# Microscopic derivation of Dirac composite fermion theory:
Aspects of noncommutativity and pairing instabilities

Dragoljub Gočanin  Faculty of Physics, University of Belgrade, Studentski Trg
12-16, 11000 Belgrade, Serbia  Sonja Predin  Institute of Information Systems,
Alfons-Goppel-Platz 1, 95030 Hof, Germany  Marija Dimitrijević Ćirić  Faculty
of Physics, University of Belgrade, Studentski Trg 12-16, 11000 Belgrade,
Serbia  Voja Radovanović  Faculty of Physics, University of Belgrade,
Studentski Trg 12-16, 11000 Belgrade, Serbia  Milica Milovanović
Corresponding author<EMAIL_ADDRESS>Scientific Computing
Laboratory, Center for the Study of Complex Systems,Institute of Physics
Belgrade, University of Belgrade, Pregrevica 118, 11080 Belgrade, Serbia

###### Abstract

Building on previous work [N. Read, Phys. Rev. B 58, Z. Dong and T. Senthil,
16262 (1998); Phys. Rev. B 102, 205126 (2020)] on the system of bosons at
filling factor $\nu=1$, we derive the Dirac composite fermion theory for a
half-filled Landau level from first principles and applying the Hartree-Fock
approach in a preferred representation. On the basis of the microscopic
formulation, in the long-wavelength limit, we propose a noncommutative field-
theoretical description, which in a commutative limit reproduces the Son’s
theory, with additional terms that may be expected on physical grounds. The
microscopic representation of the problem is also used to discuss pairing
instabilities of composite fermions. We find that a presence of a particle-
hole symmetry breaking leads to a weak (BCS) coupling $p$-wave pairing in the
lowest Landau level, and strong coupling $p$-wave pairing in the second Landau
level that occurs in a band with nearly flat dispersion, a third power
function of momentum.

##  I Introduction

The fractional quantum Hall effect (FQHE) can be explained by focusing on a
strongly correlated problem of particles (electrons) in two dimensions when
there is a commensuration between the number of particles and the number of
flux quanta of the applied, orthogonal to the two-dimensional plane, magnetic
field. Many phenomenological questions can be answered by assuming that the
most important physics takes place in a fixed Landau level (LL) with the
precise commensuration of the number of particles and the number of orbitals
in a fixed LL (and other LLs are inert). That is why a mathematical, idealized
problem of an isolated LL is so useful and relevant for the understanding of
the FQHE.

Some of the most interesting experimental phenomena occur at filling factors
(ratio of the number of electrons and the number of flux quanta)
$\nu=\tfrac{1}{2}$ and $\nu=\tfrac{5}{2}$, even-denominator fractions. The
gapless system at $\nu=\tfrac{1}{2}$ is believed to be in a Fermi liquid state
of underlying quasiparticles - composite fermions (CFs), as proposed and
described in [1] early on, while it is believed that incompressible (gapped)
FQHE at $\nu=\tfrac{5}{2}$ can be associated with some kind of $p$-wave
pairing of CFs in the second LL (sLL), as proposed in [2]. To understand more
closely these systems, one may start by focusing on an isolated half-filled
LL, the lowest LL (LLL) at $\nu=\tfrac{1}{2}$ in the case of the CF liquid
(CFL), and sLL at $\nu=\tfrac{5}{2}$ in the case of the gapped system.

One of the most interesting theoretical developments associated with the
physics in an isolated, half-filled LL is the proposal in [3] for the
description of the CFL state that is based on an assumption that the
underlying quasiparticles: CFs can be effectively described as Dirac CFs,
using an effective Dirac theory in two dimensions. This can be of general
interest: a system of interacting fermions on a noncommutative (NC) space of
an isolated LL, in which they fill half of the allowed, countable states, can
be described by an effective Dirac theory.

The proposed Dirac CF theory is a phenomenological theory, based on the
assumption that an effective theory of an isolated, half-filled LL must be
manifestly invariant under the particle- (electron-) hole transformation.
Certainly, there is a need for a microscopic derivation of the Dirac CF
theory, which can serve as a base for further understanding of this strongly
correlated system. In this paper we develop a microscopic support for the
Dirac CF theory, and provide a framework for a more detailed investigations.

To describe the physics of an isolated half-filled LL, we generalize the
approach to bosons in an isolated LL at filling factor one, of Pasquier and
Haldane [4], and later developed by Read [5], and more recently by Dong and
Senthil [6]. This approach introduces additional, vortex degrees of freedom to
efficiently capture Laughlin-Jastrow correlations. In the case of bosons, the
vortex (holelike, unphysical) degrees of freedom are fermionic, and combine
with elementary bosons to make quasiparticles of the problem - CFs. Due to
their fermionic nature, the vortex degrees of freedom are uniformly
distributed in the LL and make a uniform background. (This feature also
guarantees a necessary independence of physics under transformations in the
unphysical sector.) We generalize this description to the case of the half-
filled LL of electrons, by assuming a uniform distribution of two kinds of
unphysical degrees of freedom: holelike and electronlike. There are holelike
vortices as many as particlelike (electronlike) vortices, and they behave as
hard-core bosons among themselves, making a uniform background. To ensure the
uniform background we need to introduce constraints in the description.
Because of two kinds of unphysical degrees of freedom, the holelike vortices
combine with electrons to make CFs, and the particlelike vortices combine with
holes to make composite holes (CHs). We can choose either electrons or holes
as physical degrees of freedom of the half-filled LL, but if we want to
capture particle-hole (PH) symmetry (i.e., the symmetry under exchange of
particles and holes), we should treat them on an equal footing. Thus, we need
to include additional constraints that will preclude the simultaneous presence
of a hole and an electron in an orbital of the fixed LL, and therefore
describe them as dependent (not two independent) degrees of freedom. The
requirement of the PH symmetry also justifies our assumption on the manner in
which unphysical (vortex-like) degrees of freedom enter the description. In
this way, on the basis of two sets of constraints, we are able to formulate
the problem of the half-filled LL, that is explicitly invariant under exchange
of particles and holes. Furthermore, within this framework, which explicitly
includes the additional - vortex degrees of freedom, we are able to consider
the “preferred (form of) Hamiltonian” in the language of CFs and CHs, natural
quasiparticles that, on the level of Hartree-Fock treatment, can effectively
capture the physics of the system. Thus, a two-component fermion description,
with the CF and CH fields, necessarily and naturally appears as a consequence
of the demand for the PH symmetry, and we show that the description is of the
Dirac type (when the constraints are taken into account). This provides a
microscopic derivation of the Dirac description and explains the Dirac nature
of the fermionic quasiparticle excitations of the low-energy physics.

On the basis of this microscopic formulation, in the long-wavelength limit, we
propose a noncommutative field-theoretical description, which in a commutative
limit reproduces the Son’s theory, with additional terms that may be expected
on physical grounds. We also discuss pairing instabilities within the
developed microscopic framework, and provide a physical understanding of the
$p$-wave pairing instability in the LLL, and in the sLL. The pairing in the
LLL is of the BCS, weak-coupling kind, and this may explain the scarcity of
the pairing phenomena in the LLL. On the other hand, the pairing in the sLL is
of the strong coupling (weak pairing) kind as proposed and discussed in [7],
though we find that the Dirac CF band dispersion, $\epsilon(k)$, is flatter:
it obeys a third-power law, i.e., $\epsilon(k)\sim k^{3}$.

The section that follows is a review of the bosonic problem at $\nu=1$, in
which we also introduce a point of view of the formalism developed in [5, 4],
that will be useful for the half-filled problem of electrons. Sections III and
IV consider a (simpler) system, closely related to the one of the half-filled
LL, a special-bilayer system with two kinds of particles, parallel to the
existence of electrons and holes in the half-filled LL. A transformation into
holes of just one kind of particles in the special-bilayer system enables a
formulation of the half-filled LL problem in Sec. V, with all necessary
constraints. Following the usual approach [8] to a formulation with
constraints [that enables a Hartree-Fock (HF) treatment] we discuss a
“preferred” form of the Hamiltonian in Sec. VI, and in Sec. VII a Dirac form
of the Hamiltonian in the HF approximation. In Section VIII we describe how in
the long-wavelength limit of the microscopic formulation we can reach a field-
theoretical description with gauge fields next to the Dirac composite
fermions. In Secs. IX and X we discuss the description of possible pairing
instabilities. The structure of the proposed NC field theory is described in
Appendix A, while Appendix B concerns some specific aspects of the relevant
covariant derivatives. Conclusions are summarized in Sec. XI.

