we have

$\Psi_{\mathrm{CF}}\left(\left\\{z_{i},\bar{\eta}_{i}\right\\}\right)=\int\prod_{i}\mathrm{d}\mu_{B}^{(0)}\left(\bm{\xi}_{i}\right)\\\
\times
e^{\sum_{i}\left(z_{i}\bar{\xi}_{i}/2l_{B}^{2}+\xi_{i}\bar{\eta}_{i}/2l_{b}^{2}\right)}\tilde{\Psi}_{\mathrm{CF}}\left(\left\\{\bm{\xi}_{i}\right\\}\right).$
(16)

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
call $\Psi_{\mathrm{CF}}$ and $\tilde{\Psi}_{\mathrm{CF}}$ as wave functions
in the dipole representation and the standard representation, respectively.

###  II.3 Wave equation: the dipole representation

The theory of composite fermions often relies on intuition when constructing
wave functions. For examples, to describe a fractional quantum Hall state, one
assumes $\tilde{\Psi}_{\mathrm{CF}}$ to be a Slater determinant of the single-
particle orbits of a number of fully occupied Landau ($\Lambda$) levels [2],
or plane-wave states forming a Fermi sea for the composite Fermi liquid state
at half-filling [11]. While these choices can be justified a posteriori by
high overlaps between ansatz wave functions and wave functions obtained from
exact numerical diagonalizations [3], it is desirable to determine and justify
a priori appropriate wave functions of composite fermions for given physical
conditions, as one does for ordinary systems when solving the Schrödinger
equation. In this subsection, we take a first step to achieve this by
developing a wave equation for composite fermions in an ideal system.

Our development is based on the semi-classical phenomenological model of
composite fermions introduced in Ref. 14. According to the model, a composite
fermion is a bound state of an electron and a $2h/e$ double quantum vortex.
The electron and the vortex are coupled to the physical magnetic field
$\bm{B}$ and the emergent CS magnetic field $\bm{b}$, respectively, and
confined in their respective Landau levels. They are bounded by the harmonic
potential

$T=\frac{\hbar^{2}}{2m^{\ast}l_{B}^{2}l_{b}^{2}}\left|\bm{z}-\bm{\eta}\right|^{2},$
(17)

where $\bm{z}$ and $\bm{\eta}$ are the coordinate of the electron and the
vortex, and $l_{B}$ and $l_{b}$ are the magnetic lengths of the physical and
the CS magnetic fields, respectively. By interpreting the momentum of a
composite fermion to be $\bm{p}=\hbar\bm{n}\times(\bm{z}-\bm{\eta})/l_{B}^{2}$
[8], the binding potential becomes the kinetic energy of the composite
fermion: $T=(l_{B}^{2}/l_{b}^{2})\left|\bm{p}\right|^{2}/2m^{\ast}$, where the
effective mass of the composite fermion $m^{\ast}\equiv m^{\ast}(\nu,B)$ is a
function of the filling factor $\nu$ and the magnetic field strength $B$. The
$l_{B}^{2}/l_{b}^{2}$ factor in the kinetic energy is a peculiarity of the
model. It is due to the density-of-states correction [22] for a composite
fermion, which is subject to a uniform Berry curvature in its momentum space
[23, 12, 13]. To fulfill the requirement of the particle-hole symmetry in a
Landau level, we need to introduce such a factor when the effective mass is
assumed to be symmetric about $\nu=1/2$ [14]. The presence of such a factor
will become evident in the microscopic theory presented in Sec. IV.2.

Using the semi-classical model, we can derive a wave equation for the
composite fermion. We interpret the binding potential Eq. (17) as the
Hamiltonian of the particle. Acting on a wave function $\psi(z,\bar{\eta})$ is
defined by,

$\left[\hat{H}\psi\right]\left(z,\bar{\eta}\right)\equiv\hat{P}H\left(\bm{z},\bm{\eta}\right)\psi\left(z,\bar{\eta}\right),$
(18)

where $H(\bm{z},\bm{\eta})\equiv T(\bm{z},\bm{\eta})$, and $\hat{P}$ denotes
the projection into the Hilbert space of the composite fermion defined in Sec.
II.1. According to the rule of the projection into Landau levels, $\bar{z}$
and $\eta$ should be interpreted as operators $\hat{\bar{z}}\equiv
2l_{B}^{2}\partial_{z}$ and $\hat{\eta}\equiv
2l_{b}^{2}\partial_{\bar{\eta}}$, respectively [2]. We then obtain the
stationary-state wave equation of a composite fermion:

$\epsilon\psi(z,\bar{\eta})=-\frac{\hbar^{2}}{2m^{\ast}}\left(2\partial_{z}-\frac{\bar{\eta}}{l_{B}^{2}}\right)\left(2\partial_{\bar{\eta}}-\frac{z}{l_{b}^{2}}\right)\psi(z,\bar{\eta}),$
(19)

where an unimportant constant term due to the order of operators is ignored,
and $\epsilon$ is the eigen-energy.

The wave equation can also be derived from a variational principle $\delta
L=0$ with the Lagrangian defined by

$L=\int\mathrm{d}\mu_{B}^{(0)}(\bm{z})\mathrm{d}\mu_{b}^{(0)}(\bm{\eta})\left[\epsilon-T(\bm{z},\bm{\eta})\right]\left|\psi(z,\bar{\eta})\right|^{2},$
(20)

which is basically the negative of the energy of a composite fermion, with
$\epsilon$ serving as the Lagrange multiplier for the normalization of the
wave function

$\int\mathrm{d}\mu_{B}^{(0)}(\bm{z})\int\mathrm{d}\mu_{b}^{(0)}(\bm{\eta})\left|\psi\left(z,\bar{\eta}\right)\right|^{2}=1.$
(21)

Note that
$\int\mathrm{d}\mu_{B}^{(0)}(\bm{z})\mathrm{d}\mu_{b}^{(0)}(\bm{\eta})\delta\psi^{\ast}(z,\bar{\eta})F(\bm{\xi},\bm{\eta})=0$
for $\delta\psi^{\ast}(z,\bar{\eta})$ varying in the Segal-Bargmann space
implies $\hat{P}F(\bm{\xi},\bm{\eta})=0$.

We can transform the wave equation into a form resembling the ordinary
Schrödinger equation for a charged particle in a uniform magnetic field by
applying the transformation

$\psi(z,\bar{\eta})=\sqrt{2\pi}l_{B}\exp\left[\frac{1}{4}\left(\frac{1}{l_{B}^{2}}+\frac{1}{l_{b}^{2}}\right)z\bar{\eta}\right]\varphi(z,\bar{\eta}).$
(22)

The wave equation for
$\varphi(\bm{\xi})=\varphi(z,\bar{\eta})|_{z\rightarrow\xi,\bar{\eta}\rightarrow\bar{\xi}}$
takes the form:

$\epsilon\varphi(\bm{\xi})=-\frac{\hbar^{2}}{2m^{\ast}}\left(2\partial_{\xi}-\frac{\sigma\bar{\xi}}{2l^{2}}\right)\left(2\partial_{\bar{\xi}}+\frac{\sigma\xi}{2l^{2}}\right)\varphi(\bm{\xi}),$
(23)

with $l=\sqrt{\hbar/e|\mathcal{B}|}$ being the magnetic length the effective
magnetic field $\mathcal{B}=B-b$, and $\sigma=\mathrm{sgn}(\mathcal{B})$
indicating the direction of the effective magnetic field
$\bm{\bm{\mathcal{B}}}=-\mathcal{B}\bm{n}$.

###  II.4 Wave equation: the standard representation

We can also have a wave equation for wave functions in the standard
representation. In the case of non-interacting composite fermions, both
$\Psi_{\mathrm{CF}}$ and $\tilde{\Psi}_{\mathrm{CF}}$ are Slater determinants
of single-particle wave functions. To fulfill the relation Eq. (16), the
single-particle wave functions in the two representations should be related
by:

$\psi\left(z,\bar{\eta}\right)=\int\mathrm{d}\mu_{B}^{(0)}\left(\bm{\xi}\right)e^{z\bar{\xi}/2l_{B}^{2}+\xi\bar{\eta}/2l_{b}^{2}}\tilde{\psi}\left(\bm{\xi}\right),$
(24)

where $\tilde{\psi}(\bm{\xi})$ denotes a single-particle wave function in the
standard representation. Substituting it into Eq. (19), we obtain the wave
equation (see Appendix C.1)

$\epsilon\tilde{\varphi}(\bm{\xi})=-\frac{\hbar^{2}}{2m^{\ast}}\left(2\partial_{\xi}-\frac{\sigma\bar{\xi}}{2l^{2}}\right)\left(2\partial_{\bar{\xi}}+\frac{\sigma\xi}{2l^{2}}\right)\tilde{\varphi}(\bm{\xi}),$
(25)

and

$\tilde{\psi}(\bm{\xi})=\sqrt{2\pi}l_{B}\exp\left(\sigma\frac{\left|\xi\right|^{2}}{2l^{2}}\right)\tilde{\varphi}(\bm{\xi}).$
(26)

We see that the wave equation for $\tilde{\varphi}(\bm{\xi})$ is just the
ordinary Schrödinger equation for a charge particle in a uniform effective
magnetic field. The solutions of the equation, i.e., Landau ($\Lambda$) orbits
in the effective magnetic field, are exactly the composite fermion wave
functions adopted by the standard ansatz for constructing the wave functions
of the ideal FQH states (see Appendix A).

Using Eq. (24), we can write the orthonormal condition between two eigenstates
[see Eq. (21)] as:

$\int\mathrm{d}\bm{\xi}\leavevmode\nobreak\
\tilde{\varphi}_{i}^{\ast}(\bm{\xi})\varphi_{j}(\bm{\xi})=\delta_{ij}.$ (27)

It indicates that eigen-states in the dipole representation and the standard
representation are dual to each other, and form a biorthogonal system. Since
the wave equations for $\varphi_{i}(\bm{\xi})$ and
$\tilde{\varphi}_{i}(\bm{\xi})$, Eq. (23) and Eq. (25), respectively, have the
same form, the two wave functions can be chosen to be different only by a
constant factor: $\varphi_{i}(\bm{\xi})=C_{i}\tilde{\varphi}_{i}(\bm{\xi})$.

Our derivation confirms that the dipole picture is a suitable basis for the
theory of composite fermions. We can now determine single-particle wave
functions of composite fermions from a wave equation. With the new ansatz, the
theory predicts the same many-body wave functions as the standard theory of
composite fermions.

To have a theory which can predict not only wave functions of ideal states but
also their responses to external perturbations, we still need to incorporate
external potentials into the wave equations. In the HLR theory, this can be
done trivially by interpreting a composite fermion as an ordinary particle.
However, as shown in this section, one has to follow a different set of
principles when mapping a classical picture to its quantum counterpart for a
composite fermion. Applying these principles for developing a general quantum
theory of composite fermions is the subject of the next section.

##  III General theory

In this section, we extend our theory for more general physical circumstances.
We consider systems that are subject to not only strong uniform magnetic
fields but also spatial and temporal fluctuations of electromagnetic fields,
and have generally inhomogeneous densities. To apply the principles developed
in the last section, we first need to identify the Hilbert space of a
composite fermion in a non-uniform magnetic field. To this end, we find that
the non-uniformity of the magnetic field only modifies the weight of the
Bergman space. With the understanding, we can generalize our theory and
establish the quantum mechanics of composite fermions for general physical
circumstances.

###  III.1 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. The
Hilbert space of a composite fermion is then 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 $\bm{B}(\bm{z})=-B(\bm{z})\bm{n}$,
$B(\bm{z})=B_{0}+B(\bm{z})>0$, and $|B_{1}(\bm{z})|/B_{0}\ll 1$. The
Hamiltonian of the system, in the complex coordinates, is given by [2]

$\hat{H}=-\frac{\hbar^{2}}{2m_{\mathrm{e}}}\left(2\partial_{z}+\mathrm{i}\frac{e}{\hbar}\bar{A}\right)\left(2\partial_{\bar{z}}+\mathrm{i}\frac{e}{\hbar}A\right)+\frac{e\hbar
B(\bm{z})}{2m_{\mathrm{e}}},$ (28)

with $A\equiv A_{x}(\bm{z})+\mathrm{i}A_{y}(\bm{z})$, $\bar{A}\equiv
A^{\ast}$, and $(A_{x},A_{y})$ being the vector potentials of the magnetic
field. The first term of the Hamiltonian yields zero-energy for a state with
the wave function $\varphi(\bm{z})$ satisfying the constraint

$\left[2\partial_{\bar{z}}+\mathrm{i}\frac{e}{\hbar}A(\bm{z})\right]\varphi\left(\bm{z}\right)=0.$
(29)

All such states form the lowest Landau level in a non-uniform magnetic field
[24], and define the physical Hilbert space of the electron in the zero-
electron-mass limit $m_{\mathrm{e}}\rightarrow 0$. The second term of the
Hamiltonian, on the other hand, can be interpreted as the orbital
magnetization energy of the electron, and becomes a part of the scalar
potential experienced by composite fermions [25]. We note that for a two-
dimensional massless Dirac particle, Eq. (29) is an exact constraint for its
zero-energy Landau level, and there is no orbital magnetization.