##  II Review of the $\nu=1$ boson system and introductory remarks

A CF is a composite object, a bound state of an underlying elementary particle
with a whole number of vortices; a vortex represents an excitation of the FQHE
system due to an insertion of one flux quantum that induces a depletion of
charge. At filling factors $\nu=1/q$, where $q$ is an integer, a composite
fermion is a neutral object; a composite of an electron (fermion) and a hole
(more precisely a depletion of charge) associated with $q$ flux quanta, when
$q$ is even, and a composite of a boson and a hole associated with $q$ flux
quanta, when $q$ is odd. To simplify the terminology, we will always call the
excitation with $q$ flux quanta a vortex. These introductory remarks serve
just to remind the reader of the physical picture of the CF, and for an
elaborate introduction to the CF formulation the reader may consult [5]. We
conclude that it may be expected and natural that an operator describing
annihilation or creation of CF will carry two indices, one for the state of
the elementary particle and the other for the state of the hole in an
orthonormal basis. In the following we will introduce the two-index formalism
that was firstly proposed in [4] and further elaborated in [5] and [6].

We start from an enlarged space with (composite) fermion $c_{mn}^{\dagger}$
with two indices, each corresponding to an orbital in the LLL (fixed LL):
$m,n=1,\dots,N_{\phi}\equiv N$ such that

$\\{c_{nm},c_{m^{\prime}n^{\prime}}^{\dagger}\\}=\delta_{n,n^{\prime}}\delta_{m,m^{\prime}}.$
(1)

Each $c_{nm}$ fermion represents a composite object. We define physical
subspace of bosonic states in the LLL by

$|n_{1},\dots,n_{N}\rangle=\sum\limits_{m_{1},\dots,m_{N}}^{N_{\phi}}\varepsilon^{m_{1}\cdots
m_{N}}c_{m_{1}n_{1}}^{\dagger}\cdots c_{m_{N}n_{N}}^{\dagger}|0\rangle$ (2)

where $\varepsilon^{m_{1}\cdots m_{N}}$ is the Levi-Civita symbol. In this
way, bosonic physical states have a property, defined by

$\rho_{mm^{\prime}}^{R}=\sum_{n}c_{mn}^{\dagger}c_{nm^{\prime}},$ (3)

that

$\rho_{mm}^{R}|n_{1},\dots,n_{N}\rangle=1\cdot|n_{1},\dots,n_{N}\rangle,$ (4)

expressing a single occupancy of each unphysical orbital $m$. The physical
states are defined by this property so that unphysical orbitals make an
uniformly occupied background. They furnish a spin-singlet representation of
$SU(N)$ group,

$m\neq m^{\prime},\;\;\;\rho_{mm^{\prime}}^{R}|n_{1},\dots,n_{N}\rangle=0,$
(5)

which is in agreement with the requirement that the physics should not depend
on the choice of basis in the unphysical sector $R$.

But we may reformulate this requirement by demanding that (1)
$\rho_{mm}^{R}=1$ and that (2) “unphysical” particles (objects), which are
uniformly distributed in the LLL, are fermions. This will automatically lead
to (5). In other words, introducing a certain type of statistics in the
unphysical sector is a way of specifying physical states together with the
demand for $\rho_{mm}^{R}=1$. In the following we would like to further this
view and introduce an approach, which will make a basis for our description of
the $\nu=\tfrac{1}{2}$ problem, and contrast it with respect to the previous
construction of the bosonic $\nu=1$ state.

First, we will recapitulate the Pasquier-Haldane construction, and the way
vortex statistics enters the description. The Pasquier-Haldane enlarged space
consists of states,

$c_{m_{1}n_{1}}^{\dagger}\cdots c_{m_{N}n_{N}}^{\dagger}|0\rangle,$ (6)

which describe the presence of $N$ composite objects. To each object we
associate two single-particle LL states, the state $n_{i}$ of the elementary,
physical particle - boson, and state $m_{i}$ of vortex. By exchanging states,
$n_{i}\leftrightarrow n_{j}$, or $m_{i}\leftrightarrow m_{j}$ , we do not get
the same state up to a sign, i.e., we cannot speak about definite statistics.
But if we trace out one of the (two) degrees of freedom, like by tracing out
in an anti-symmetric way vortices in (2), we can speak about definite
statistics; we have to do an ordinary (symmetric) trace in all $n$ indices to
get a state with definite statistics in the unphysical sector - a single
Slater (Vandermonde) determinant of fermionic vortices in this $\nu=1$ case.

This motivates our approach: we consider an enlarged space (a subspace of the
Pasquier-Haldane space) in which all $m$’s (in the unphysical sector) are
different (i.e., $m_{i}\neq m_{j}$ for $i\neq j;i,j=1,\ldots,N).$ Physically,
we may understand this as a modeling of a uniformly distributed vortex
background. The $SU(N)$ invariance in the $R$ sector is present, and acts
trivially, because each generator, (3) with $m\neq m^{\prime}$ will map out of
this restricted space, and thus effectively (considering the necessary
projection) annihilate state in the restricted space. If we further require
that the states of the restricted space have definite (fermionic) statistics
of vortices we will have a unique realization of the $SU(N)$ symmetry, given
by (2). Thus, we do not constrain the unphysical degrees of freedom by a
single symmetry requirement (as in the approach to the bosonic $\nu=1$ in [6],
and which is possible, allowed in that system), but primarily by a physical
requirement of the uniform vortex (unphysical degrees of freedom) density.
Thus, we consider a description with an (additional but necessary) constraint
in the unphysical sector, which is present automatically in the bosonic
$\nu=1$ case, because of the fermionic statistics of the unphysical degrees of
freedom. The constraint of the uniform vortex density is a way to introduce
bosonic vortices into a theory, and still maintain the $SU(N)$ invariance in
the unphysical (vortex) sector, i.e., invariance of the physics under change
of basis (transformations) in the unphysical sector. We will consider examples
of this at the end of this section and in the following section when we
discuss a special bilayer problem as a preparation for the half-filled LL
problem. Furthermore, we will see that in the half-filled LL problem,
unphysical degrees of freedom are associated with both $L$ and $R$ sectors,
and thus we see that in that case also, the requirement of the uniform density
of the unphysical degrees of freedom is the most natural to constrain these
degrees of freedom, and maintain the invariance under change of basis [not by
a spin-singlet realization of $SU(N)$ symmetry in one ($R$) sector].
Therefore, we can specify physical states by demanding the uniform
distribution of unphysical degrees of freedom in the states that enter
description of the physical states. These states form a space that is
effectively invariant under $SU(N)$ transformations. The demand for a definite
statistics of unphysical degrees of freedom in these states leads to physical
states.

Thus, in principle, we can discuss a possible physical sector (system) in an
enlarged theory for which $\rho_{mm}^{R}=1$, but with the unphysical particles
being (“hard-core”) bosons, and the physical sector being a $\nu=1$ fermionic
system. The old requirement (5) would be fulfilled effectively (= under the
necessary projection) in a restricted space for which $\rho_{mm}^{R}=1$, i.e.,
space defined by hard-core vortex configurations (of $N$ of vortices). The
physical space is defined by a hard-core configuration of vortices that
correlate as bosons. In this case

$|n_{1},\dots,n_{N}\rangle_{f}=\sum\limits_{m_{1},\dots,m_{N}}^{N_{\phi}}s^{m_{1}\cdots
m_{N}}c_{m_{1}n_{1}}^{\dagger}\cdots c_{m_{N}n_{N}}^{\dagger}|0\rangle,$ (7)

where $s^{m_{1}\cdots m_{N}}=1$ only if no index is equal to any other index,
but otherwise zero. But as we know, this would not lead to a plausible
Hartree-Fock description, i.e., a good representation in which the Hartree-
Fock approach to the system of composite fermions $c_{nm}$ would make a good
starting point for more refined descriptions.

Therefore, in principle, we can consider both, bosonic and fermionic
realizations of the $SU(N)$ group (that works in the unphysical sector), by
considering the theory with CFs as building blocks of the effective
description, at a particular fraction, and only one will realize as a physical
theory. [We should choose composite bosons (CBs) in the system in which the CF
realization fails.] The single occupancy in the unphysical sector will lead to
definite statistics (fermionic or bosonic) in the unphysical sector, because
the algebra of the density operators (that the theory is built on), in a LL
basis, is

$[\rho^{R}_{mm^{\prime}},\rho^{R}_{ll^{\prime}}]=\rho^{R}_{ml^{\prime}}\delta_{m^{\prime},l}-\rho^{R}_{lm^{\prime}}\delta_{m,l^{\prime}},$
(8)

i.e., the algebra of $SU(N)$ generators that can be realized either by
fermions or bosons. The constant density and demand for fermionic statistics
will coincide with a simple $SU(N)$ invariance: the group action will map the
state in (2) into itself. The constant density requirement and demand for
bosonic statistics will coincide with a special $SU(N)$ invariance (a
maintenance of the symmetry under the projection to the constant density): the
group action will map the state in (7) into the same state up to a number (a
coefficient because of the projection) and only the symmetric group $S_{n}$,
for $n=N$, a subgroup of $SU(N)$, represented by signed permutation matrices,
will map the state into itself, up to a sign. Thus, by fixing the density, we
will have either fermionic or bosonic realization in the unphysical sector,
and in the following we will emphasize which realization (statistics) we
choose.