To fulfill the constraint, a wave function in the Hilbert space must take the
form

$\varphi(\bm{z})=\psi(z)\exp\left[-\frac{1}{2}f_{B}(\bar{z},z)\right],$ (30)

where $\psi(z)$ is a holomorphic function in $z$, and $f_{B}(\bar{z},z)$ is
determined by the equation

$\partial_{\bar{z}}f_{B}(\bar{z},z)=\mathrm{i}\frac{e}{\hbar}A(\bar{z},z).$
(31)

Fixing the vector potential in the Coulomb gauge, we have
$\partial_{z}A=-\partial_{\bar{z}}\bar{A}=-\mathrm{i}B(\bm{z})/2,$ and

$\partial_{z}\partial_{\bar{z}}f_{B}(\bar{z},z)=\frac{e}{2\hbar}B(\bm{z}).$
(32)

We can then choose $f_{B}(\bar{z},z)$ to be a real solution of the equation.

The Hilbert space of the electron is therefore a weighted Bergman space
consisting all holomorphic functions that are normalized by the condition
$\int\mathrm{d}\mu_{B}(z)|\psi(z)|^{2}=1$, where the integral measure is
modified from that of a Segal-Bargmann space to

$\mathrm{d}\mu_{B}(\bm{z})=w_{B}\left(\bm{z}\right)\mathrm{d}\bm{z}\equiv\frac{\mathrm{d}\bm{z}}{2\pi
l_{B}^{2}}\exp\left[-f_{B}(\bar{z},z)\right],$ (33)

with $w_{B}(\bm{z})$ being the weight of the Bergman space, and
$l_{B}\equiv\sqrt{\hbar/eB_{0}}$. We can choose the constant of integration
for $f_{B}$ to normalize the measure: $\int\mathrm{d}\mu_{B}(\bm{z})=1$.

Similarly, for a vortex in a non-uniform CS magnetic field
$\bm{b}(\bm{\eta})=b(\bm{\eta})\bm{n}$, $b(\bm{\eta})>0$, its Hilbert space is
a Bergman space consisting all anti-holomorphic functions in $\bar{\eta}$ with
the modified integral measure

$\mathrm{d}\mu_{b}(\bm{\eta})=w_{b}\left(\bm{\eta}\right)\mathrm{d}\bm{\eta}=\frac{\mathrm{d}\bm{\eta}}{2\pi
l_{b}^{2}}\exp\left[-f_{b}(\bar{\eta},\eta)\right],$ (34)

where $f_{b}(\bar{\eta},\eta)$ is a real solution of the equation

$\partial_{\bar{\eta}}\partial_{\eta}f_{b}(\bar{\eta},\eta)=\frac{e}{2\hbar}b(\bm{\eta}).$
(35)

The counterpart of Eq. (31) for the vortex is

$\partial_{\eta}f_{b}(\bar{\eta},\eta)=\mathrm{i}\frac{e}{\hbar}\bar{a}(\bar{\eta},\eta).$
(36)

where $\bar{a}\equiv a_{x}-\mathrm{i}a_{y}$ denotes the complex-conjugated
component of the vector potential $(a_{x},a_{y})$ of the CS magnetic field.

As in the ideal case, wave functions of a composite fermion are bi-variate
functions which are holomorphic in the coordinate of the electron and anti-
holomorphic in the coordinate of the vortex.

###  III.2 Quantization in a Bergman space

In this subsection, we discuss the quantization in a weighted Bergman space.

We first define the reproducing kernel for the weighted Bergman space of an
electron. It is the coordinate representation of the identity operator which
transforms an arbitrary holomorphic function $\psi(z)$ to itself:

$\psi(z)=\int\mathrm{d}\mu_{B}(\bm{\xi})K_{B}(z,\bar{\xi})\psi(\xi).$ (37)

The relation is basically a generalization of Eq. (9) with the modified
integral measure. The reproducing kernel can generally be constructed by
$K_{B}(z,\bar{\xi})=\sum_{n}\phi_{n}(z)\phi_{n}^{\ast}(\xi)$ using a complete
orthonormal basis set $\\{\phi_{n}(z)\\}$ of the space with
$\int\mathrm{d}\mu_{B}(\bm{z})\phi_{m}^{\ast}(z)\phi_{n}(z)=\delta_{mn}$ [21].
We can express the reproducing kernel in an exponential form

$K_{B}(z,\bar{\xi})\equiv e^{F_{B}(\bar{\xi},z)}$ (38)

by introducing the function $F_{B}(\bar{\xi},z)$. In analogy to Eq. (31), we
define a starred vector potential which is related to $F_{B}(\bar{\xi},z)$ by

$\partial_{z}F_{B}(\bar{\xi},z)=-\mathrm{i}\frac{e}{\hbar}\bar{A}^{\bigstar}(\bar{\xi},z).$
(39)

The starred vector potential $\bar{A}^{\bigstar}$ can be related to the
physical vector potential $\bar{A}$, see Appendix B.

Using the reproducing kernel as the projector operator into the Hilbert space,
we can quantize an arbitrary function $H(\bar{z},z)$ to an operator $\hat{H}$.
In analogy to Eq. (18), the action of $\hat{H}$ to a wave function $\psi(z)$
is defined by:

$\left[\hat{H}\psi\right]\left(z\right)=\int\mathrm{d}\mu_{B}(\bm{\xi})K_{B}(z,\bar{\xi})H(\bar{\xi},\xi)\psi(\xi).$
(40)

With the modified integral measure in the definition, the simple quantization
rule of substituting $\bar{z}$ with $2l_{B}^{2}\partial_{z}$ for a Segal-
Bargmann space is not valid anymore. Instead, the operator $\hat{\bar{z}}$ is
in general a function of $\partial_{z}$ and $z$ satisfying

$\hat{\bar{z}}\left(\partial_{z},z\right)K_{B}\left(z,\bar{\xi}\right)=\bar{\xi}K_{B}\left(z,\bar{\xi}\right).$
(41)

With the operator $\hat{\bar{z}}$, the arbitrary function $H(\bar{z},z)$ can
be quantized as

$\hat{H}=N_{+}\left[H\left(\hat{\bar{z}},z\right)\right],$ (42)

where $N_{+}[\dots]$ denotes the normal ordering that places $\hat{\bar{z}}$
on the left of all $z$’s in $H(\bar{z},z)$. We can also define an alternative
form of the quantization in which $\hat{H}$ can be expressed as the anti-
normal ordering of a function $H^{\bigstar}(\hat{\bar{z}},z)$. See Appendix B.

To obtain an explicit form of the $\hat{\bar{z}}$ operator, we need to solve
Eq. (41). For the case of a uniform magnetic field with the reproducing kernel
Eq. (7), it is easy to see that $\hat{\bar{z}}=2l_{B}^{2}\partial_{z}$ does
satisfy Eq. (41). For general systems, we substitute Eq. (38) into Eq. (41),
apply Eq. (39), and obtain

$\hat{\bar{z}}=2l_{B}^{2}\left(\partial_{z}+\frac{\mathrm{i}e}{\hbar}N_{-}\left[\bar{A}_{1}^{\bigstar}\left(\hat{\bar{z}},z\right)\right]\right),$
(43)

where we decompose the starred vector potential into
$\bar{A}^{\bigstar}(\bar{\xi},z)=\mathrm{i}B_{0}\bar{\xi}/2+\bar{A}_{1}^{\bigstar}(\bar{\xi},z)$,
and $N_{-}[\dots]$ denotes the anti-normal ordering that places
$\hat{\bar{z}}$’s on the right of all $z$’s. Alternatively, the equation can
be written as

$\hat{\bar{z}}=2l_{B}^{2}\left(\partial_{z}+\frac{\mathrm{i}e}{\hbar}N_{+}\left[\bar{A}_{1}\left(\hat{\bar{z}},z\right)\right]\right)$
(44)

because $\bar{A}_{1}$ and $\bar{A}_{1}^{\bigstar}$ are related. These
equations can be solved iteratively.

The vortex degree of freedom can be quantized similarly. We define the
reproducing kernel
$K_{b}(\bar{\eta},\eta^{\prime})\equiv\exp[F_{b}(\bar{\eta},\eta^{\prime})]$
and the corresponding starred vector potential:
$\partial_{\bar{\eta}}F_{b}(\bar{\eta},\eta^{\prime})=-\mathrm{i}ea^{\bigstar}(\bar{\eta},\eta^{\prime})/\hbar$.
Making substitutions $z\rightarrow\bar{\eta}$,
$\hat{\bar{z}}\rightarrow\hat{\eta}$, $B\rightarrow b$ and
$\bar{A}^{\bigstar}\rightarrow a^{\bigstar}$, we can obtain a similar set of
relations for quantizing the vortex degree of freedom. We have

$\displaystyle\hat{\eta}$
$\displaystyle=2l_{b}^{2}\left(\partial_{\bar{\eta}}+\frac{\mathrm{i}e}{\hbar}N_{-}\left[a_{1}^{\bigstar}\left(\bar{\eta},\hat{\eta}\right)\right]\right)$
(45)
$\displaystyle=2l_{b}^{2}\left(\partial_{\bar{\eta}}+\frac{\mathrm{i}e}{\hbar}N_{+}\left[a_{1}\left(\bar{\eta},\hat{\eta}\right)\right]\right).$
(46)

The (anti-)normal ordering should be re-interpreted accordingly to place
$\hat{\eta}$’s on the left (right) of all $\bar{\eta}$’s.

###  III.3 Wave function ansatz

Using the modified integral measure Eq. (34), the wave function ansatz Eq.
(12) can be generalized straightforwardly:

$\Psi\left(\left\\{z_{i}\right\\}\right)=\int\prod_{i}\mathrm{d}\mu_{b}\left(\bm{\eta}_{i}\right)J\left(\left\\{\eta_{i}\right\\}\right)\Psi_{\mathrm{CF}}\left(\left\\{z_{i},\bar{\eta}_{i}\right\\}\right).$
(47)

Here, we only change the integral measures for $\\{\bm{\eta}_{i}\\}$, and
assume that the wave function of vortices in the physical subspace is still
Eq. (13).

Due to the change of the weight of the $\eta$–Bergman space, vortices are
actually in a deformed bosonic Laughlin state with an inhomogeneous density.
The joint density distribution of vortices is proportional to

$e^{\sum_{i<j}4\ln\left|\eta_{i}-\eta_{j}\right|-\sum_{i}f_{b}\left(\bar{\eta}_{i},\eta_{i}\right)}.$
(48)

Using Laughlin’s plasma analogy [26], 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 a “charge” density
$\partial_{\eta}\partial_{\bar{\eta}}f_{b}(\bar{\eta},\eta)/\pi$. Such a
system is expected to be nearly “charge-neutral” everywhere. It implies that
the single-particle density of vortices should be

$2\rho_{\mathrm{v}}\left(\bm{\eta}\right)\approx\frac{1}{\pi}\partial_{\eta}\partial_{\bar{\eta}}f_{b}\left(\bar{\eta},\eta\right)=\frac{e}{h}b\left(\bm{\eta}\right),$
(49)

where we make use of Eq. (35). We see that the CS self-consistent condition,
which relates the vortex density with the strength of the CS magnetic field,
emerges as a result of the constraint of the physical subspace.

The standard ansatz can also be generalized to

$\Psi\left(\left\\{z_{i}\right\\}\right)=\int\prod_{i}\mathrm{d}\mu_{B}\left(\bm{\xi}_{i}\right)\prod_{i}K_{B}\left(z_{i},\bar{\xi}_{i}\right)\\\
\times
J\left(\left\\{\xi_{i}\right\\}\right)\tilde{\Psi}_{\mathrm{CF}}\left(\left\\{\bm{\xi}_{i}\right\\}\right)$
(50)

with the modified integral measure. It is nothing but Eq. (1) with the
projection operator $\hat{P}_{\mathrm{LLL}}$ expressed in an integral form
using the reproducing kernels of the Bergman spaces of electrons.

By generalizing the derivation shown in Eq. (15), it is easy to show that
$\tilde{\Psi}_{\mathrm{CF}}$ is related to $\Psi_{\mathrm{CF}}$ by the
transformation

$\Psi_{\mathrm{CF}}\left(\left\\{z_{i},\bar{\eta}_{i}\right\\}\right)=\int\prod_{i}\mathrm{d}\mu_{B}\left(\bm{\xi}_{i}\right)\\\
\times\left[\prod_{i}K_{B}\left(z_{i},\bar{\xi}_{i}\right)K_{b}\left(\bar{\eta}_{i},\xi_{i}\right)\right]\tilde{\Psi}_{\mathrm{CF}}\left(\left\\{\bm{\xi}_{i}\right\\}\right).$
(51)

###  III.4 Wave equation: the dipole representation

In this subsection, we extend the wave equation Eq. (19) for systems subject
to fluctuating electromagnetic fields. We assume that the external magnetic
field is nearly uniform and varies slowly over space with
$|B_{1}(\bm{z})|/B_{0}\ll 1$, $|\nabla B(\bm{z})|l_{B}/B_{0}\ll 1$, and the
strength of the external electric field is weak and does not induce inter-
Landau-level transitions. In the derivation, we will ignore all corrections
proportional to the gradients of the strengths of the magnetic and electric
fields as well as contributions beyond the linear order of $B_{1}(\bm{z})$ and
the external electric field. The resulting theory will be adequate for
predicting long-wavelength responses to electromagnetic fields with $ql_{B}\ll
1$, where $q$ is the wave-number of the electromagnetic fields [14]. In this
limit, we can establish a 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. IV.