##  III Towards $\nu=\tfrac{1}{2}$ fermions - a special bilayer system

As a preparation for a set-up of the $\nu=\tfrac{1}{2}$ (i.e. the half-filled
LL) problem in the language of composite quasiparticles, we will discuss a
set-up for a special quantum Hall bilayer system. The special bilayer is
characterized by two layers of electrons, each at filling factor
$\nu=\tfrac{1}{2}$ of a fixed LL, but what is special is that an electron in
one layer cannot be in the same (LL) orbital with an electron from the other
layer. Thus, we may speak about two kinds of composite objects, composite
fermions $c$ and $d$, which are overall neutral objects consisting of
elementary particle - electron of a given layer, and unphysical (vortex)
object of opposite charge. Therefore, we consider two CFs, $c^{\dagger}_{mn}$
and $d^{\dagger}_{mn}$, for the two layers, and an enlarged space with states

$c_{m_{1}n_{1}}^{\dagger}\cdots
c_{m_{N/2}n_{N/2}}^{\dagger}d_{m^{\prime}_{1}n^{\prime}_{1}}^{\dagger}\cdots
d_{m^{\prime}_{N/2}n^{\prime}_{N/2}}^{\dagger}|0\rangle,$ (9)

i.e., always there are $N/2$ $c$-fermions and $N/2$ $d$-fermions, where $N$,
as before, is the number of available orbitals in the LL.

The density of the unphysical objects, vortices (of the opposite charge with
respect to electrons), may be expressed in the analogous way as in the
previous section:

$\rho_{mm^{\prime}}^{R(c)}=\sum_{n}c_{mn}^{\dagger}c_{nm^{\prime}},$ (10)

and

$\rho_{mm^{\prime}}^{R(d)}=\sum_{n}d_{mn}^{\dagger}d_{nm^{\prime}}.$ (11)

Following the discussion in the previous section, we require

$\rho_{mm}^{R(c)}+\rho_{mm}^{R(d)}=1,$ (12)

i.e., we uniformly distribute particles in the unphysical sector. Furthermore,
we choose them to be bosons and mutual bosons. The requirement (12) may be
associated with special (identical for both $c$-fermions and $d$-fermions)
transformations of the LL basis in the unphysical sector, which we denote by
$SU^{R}_{c}(N)$ (where $c$ stands for charge), a transformation that is
realized identically on both $c$-fermions and $d$-fermions by affecting their
unphysical index. These transformations would leave the physical states
unchanged if we work in the restricted subspace with (unphysical) hard-core
bosons

$m\neq
m^{\prime},\;\left(\rho_{mm^{\prime}}^{R(c)}+\rho_{mm^{\prime}}^{R(d)}\right)|n_{1},\dots,n_{N/2}n^{\prime}_{1},\dots,n^{\prime}_{N/2}\rangle=0,$
(13)

where the equality is the result of the projection to the restricted state,
and thus, by assuming (12) and bosonic correlations in the unphysical sector,
we have that

$\displaystyle|n_{1},\dots,n_{N/2},n^{\prime}_{1},\dots,n^{\prime}_{N/2}\rangle$
$\displaystyle=\sum\limits_{m_{1},\dots,m_{N/2},m^{\prime}_{1},\dots,m^{\prime}_{N/2}}^{N}s^{m_{1}\cdots
m_{N/2}m^{\prime}_{1},\dots,m^{\prime}_{N/2}}\times$ $\displaystyle
c_{m_{1}n_{1}}^{\dagger}\cdots
c_{m_{N/2}n_{N/2}}^{\dagger}d_{m^{\prime}_{1}n^{\prime}_{1}}^{\dagger}\cdots
d_{m^{\prime}_{N/2}n^{\prime}_{N/2}}^{\dagger}|0\rangle,$ (14)

where $s^{m_{1}\cdots m_{N/2}m^{\prime}_{1},\dots,m^{\prime}_{N/2}}$ is non-
zero, equal to one only if no index is equal to any other index.

In (14) we have not only required that the unphysical - bosonic degrees of
freedom are uniformly distributed (12), but that they correlate mutually in a
symmetric way. Thus, we chose a sector of definite statistics in the enlarged
space.

If $N=2$, we have the following candidates for physical states:

$|n,n^{\prime}\rangle=(c_{1n}^{\dagger}d_{2n^{\prime}}^{\dagger}+c_{2n}^{\dagger}d_{1n^{\prime}}^{\dagger})|0\rangle,$
(15)

where $n,n^{\prime}=1,2$. Additionally, as a part of the definition, we
require that in the special bilayer system, i.e., two half-filled LL system,
LL orbitals in the physical states cannot be doubly occupied. Thus, what is
needed is to suppress the unwanted states (those with $(n,n^{\prime})=(1,1)$
or $(2,2)$ in the $N=2$ example) of double occupancy (with an eye on the half-
filled problem). Therefore, we need also

$\rho_{nn}^{L(c)}+\rho_{nn}^{L(d)}=1,$ (16)

This leads to $|\text{phy}\rangle$, physical states for which

$n\neq
n^{\prime},\;\;\left(\rho_{nn^{\prime}}^{L(c)}\pm\rho_{nn^{\prime}}^{L(d)}\right)|\text{phy}\rangle=0.$
(17)

We may associate the plus combination with $SU^{L}_{c}(N)$, and the minus
combination with $SU^{L}_{s}(N)$ \- “spin” transformations - which are inverse
in the $d$-sector with respect to the ones in the $c$-sector. Together, (16)
and (17) with the plus sign lead to conclusion that $|\text{phy}\rangle$
states are spin singlet(s) under $SU^{L}_{c}(N)$. On the other hand, in the
physical states, the generators of $SU^{L}_{s}(N)$ transformations,

$\rho_{nn}^{L(c)}-\rho_{nn}^{L(d)},$ (18)

may have expectation values from the interval $[-1,1]$. At the beginning,
before the requirement (16), operator sets, $\\{\rho_{nn^{\prime}}^{L(c)}\\}$
and $\\{\rho_{nn^{\prime}}^{L(d)}\\}$, with the constraints
$\sum\rho_{nn}^{L(c)}=\sum\rho_{nn}^{L(d)}=N/2$, furnished two adjoint
representations of $SU(N)$ group. However, with the hard-core constraint (16),
we have only one non-trivial representation of $SU(N)$ group, $SU^{L}_{s}(N)$.
The physical states are invariant under global $U_{s}(1)$ transformation
because

$\sum_{n}\rho_{nn}^{L(c)}=\sum_{n}\rho_{nn}^{L(d)}=N/2,$ (19)

and so are the unphysical ($R$-sector) states,

$\sum_{n}\rho_{nn}^{R(c)}=\sum_{n}\rho_{nn}^{R(d)}=N/2,$ (20)

by definition.

This completes a constraint [(12), (16), (19), (20)] plus statistics set-up
for the description in an enlarged space of the problem that concerns a
special bilayer at $\nu_{\rm{tot}}=1$ with two kinds of electrons, i.e.,
composite $c$-fermions and $d$-fermions, which cannot occupy the same orbital
in the restricted space of a fixed LL.

##  IV The special bilayer system and its transformation into the half-filled
system

We may consider the previous formulation of the special bilayer as a starting
point for the formulation of the half-filled problem. To reach the half-filled
problem, one kind of electrons (in one of the layers) should transform, i.e.,
become (elementary, physical) holes. If we consider a layer with composite
$d$-fermion, under the special electrons into holes transformation, what was
the density of “physical” electrons, $\rho^{L(d)}$, should become the
description of the density of unphysical vortices, quasielectrons, and be a
part of the constraints and description in the unphysical sector, i.e.,
$\rho^{R(d)}\rightarrow\rho^{L(d)}$. And similarly, what was the density of
unphysical vortices (with charge opposite to the one of electron)
$\rho^{R(d)}$, should become the description of the density of “physical”
holes, and be a part of constraints and description in the physical sector,
i.e., $\rho^{L(d)}\rightarrow\rho^{R(d)}$.

Thus, by making a PH transformation, i.e., charge conjugation in one of the
layers, described by composite object $d$, it is appropriate to call this
object composite hole, because now the physical degree of freedom is a hole.

We may make this discussion more concrete by considering the special bilayer
description in the inverse space and fixing the notation that will be in place
also for the half-filled case. Also, we will discuss possible particle-hole
transformations in the special bilayer case and derive again the necessary
transformations that transform the special bilayer problem into the one of the
half-filled LL.

We consider a representation of $c$ and $d$ composite fermions of the special
bilayer in the inverse (momentum) space. Following the previous studies on the
$\nu=1$ bosonic problem, we introduce the following decompositions:

$c_{nm}=\int\frac{d\boldsymbol{k}}{(2\pi)^{\frac{3}{2}}}\langle
n|\tau_{\boldsymbol{k}}|m\rangle c_{\boldsymbol{k}},$ (21)

and

$d_{nm}=\int\frac{d\boldsymbol{k}}{(2\pi)^{\frac{3}{2}}}\langle
n|\tau_{\boldsymbol{k}}|m\rangle d_{\boldsymbol{k}},$ (22)

with
$\tau_{\boldsymbol{k}}=\exp\left(i\boldsymbol{k}\cdot\boldsymbol{R}\right)$,
where $\boldsymbol{R}$ is a guiding-center coordinate of a single particle in
the external magnetic field,

$[R_{x},R_{y}]=-i,$ (23)

we took $l_{B}$ (magnetic length) $=1$, and $\\{|n\rangle\\}$ are single-
particle states (orbitals) in a fixed LL.