Our derivation is based on a straightforward generalization of the semi-
classical phenomenological model adopted in Sec. II.3. The Lagrangian of the
model for a set of composite fermions, in terms of single-particle wave-
functions $\\{\psi_{i}(z,\bar{\eta})\\}$, can be written as:

$L=\sum_{i}\int\mathrm{d}\mu_{B}\left(\bm{z}\right)\mathrm{d}\mu_{b}(\bm{\eta})\biggr{\\{}\epsilon_{i}\left|\psi_{i}\left(z,\bar{\eta}\right)\right|^{2}-\frac{\hbar^{2}}{2m^{\ast}}\\\
\times\int\mathrm{d}\mu_{b}(\bm{\eta}^{\prime})\psi_{i}^{\ast}(z,\bar{\eta})\frac{\left(\bar{z}-\bar{\eta}\right)\left(z-\eta^{\prime}\right)}{l_{b}^{2}(\bm{z})l_{B}^{2}(\bm{z})}K_{b}(\bar{\eta},\eta^{\prime})\psi_{i}(z,\bar{\eta}^{\prime})\biggr{\\}}\\\
-\int\mathrm{d}\bm{z}\Phi\left(\bm{z}\right)\rho_{\mathrm{e}}(\bm{z})-E_{\mathrm{xc}}\left[\rho_{\mathrm{e}}\right]\\\
-\frac{e^{2}}{8\pi\varepsilon}\int\mathrm{d}\bm{z}\mathrm{d}\bm{z}^{\prime}\frac{\left[\rho_{\mathrm{e}}(\bm{z})-\rho_{0}\right]\left[\rho_{\mathrm{e}}(\bm{z}^{\prime})-\rho_{0}\right]}{\left|\bm{z}-\bm{z}^{\prime}\right|}\\\
-\int\mathrm{d}\bm{\eta}\phi\left(\bm{\eta}\right)\left[\rho_{\mathrm{v}}\left(\bm{\eta}\right)-\frac{e}{2h}b\left(\bm{\eta}\right)\right],$
(52)

where the summation is over occupied states of composite fermions, and
$\epsilon_{i}$ is the Lagrange multiplier for the normalization of the wave
function. The second term is the kinetic energy, which is basically the
harmonic binding potential Eq. (17) written in the star-transformed form (see
Appendix B) with a space-dependent coefficient parametrized in the space-
dependent magnetic length of the external field
$l_{B}(\bm{z})=\sqrt{\hbar/eB(\bm{z})}$ and the CS magnetic length
$l_{b}(\bm{z})=\sqrt{\hbar/eb(\bm{z})}$. The third term is the energy for the
single-body physical scalar potential $\Phi(\bm{z})$, which includes the
scalar potential of the external electromagnetic fields as well as the orbital
magnetization energy discussed in Sec. III.1, and

$\rho_{\mathrm{e}}(\bm{z})=w_{B}\left(\bm{z}\right)\sum_{i}\int\mathrm{d}\mu_{b}\left(\bm{\eta}\right)\left|\psi_{i}\left(z,\bar{\eta}\right)\right|^{2}$
(53)

is the local density of electrons. The next two terms are the Coulomb energy
and an exchange-correlation energy functional
$E_{\mathrm{xc}}[\rho_{\mathrm{e}}]$ which accounts for the remaining
contributions of exchange and correlation effects. The last term imposes the
CS constraint which relates the local density of vortices

$\rho_{\mathrm{v}}(\bm{\eta})=w_{b}\left(\bm{\eta}\right)\sum_{i}\int\mathrm{d}\mu_{B}\left(\bm{z}\right)\left|\psi_{i}\left(z,\bar{\eta}\right)\right|^{2}.$
(54)

to the local strength of the CS magnetic field $b(\bm{\eta})$, with
$\phi(\bm{\eta})$ serving as the Lagrange multiplier for imposing the
constraint. The effects of the non-uniform physical and CS magnetic fields are
included implicitly in the integral measures $\mathrm{d}\mu_{B}$ and
$\mathrm{d}\mu_{b}$, respectively. The Lagrangian can be derived in a
microscopic theory, see Sec. IV. For the moment, we consider only time-
independent systems. The generalization for time-dependent systems will be
presented in Sec. III.6.

Differentiating the Lagrangian with respect to $\psi_{i}^{\ast}$, we obtain a
generalized wave equation for the stationary state of composite fermion:

$\epsilon\psi\left(z,\bar{\eta}\right)=\hat{H}_{\psi}\psi\left(z,\bar{\eta}\right),$
(55)

where we drop the state index subscripts for brevity, and the Hamiltonian is

$\hat{H}_{\psi}=\frac{\hbar^{2}}{2m^{\ast}}\frac{1}{l_{b}^{2}(\bm{z})l_{B}^{2}(\bm{z})}\left(\hat{\bar{z}}-\bar{\eta}\right)\left(z-\hat{\eta}\right)\\\
+N_{+}\left[\Phi_{\mathrm{eff}}\left(\hat{\bar{z}},z\right)+\phi\left(\bar{\eta},\hat{\eta}\right)\right].$
(56)

Here, we use the $\hat{\bar{z}}$ and $\hat{\eta}$ operators defined in Sec.
III.2, and ignore the non-commutativity between the operators and the magnetic
lengths, whose corrections are proportional to the spatial gradients of the
magnetic fields. The form of the Lagrangian Eq. (52) implies the anti-normal
ordering of $\bar{\eta}$ and $\hat{\eta}$ in the kinetic energy (see Appendix
B). The effective scalar potential for electrons is

$\Phi_{\mathrm{eff}}(\bm{z})=\Phi(\bm{z})+\frac{e^{2}}{4\pi\varepsilon}\int\mathrm{d}\bm{z}^{\prime}\frac{\rho_{\mathrm{e}}(\bm{z}^{\prime})-\rho_{0}}{\left|\bm{z}-\bm{z}^{\prime}\right|}+v_{\mathrm{xc}}[\rho_{\mathrm{e}}](\bm{z}),$
(57)

with the exchange-correlation potential
$v_{\mathrm{xc}}[\rho_{\mathrm{e}}]\equiv\delta
E_{\mathrm{xc}}[\rho_{\mathrm{e}}]/\delta\rho_{\mathrm{e}}+\tau_{\mathrm{xc}}(\bm{z})$,
which besides the contribution from $E_{\mathrm{xc}}[\rho_{\mathrm{e}}]$, also
contains a contribution from differentiating the factor
$1/l_{b}^{2}(\bm{z})=4\pi\rho_{\mathrm{v}}(\bm{z})\approx
4\pi\rho_{\mathrm{e}}(\bm{z})$ in the kinetic energy with respect to the
electron density:

$\tau_{\mathrm{xc}}(\bm{z})=\frac{2\pi\hbar^{2}}{m^{\ast}}\int\mathrm{d}\mu_{b}(\bm{\eta})\sum_{i}\left|\frac{z-\hat{\eta}}{l_{B}(\bm{z})}\psi_{i}(z,\bar{\eta})\right|^{2}.$
(58)

$\phi(\bm{\eta})$ can be interpreted as the scalar potential for vortices.
Eigenstates of the wave equation satisfy the orthonormal condition

$\int\mathrm{d}\mu_{B}(\bm{z})\int\mathrm{d}\mu_{b}(\bm{\eta})\psi_{i}^{\ast}\left(z,\bar{\eta}\right)\psi_{j}\left(z,\bar{\eta}\right)=\delta_{ij}.$
(59)

In the long-wavelength limit, we can determine approximate forms of the
$\hat{\bar{z}}$ and $\hat{\eta}$ operators. Since the bound state of the
vortex and electron has a length scale of the magnetic length (see Appendix
A), which is much smaller than the wavelength of the fluctuation of the
electromagnetic fields, the expectation value of $\hat{\bar{z}}$ should be
close to $\bar{\eta}$. We can thus expand the vector potential around
$\bar{\eta}$:
$N_{+}[\bar{A}_{1}(\hat{\bar{z}},z)]\approx\bar{A}_{1}(\bar{\eta},z)+(\hat{\bar{z}}-\bar{\eta})[\partial_{\bar{\eta}}\bar{A}_{1}(\bar{\eta},z)]$.
Substituting it into Eq. (44), we have

$\hat{\bar{z}}-\bar{\eta}\approx
2\left[\partial_{z}+\mathrm{i}\frac{e}{\hbar}\bar{A}\left(\bar{\eta},z\right)\right]l_{B}^{2}(\bar{\eta},z).$
(60)

Similarly, we can approximate the $\hat{\eta}$ operator as

$\hat{\eta}-z\approx
2\left[\partial_{\bar{\eta}}+\mathrm{i}\frac{e}{\hbar}a\left(\bar{\eta},z\right)\right]l_{b}^{2}\left(\bar{\eta},z\right).$
(61)

Accordingly, the scalar potentials are approximated as

$\displaystyle N_{+}[\Phi_{\mathrm{eff}}]$
$\displaystyle\approx\Phi_{\mathrm{eff}}(\bar{\eta},z)-V(\bar{\eta},z)\left[\mathrm{i}\hbar\partial_{z}-e\bar{A}(\bar{\eta},z)\right],$
(62) $\displaystyle N_{+}[\phi]$
$\displaystyle\approx\phi(\bar{\eta},z)-\bar{v}(\bar{\eta},z)\left[-\mathrm{i}\hbar\partial_{\bar{\eta}}-ea(\bar{\eta},z)\right],$
(63)

where
$V(\bar{\eta},z)=2\mathrm{i}\partial_{\bar{\eta}}\Phi_{\mathrm{eff}}(\bar{\eta},z)/eB(\bar{\eta},z)$
and
$\bar{v}(\bar{\eta},z)=2\mathrm{i}\partial_{z}\phi(\bar{\eta},z)/eb(\bar{\eta},z)$
are the complex components of the drift velocities
$\bm{V}=\bm{E}\times\bm{B}/B^{2}$ and
$\bm{v}=\bm{E}_{\mathrm{v}}\bm{\times\bm{b}}/b^{2}$ in the presence of the
electric fields $\bm{E}\equiv e^{-1}\bm{\nabla}\Phi_{\mathrm{eff}}$,
$\bm{E}_{\mathrm{v}}\equiv e^{-1}\bm{\nabla}\phi$ for electrons and vortices,
respectively.

Combining all, we have an approximate form of the Hamiltonian:

$\hat{H}_{\psi}\approx\frac{1}{2m^{\ast}}\left[-2\mathrm{i}\hbar\partial_{z}+2e\bar{A}\left(\bar{\eta},z\right)+m^{\ast}\bar{v}\left(\bar{\eta},z\right)\right]\\\
\times\left[-2\mathrm{i}\hbar\partial_{\bar{\eta}}+2ea\left(\bar{\eta},z\right)+m^{\ast}V\left(\bar{\eta},z\right)\right]\\\
+\Phi_{\mathrm{eff}}\left(\bar{\eta},z\right)+\phi\left(\bar{\eta},z\right).$
(64)

where we approximate the magnetic lengths as $l_{B}^{2}(\bm{z})\approx
l_{B}^{2}(\bar{\eta},z)$ and $l_{b}^{2}(\bm{z})\approx
l_{b}^{2}(\bar{\eta},z)$, and ignore corrections which are proportional to the
gradients of the magnetic fields.

In order to obtain a wave equation resembling the ordinary Schrödinger
equation, we introduce the transformation

$\psi\left(z,\bar{\eta}\right)=\sqrt{2\pi}l_{B}\exp\left[\frac{f_{B}(\bar{\eta},z)+f_{b}(\bar{\eta},z)}{2}\right]\varphi\left(z,\bar{\eta}\right),$
(65)

The wave equation for
$\varphi(\bm{\xi})\equiv\varphi(z,\bar{\eta})|_{z\rightarrow\xi,\bar{\eta}\rightarrow\bar{\xi}}$
can then be written as

$\epsilon\varphi(\bm{\xi})=\hat{H}\varphi(\bm{\xi}),$ (66)

with

$\hat{H}=-\frac{\hbar^{2}}{2m^{\ast}}\left[2\partial_{\xi}+\mathrm{i}\frac{e}{\hbar}\bar{\mathcal{A}}\left(\bm{\xi}\right)+\mathrm{i}\frac{m^{\ast}}{\hbar}\bar{v}(\bm{\xi})\right]\\\
\times\left[2\partial_{\bar{\xi}}+\mathrm{i}\frac{e}{\hbar}\mathcal{A}\left(\bm{\xi}\right)+\mathrm{i}\frac{m^{\ast}}{\hbar}V(\bm{\xi})\right]+\Phi_{\mathrm{eff}}\left(\bm{\xi}\right)+\phi\left(\bm{\xi}\right),$
(67)

where $(\bar{\mathcal{A}},\mathcal{A})$ denotes the effective vector potential
experienced by composite fermions:

$\bm{\mathcal{A}}=\bm{a}+\bm{A}.$ (68)

We see that the wave equation resembles the ordinary Schrödinger equation for
an charge particle but has corrections from the drift velocities $V$ and
$\bar{v}$. The corrections are the manifestation of the dipole nature of the
composite fermion [14], and not present in the HLR theory.

The wave equation is complemented by the CS self-consistent conditions, which
are obtained by differentiating the Lagrangian Eq. (52) with respect to the CS
potentials $\bm{a}$ and $\phi$. We have

$\displaystyle b(\bm{\eta})$
$\displaystyle=\frac{2h}{e}\rho_{\mathrm{v}}(\bm{\eta}),$ (69)
$\displaystyle\bm{E}_{\mathrm{v}}(\bm{\eta})$
$\displaystyle=\frac{2h}{e}\bm{n}\times\bm{j}_{\mathrm{v}}(\bm{\eta}),$ (70)

where $\bm{j}_{\mathrm{v}}(\bm{\eta})\equiv(-e)^{-1}\delta
L_{\mathrm{CF}}/\delta\bm{a}(\bm{\eta})$ denotes the current density of
vortices, and $L_{\mathrm{CF}}$ is the Lagrangian Eq. (52) with the CS term
removed.