With these decompositions we find that

$\rho_{nn^{\prime}}^{L(c)}=\sum_{m}c_{mn}^{\dagger}c_{n^{\prime}m}=\int\frac{d\boldsymbol{q}}{2\pi}\langle
n^{\prime}|\tau_{\boldsymbol{q}}|n\rangle\rho_{\boldsymbol{q}}^{L(c)},$ (24)

where

$\rho_{\boldsymbol{q}}^{L(c)}=\int\frac{d\boldsymbol{k}}{(2\pi)^{2}}c_{\boldsymbol{k}-\boldsymbol{q}}^{\dagger}c_{\boldsymbol{k}}\exp\left(i\frac{\boldsymbol{k}\times\boldsymbol{q}}{2}\right),$
(25)

and similarly for $\rho_{nn^{\prime}}^{L(d)}$,

$\rho_{nn^{\prime}}^{L(d)}=\sum_{m}d_{mn}^{\dagger}d_{n^{\prime}m}=\int\frac{d\boldsymbol{q}}{2\pi}\langle
n^{\prime}|\tau_{\boldsymbol{q}}|n\rangle\rho_{\boldsymbol{q}}^{L(d)},$ (26)

where

$\rho_{\boldsymbol{q}}^{L(d)}=\int\frac{d\boldsymbol{k}}{(2\pi)^{2}}d_{\boldsymbol{k}-\boldsymbol{q}}^{\dagger}d_{\boldsymbol{k}}\exp\left(i\frac{\boldsymbol{k}\times\boldsymbol{q}}{2}\right).$
(27)

Note the inverse order of indices, $n$ and $n^{\prime}$, on the left- and
right-hand sides of (24) and (26). Similarly,

$\rho_{mm^{\prime}}^{R(c)}=\sum_{n}c_{mn}^{\dagger}c_{nm^{\prime}}=\int\frac{d\boldsymbol{q}}{2\pi}\langle
m|\tau_{\boldsymbol{q}}|m^{\prime}\rangle\rho_{\boldsymbol{q}}^{R(c)},$ (28)

where

$\rho_{\boldsymbol{q}}^{R(c)}=\int\frac{d\boldsymbol{k}}{(2\pi)^{2}}c_{\boldsymbol{k}-\boldsymbol{q}}^{\dagger}c_{\boldsymbol{k}}\exp\left(-i\frac{\boldsymbol{k}\times\boldsymbol{q}}{2}\right),$
(29)

and analogously for $\rho_{nn^{\prime}}^{R(d)}$,

$\rho_{mm^{\prime}}^{R(d)}=\sum_{n}d_{mn}^{\dagger}d_{nm^{\prime}}=\int\frac{d\boldsymbol{q}}{2\pi}\langle
m|\tau_{\boldsymbol{q}}|m^{\prime}\rangle\rho_{\boldsymbol{q}}^{R(d)},$ (30)

where

$\rho_{\boldsymbol{q}}^{R(d)}=\int\frac{d\boldsymbol{k}}{(2\pi)^{2}}d_{\boldsymbol{k}-\boldsymbol{q}}^{\dagger}d_{\boldsymbol{k}}\exp\left(-i\frac{\boldsymbol{k}\times\boldsymbol{q}}{2}\right),$
(31)

Note the positions of indices $m$ and $m^{\prime}$ on the left- and right-hand
sides of (28) and (30).

We have

$[\rho_{\boldsymbol{q}}^{L},\rho_{\boldsymbol{q}^{\prime}}^{L}]=2i\sin\left(\frac{\boldsymbol{q}\times\boldsymbol{q}^{\prime}}{2}\right)\rho_{\boldsymbol{q}+\boldsymbol{q}^{\prime}}^{L},$
(32)

and

$[\rho_{\boldsymbol{q}}^{R},\rho_{\boldsymbol{q}^{\prime}}^{R}]=-2i\sin\left(\frac{\boldsymbol{q}\times\boldsymbol{q}^{\prime}}{2}\right)\rho_{\boldsymbol{q}+\boldsymbol{q}^{\prime}}^{R},$
(33)

i.e., Girvin-MacDonald-Plazmann (GMP) algebra for two kinds of particles -
particles with opposite electric charge.

We may introduce a particle-hole transformation in the $d$-sector by taking

$d_{\boldsymbol{k}}\rightarrow d_{-\boldsymbol{k}}^{\dagger}\hskip
5.69046pt(d_{\boldsymbol{k}}^{\dagger}\rightarrow d_{-\boldsymbol{k}}).$ (34)

This implies

$d_{mn}\rightarrow d_{mn}^{\dagger}\hskip
5.69046pt(d_{mn}^{\dagger}\rightarrow d_{mn}),$ (35)

and also for

$\rho_{nn^{\prime}}^{L(d)}=\sum_{m}d_{mn}^{\dagger}d_{n^{\prime}m},\;\;n\neq
n^{\prime}$ (36)

we have

$\rho_{nn^{\prime}}^{L(d)}\rightarrow\sum_{m}d_{mn}d_{n^{\prime}m}^{\dagger}=-\sum_{m}d_{n^{\prime}m}^{\dagger}d_{mn}=-\rho_{n^{\prime}n}^{R(d)}.$
(37)

In the inverse space this implies

$\rho^{L(d)}({\boldsymbol{q}})\rightarrow-\rho^{R(d)}(\boldsymbol{q}),$ (38)

which is consistent with our expectation of what a particle-hole
transformation should imply on the physical density in the $d$-sector; it
should induce a density of particles of opposite charge in the magnetic field
($L\rightarrow R$) (and a minus sign that is always accompanied with such a
transformation). We can reach the same result by considering
$\rho^{L(d)}(\boldsymbol{q})$ for ${\boldsymbol{q}}\neq 0$,

$\rho^{L(d)}({\boldsymbol{q}})=\int\frac{d{\boldsymbol{k}}}{(2\pi)^{2}}d_{{\boldsymbol{k}}-{\boldsymbol{q}}}^{\dagger}d_{{\boldsymbol{q}}}\exp\left(i\frac{{\boldsymbol{k}}\times{\boldsymbol{q}}}{2}\right),$
(39)

and applying the transformation $d_{\boldsymbol{k}}\rightarrow
d_{-\boldsymbol{k}}^{\dagger}$ $(d_{\boldsymbol{k}}^{\dagger}\rightarrow
d_{-\boldsymbol{k}})$.

Therefore, this transformation may be identified to be the one that
corresponds (in the enlarged space) to the particle-hole transformation on the
elementary (fundamental) degrees of freedom, $e_{d}$, the second kind of
electrons in the special bilayer: $e_{d}\rightarrow h_{d}^{\dagger}$,
$e_{d}^{\dagger}\rightarrow h_{d}$ (where $e_{d}$, $e_{d}^{\dagger}$, $h_{d}$,
$h_{d}^{\dagger}$ are annihilation and creation operators).

Above we introduced the effect of the particle-hole transformation (on
electrons $e_{d}$) in the $d$-sector on composite fermion operators, while
putting aside the question of the diagonal terms, $\rho_{nn}^{L(d)}$, and the
necessary existence of a constant term, equal to $N$, due to the anti-
commutation relation of $d_{mn}$’s. This would imply an additional delta-
function contribution ($\sim\delta^{2}(\boldsymbol{q})$) in the inverse space
for the particle-hole transformation of $\rho^{L(d)}(\boldsymbol{q})$.

To comply with the restrictions of physical spaces in the enlarged spaces of
composite fermion operators, $d_{nm}$ and $c_{nm}$, of the half-filled and
special bilayer problem, we expect

$\rho_{nn}^{L(d)}\rightarrow 1-\rho_{nn}^{R(d)},$ (40)

because the summation on $n$ on both sides, and the restrictions

$\sum_{n}\rho_{nn}^{L(d)}=\sum_{n}\rho_{nn}^{R(d)}=\frac{N}{2},$ (41)

would be consistent with the particle-hole (single-layer) symmetry of the
physical system and restrictions on the special bilayer system.

Thus, in order to project the transformation $d_{mn}\rightarrow
d_{mn}^{\dagger}$ on the physical spaces of half-filled and special bilayer
problems we demand

$\rho_{nn}^{L(d)}\rightarrow 1-\rho_{nn}^{R(d)},$ (42)

which requires an additional subtraction of a constant term $(N-1)$ after the
$d_{mn}\rightarrow d_{mn}^{\dagger}$ $(d_{mn}^{\dagger}\rightarrow d_{mn})$
transformation, in order to project out the unphysical degrees of freedom. In
the inverse space this affects the delta function contribution; thus, for
${\boldsymbol{q}}\neq 0$ we still have
$\rho^{L(d)}({\boldsymbol{q}})\rightarrow-\rho^{R(d)}(\boldsymbol{q})$.