The current density of vortices can be written as (see Appendix D)

$\bm{j}_{\mathrm{v}}(\bm{\eta})=\frac{\rho_{\mathrm{v}}(\bm{\eta})}{b(\bm{\eta})}\bm{E}_{\mathrm{v}}(\bm{\eta})\times\bm{n}+\frac{\hbar}{m^{\ast}}w_{b}(\bm{\eta})\sum_{i}\\\
\times\int\mathrm{d}\mu_{B}\left(\bm{z}\right)\psi_{i}^{\ast}\left(z,\bar{\eta}\right)\frac{\bm{n}\times(\bm{z}-\bm{\eta})}{l_{B}^{2}(\bm{z})}\psi_{i}\left(z,\bar{\eta}\right).$
(71)

After applying the self-consistent condition Eq. (69), the first term becomes
an anomalous Hall current with a half-quantized Hall conductance
$\sigma_{xy}^{(\mathrm{v})}=-e^{2}/2h$. Comparing it with the second CS
constraint Eq. (70), we have

$\bm{P}(\bm{\eta})\equiv
w_{b}(\bm{\eta})\sum_{i}\int\mathrm{d}\mu_{B}(\bm{z})(\bm{z}-\bm{\eta})\left|\psi_{i}\left(z,\bar{\eta}\right)\right|^{2}\approx
0,$ (72)

where we define the dipole density $\bm{P}(\bm{\eta})$. The CS constraints
imply that the dipole density always vanishes. It also suggests that the
coordinate of an electron, on average, always coincides with the coordinate of
the vortex that it bounds to.

Equations (65–67) as well as the self-consistent conditions (69, 70) define
the quantum mechanics in the fictitious world of composite fermions.

###  III.5 Wave equation: the standard representation

For Slater determinant wave functions $\Psi_{\mathrm{CF}}$ and
$\tilde{\Psi}_{\mathrm{CF}}$ related by Eq. (51), single-particle wave
functions in the two representations are related by the transformation

$\psi\left(z,\bar{\eta}\right)=\int\mathrm{d}\mu_{B}\left(\bm{\xi}\right)K_{B}\left(z,\bar{\xi}\right)K_{b}\left(\bar{\eta},\xi\right)\tilde{\psi}\left(\bm{\xi}\right).$
(73)

We can map operators in the dipole representation to their counterparts in the
standard representation. See Appendix C.2 for the rules of mapping.

Applying the mapping rules and the approximate forms of operators Eqs. (60,
61), we can obtain a wave equation in the standard representation (see
Appendix C.3):

$\epsilon\tilde{\varphi}\left(\bm{\xi}\right)=\hat{H}^{\dagger}\tilde{\varphi}\left(\bm{\xi}\right),$
(74)

with

$\tilde{\psi}(\bm{\xi})=\sqrt{2\pi}l_{B}\exp\left[\frac{f_{B}(\bm{\xi})-f_{b}(\bm{\xi})}{2}\right]\tilde{\varphi}\left(\bm{\xi}\right).$
(75)

We see that the Hamiltonian governing the wave-equation for
$\tilde{\varphi}(\bm{\xi})$ is complex conjugate to that for
$\varphi(\bm{\xi})$.

As in ideal systems, $\\{\varphi_{i}\\}$ and $\\{\tilde{\varphi}_{i}\\}$ are
dual to each other, and form a biorthogonal system. The orthonormal condition
Eq. (59) can be written as

$\int\mathrm{d}\bm{\xi}\tilde{\varphi}_{i}^{\ast}(\bm{\xi})\varphi_{j}(\bm{\xi})=\delta_{ij}.$
(76)

The effective Lagrangian for a composite fermion can be written as:

$L_{\mathrm{CF}}=\sum_{i}\int\mathrm{d}\bm{\xi}\tilde{\varphi_{i}}^{\ast}(\bm{\xi})\left[\epsilon_{i}-\hat{H}\right]\varphi_{i}(\bm{\xi}).$
(77)

Differentiating it with respect to $\tilde{\varphi}_{i}^{\ast}$ and
$\varphi_{i}$ gives rise to the wave equations Eq. (66) and (74),
respectively.

Differentiating the Lagrangian with respect to
$\left(\Phi_{\mathrm{eff}},\bm{A}\right)$ and $(\phi,\bm{a})$, we can obtain
the particle and current densities of electrons and vortices, respectively.
For stationary states, we have:
$\bm{j}_{\mathrm{e}}(\bm{\xi})\approx\bm{j}_{\mathrm{v}}(\bm{\xi})\approx\bm{j}(\bm{\xi)}$,
and

$\displaystyle j(\bm{\xi})\approx$
$\displaystyle\sum_{i}\tilde{\varphi}_{i}^{\ast}(\bm{\xi})\left[\frac{-2\mathrm{i}\hbar\partial_{\bar{\xi}}+e\mathcal{A}+m^{\ast}V}{m^{\ast}}\varphi_{i}(\bm{\xi})\right],$
(78) $\displaystyle\bar{j}(\bm{\xi})\approx$
$\displaystyle\sum_{i}\left[\frac{-2\mathrm{i}\hbar\partial_{\bar{\xi}}+e\mathcal{A}+m^{\ast}v}{m^{\ast}}\tilde{\varphi}_{i}(\bm{\xi})\right]^{\ast}\varphi_{i}(\bm{\xi}),$
(79) $\displaystyle\rho_{\mathrm{e}}(\bm{\xi})$
$\displaystyle\approx\sum_{i}\tilde{\varphi}_{i}^{\ast}(\bm{\xi})\varphi_{i}(\bm{\xi})-\partial_{\bar{\xi}}\bar{P}(\bm{\xi}),$
(80) $\displaystyle\rho_{\mathrm{v}}(\bm{\xi})$
$\displaystyle\approx\sum_{i}\tilde{\varphi}_{i}^{\ast}(\bm{\xi})\varphi_{i}(\bm{\xi})+\partial_{\xi}P(\bm{\xi}),$
(81)

where $P$ and $\bar{P}$ are the complex components of the dipole density
defined in Eq. (72), and $j\equiv 2(-e)^{-1}\delta
L_{\mathrm{CF}}/\delta\mathcal{\bar{A}}$, $\bar{j}\equiv 2(-e)^{-1}\delta
L_{\mathrm{CF}}/\delta\mathcal{A}$. In terms of $\varphi(\bm{\xi})$ and
$\tilde{\varphi}(\bm{\xi})$, the dipole density can be approximated as

$\displaystyle P(\bm{\xi})$ $\displaystyle\approx-
l_{b}^{2}(\bm{\xi})\sum_{i}\tilde{\varphi}_{i}^{\ast}(\bm{\xi})\left(2\partial_{\bar{\xi}}+\mathrm{i}\frac{e}{\hbar}\mathcal{A}\right)\varphi_{i}(\bm{\xi})$
(82) $\displaystyle\approx-
l_{B}^{2}(\bm{\xi})\sum_{i}\varphi_{i}^{\ast}(\bm{\xi})\left(2\partial_{\bar{\xi}}+\mathrm{i}\frac{e}{\hbar}\mathcal{A}\right)\tilde{\varphi}_{i}(\bm{\xi}).$
(83)

Note that according to Eq. (72), the dipole density always vanishes.

###  III.6 Time-dependent systems

For time-dependent systems, the term proportional to $\epsilon_{i}$ in the
Lagrangian Eq. (52) should be replaced with

$\sum_{i}\int\mathrm{d}\mu_{B}(\bm{z})\mathrm{d}\mu_{b}(\bm{\eta})\psi_{i}^{\ast}(z,\bar{\eta};t)\\\
\times\left\\{\mathrm{i}\hbar\frac{\partial}{\partial
t}-\frac{\mathrm{i}\hbar}{2}\left[\frac{\partial f_{B}(\bm{z},t)}{\partial
t}+\frac{\partial f_{b}(\bm{\eta},t)}{\partial
t}\right]\right\\}\psi_{i}(z,\bar{\eta};t),$ (84)

where the second term in the braces originates from the exponential factor in
Eq. (30) and its counterpart for vortices. The effective action of composite
fermions can be written as

$S_{\mathrm{CF}}=\sum_{i}\int\mathrm{d}t\int\mathrm{d}\mu_{B}(\bm{z})\mathrm{d}\mu_{b}(\bm{\eta})\psi_{i}^{\ast}(z,\bar{\eta};t)\\\
\times\left(\mathrm{i}\hbar\frac{\partial}{\partial
t}-\hat{H}_{\psi}\right)\psi_{i}(z,\bar{\eta};t),$ (85)

where $\hat{H}_{\psi}$ is Eq. (56) but with the substitutions

$\displaystyle\Phi(\bm{z})$
$\displaystyle\rightarrow\Phi(\bm{z},t)+\frac{\mathrm{i}\hbar}{2}\partial_{t}f_{B}(\bm{z},t),$
(86) $\displaystyle\phi(\bm{\eta})$
$\displaystyle\rightarrow\phi(\bm{\eta},t)+\frac{\mathrm{i}\hbar}{2}\partial_{t}f_{b}(\bm{\eta},t).$
(87)

For the long-wavelength limit, we repeat the derivation leading to Eq. (66),
and make use of the relations Eqs. (65, 73, 75). We find that the effective
action of a time-dependent composite fermion system can be written as

$S_{\mathrm{CF}}=\int\mathrm{d}t\int\mathrm{d}\bm{\xi}\tilde{\varphi}^{\ast}(\bm{\xi},t)\left(\mathrm{i}\hbar\frac{\partial}{\partial
t}-\hat{H}\right)\varphi(\bm{\xi},t),$ (88)

where $\hat{H}$ has a form identical to the stationary state Hamiltonian Eq.
(67), but the electric fields $\bm{E}$ and $\bm{E}_{\mathrm{v}}$, which
determine the drift velocities, are replaced by their gauge-invariant form
$\bm{E}=e^{-1}\bm{\nabla}\Phi_{\mathrm{eff}}-\partial_{t}\bm{A}$ and
$\bm{E}_{\mathrm{v}}=e^{-1}\bm{\nabla}\phi-\partial_{t}\bm{a}$. Note that the
substitutions Eqs. (86, 87) do not apply to the potentials in $\hat{H}$.

Differentiating the action with respect to
$\tilde{\varphi}^{\ast}(\bm{\xi},t)$ and $\varphi(\bm{\xi},t)$, we obtain the
set of time-dependent wave equations

$\displaystyle\mathrm{i}\hbar\frac{\partial\varphi(\bm{\xi},t)}{\partial t}$
$\displaystyle=\hat{H}\varphi(\bm{\xi},t),$ (89)
$\displaystyle\mathrm{i}\hbar\frac{\partial\tilde{\varphi}(\bm{\xi},t)}{\partial
t}$ $\displaystyle=\hat{H}^{\dagger}\tilde{\varphi}(\bm{\xi},t).$ (90)

Differentiating the action with respect to the vector and scalar fields, we
can obtain number and current densities of a time-dependent system. We find
that the current densities are modified by displacement currents:

$\displaystyle\bar{j}_{\mathrm{e}}(\bm{\xi,t})$
$\displaystyle=\bar{j}(\bm{\xi},t)+\partial_{t}\bar{P}(\bm{\xi},t),$ (91)
$\displaystyle j_{\mathrm{e}}(\bm{\xi,t})$ $\displaystyle=j(\bm{\xi},t),$ (92)
$\displaystyle\bar{j}_{\mathrm{v}}(\bm{\xi,t})$
$\displaystyle=\bar{j}(\bm{\xi},t),$ (93) $\displaystyle
j_{\mathrm{v}}(\bm{\xi},t)$
$\displaystyle=j(\bm{\xi},t)-\partial_{t}P(\bm{\xi},t).$ (94)

##  IV Microscopic underpinning

In this section, we derive the phenomenological dipole model underlying 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

$L_{\mathrm{M}}=\Braket{\Psi}{E-V_{\mathrm{ee}}-\Phi}{\Psi},$ (95)

where $E$ is a Lagrange multiplier for the normalization of $|\Psi\rangle$,
$V_{\mathrm{ee}}=(e^{2}/4\pi\varepsilon)\sum_{i<j}|z_{i}-z_{j}|^{-1}+V_{\mathrm{B}}$
denotes the Coulomb interaction between electrons with $V_{\mathrm{B}}$ being
the potential from a uniform neutralizing positive charge background, and
$\Phi\equiv\sum_{i}\Phi(\bm{z}_{i})$ denotes the externally applied scalar
potential. The kinetic energy is ignored since it is completely quenched in a
Landau level.