In the following we would like to examine how this particle-hole
transformation affects the constraints imposed on the bilayer system in order
to see how they look like in the (enlarged) space of the special bilayer
system. The two “hard-core” constraints, $\rho_{nn}^{R(c)}+\rho_{nn}^{R(d)}=1$
in (12) and $\rho_{nn}^{L(c)}+\rho_{nn}^{L(d)}=1$ in (16) become

$\rho_{nn}^{R(c)}=\rho_{nn}^{L(d)},$ (43)

and

$\rho_{nn}^{L(c)}=\rho_{nn}^{R(d)},$ (44)

respectively, which is consistent with the view that now the $d$-sector is
described by hole degrees of freedom. On the other hand, the operator
$\rho_{nn}^{L(c)}-\rho_{nn}^{L(d)}$ transforms into
$\rho_{nn}^{L(c)}+\rho_{nn}^{R(d)}-1$, i.e.,
$\rho_{nn}^{L(c)}+\rho_{nn}^{R(d)}$ acquires expectation values in the
physical states ranging from $0$ to $2$. Thus, we introduced the hole view in
the $d$-sector and this increased the allowed occupancy of $c$-particles and
$d$-holes of a single site to $2$. _But, we want to introduce a description in
terms of holes not in the way of change of variables but in the way of a real
change in the $d$-sector: where there are particles there should be holes and
vice versa._

Figure 1: An illustration of the particle-hole transformation in the special
bilayer problem. The transformation is done in the layer with particles
(electrons) 2 and thus, also on the associated composite $d$-fermion. Figure
2: An illustration of the “active” particle-hole transformation, i.e., the
particle-hole conjugation on particles (electrons) 2 which become holes. In
this way the special bilayer problem is transformed into the half-filled LL
problem.

Thus, $\rho^{L(d)}\rightarrow\rho^{R(d)}$ and
$\rho^{R(d)}\rightarrow\rho^{L(d)}$ in accordance with the discussion at the
beginning of this section. In this way the operator
$\rho^{L(c)}_{nn}-\rho^{L(d)}_{nn}$ becomes
$\rho^{L(c)}_{nn}-\rho^{R(d)}_{nn}$ and describes fluctuating charge of the
half-filled LL. Also, the following action on operators $d_{mn}$ and
$d_{mn}^{\dagger}$ is implied:

$d_{mn}\rightarrow d_{nm}\;\;(d_{mn}^{\dagger}\rightarrow d_{nm}^{\dagger}),$
(45)

[compare the definitions of the density operators in (26) and (30), in $L$ and
$R$ sectors]. Fig. 1 and Fig. 2 illustrate the difference between the
particle-hole transformation as a change of variables, and one that is an
active transformation that transforms the special bilayer problem into the one
of the half-filled LL.

##  V The formulation of the half-filled problem

On the basis of the discussion in the previous section, we can conclude that
the charge fluctuations around mean density $\frac{1}{2}(\frac{1}{2\pi
l_{B}^{2}})$, where $l_{B}$ is the magnetic length, can be expressed by
$(\rho_{nn}^{L(c)}-\rho_{nn}^{R(d)})/2$. Thus, the Hamiltonian that can be
considered together with the hard-core constraints
$\rho_{nn}^{R(c)}+\rho_{nn}^{L(d)}=1$ and
$\rho_{nn}^{L(c)}+\rho_{nn}^{R(d)}=1$ is

$\displaystyle{\cal H}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\int
d{\boldsymbol{q}}\;V(|{\boldsymbol{q}}|)$
$\displaystyle\times\frac{\rho^{L(c)}({\boldsymbol{q}})-\rho^{R(d)}({\boldsymbol{q}})}{2}\frac{\rho^{L(c)}(-{\boldsymbol{q}})-\rho^{R(d)}(-{\boldsymbol{q}})}{2}.$

The charge operator
$(\rho^{L(c)}({\boldsymbol{q}})-\rho^{R(d)}({\boldsymbol{q}}))/2$ does not
satisfy the GMP algebra because of the doubling of the degrees of freedom
(extra 2 in the GMP algebra). But together with the constraint
$\rho^{L(c)}({\boldsymbol{q}})+\rho^{R(d)}({\boldsymbol{q}})=0$ it does,
because $(\rho^{L(c)}({\boldsymbol{q}})-\rho^{R(d)}({\boldsymbol{q}}))/2$
becomes $\rho^{L(c)}({\boldsymbol{q}})$ (or $-\rho^{R(d)}({\boldsymbol{q}})$
due to the PH symmetry) with the constraint, and represents the physical
charge that satisfies the GMP algebra, expressed as a change of the electron
density $\rho^{L(c)}({\boldsymbol{q}})$ or a negative change in the hole
density $-\rho^{R(d)}({\boldsymbol{q}})$.

In the form of the Hamiltonian in (V) we incorporated the PH symmetry, by
including both particles and holes with equal weights (on an equal footing) in
the description of the change of the physical (electric) charge from the mean
value. The Hamiltonian is manifestly invariant under the transformation:
$\rho^{L(c)}\rightarrow\rho^{R(d)}$ and $\rho^{R(d)}\rightarrow\rho^{L(c)}$,
which represents an effective PH transformation. It consists of a charge
conjugation, which transforms CFs, i.e., $c$’s into CHs, i.e., $d$’s, and vice
versa, and an anti-unitary, i.e., time-reversal transformation that transforms
phases into complex-conjugated ones $(R\rightarrow L$ and $R\rightarrow L$).
Also, the transformation incorporates the reversal of momenta,
${\boldsymbol{k}}\rightarrow-{\boldsymbol{k}}$, that can be associated with
the time reversal, and thus,
$\rho^{L(c)}({\boldsymbol{q}})\rightarrow\rho^{R(d)}(-{\boldsymbol{q}})$ and
$\rho^{R(d)}({\boldsymbol{q}})\rightarrow\rho^{L(c)}(-{\boldsymbol{q}})$. The
charge conjugation and time reversal constitute the usual definition of the PH
transformation in the presence of the magnetic field, under which the physics
should be invariant. We should note that the charge conjugation in our case is
not uniquely defined on $c$’s and $d$’s and we get the same transformations of
$\rho^{L(c)}$ and $\rho^{R(d)}$ by considering
$c_{{\boldsymbol{k}}}\rightarrow\alpha d_{-{\boldsymbol{k}}}$ and
$d_{{\boldsymbol{k}}}\rightarrow\beta c_{-{\boldsymbol{k}}}$, where $\alpha$
and $\beta$ are constants, and $|\alpha|=|\beta|=1$.

We will recapitulate the necessary constraints in the formulation of the half-
filled LL problem. We summarize that

$\rho_{nn}^{R(c)}+\rho_{nn}^{L(d)}=1,$ (47)

together with the definite statistics requirement in the unphysical sector,
and

$\rho_{nn}^{L(c)}+\rho_{nn}^{R(d)}=1,$ (48)

with global constraints,

$\sum_{n}\rho_{nn}^{L(c)}=\sum_{n}\rho_{nn}^{R(d)}=N/2,$ (49)

and

$\sum_{n}\rho_{nn}^{R(c)}=\sum_{n}\rho_{nn}^{L(d)}=N/2,$ (50)

form a set of constraints that define the half-filled LL problem.

In this case physical states are

$\displaystyle|n_{1},\dots,n_{N/2},n^{\prime}_{1},\dots,n^{\prime}_{N/2}\rangle$
$\displaystyle=\sum\limits_{m_{1},\dots,m_{N/2},m^{\prime}_{1},\dots,m^{\prime}_{N/2}}^{N}s^{m_{1}\cdots
m_{N/2}m^{\prime}_{1},\dots,m^{\prime}_{N/2}}$ $\displaystyle\times
c_{m_{1}n_{1}}^{\dagger}\cdots
c_{m_{N/2}n_{N/2}}^{\dagger}d_{n^{\prime}_{1}m^{\prime}_{1}}^{\dagger}\cdots
d_{n^{\prime}_{N/2}m^{\prime}_{N/2}}^{\dagger}|0\rangle,$ (51)

where $s^{m_{1}\cdots m_{N/2}m^{\prime}_{1},\dots,m^{\prime}_{N/2}}$ is non-
zero, equal to one only if no index is equal to any other index. Compare with
(14), and the discussion and (45) at the end of the previous section. Also, in
(51), $n_{i}\neq n^{\prime}_{j}$ for any $i,j=1,\dots,N/2$.

##  VI Preferred form of the Hamiltonian

The most natural binding in ${\cal H}$ is the Cooper pair binding $\langle
c_{\boldsymbol{k}}d_{-{\boldsymbol{k}}}\rangle\neq 0$ in the $s$-wave channel.
In a Hartree-Fock treatment, the mean-field description would have kinetic
terms with quadratic dispersions, for $c$ and $d$ degrees of freedom, but this
description of these objects does not conform to our expectation that they are
dipoles - distinct dipole objects that pair, and that their dispersion comes
from the polarization energy due to their dipole moments in a Hartree
contribution as emphasized in [6]. In this way, we see a reason why the so-
called PH Pfaffian, connected with $s$-wave pairing, is absent in a fixed LL
[12, 9, 10, 11].