We base our derivation on the general variational principle of quantum
mechanics. Trial electron wave functions for $\ket{\Psi}$ are constructed
using the ansatz form Eq. (47), and assuming $\Psi_{\mathrm{CF}}$ to be a
Slater determinant of a set of single-body trial wave-functions
$\\{\psi_{i}\\}$ of composite fermions. We shall show that the Lagrangian Eq.
(52) can be derived from the microscopic Lagrangian Eq. (95). The wave
equation as well as the CS constraints then follow by applying the variational
principle

$\delta L=0.$ (96)

###  IV.1 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 the conditions
Eqs. (69, 70). 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

$\Psi_{\mathrm{CF}}\left(\left\\{z_{i},\bar{\eta}_{i}\right\\}\right)=\frac{1}{\sqrt{N!}}\mathrm{det\left[\psi_{j}(z_{i},\bar{\eta}_{i})\right]},$
(97)

two sets of single particle trial wave functions $\\{\psi_{i}\\}$ and
$\\{\psi_{i}^{\prime}\\}$ which are related by a non-singular linear
transformation yield the same physical wave function after applying Eq. (12)
or Eq. (47). To eliminate the redundancy, it is necessary to impose the
orthonormal condition:

$\int\mathrm{d}\mu_{B}(\bm{\xi})\mathrm{d}\mu_{b}(\bm{\eta})\psi_{i}^{\ast}(\xi,\bar{\eta})\psi_{j}(\xi,\bar{\eta})=\delta_{ij}.$
(98)

We note that the orthonormality depends on the weight in $\mathrm{d}\mu_{b}$,
which is not yet determined 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 [27], we introduce a test particle which 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 a test
particle can be written as

$\displaystyle\Psi^{\mathrm{t}}\left(z,\left\\{z_{i}\right\\}\right)=$
$\displaystyle\int\mathrm{d}\mu_{b}\left(\bm{\eta}\right)\Psi_{\eta}^{\mathrm{v}}\left(\left\\{z_{i}\right\\}\right)\psi\left(z,\bar{\eta}\right)$
(99)
$\displaystyle\Psi_{\eta}^{\mathrm{v}}\left(\left\\{z_{i}\right\\}\right)=$
$\displaystyle\int\prod_{i=1}^{N}\mathrm{d}\mu_{b}(\bm{\eta}_{i})\prod_{i=1}^{N}(\eta-\eta_{i})^{2}$
$\displaystyle\times
J\left(\left\\{\eta_{i}\right\\}\right)\Psi_{\mathrm{CF}}\left(\left\\{z_{i},\bar{\eta}_{i}\right\\}\right),$
(100)

where the test particle has the wave function $\psi(z,\bar{\eta}$), and
correlates with other composite fermions via the Bijl-Jastrow factor, as shown
in Eq. (100). Because the test particle has no exchange symmetry with other
composite fermions, it can occupy any state, including those already occupied
in $\Psi_{\mathrm{CF}}$. Our approximation is to assume that the set of single
particle wave functions for constructing $\Psi_{\mathrm{CF}}$ can be chosen
from eigen-wave-functions of the test particle.

With the approximation, we can determine the weight of $\mathrm{d}\mu_{b}$,
self-consistently, by requiring that the orthonormality Eq. (98) in the
Hilbert space of composite fermions is consistent with that of the physical
Hilbert space. This is to require

$\Braket{\Psi_{i}^{\mathrm{t}}}{\Psi_{j}^{\mathrm{t}}}=\delta_{ij},$ (101)

where $\Psi_{i}^{\mathrm{t}}$ and $\Psi_{j}^{\mathrm{t}}$ denote two physical
states corresponding to $\psi=\psi_{i}$ and $\psi=\psi_{j}$ in Eq. (99),
respectively, and $\psi_{i}$ and $\psi_{j}$ satisfy the orthonormal condition
Eq. (98). Equation (101) can be rewritten as

$\int\mathrm{d}\mu_{B}(\bm{z})\mathrm{d}\mu_{b}(\bm{\eta})\mathrm{d}\mu_{b}(\bm{\eta}^{\prime})\psi_{i}^{\ast}(z,\bar{\eta})\\\
\times
K_{b}(\bar{\eta},\eta^{\prime})\psi_{j}(z,\bar{\eta}^{\prime})=\delta_{ij},$
(102)

with

$K_{b}(\bar{\eta},\eta^{\prime})=\Braket{\Psi_{\eta}^{\mathrm{v}}}{\Psi_{\eta^{\prime}}^{\mathrm{v}}}.$
(103)

To fulfill it, we can adjust the weight of $\mathrm{d}\mu_{b}$ to make
$K_{b}(\bar{\eta},\eta^{\prime})$ the reproducing kernel corresponding to the
weight. It can then be reduced to Eq. (98) by integrating out
$\bm{\eta}^{\prime}$.

The requirement gives rise to the CS constraint Eq. (69) in the long-
wavelength limit. To see this, we rewrite Eq (103) as
$K_{b}(\bar{\eta},\eta^{\prime})=\langle e^{\mathcal{F}}\rangle$, with

$\langle
e^{\mathcal{F}}\rangle\equiv\int\prod_{i=1}^{N}\mathrm{d}\mu_{B}(\bm{z}_{i})\mathrm{d}\mu_{b}(\bm{\eta}_{i})\mathrm{d}\mu_{b}(\bm{\eta}_{i}^{\prime})e^{\mathcal{F}(\bar{\eta},\eta^{\prime},\\{\bar{\eta}_{i},\eta_{i}^{\prime}\\})}\\\
\times
J^{\ast}(\\{\eta_{i}\\})J(\\{\eta_{i}^{\prime}\\})\Psi_{\mathrm{CF}}^{\ast}(\\{z_{i},\bar{\eta}_{i}\\})\Psi_{\mathrm{CF}}(\\{z_{i},\bar{\eta}_{i}^{\prime}\\})$
(104)

and
$\mathcal{F}(\bar{\eta},\eta^{\prime},\\{\bar{\eta}_{i},\eta_{i}^{\prime}\\})\equiv
2\sum_{i}\ln(\bar{\eta}-\bar{\eta}_{i})(\eta^{\prime}-\eta_{i}^{\prime})$.
Using the cumulant expansion, we can determine
$F_{b}(\bar{\eta},\eta^{\prime})\equiv\ln K_{b}(\bar{\eta},\eta^{\prime})$
approximately:

$F_{b}\left(\bar{\eta},\eta^{\prime}\right)\approx\left\langle\mathcal{F}\right\rangle+\frac{1}{2}\left\langle\left(\mathcal{F}-\left\langle\mathcal{F}\right\rangle\right)^{2}\right\rangle+\cdots,$
(105)

In the lowest order, we ignore the fluctuation and higher order corrections,
and approximate $F_{b}$ as
$\langle\mathcal{F}\rangle=2\sum_{i}\langle\ln(\bar{\eta}-\bar{\eta}_{i})(\eta^{\prime}-\eta_{i}^{\prime})\rangle$.
To evaluate the $i$-th term of the summation, we expand the Slater determinant
Eq. (97) along its $i$-th row, substitute the expansion into Eq. (104), and
ignore contributions involving particle exchanges. We obtain

$F_{b}\left(\bar{\eta},\eta^{\prime}\right)\approx\sum_{i}\int\mathrm{d}\mu_{B}\left(\bm{z}_{i}\right)\mathrm{d}\mu_{b}\left(\bm{\eta}_{i}\right)\mathrm{d}\mu_{b}\left(\bm{\eta}_{i}^{\prime}\right)\\\
\times
2\left[\ln(\bar{\eta}-\bar{\eta}_{i})+\ln(\eta^{\prime}-\eta_{i}^{\prime})\right]\\\
\times\frac{1}{N}\sum_{a}\psi_{a}^{\ast}\left(z_{i},\bar{\eta}_{i}\right)K_{b}^{(a)}\left(\bar{\eta}_{i},\eta_{i}^{\prime}\right)\psi_{a}\left(z_{i},\bar{\eta}_{i}^{\prime}\right),$
(106)

where $K_{b}^{(a)}$ is defined by Eq. (103) but with $\Psi_{\mathrm{CF}}$
being a Slater determinant of $N-1$ composite fermions with the state
$\psi_{a}$ unoccupied. We assume that the effect of removing a composite
fermion from the effective medium of $N$ composite fermions is negligible,
therefore have

$K_{b}^{(a)}(\bar{\eta}_{i},\eta_{i}^{\prime})\approx
K_{b}(\bar{\eta}_{i},\eta_{i}^{\prime}).$ (107)

After integrating out $\bm{\eta}_{i}^{\prime}$, we obtain

$F_{b}\left(\bar{\eta},\eta\right)\approx
2\int\mathrm{d}^{2}\bm{\eta}_{1}\ln\left(\left|\eta-\eta_{1}\right|^{2}\right)\rho_{\mathrm{v}}\left(\bm{\eta}_{1}\right).$
(108)

where $\rho_{\mathrm{v}}(\bm{\eta}_{1})$ is the vortex density defined in Eq.
(54). Applying the identity
$\partial_{\eta}\partial_{\bar{\eta}}\ln(|\eta-\eta_{1}|^{2})=\pi\delta(\bm{\eta}-\bm{\eta}_{1})$,
we have

$\partial_{\eta}\partial_{\bar{\eta}}F_{b}\left(\bar{\eta},\eta\right)=2\pi\rho_{\mathrm{v}}\left(\bm{\eta}\right).$
(109)

In the long-wavelength limit, we have $F_{b}(\bar{\eta},\eta)\approx
f_{b}(\bm{\eta})-\ln[l_{b}^{2}(\bm{\eta})/l_{b}^{2}]$ [see Eq. (147)].
Substituting the relation into Eq. (109) and ignoring the spatial gradient of
the magnetic length, the equation becomes the CS constraint Eq. (69).

We can then replace the normalization constraint $\braket{\Psi}{\Psi}=1$ in
Eq. (95) with normalization constraints of the single-body wave-functions as
well as the CS constraint Eq. (69), and introduce $\epsilon_{i}$ and
$\phi(\bm{\eta})$ as respective Lagrange multipliers. The Lagrangian becomes

$L=\int\mathrm{d}\mu_{B}(\bm{z})\mathrm{d}\mu_{b}(\bm{\eta})\sum_{i}\epsilon_{i}\left|\psi_{i}(z,\bar{\eta})\right|^{2}-\int\mathrm{d}\bm{\eta}\phi(\bm{\eta})\\\
\times\left[\rho_{\mathrm{v}}\left(\bm{\eta}\right)-\frac{e}{2h}b\left(\bm{\eta}\right)\right]-\Braket{\Psi}{V_{\mathrm{ee}}+\Phi}{\Psi},$
(110)

###  IV.2 Energy

In this subsection, we determine the expectation value
$\Braket{\Psi}{V_{\mathrm{ee}}+\Phi}{\Psi}.$ We shall show how the kinetic
energy of a composite fermion, i.e., the electron-vortex binding energy, would
emerge out of the Coulomb interaction as a correlation effect.

We first determine the expectation value of the scalar potential
$\braket{\Psi}{\Phi}{\Psi}$. Similar to Eq. (106), we have

$\Braket{\Psi}{\Phi}{\Psi}\approx\sum_{i}\int\mathrm{d}\mu_{B}\left(\bm{z}_{i}\right)\mathrm{d}\mu_{b}\left(\bm{\eta}_{i}\right)\mathrm{d}\mu_{b}\left(\bm{\eta}_{i}^{\prime}\right)\Phi(\bm{z}_{i})\\\
\times\frac{1}{N}\sum_{a}\psi_{a}^{\ast}\left(z_{i},\bar{\eta}_{i}\right)K_{b}^{(a)}\left(\bar{\eta}_{i},\eta_{i}^{\prime}\right)\psi_{a}\left(z_{i},\bar{\eta}_{i}^{\prime}\right).$
(111)

Applying the approximation Eq. (107), we obtain

$\Braket{\Psi}{\Phi}{\Psi}\approx\int\mathrm{d}\bm{z}\Phi(\bm{z})\rho_{\mathrm{e}}(\bm{z}).$
(112)

Next, we determine the expectation value of the Coulomb interaction energy. It
can be written as

$\left\langle
V_{\mathrm{ee}}\right\rangle=\frac{e^{2}}{8\pi\varepsilon}\int\mathrm{d}\bm{z}\mathrm{d}\bm{z}^{\prime}\frac{\rho_{2}(\bm{z},\bm{z}^{\prime})-2\rho_{\mathrm{e}}(\bm{z})\rho_{0}+\rho_{0}^{2}}{\left|z-z^{\prime}\right|},$
(113)

where $\rho_{2}(\bm{z},\bm{z}^{\prime})=\braket{\Psi}{\sum_{i\neq
j}\delta(\bm{z}-\bm{z}_{i})\delta(\bm{z}^{\prime}-\bm{z}_{j})}{\Psi}$ is the
two-particle reduced density of electrons. We decompose $\langle
V_{\mathrm{ee}}\rangle$ into two parts. The first part is the mean-field
contribution of the Coulomb interaction

$\bar{V}_{\mathrm{ee}}=\frac{e^{2}}{8\pi\varepsilon}\int\mathrm{d}\bm{z}\mathrm{d}\bm{z}^{\prime}\frac{\left[\rho_{\mathrm{e}}(\bm{z})-\rho_{0}\right]\left[\rho_{\mathrm{e}}(\bm{z}^{\prime})-\rho_{0}\right]}{\left|\bm{z}-\bm{z}^{\prime}\right|},$
(114)

which gives rise to the Coulomb energy term of Eq. (52). The second part is
the correlation contribution

$T=\frac{e^{2}}{8\pi\varepsilon}\int\mathrm{d}\bm{z}\mathrm{d}\bm{z}^{\prime}\frac{\rho_{2}(\bm{z},\bm{z}^{\prime})-\rho_{\mathrm{e}}(\bm{z})\rho_{\mathrm{e}}(\bm{z}^{\prime})}{\left|\bm{z}-\bm{z}^{\prime}\right|},$
(115)

which gives rise to the binding energy between electrons and vortices.

We determine the two-particle reduced density by applying the Hartree-like
approximation discussed in the last subsection. It is related to the physical
wave function by
$\rho_{2}(\bm{z},\bm{z}^{\prime})=N(N-1)w_{B}(\bm{z})w_{B}(\bm{z}^{\prime})\int\prod_{i=3}^{N}\mathrm{d}\mu_{B}(\bm{z}_{i})|\Psi(\\{z_{i}\\}|^{2}$
with $z_{1}=z$ and $z_{2}=z^{\prime}$. We treat the first particle
($\bm{z}_{1}$) as a test particle, and the ensemble of other $N-1$ particles
as an effective medium. By expanding the Slater determinant Eq. (97) along its
first row, ignoring exchange terms in $|\Psi(\\{z_{i}\\})|^{2}$, and replacing
the $N-1$ particle effective medium with the $N$-particle one as in Eq. (99),
we can approximate $\rho_{2}$ as

$\displaystyle\rho_{2}(\bm{z},\bm{z}_{1})\approx$
$\displaystyle\leavevmode\nobreak\
w_{B}(\bm{z})\int\mathrm{d}\mu_{b}(\bm{\eta})\mathrm{d}\mu_{b}(\bm{\eta}^{\prime})K_{b}(\bar{\eta},\eta^{\prime})$
$\displaystyle\times\sum_{a}\psi_{a}^{\ast}(z,\bar{\eta})\psi_{a}(z,\bar{\eta}^{\prime})\rho_{\mathrm{c}}(\bm{z}_{1};\bar{\eta},\eta^{\prime}),$
(116) $\displaystyle\rho_{\mathrm{c}}(\bm{z}_{1};\bm{\eta})=$ $\displaystyle
w_{B}(\bm{z}_{1})N\int\prod_{i=2}^{N}\mathrm{d}\mu_{B}(\bm{z}_{i})\frac{\left|\Psi_{\eta}^{\mathrm{v}}\left(\left\\{z_{i}\right\\}\right)\right|^{2}}{\Braket{\Psi_{\eta}^{\mathrm{v}}}{\Psi_{\eta}^{\mathrm{v}}}},$
(117)

and
$\rho_{\mathrm{c}}(\bm{z}_{1};\bar{\eta},\eta^{\prime})=\rho_{\mathrm{c}}(\bm{z}_{1},\bm{\eta})|_{\eta\rightarrow\eta^{\prime}}$.
$\rho_{\mathrm{c}}(\bm{z}_{1};\bm{\eta})$ is the density profile of electrons
surrounding a vortex at $\bm{\eta}$. The vortex would suppress the electron
density locally, creating a correlation hole.

The Coulomb attraction between the test (first) electron and the correlation
hole gives rise to the binding energy of a composite fermion. Substituting Eq.
(116) and (53) into Eq. (115), we obtain the binding energy

$T\approx\int\mathrm{d}\mu_{B}(\bm{z})\mathrm{d}\mu_{b}(\eta)\mathrm{d}\mu_{b}(\eta^{\prime})K_{b}(\bar{\eta},\eta^{\prime})\\\
\times\epsilon_{\mathrm{b}}^{\bigstar}(\bm{z};\bar{\eta},\eta^{\prime})\sum_{a}\psi_{a}^{\ast}(z,\bar{\eta})\psi_{a}(z\bar{\eta}^{\prime}),$
(118)

with
$\epsilon_{\mathrm{b}}^{\bigstar}(\bm{z};\bar{\eta},\eta^{\prime})\equiv\epsilon_{\mathrm{b}}^{\bigstar}(\bm{z};\bm{\eta})|_{\eta\rightarrow\eta^{\prime}}$,
and

$\epsilon_{\mathrm{b}}^{\bigstar}(\bm{z};\bm{\eta})=\frac{e^{2}}{8\pi\varepsilon}\int\mathrm{d}\bm{z}_{1}\frac{\rho_{\mathrm{e}}(\bm{z}_{1})h(\bm{z}_{1};\bm{\eta})}{|z-z_{1}|},$
(119)

where we define the electron-vortex pair correlation function

$h(\bm{z}_{1};\bm{\bm{\eta}})=\frac{\rho_{\mathrm{c}}(\bm{z}_{1};\bm{\eta})}{\rho_{\mathrm{e}}(\bm{z}_{1})}-1.$
(120)

$\epsilon_{\mathrm{b}}^{\bigstar}(\bm{z};\bm{\eta})$ is the binding energy of
a composite fermion.

The electron-vortex correlation function describes the deviation of the
electron density from its local average $\rho_{\mathrm{e}}(\bm{z}_{1})$ in the
proximity of a vortex. We expect that the electron density is suppressed near
the center of the vortex, and recovers in a length scale $\sim l_{B}$. We thus
have $h(\bm{\bm{z}_{1}};\bm{\eta})<0$ for $\bm{z}_{1}\rightarrow\bm{\eta}$ and
$h(\bm{z}_{1};\bm{\eta})\rightarrow 0$ for $|\bm{z}_{1}-\bm{\eta}|\gg l_{B}$,
where we assume that the void of electrons created by the vortex is centered
$\bm{\eta}$,. This is true because the coordinate of an electron on average
always coincide with the vortex that it binds to, as indicated by Eq. (72).
Furthermore, we have the sum rule

$\int h(\bm{z}_{1};\bm{\eta})\mathrm{d}\bm{z}_{1}=-4\pi l_{B}^{2}(\bm{\eta}).$
(121)

This is because the insertion of a $2h/e$ vortex should induce a void of
electrons with a total charge $2\nu e$, where $\nu\equiv 2\pi
l_{B}^{2}(\bm{\eta})\rho_{\mathrm{e}}(\bm{\eta})$ is the local filling factor,
and we assume that the density varies slowly over space. In the long-
wavelength limit, the system is locally homogeneous. We can approximate
$h(\bm{z}_{1};\bm{\eta})$ as the pair correlation function of a homogeneous
system: $h(\bm{z}_{1};\bm{\eta})\approx
h_{0}(|\bm{z}_{1}-\bm{\eta}|/l_{B}(\bm{\eta}))$. For $h_{0}(r)$, the sum rule
can be written as

$\int_{0}^{\infty}\mathrm{d}rh_{0}(r)r=-2.$ (122)

It is notable that the sum rule is independent of the density. It is
reasonable to expect that $h_{0}(r)$ only weakly depends on the density since
it should have the same asympotic behavior for different densities while
constrained by a sum rule independent of the density.

The binding energy of a composite fermion can then be written as

$\displaystyle\epsilon_{\mathrm{b}}^{\bigstar}(\bm{z};\bm{\eta})$
$\displaystyle\approx\frac{e^{2}l_{B}(\bm{z})}{8\pi\varepsilon}\rho_{\mathrm{e}}(\bm{z})u\left(\frac{|z-\eta|}{l_{B}(\bm{z})}\right),$
(123) $\displaystyle u(|x|)$
$\displaystyle=\int\mathrm{d}\bm{y}\frac{h_{0}(|\bm{y}|)}{|\bm{x+\bm{y}}|}.$
(124)

We see that the binding energy is proportional to the local electron density
$\rho_{\mathrm{e}}(\bm{z})\equiv 1/4\pi l_{b}^{2}(\bm{z})$. It gives rise to
the density-of-sates correction of the kinetic energy (see Sec. II.3). The
peculiar feature of the phenomenological model turns out to a natural result
of the interaction origin of the binding energy.

We expect that the function $u(r)$ is approximately a universal function, and
only weakly depends on the filling factor or the density since it is
determined by $h_{0}(r)$. We thus estimate $u(r)$ using the filling factor
$\nu=1/3$, for which we can complete the integrals with respect to
$\\{\bm{\eta}_{i}\\}$ in Eq. (100), and have

$\Psi_{\eta=0}^{\mathrm{v}}(\\{z_{i}\\})=\prod_{i}z_{i}^{2}\prod_{i<j}(z_{i}-z_{j})^{3},$
(125)

where we place the vortex at the origin of the space without loss of
generality. The density profile of electrons surrounding the vortex can be
determined numerically using the Monte-Carlo method. We find that $u(r)$ can
be well fitted by the quadratic function $u(r)=c_{0}+c_{1}r^{2}$ in the region
$r\in[0,2]$ with $c_{0}=-12.6$ and $c_{1}=1$ 111Hao Jin, private
communications..

The binding energy of a composite fermion can then be approximated as

$\epsilon_{\mathrm{b}}^{\bigstar}(\bm{z};\bm{\eta})\approx-
g_{0}\frac{e^{2}l_{B}(\bm{z})}{\varepsilon}\rho_{\mathrm{e}}(\bm{z})+\frac{\hbar^{2}}{2m^{\ast}}\frac{\left|\bm{z}-\bm{\eta}\right|^{2}}{l_{B}^{2}(\bm{z})l_{b}^{2}(\bm{z})},$
(126)

with the coefficients

$\displaystyle g_{0}$ $\displaystyle=-\frac{c_{0}}{8\pi},$ (127)
$\displaystyle\frac{\hbar^{2}}{m^{\ast}}$
$\displaystyle=\left(\frac{c_{1}}{4\pi}\right)\frac{e^{2}l_{B}(\bm{z})}{4\pi\varepsilon}.$
(128)

We have $g_{0}\approx 0.5$, and $c_{1}/4\pi\approx 0.08$. The estimated
effective mass is roughly four times larger than that commonly adopted
($\hbar^{2}/m^{\ast}\approx 0.3e^{2}l_{B}/4\pi\varepsilon$) in the literature
[4, 29]. On the other hand, effective masses determined in experiments vary
with measurement methods [2]. Our estimation is actually close to the
cyclotron effective mass measured by Kukushkin et al. [30].

We can collect the contribution of the first term of Eq. (126), and define an
exchange-correlation functional

$E_{\mathrm{xc}}[\rho_{\mathrm{e}}]=g_{0}\frac{e^{2}}{\varepsilon}\int\mathrm{d}\bm{z}l_{B}(\bm{z})\rho_{\mathrm{e}}^{2}(\bm{z})+\dots\leavevmode\nobreak\
,$ (129)

where the ellipsis denotes all contributions to the energy which are ignored
in our derivation, in particular the effect of particle exchanges. In the
spirit of Kohn-Sham approach of the density functional theory, we could define
$E_{\mathrm{xc}}[\rho_{\mathrm{e}}]$ as the difference between the exact
ground state energy of a system with a uniform density $\rho_{\mathrm{e}}$ and
the total kinetic energy of non-interacting composite fermions at the same
density [31, 29].

Combining all, we obtain the Lagrangian Eq. (52).

##  V Summary and discussion

In summary, we present a reformulation of the theory of composite fermions
based on the dipole picture. We show that: (i) the standard wave function
ansatz can be recast and derived from the dipole picture by interpreting the
form as a projection for eliminating the auxiliary vortex degrees of freedom;
(ii) the responses of a composite fermion system to external electromagnetic
perturbations can be determined by solving a wave-equation resembling the
ordinary Schrödinger equation but with drift velocity corrections. The
reformulation provides a unified basis for the two complementary parts of the
theory, and extends its applicability: we now have a set of wave equations and
a generalized wave-function ansatz which are valid for general systems that
may be subject to non-uniform electromagnetic fields and have inhomogeneous
densities. We further clarify the microscopic underpinning of the dipole
picture by showing how the effective Lagrangian can be derive from the
microscopic Lagrangian, and the CS fields and the kinetic energy of a
composite fermion could emerge.

Our reformulation indicates that the the Bijl-Jastrow factor appeared in the
ansatz wave-function should be interpreted as the complex conjugate of the
wave function of vortices, instead of the numerator of the singular CS
transformation. The interpretation naturally leads to the general wave
function ansatz. Such an interpretation could potentially be justified
microscopically as already done for the $\nu=1$ bosonic state in Refs. 15, 16.

We also establish the general principle of quantum mechanics for the
fictitious world of composite fermions based on wave functions defined in the
Bergman space. It has a set of rules of quantization distinct from that for
ordinary particles due to the fact that both electrons and vortices are
confined in Landau levels. The principle is actually more general than what we
present in this study for non-interacting systems of composite fermions. For
systems in which correlations between composite fermions are important (e.g.
$\nu=5/2$), it would not be difficult to establish appropriate wave-equations
by applying the same principle. It is important to note that the intuition
which we develop for the real world may not always apply well in the
fictitious world of composite fermions.

###### Acknowledgements.

The author acknowledges Hao Jin for the help of determining the binding energy
of composite fermion in $\nu=1/3$, and Yinhan Zhang, Guangyue Ji, Bo Yang, Di
Xiao and Yue Yu for useful discussions. He is supported by the National Key
R&D Program of China under Grand Nos. 2021YFA1401900 and 2018YFA0305603, and
the National Science Foundation of China under Grant No. 12174005.

##  Appendix A $\bm{\Lambda}$-levels of the fractional quantum Hall states

The wave equation (23) is the same as that for an ordinary charge particle in
an effective magnetic field $\bm{\mathcal{B}}$ except for a constant
magnetization energy term. Therefore, the wave functions of $\Lambda$-levels
are identical to those of ordinary Landau levels, which can be written as [2]

$\varphi_{n,m}(\bm{\xi})\propto\frac{e^{-|\xi|^{2}/4l^{2}}}{\sqrt{2\pi}l}f_{n,m}(\bm{\xi}),$
(130)

with

$f_{n,m}(\bm{\xi})=c_{nm}l^{2n+m}e^{|\xi|^{2}/2l^{2}}\\\
\times\begin{cases}\partial_{\xi}^{n}\partial_{\bar{\xi}}^{m+n}e^{-|\xi|^{2}/2l^{2}}&\nu<\frac{1}{2}\\\
\partial_{\bar{\xi}}^{n}\partial_{\xi}^{m+n}e^{-|\xi|^{2}/2l^{2}}&\nu>\frac{1}{2}\end{cases}$
(131)

and $c_{nm}\equiv\sqrt{2^{2n+m}/n!(m+n)!}$, $m\geq-n$.
$\psi_{mn}(z,\bar{\eta})$ is related to
$\varphi_{n,m}(z,\bar{\eta})\equiv\varphi_{n,m}(\bm{\xi})|_{\xi\rightarrow
z,\bar{\xi}\rightarrow\bar{\eta}}$ by Eq. (22), and normalized by Eq. (21). We
thus have

$\psi_{n,m}(z,\bar{\eta})=f_{n,m}(z,\bar{\eta})\begin{cases}\frac{l_{B}}{l}\left(\frac{l_{B}}{l_{b}}\right)^{n}e^{z\bar{\eta}/2l_{b}^{2}}&\nu<\frac{1}{2}\\\
\frac{l_{b}}{l}\left(\frac{l_{b}}{l_{B}}\right)^{n}e^{z\bar{\eta}/2l_{B}^{2}}&\nu>\frac{1}{2}\end{cases},$
(132)

The corresponding eigen-energies are

$\epsilon_{n,m}=\hbar\omega_{c}^{\ast}\begin{cases}n&\nu<1/2\\\
n+1&\nu>1/2\end{cases},$ (133)

with $\omega_{c}^{\ast}\equiv e|\mathcal{B}|/m^{\ast}$.