As in the $\nu=1$ bosonic case we may wonder whether there exists a
“preferred” form of the Hamiltonian, i.e., the Hamiltonian with some of
constraints included in its formulation but with the same description (and
action) as the original one in the physical space. The “preferred” form should
capture the basic physics in the most efficient way, enabling the description
of the basic physics in a Hartree-Fock treatment.

It is not hard to see that a unique low-momentum possibility for a kinetic
(non-pairing) term can be reached by an addition of the following term,

$\displaystyle{\cal H}$ $\displaystyle\rightarrow{\cal H}+\frac{1}{2}\int
d{\boldsymbol{q}}\;V(|{\boldsymbol{q}}|)$
$\displaystyle\times\frac{\rho^{R(c)}({\boldsymbol{q}})+\rho^{L(d)}({\boldsymbol{q}})}{2}\frac{\rho^{R(c)}(-{\boldsymbol{q}})+\rho^{L(d)}(-{\boldsymbol{q}})}{2},$

which uses the following constraint,

$\rho^{R(c)}({\boldsymbol{q}})+\rho^{L(d)}({\boldsymbol{q}})=0,$ (53)

in the physical sector for the unphysical degrees of freedom that directly
follows from the requirement (47).

In this way we removed the cause for the $s$-wave Cooper pairing and modified
the relevant term from

$\sim\int
d{\boldsymbol{q}}\left[-\rho^{L(c)}({\boldsymbol{q}})\rho^{R(d)}(-{\boldsymbol{q}})\right]V(|{\boldsymbol{q}}|)$
(54)

to

$\displaystyle\sim\int
d{\boldsymbol{q}}\left[-\rho^{L(c)}({\boldsymbol{q}})\rho^{R(d)}(-{\boldsymbol{q}})+\rho^{R(c)}({\boldsymbol{q}})\rho^{L(d)}(-{\boldsymbol{q}})\right]V(|{\boldsymbol{q}}|)$
$\displaystyle\sim\int d{\boldsymbol{q}}\int d{\boldsymbol{k}}_{1}\int
d{\boldsymbol{k}}_{2}\;c_{{{\boldsymbol{k}}_{1}}-{\boldsymbol{q}}}^{\dagger}c_{{\boldsymbol{k}}_{1}}d_{{{\boldsymbol{k}}_{2}}+{\boldsymbol{q}}}^{\dagger}d_{{\boldsymbol{k}}_{2}}$
(55) $\displaystyle\hskip
128.0374pt\times\left[i({\boldsymbol{k}}_{1}+{\boldsymbol{k}}_{2})\times{\boldsymbol{q}}\right]V(|{\boldsymbol{q}}|).$

Clearly, this term in the Hartree-Fock treatment can lead only to a $\langle
c_{\boldsymbol{k}}^{\dagger}d_{{\boldsymbol{k}}}\rangle\neq 0$ instability and
a Dirac-type description of the low-momentum physics.

##  VII The Dirac theory from the mean field

Thus, we apply the Hartree-Fock approach to the relevant part of the
Hamiltonian (we neglect the quadratic contributions from the other terms),

$\displaystyle{\cal
H}_{D}=\int\frac{d{\boldsymbol{q}}}{4}V(|{\boldsymbol{q}}|)\Big{[}-\rho^{L(c)}({\boldsymbol{q}})\rho^{R(d)}(-{\boldsymbol{q}})+\rho^{R(c)}({\boldsymbol{q}})\rho^{L(d)}(-{\boldsymbol{q}})\Big{]}$
$\displaystyle\approx\int d{\boldsymbol{q}}\int d{\boldsymbol{k}}_{1}\int
d{\boldsymbol{k}}_{2}\;\frac{V(|{\boldsymbol{q}}|)}{4(2\pi)^{4}}[i({\boldsymbol{k}}_{1}+{\boldsymbol{k}}_{2})\times{\boldsymbol{q}}]$
$\displaystyle\times\Big{[}\langle
c_{{{\boldsymbol{k}}_{1}}-{\boldsymbol{q}}}^{\dagger}d_{{\boldsymbol{k}}_{2}}\rangle
d_{{{\boldsymbol{k}}_{2}}+{\boldsymbol{q}}}^{\dagger}c_{{\boldsymbol{k}}_{1}}$
$\displaystyle+c_{{{\boldsymbol{k}}_{1}}-{\boldsymbol{q}}}^{\dagger}d_{{\boldsymbol{k}}_{2}}\langle
d_{{{\boldsymbol{k}}_{2}}+{\boldsymbol{q}}}^{\dagger}c_{{\boldsymbol{k}}_{1}}\rangle-\langle
c_{{{\boldsymbol{k}}_{1}}-{\boldsymbol{q}}}^{\dagger}d_{{\boldsymbol{k}}_{2}}\rangle\langle
d_{{{\boldsymbol{k}}_{2}}+{\boldsymbol{q}}}^{\dagger}c_{{\boldsymbol{k}}_{1}}\rangle\Big{]}$
$\displaystyle=\int\frac{d{\boldsymbol{k}}}{(2\pi)^{2}}\left(\Delta_{\boldsymbol{k}}^{*}d_{\boldsymbol{k}}^{\dagger}c_{\boldsymbol{k}}+\Delta_{\boldsymbol{k}}c_{\boldsymbol{k}}^{\dagger}d_{\boldsymbol{k}}\right)+{\cal
C},$ (56)

where ${\cal C}$ is a constant and

$\Delta_{\boldsymbol{k}}=|{\boldsymbol{k}}|\int
d{\boldsymbol{q}}\;\frac{V(|{\boldsymbol{q}}|)}{2(2\pi)^{2}}(i{\hat{\boldsymbol{k}}}\times{\boldsymbol{q}})\langle
d_{{\boldsymbol{k}}+{\boldsymbol{q}}}^{\dagger}c_{{\boldsymbol{k}}+{\boldsymbol{q}}}\rangle.$
(57)

We diagonalize ${\cal H}_{D}$ by introducing $\alpha_{\boldsymbol{k}}$ and
$\beta_{\boldsymbol{k}}$ operators,

$\left[\begin{array}[]{c}c_{\boldsymbol{k}}\\\ d_{\bf k}\\\
\end{array}\right]=\frac{1}{\sqrt{2}}\left[\begin{array}[]{cc}1&-\exp\\{-i\delta_{\boldsymbol{k}}\\}\\\
\exp\\{i\delta_{\boldsymbol{k}}\\}&1\\\
\end{array}\right]\left[\begin{array}[]{c}\alpha_{\boldsymbol{k}}\\\
\beta_{\bf k}\\\ \end{array}\right],$ (58)

where $\delta_{\bf k}$ is defined by $\Delta_{\bf k}=|\Delta_{\bf
k}|\exp\\{-i\delta_{\bf k}\\}$.

The very important question is how we choose the occupation of the momentum
${\boldsymbol{k}}$ states in the ground state that is subjected to the
constraints,

$\rho_{nn^{\prime}}^{R(c)}+\rho_{n^{\prime}n}^{L(d)}=\delta_{nn^{\prime}},$
(59)

and

$\rho_{nn^{\prime}}^{L(c)}+\rho_{n^{\prime}n}^{R(d)}=\delta_{nn^{\prime}}.$
(60)

In a mean field treatment we expect that at least the global constraints,

$\int\frac{d{\boldsymbol{k}}}{(2\pi)^{2}}c^{\dagger}_{\boldsymbol{k}}c_{\boldsymbol{k}}=\int\frac{d{\boldsymbol{k}}}{(2\pi)^{2}}d^{\dagger}_{\boldsymbol{k}}d_{\boldsymbol{k}}=\bar{\rho}_{e}=\frac{1}{2}\frac{1}{2\pi
l_{B}^{2}},$ (61)

will be satisfied.

In the $\alpha_{\boldsymbol{k}}$, $\beta_{\boldsymbol{k}}$ language this
implies

$\int
d{\boldsymbol{k}}\left(e^{-i\delta_{\boldsymbol{k}}}\alpha_{\boldsymbol{k}}^{\dagger}\beta_{\boldsymbol{k}}+e^{i\delta_{\boldsymbol{k}}}\beta_{\boldsymbol{k}}^{\dagger}\alpha_{\boldsymbol{k}}\right)=0.$
(62)

This is a complex constraint and we may try to satisfy the requirement on
$\alpha$’s and $\beta$’s, by demanding that also the number of $\alpha$’s and
$\beta$’s is conserved. We might expect,

$\int\frac{d{\boldsymbol{k}}}{(2\pi)^{2}}\alpha^{\dagger}_{\boldsymbol{k}}\alpha_{\boldsymbol{k}}=\int\frac{d{\boldsymbol{k}}}{(2\pi)^{2}}\beta^{\dagger}_{\boldsymbol{k}}\beta_{\boldsymbol{k}}=\bar{\rho}_{e}.$
(63)

This seems a very crude “translation” of (62), but it incorporates the basic
idea of our approach: to treat the particles and holes in an equal way, with
their dynamics not independent but constrained, and in this way duplicated in
a theory. The constraint implies two sectors, $\alpha$ and $\beta$, in the
ground-state configuration: half-filled $\alpha$-sector and half-empty
$\beta$-sector.