For the special case $\nu=1/2$, $B=b$, we have $\mathcal{B}=0$. The wave
function is plane-wave like:

$\psi_{\bm{k}}(z,\bar{\eta})=\frac{l_{B}}{\sqrt{2\pi}}e^{\mathrm{i}\frac{\bar{k}z+k\bar{\eta}}{2}+\frac{z\bar{\eta}}{2l_{B}^{2}}-\frac{\left|k\right|^{2}l_{B}^{2}}{4}},$
(134)

where $\bm{k}\equiv(k_{x},k_{y})$ denotes the wave-vector of the state, and
$k\equiv k_{x}+\mathrm{i}k_{y}$, $\bar{k}=k^{\ast}$. The wave function is
normalized by
$\int\mathrm{d}\mu_{B}(\bm{z})\mathrm{d}\mu_{b}(\bm{\eta})\psi_{\bm{k}}^{\ast}(z,\bar{\eta})\psi_{\bm{k}^{\prime}}(z,\bar{\eta})=\delta(\bm{k}-\bm{k}^{\prime})$.

It is easy to show that these wave functions do describe bound states of an
electron and a vortex. Their spatial distributions can be written as

$\left|\psi_{n,m}\left(z,\bar{\eta}\right)\right|^{2}e^{-\left|z\right|^{2}/2l_{B}^{2}-\left|\eta\right|^{2}/2l_{b}^{2}}\\\
\propto\begin{cases}e^{-\left|z\right|^{2}/2l^{2}-\left|z-\eta\right|^{2}/2l_{b}^{2}}&\nu<1/2\\\
e^{-\left|\eta\right|^{2}/2l^{2}-\left|z-\eta\right|^{2}/2l_{B}^{2}}&\nu>1/2\end{cases}.$
(135)

We see that the electron and the vortex are bound by a Gaussian factor with a
length scale $l_{b}$ ($l_{B}$) for $\nu<1/2$ ($\nu>1/2$).

We can also directly solve Eq. (25), and obtain wave functions in the standard
representation:

$\tilde{\varphi}_{n,m}\left(\bm{\xi}\right)=\frac{e^{-\frac{\left|\xi\right|^{2}}{4l^{2}}}}{\sqrt{2\pi}l}f_{n,m}\left(\bm{\xi}\right)\begin{cases}\left(\frac{l_{b}}{l_{B}}\right)^{n}&\nu<\frac{1}{2}\\\
\left(\frac{l_{B}}{l_{b}}\right)^{n+1}&\nu>\frac{1}{2}\end{cases}.$ (136)

where the normalization constants are fixed using Eq. (76).
$\tilde{\psi}_{n,m}(\bm{\xi})$ is related to $\tilde{\varphi}_{n,m}$ via Eq.
(26). It is straightforward to verify that $\psi_{n,m}(z,\bar{\eta})$ and
$\tilde{\psi}_{n,m}(\bm{\xi})$ are related by Eq. (24).

##  Appendix B Alternative form of the quantization

There are two alternative ways of defining the quantization. The first one is
defined in Eq. (40). With the $\hat{\bar{z}}$ operator defined in Eq. (41),
the $\hat{H}$ operator can be written has $\hat{H}=N_{+}[H(\hat{\bar{z}},z)]$,
where $N_{+}$ denotes the normal ordering that places $\hat{\bar{z}}$ to the
left of all $z$’s. Alternatively, we can define

$\left[\hat{H}\psi\right]\left(z\right)=\int\mathrm{d}\mu_{B}(\bm{\xi})K_{B}\left(z,\bar{\xi}\right)H^{\bigstar}\left(z,\bar{\xi}\right)\psi\left(\xi\right),$
(137)

which corresponds $\hat{H}=N_{-}\left[H^{\bigstar}(z,\hat{\bar{z}})\right]$
with the anti-normal ordering $N_{-}$ that places $\hat{\bar{z}}$ to the right
of all $z$’s. For a given $\hat{H}$ operator, the $H$ and $H^{\bigstar}$
functions should be related by the transformation

$H^{\bigstar}\left(z,\bar{\xi}\right)=\frac{1}{K_{B}\left(z,\bar{\xi}\right)}\\\
\times\int\mathrm{d}\mu_{B}\left(\bm{\zeta}\right)K_{B}\left(z,\bar{\zeta}\right)H\left(\bar{\zeta},\zeta\right)K_{B}\left(\zeta,\bar{\xi}\right).$
(138)

It is easy to verify that Eq. (137) reduces to Eq. (40) after substituting Eq.
(138).

For a ideal system with the reproducing kernel Eq. (7), the transformation is
just a Gaussian convolution:

$H^{\bigstar}\left(z,\bar{z}\right)=\int\frac{\mathrm{d}^{2}\bm{\xi}}{2\pi
l_{B}^{2}}e^{-\left|z-\xi\right|^{2}/2l_{B}^{2}}H\left(\bar{\xi},\xi\right),$
(139)

and
$H^{\bigstar}(z,\bar{\xi})=H^{\bigstar}(z,\bar{z})|_{\bar{z}\rightarrow\bar{\xi}}$.
For a function $H(\bar{\xi},\xi)$ which varies slowly over space in a length
scale much larger than $l_{B}$, we have

$H^{\bigstar}(z,\bar{\xi})=\left.H(\bar{z},z)\right|_{\bar{z}\rightarrow\bar{\xi}}+\mathcal{O}\left(l_{B}^{2}\nabla^{2}H\right).$
(140)

The vector potential $\bar{A}$ and its starred counterpart
$\bar{A}^{\bigstar}$ are also related by the transformation. To see that, we
note that $\partial_{z}\psi(z)$ can be written in the two alternative forms:

$\displaystyle\partial_{z}\psi(z)$
$\displaystyle=\int\mathrm{d}\tilde{\mu}_{B}\left(\xi\right)K_{B}\left(z,\bar{\xi}\right)\partial_{\xi}\psi\left(\xi\right)$
(141)
$\displaystyle=\int\mathrm{d}\tilde{\mu}_{B}\left(\xi\right)\left[\partial_{z}K_{B}\left(z,\bar{\xi}\right)\right]\psi\left(\xi\right).$
(142)

Applying integral by parts to the first form, we have

$\int\mathrm{d}\tilde{\mu}_{B}\left(\xi\right)K_{B}\left(z,\bar{\xi}\right)\bar{A}^{\bigstar}\left(z,\bar{\xi}\right)\psi\left(\xi\right)\\\
=\int\mathrm{d}\tilde{\mu}_{B}\left(\xi\right)K_{B}\left(z,\bar{\xi}\right)\bar{A}\left(\bar{\xi},\xi\right)\psi\left(\xi\right).$
(143)

The two sides of the equation correspond to the two alternative quantization
forms of the vector potential. Therefore, $\bar{A}$ and $\bar{A}^{\bigstar}$
are related by the transformation Eq. (138).

From Eq. (140), it is easy to show

$\bar{A}^{\bigstar}(\bar{\xi},z)\approx\left.\bar{A}(\bar{z},z)\right|{}_{\bar{z}\rightarrow\bar{\xi}}$
(144)

for a nearly uniform magnetic field in the long-wavelength limit. The lowest-
order correction to the approximation is proportional to the first gradient of
the magnetic field. To see that, we decompose the vector potentials as
$\bar{A}(\bar{z},z)=\mathrm{i}B_{0}\bar{z}/2+\bar{A}_{1}(\bar{z},z)$ and
$\bar{A}^{\bigstar}(z,\bar{\xi})=\mathrm{i}B_{0}\bar{\xi}/2+\bar{A}_{1}^{\bigstar}(z,\bar{\xi)}$.
The first parts can be related exactly by the transformation Eq. (138). The
second parts, which are assumed to be small corrections to the uniform part of
the magnetic field, can be related approximately by Eq. (139) and (140).
Combining the two parts, we get the approximate relation.

We can also related $f_{B}$ with $F_{B}$. In the long-wavelength limit, Eq.
(37) can be approximated as

$\psi(z)\approx\int\mathrm{d}\mu^{(0)}(\bm{z})K_{B}^{(0)}(z,\bar{\xi})\\\
\times\left[1-f_{B}^{(1)}(\bm{\xi})+F_{B}^{(1)}(\bar{\xi},z)\right]\psi(\xi),$
(145)

where $f_{B}^{(1)}$ and $F_{B}^{(1)}$ denote the corrections to the
exponential factors of the weight and the reproducing kernel from the non-
uniform part of the magnetic field, respectively. We expand $f_{B}^{(1)}$ and
$F_{B}^{(1)}$ to the linear order of $\bar{\xi}$ around a point
$\bar{\eta}\approx\bar{z}$, and complete the integral. We have

$\psi(z)\approx\left[1-f_{B}^{(1)}(\bar{\eta},z)+F_{B}^{(1)}(\bar{\eta},z)\right]\psi(z)\\\
-\left[2l_{B}^{2}\partial_{z}\partial_{\bar{\eta}}f_{B}^{(1)}(\bar{\eta},z)\right]\psi(z).$
(146)

We can then obtain the relation

$F_{B}(z,\bar{\eta})\approx\left.\left[f_{B}(\bm{z})-\ln\frac{l_{B}^{2}(\bm{z})}{l_{B}^{2}}\right]\right|_{\bar{z}\rightarrow\bar{\eta}}.$
(147)

The quantization of the vortex degree of freedom and its alternative form can
be defined similarly.

##  Appendix C Operators in the standard representation

###  C.1 Ideal systems

In ideal systems, wave functions of composite fermions in the dipole
representation and the standard representation are related by Eq. (24).
Transforming to the standard representation, we have

$\displaystyle\bar{z}\psi\left(z,\bar{\eta}\right)\rightarrow$ $\displaystyle
2l_{B}^{2}\partial_{z}\psi\left(z,\bar{\eta}\right)$ $\displaystyle=$
$\displaystyle\int\mathrm{d}\mu_{B}^{(0)}\left(\bm{\xi}\right)e^{z\bar{\xi}/2l_{B}^{2}+\xi\bar{\eta}/2l_{b}^{2}}\bar{\xi}\tilde{\psi}\left(\bm{\xi}\right),$
(148) $\displaystyle z\psi\left(z,\bar{\eta}\right)=$
$\displaystyle\int\mathrm{d}\mu_{B}^{(0)}\left(\bm{\xi}\right)\left[2l_{B}^{2}\partial_{\bar{\xi}}e^{z\bar{\xi}/2l_{B}^{2}+\xi\bar{\eta}/2l_{b}^{2}}\right]\tilde{\psi}\left(\bm{\xi}\right)$
$\displaystyle=$
$\displaystyle\int\mathrm{d}\mu_{B}^{(0)}\left(\bm{\xi}\right)e^{z\bar{\xi}/2l_{B}^{2}+\xi\bar{\eta}/2l_{b}^{2}}$
$\displaystyle\times\left(-2l_{B}^{2}\partial_{\bar{\xi}}+\xi\right)\tilde{\psi}\left(\bm{\xi}\right),$
(149)

and similar expressions for $\eta$ and $\bar{\eta}$. Therefore, in the
standard representation, we should apply the mappings

$\displaystyle z$
$\displaystyle\rightarrow-2l_{B}^{2}\partial_{\bar{\xi}}+\xi,$ (150)
$\displaystyle\bar{z}$ $\displaystyle\rightarrow\bar{\xi},$ (151)
$\displaystyle\eta$ $\displaystyle\rightarrow\xi,$ (152)
$\displaystyle\bar{\eta}$
$\displaystyle\rightarrow-2l_{b}^{2}\partial_{\xi}+\left(l_{b}^{2}/l_{B}^{2}\right)\bar{\xi}.$
(153)

We then apply the mappings to Eq. (17) and obtain a wave equation for
$\tilde{\psi}(\bm{\xi})$. After applying the transformation Eq. (26), we
obtain Eq. (25).