Figure 3: A schematic illustration of the implementation of the global
constraint in (61) via (63), i.e., half-filled positive energy sector and
half-empty negative energy sector.

In this we implicitly assumed the finiteness of the available volume of
${\boldsymbol{k}}$: the number of available ${\boldsymbol{k}}$’s is $N$, the
number of orbitals in the fixed LL. We expect that the description is
duplicated by treating particles and holes in an equal way, and, in the first
(mean-field) approximation, the dynamics of $\alpha$ and $\beta$ are separate
and independent, and we may consider one or the other sector as a description
of the problem.

Thus, for $\Delta_{\boldsymbol{k}}$ we get, by self-consistency,

$\Delta_{\boldsymbol{k}}=|{\boldsymbol{k}}|e^{-i\phi_{\boldsymbol{k}}}\int_{|{\boldsymbol{q}}|\leq
q_{F}}d{\boldsymbol{q}}\frac{V(|{\boldsymbol{q}}|)}{4(2\pi)^{2}}(i\hat{\boldsymbol{k}}\times{\boldsymbol{q}})e^{-i\phi_{{\boldsymbol{k}}+{\boldsymbol{q}}}+i\phi_{\boldsymbol{k}}},$
(64)

where $q_{F}=1/l_{B}(=1)$. In this expression for $\Delta_{\boldsymbol{k}}$,
because of the Gaussian in $V(|{\boldsymbol{q}}|)$, and the long-wavelength,
$|{\boldsymbol{k}}|\sim 0$, approximation, the contribution of the $\alpha$
sector is taken into account and we neglected the contribution from the
$\beta$ sector. We choose $\delta_{\boldsymbol{k}}$ to describe a definite
momentum state, $\delta_{\boldsymbol{k}}=\phi_{\boldsymbol{k}}$, where
$\phi_{\boldsymbol{k}}$ is the phase of the complex variable,
$k=k_{x}+ik_{y}$. It follows that

$|\Delta_{\boldsymbol{k}}|=|{\boldsymbol{k}}|\frac{\pi}{4(2\pi)^{2}}\int_{0}^{q_{F}}dq\;q^{2}V(q).$
(65)

The strength of the amplitude is zero for higher angular momenta (other than
angular momentum one).

Thus, by applying the Hartree-Fock approach to the preferred form of the
Hamiltonian, (VI), we reached a low-energy description of the problem in terms
of

${\cal
H}_{D}=\int\frac{d{\boldsymbol{k}}}{(2\pi)^{2}}\left(\Delta_{\boldsymbol{k}}^{*}d_{\boldsymbol{k}}^{\dagger}c_{\boldsymbol{k}}+\Delta_{\boldsymbol{k}}c_{\boldsymbol{k}}^{\dagger}d_{\boldsymbol{k}}\right),$
(66)

where $\Delta_{\boldsymbol{k}}=(k_{x}-ik_{y})\Delta$ with
$\Delta=\frac{\pi}{4(2\pi)^{2}}\int_{0}^{q_{F}}dq\;q^{2}V(q)$, at the finite
density of the Dirac system.

The system that is described by the Dirac Hamiltonian, at finite density,
represents a Fermi liquid. The time reversal in this system transforms a state
at momentum ${\boldsymbol{k}}$ into a one with momentum $-{\boldsymbol{k}}$.
Concretely, the state with momentum ${\boldsymbol{k}}$:

$\frac{1}{\sqrt{2}}\left[\begin{array}[]{c}\exp\\{-i\phi_{\boldsymbol{k}}\\}\\\
1\\\ \end{array}\right],$ (67)

where $\phi_{\boldsymbol{k}}$ is the phase of $k_{+}=k_{x}+ik_{y}$, is
transformed, under the time reversal transformation, $U=-i\sigma_{y}K$, where
$\sigma_{y}$ is the Pauli matrix, and $K$ denotes the complex conjugation,
into

$\frac{1}{\sqrt{2}}\left[\begin{array}[]{c}-1\\\
\exp\\{i\phi_{\boldsymbol{k}}\\}\\\ \end{array}\right],$ (68)

i.e., the state of the same energy but opposite momentum, $-{\boldsymbol{k}}$.
In the language of the basic PH transformation on the system (that acts on
elementary particles, electrons and holes), more precisely its realization on
the particular description of the problem that we introduced, the time
reversal transformation, in the Dirac description, corresponds to the
following transformation on $c$’s and $d$’s, $c_{{\boldsymbol{k}}}\rightarrow-
d_{-{\boldsymbol{k}}}$ and $d_{{\boldsymbol{k}}}\rightarrow
c_{-{\boldsymbol{k}}}$ (or $c_{{\boldsymbol{k}}}\rightarrow
d_{-{\boldsymbol{k}}}$ and $d_{{\boldsymbol{k}}}\rightarrow-
c_{-{\boldsymbol{k}}}$, compare with the description in Section V).

##  VIII The inclusion of the gauge invariance in the effective Dirac theory

The original $SU(N)$ gauge invariance (i.e., invariance under a change of
basis in the fixed LL) is broken down [6] to $U(1)$ in the mean-field
(Hartree-Fock, averaged) description in (66). As we already detailed, in the
microscopic approach the $SU(N)$ gauge invariance is realized by the following
two constraints,

$\rho_{nn}^{R(c)}+\rho_{nn}^{L(d)}=1,$ (69)

i.e., equal charge distribution of unphysical degrees of freedom, and

$\rho_{nn}^{L(c)}+\rho_{nn}^{R(d)}=1,$ (70)

i.e., the exclusion of the double occupancy between particles and holes (extra
unphysical degrees of freedom).

To include fluctuations beyond the Hartree-Fock (mean-field) level in the
long-wavelength domain, Dong and Senthil reinstated the $SU(N)$ invariance in
the description of the boson problem in terms of CFs, by introducing composite
fermion fields on noncommutative (deformed) space,

$c({\boldsymbol{R}},\tau)=\int\frac{d^{2}\boldsymbol{k}}{(2\pi)^{\frac{3}{2}}}\exp\left(i\boldsymbol{k}\cdot\boldsymbol{R}\right)c_{\boldsymbol{k},\tau},$
(71)

where ${\boldsymbol{R}}$ is the noncommutative (guiding center) coordinate,
and $\tau$ is imaginary time. The connection with the microscopic description
is the following:

$c_{nm}=\langle
n|c({\boldsymbol{R}},\tau)|m\rangle=\int\frac{d^{2}\boldsymbol{k}}{(2\pi)^{\frac{3}{2}}}\langle
n|\tau_{\boldsymbol{k}}|m\rangle c_{\boldsymbol{k},\tau}.$ (72)

Physically, the state of the CF with vortex orbital $m$ and electron orbital
$n$, can be described by a superposition of the (commutative) momentum
${\boldsymbol{k}}$ states, the weights of which depend on the effective
distance between orbitals (the size of the dipole), $|{\boldsymbol{k}}_{\rm
eff}|$, because
$\tau_{\boldsymbol{k}}=\exp\left(i\boldsymbol{k}\cdot\boldsymbol{R}\right)$ is
the translation operator. For more details see [6]. In this way, the $L$ and
$R$ transformations on $c_{nm}$,

$c_{nm}\rightarrow U^{L}_{n^{\prime}n}c_{nm}U^{R}_{mm^{\prime}},$ (73)

can be represented by noncommutative Moyal-Weyl star-product of fields on
ordinary commutative space,

$c({\boldsymbol{x}},\tau)\rightarrow U^{L}({\boldsymbol{x}},\tau)\star
c({\boldsymbol{x}},\tau)\star U^{R}({\boldsymbol{x}},\tau),$ (74)

the star-product being defined by

$a({\boldsymbol{x}},\tau)\star
b({\boldsymbol{y}},\tau)=e^{-\frac{i}{2}l^{2}_{B}\epsilon^{ij}\frac{\partial}{\partial
x^{i}}\frac{\partial}{\partial
y^{j}}}a({\boldsymbol{x}},\tau)b({\boldsymbol{y}},\tau)|_{{\boldsymbol{y}}\rightarrow{\boldsymbol{x}}}.$
(75)

Two-dimensional Levi-Civita ($i,j=1,2$) is defined by
$\epsilon^{12}=-\epsilon^{21}=1$. See Appendix A for a more elaborate account.

The two constraints in our problem, (69) and (70), imply, in the effective
description, simultaneous $SU(N)$ transformations in $L$ and $R$ sectors to
which we may associate, gauge fields $a_{\mu}^{(1)}$ and $a_{\mu}^{(2)}$. We
may also consider a (background) field $A_{\mu}$ that is associated with
physical degrees of freedom (particles and holes),
$\rho_{nn}^{L(c)}-\rho_{nn}^{R(d)}$.