###  C.2 General systems

For general systems, wave functions in the dipole representation and the
standard representation are related by Eq. (73). With the the $\hat{\bar{z}}$
operator defined in Eq. (41), we have

$\hat{\bar{z}}\psi\left(z,\bar{\eta}\right)=\int\mathrm{d}\mu_{B}\left(\bm{\xi}\right)K_{B}\left(z,\bar{\xi}\right)K_{b}\left(\bar{\eta},\xi\right)\bar{\xi}\tilde{\psi}\left(\bm{\xi}\right).$
(154)

On the other hand, applying complex conjugation to Eq. (41) and exchanging
$\bm{z}$ and $\bm{\xi}$, we obtain

$\hat{\bar{z}}^{\ast}\left(\partial_{\xi},\xi\right)K_{B}\left(z,\bar{\xi}\right)=zK_{B}\left(z,\bar{\xi}\right).$
(155)

Here, we make use of the relation
$[K_{B}(z,\bar{\xi})]^{\ast}=K_{B}(\xi,\bar{z})$. We thus have

$\displaystyle z\psi\left(z,\bar{\eta}\right)=$
$\displaystyle\int\mathrm{d}\mu_{B}\left(\xi\right)\left[\hat{\bar{z}}^{\ast}\left(\partial_{\xi},\xi\right)K_{B}\left(z,\bar{\xi}\right)\right]$
$\displaystyle\times
K_{b}\left(\bar{\eta},\xi\right)\tilde{\psi}\left(\bm{\xi}\right)$ (156)
$\displaystyle=$
$\displaystyle\int\mathrm{d}\mu_{B}\left(\xi\right)K_{B}\left(z,\bar{\xi}\right)K_{b}\left(\bar{\eta},\xi\right)$
$\displaystyle\times\hat{\bar{z}}^{\dagger}\left(\partial_{\xi}-\frac{\mathrm{i}e}{\hbar}\bar{A}\left(\bm{\xi}\right),\xi\right)\tilde{\psi}\left(\bm{\xi}\right)$
(157)

Here we apply integral by parts and make use of Eq. (31). Note
$(\partial_{\xi})^{\dagger}=-\partial_{\bar{\xi}}$. The mappings of $\eta$ and
$\bar{\eta}$ can be obtained similarly.

All summarized, we have the following mapping rules in the standard
representation:

$\displaystyle\bar{z}$ $\displaystyle\rightarrow\bar{\xi},$ (158)
$\displaystyle\eta$ $\displaystyle\rightarrow\xi,$ (159) $\displaystyle z$
$\displaystyle\rightarrow\hat{\bar{z}}^{\dagger}\left(\partial_{\xi}-\frac{\mathrm{i}e}{\hbar}\bar{A}(\bm{\xi}),\xi\right),$
(160) $\displaystyle\bar{\eta}$
$\displaystyle\rightarrow\hat{\eta}^{\dagger}\left(\partial_{\bar{\xi}}+\frac{\mathrm{i}e}{\hbar}A(\bm{\xi}),\xi\right).$
(161)

###  C.3 Hamiltonian in the standard representation

Using Eq. (60) and (61), we can determine approximate forms of
$\hat{\bar{z}}^{\dagger}$ and $\hat{\eta}^{\dagger}$:

$\displaystyle\hat{\bar{z}}^{\dagger}\left(\partial_{\xi}-\frac{\mathrm{i}e}{\hbar}\bar{A},\xi\right)-\xi$
$\displaystyle\approx-2l_{B}^{2}(\bm{\xi})\partial_{\bar{\xi}},$ (162)
$\displaystyle\hat{\eta}^{\dagger}\left(\partial_{\xi}+\frac{\mathrm{i}e}{\hbar}A,\xi\right)-\bar{\xi}$
$\displaystyle\approx-2l_{b}^{2}(\bm{\xi})\left[\partial_{\xi}+\mathrm{i}\frac{e}{\hbar}\bar{\mathcal{A}}\right].$
(163)

Applying Eqs. (158–161), we map the kinetic energy to

$\frac{\hbar^{2}}{2m^{\ast}}\frac{1}{l_{b}^{2}(\bm{\eta})l_{B}^{2}(\bm{z})}\left(\hat{\bar{z}}-\bar{\eta}\right)\left(z-\hat{\eta}\right)\\\
\rightarrow-\frac{\hbar^{2}}{2m^{\ast}}\left(2\partial_{\xi}+\frac{2\mathrm{i}e}{\hbar}\bar{\mathcal{A}}\right)2\partial_{\bar{\xi}}.$
(164)

The potentials are mapped to

$N_{+}\left[\Phi_{\mathrm{eff}}\left(\hat{\bar{z}},z\right)\right]\rightarrow
N_{+}\left[\Phi_{\mathrm{eff}}\left(\bar{\xi},\xi-2l_{B}^{2}(\bm{\xi})\partial_{\bar{\xi}}\right)\right]\\\
\approx\Phi_{\mathrm{eff}}\left(\bm{\xi}\right)-2l_{B}^{2}(\bm{\xi})\partial_{\xi}\Phi_{\mathrm{eff}}\left(\bm{\xi}\right)\partial_{\bar{\xi}},$
(165) $N_{+}\left[\phi\left(\bar{\eta},\hat{\eta}\right)\right]\rightarrow
N_{+}\left[\phi\left(\bar{\xi}-2l_{b}^{2}(\bm{\xi})\left(\partial_{\xi}+\mathrm{i}\frac{e}{\hbar}\bar{\mathcal{A}}\right),\xi\right)\right]\\\
\approx\phi\left(\bm{\xi}\right)-2l_{b}^{2}(\bm{\xi})\partial_{\bar{\xi}}\phi\left(\bm{\xi}\right)\left(\partial_{\xi}+\frac{\mathrm{i}e}{\hbar}\bar{\mathcal{A}}\right).$
(166)

We can then have the wave equation
$\epsilon\tilde{\psi}(\bm{\xi})=\hat{\tilde{H}}\tilde{\psi}_{\psi}(\bm{\xi})$
for $\tilde{\psi}(\bm{\xi})$, with the Hamiltonian:

$\hat{\tilde{H}}_{\psi}\approx-\frac{\hbar^{2}}{2m^{\ast}}\left(2\partial_{\xi}+\frac{2\mathrm{i}e}{\hbar}\bar{\mathcal{A}}+\frac{2m^{\ast}l_{B}^{2}}{\hbar^{2}}\partial_{\xi}\Phi_{\mathrm{eff}}\right)\\\
\times\left(2\partial_{\bar{\xi}}+\frac{2m^{\ast}l_{b}^{2}}{\hbar^{2}}\partial_{\bar{\xi}}\phi\right)+\Phi_{\mathrm{eff}}+\phi,$
(167)

where we ignore all terms proportional to the gradients of the strengths of
the electric and magnetic fields.

Applying the transformation Eq. (75), we obtain the wave equation Eq. (74).

##  Appendix D Particle and current densities

###  D.1 Current density in a Landau levels

In a Landau level, the particle density of a state $\psi$ can be defined as

$\rho(\bm{z},t)=w(\bm{z},t)\left|\psi(z,t)\right|^{2},$ (168)

where $w(\bm{z})$ denotes the weight of the Bergman space. The wave-equation
in the space can be in general written as

$\mathrm{i}\hbar\frac{\partial\psi}{\partial
t}=\int\mathrm{d}\mu(\bm{\xi})K(z,\bar{\xi})H^{\bigstar}(\bar{\xi},z)\psi(\xi,t).$
(169)

We can determine the current density by establishing a continuity equation for
$\rho(\bm{z})$. We have

$\frac{\partial\rho(\bm{z},t)}{\partial
t}=\frac{1}{\mathrm{i}\hbar}\int\mathrm{d}\mu(\bm{\xi}_{1})\mathrm{d}\mu(\bm{\xi}_{2})\psi^{\ast}(\xi_{1},t)K(\xi_{1},\bar{\xi}_{2})\\\
\times
H^{\bigstar}(\bar{\xi}_{2},\xi_{1})\psi(\xi_{2},t)\left[\delta\left(\bm{z}-\bm{\xi}_{1}\right)-\delta\left(\bm{z}-\bm{\xi}_{2}\right)\right].$
(170)

We then substitute the expansion

$\delta(\bm{z}-\bm{\xi}_{1})-\delta(\bm{z}-\bm{\xi_{2}})=-\frac{1}{2}\sideset{}{{}^{\prime}}{\sum}_{n,m=0}^{\infty}\biggr{[}\frac{1}{m!n!}(\xi_{1}-\xi_{2})^{m}\\\
\times(\bar{\xi}_{1}-\bar{\xi}_{2})^{n}\partial_{z}^{m}\partial_{\bar{z}}^{n}\delta(\bm{z}-\bm{\xi}_{1})-\mathrm{(1\leftrightarrow
2)}\biggr{]},$ (171)

where the summation excludes $(m,n)=(0,0)$. We obtain the continuity equation

$\frac{\partial\rho(\bm{z},t)}{\partial
t}+\bm{\nabla}\cdot\bm{j}(\bm{z},t)=0,$ (172)

with the current density $j\equiv j_{x}+\mathrm{i}j_{y}$:

$\displaystyle j(\bm{z},t)=$ $\displaystyle\leavevmode\nobreak\
j_{0}(\bm{z},t)-2\mathrm{i}\partial_{\bar{z}}m(\bm{z},t),$ (173)
$\displaystyle j_{0}(\bm{z},t)=$ $\displaystyle\leavevmode\nobreak\
\frac{\mathrm{1}}{\hbar}w(\bm{z},t)\int\mathrm{d}\mu(\bm{\xi})(z-\xi)$
$\displaystyle\times\mathrm{Im}\left[\psi^{\ast}(z,t)K(z,\bar{\xi})H^{\bigstar}(\bar{\xi},z)\psi(\xi,t)\right],$
(174) $\displaystyle m(\bm{z},t)=$ $\displaystyle\leavevmode\nobreak\
\frac{1}{4\hbar}\mathrm{Re}\sum_{m,n=1}^{\infty}\frac{\partial_{z}^{m-1}\partial_{\bar{z}}^{n-1}}{m!n!}w(\bm{z},t)\int\mathrm{d}\mu(\xi)$
$\displaystyle\times\psi^{\ast}(z,t)K(z,\bar{\xi})H^{\bigstar}(\bar{\xi},z)\psi(\xi,t)$
$\displaystyle\times(z-\xi)^{m}(\bar{z}-\bar{\xi})^{n}.$ (175)

where $m(\bm{z},t)$ is the orbital magnetization density.

Applying the quantization rules shown in Sec. III.2 and Appendix B, we can
rewrite the equations in operator forms

$\displaystyle\bm{j}_{0}(\bm{z},t)=$ $\displaystyle\leavevmode\nobreak\
w(\bm{z},t)\mathrm{Re}\left\\{\psi^{\ast}(z,t)\left[\hat{\bm{v}}\psi\right](z,t)\right\\},$
(176) $\displaystyle m(\bm{z},t)=$ $\displaystyle\leavevmode\nobreak\
\mathrm{Re}\sum_{m,n=1}^{\infty}\frac{\partial_{z}^{m-1}\partial_{\bar{z}}^{n-1}}{m!n!}w(\bm{z},t)\psi^{\ast}(z,t)$
$\displaystyle\times\left[\hat{m}_{mn}\psi\right](z,t),$ (177)

with

$\displaystyle\hat{\bm{v}}$
$\displaystyle\equiv\frac{1}{\mathrm{i}\hbar}\left[\hat{\bm{z}},\hat{H}\right],$
(178) $\displaystyle\hat{m}_{mn}$
$\displaystyle\equiv\frac{1}{4\hbar}[\underset{n}{\underbrace{\hat{\bar{z}}\dots,[\hat{\bar{z}}}},[\underset{m}{\underbrace{\hat{z},\dots[\hat{z}}},\hat{H}]\dots]]\dots].$
(179)

In the long-wavelength limit, we can keep only the contribution of
$j_{0}(\bm{z},t)$, and ignore the magnetization current.

###  D.2 Current densities of a composite fermion system

The result derived in the last subsection can be applied to composite fermions
with straightforward generalization. The electron and vortex current densities
for a state $\psi(z,\bar{\eta})$ can be written as

$\displaystyle\bm{j}_{\mathrm{e}}(\bm{z})$
$\displaystyle=w_{B}(\bm{z})\mathrm{Re}\int\mathrm{d}\mu_{b}(\bm{\eta})\psi^{\ast}(z,\bar{\eta})\left[\hat{\bm{U}}\psi\right](z,\bar{\eta}),$
(180) $\displaystyle\bm{j}_{\mathrm{v}}(\bm{\eta})$
$\displaystyle=w_{b}(\bm{z})\mathrm{Re}\int\mathrm{d}\mu_{B}(\bm{z})\psi^{\ast}(z,\bar{\eta})\left[\hat{\bm{u}}\psi\right](z,\bar{\eta}),$
(181)

where $\hat{\bm{U}}\equiv[\hat{\bm{z}},\hat{H}_{\psi}]/\mathrm{i}\hbar$ and
$\hat{\bm{u}}\approx[\hat{\bm{\eta}},\hat{H_{\psi}}]/\mathrm{i}\hbar$ are the
electron and vortex velocity operators, respectively, $\hat{H}_{\psi}$ is the
effective Hamiltonian shown in Eq. (56), and we ignore the orbital
magnetization contribution.

In the long-wavelength limit, we can apply the approximate commutators

$\displaystyle\left[\hat{\bar{z}},\hat{z}\right]$ $\displaystyle\approx
2l_{B}^{2}(\bm{z}),$ (182)
$\displaystyle\left[\hat{\eta},\hat{\bar{\eta}}\right]$ $\displaystyle\approx
2l_{b}^{2}(\bm{z}),$ (183)

to determine the velocity operators

$\displaystyle\hat{\bm{U}}$
$\displaystyle\approx\frac{\hbar}{m^{\ast}}\frac{\bm{n}\times\left(\hat{\bm{z}}-\hat{\bm{\eta}}\right)}{l_{b}^{2}(\bm{z})}+\bm{V},$
(184) $\displaystyle\hat{\bm{u}}$
$\displaystyle\approx\frac{\hbar}{m^{\ast}}\frac{\bm{n}\times\left(\hat{\bm{z}}-\hat{\bm{\eta}}\right)}{l_{B}^{2}(\bm{z})}+\bm{v}.$
(185)

Substituting Eq. (185) into Eq. (181) and summing over occupied states, we
obtain the current density of vortices Eq. (71).

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