Thus, we expect the corresponding covariant derivatives of the following form,

$\displaystyle D_{\mu}c$
$\displaystyle=\partial_{\mu}c-i(a_{\mu}^{(1)}+A_{\mu})\star c-ic\star
a_{\mu}^{(2)},$ (76) $\displaystyle D_{\mu}d$
$\displaystyle=\partial_{\mu}d-ia_{\mu}^{(2)}\star
d-id\star(a_{\mu}^{(1)}-A_{\mu}),$ (77)

in the noncommutative description of the low-energy physics, following the
considerations in [6] for the $\nu=1$ system of bosons.

But there is a problem with this proposal for a noncommutative description:
the structure of these derivatives is not consistent with unitarity of gauge
transformations, see the Appendix B for details.

We have to step back to understand why this problem occurs. The formulation of
the half-filled LL with constraints (69) and (70) is different from the case
of bosons at $\nu=1$ filling, in which there is a clear distinction between
$L$ and $R$, physical and unphysical sector, The formulation of the half-
filled LL system is more intricate since it includes an exchange of $L$ and
$R$ sectors, which complicates the distinction between them.

Having this in mind, we may reconsider the question of constraints and gauge
invariance in an effective, long-wavelength theory that we are looking for, a
theory that will nevertheless include some noncommutative aspects of the
physical system. In order to get a gauge-invariant description, we will also
apply the long-wavelength limit on the constraints (not just in the derivation
of the Hamiltonian). In the ensuing long-wavelength description there is no
distinction between physical (particles or holes) and unphysical (quantum of
flux excitation) degrees of freedom, constraints (69) and (70) become one
[compare (25) and (29) in the small-$q$ limit, etc.]. We may also expect that
$A_{\mu}$ (background field) couples symmetrically to the unphysical and
physical sector. Thus, we may define covariant derivatives $D_{\mu}c$ and
$D_{\mu}d$ in a way that ensures gauge invariance of the theory,

$\displaystyle D_{\mu}c$ $\displaystyle=\partial_{\mu}c-iA_{\mu}\star
c-ic\star(a_{\mu}-A_{\mu}),$ (78) $\displaystyle D_{\mu}d$
$\displaystyle=\partial_{\mu}d-iA_{\mu}\star d-id\star(a_{\mu}-A_{\mu}).$ (79)

The $R$ gauge field is introduced in a decomposed way, $a_{\mu}-A_{\mu}$, that
will be natural in the commutative limit.

The change that we introduced by going from (76) and (77) to (78) and (79) is
certainly drastic; the change is a departure from the microscopic description
that we found based on the view of quasiparticles as neutral composites. In
the case of bosons at filling factor one, Dong and Senthil showed that, on the
basis of the Hartree-Fock description that is modified to include the $SU(N)$
invariance, in the manner of noncommutative field theory, an approximate
commutative field theory description can be reached (via Seiberg-Witten map)
in the form of Halperin-Lee-Read (HLR) [1] description with a Chern-Simons
(CS) term. This description cannot be viewed as a HLR theory (although of the
same form), but as one that is based on the neutral composites (CFs) and
limited to a LL, a description that is analogous to the description by
composite bosons of the Laughlin case in [13]. In the case of electrons, in
the half-filled LL, for which the Son’s theory is relevant, one expects that,
due to the requirement for the PH symmetry, a microscopic description will not
generate a Chern-Simons term in an effective description. We encountered
difficulties in the application of the program proposed by Dong and Senthil -
to maintain the gauge invariance we consider constraints in the long-
wavelength limit, which certainly implies changes in the microscopic physics,
i.e., ultra-violet domain. We will find that the ensuing field theory will
have a Chern-Simons term, but one may argue that now Pauli-Villars type of
regularization is associated with the field theory, i.e., we have to treat the
high-energy physics in a different way as opposed to the version of the Son’s
theory that was firstly proposed by Son, and assumes the dimensional
regularization. Therefore, although we have done a drastic change in the
microscopic domain, the theory may still make sense as a theory based on
another kind of quasiparticle [12, 14] and give a version of the Son’s theory
described in [15, 16, 17].

The covariant derivatives (78) and (79) can define an NC description, and it
should be checked whether in the commutative limit via the Seiberg-Witten map
we can recover the Son’s theory to the linear order in the small parameter
$\theta=-l_{B}^{2}$. Because of the simultaneous presence of small-$\theta$
and long-wavelength expansions, we will seek an effective description by
considering only lowest order terms. To find the first correction to the
commutative limit we start with the (Euclidean) NC action of the form:

$\displaystyle S_{NC}=\int d\tau d^{2}\boldsymbol{x}\;\Big{(}c^{\dagger}\star
D_{\tau}c+d^{\dagger}\star D_{\tau}d$ (80)
$\displaystyle+c^{\dagger}\star(iD_{x}+D_{y})d+d^{\dagger}\star(iD_{x}-D_{y})c+i(a_{0}-A_{0})\bar{\rho}_{e}\Big{)}.$

In $S_{NC}$ we have a constraint term, linear in NC field $a_{0}-A_{0}$, that
fixes the total number of $c$ and $d$ fermions. Recall that our description is
for $0\leq|{\boldsymbol{k}}|$ in the upper half of Fig. 3, and thus, we have

$\int\frac{d^{2}{\boldsymbol{k}}}{(2\pi)^{2}}\left(c^{\dagger}_{\boldsymbol{k}}c_{\boldsymbol{k}}+d^{\dagger}_{\boldsymbol{k}}d_{\boldsymbol{k}}\right)|_{\text{upper
half}}=\bar{\rho}_{e}.$ (81)

In Appendix A we detail the small-$\theta$ expansion in terms of commutative
fields $\hat{c},\hat{d},\hat{a}_{\mu},\hat{A}_{\mu}$ (denoted by a hat
symbol). We find $S_{NC}=S^{(0)}+S^{(1)}+\dots$, where the classical limit
($\theta=0$) is simply

$\displaystyle S^{(0)}$ $\displaystyle=\int d\tau
d^{2}\boldsymbol{x}\;\Big{(}\hat{c}^{\dagger}D_{\tau}\hat{c}+\hat{d}^{\dagger}D_{\tau}\hat{d}$
(82)
$\displaystyle+\hat{c}^{\dagger}(iD_{x}+D_{y})\hat{d}+\hat{d}^{\dagger}(iD_{x}-D_{y})\hat{c}+i(\hat{a}_{0}-\hat{A}_{0})\bar{\rho}_{e}\Big{)},$

and the linear NC correction reads as (see Appendix A for details):

$\displaystyle S^{(1)}=$
$\displaystyle\frac{i\bar{\rho}_{e}\theta}{2}\epsilon^{\alpha\beta\gamma}\int
d\tau
d^{2}\boldsymbol{x}\;(\hat{a}_{\alpha}-\hat{A}_{\alpha})\partial_{\beta}(\hat{a}_{\gamma}-\hat{A}_{\gamma})$
$\displaystyle+$ $\displaystyle\theta\int{\rm d}\tau{\rm
d}^{2}\boldsymbol{x}\left[\hat{c}^{\dagger}\left(\frac{1}{2}\hat{f}_{12}-\hat{F}_{12}\right)D_{\tau}\hat{c}-\hat{c}^{\dagger}\left(\frac{1}{2}\hat{f}_{10}-\hat{F}_{10}\right)D_{y}\hat{c}+\hat{c}^{\dagger}\left(\frac{1}{2}\hat{f}_{20}-\hat{F}_{20}\right)D_{x}\hat{c}\right]$
$\displaystyle+$ $\displaystyle\theta\int{\rm d}\tau{\rm
d}^{2}\boldsymbol{x}\left[\hat{d}^{\dagger}\left(\frac{1}{2}\hat{f}_{12}-\hat{F}_{12}\right)D_{\tau}\hat{d}-\hat{d}^{\dagger}\left(\frac{1}{2}\hat{f}_{10}-\hat{F}_{10}\right)D_{y}\hat{d}+\hat{d}^{\dagger}\left(\frac{1}{2}\hat{f}_{20}-\hat{F}_{20}\right)D_{x}\hat{d}\right],$
(83)

where we introduced classical (commutative) gauge field strengths,

$\displaystyle\hat{F}_{\mu\nu}$
$\displaystyle=\partial_{\mu}\hat{A}_{\nu}-\partial_{\nu}\hat{A}_{\mu},$ (84)
$\displaystyle\hat{f}_{\mu\nu}$
$\displaystyle=\partial_{\mu}\hat{a}_{\nu}-\partial_{\nu}\hat{a}_{\mu},$ (85)

and $\varepsilon^{\alpha\beta\gamma}$ is the Levi-Civita symbol.

The classical action $S^{(0)}$ and the Chern-Simons term in $S^{(1)}$ give the
description of a version of the Son’s theory [3] of the Dirac composite
fermion [15, 16, 17] that assumes the Pauli-Villars type of regularization
[12, 14] because of the presence of the CS term for field $a_{\mu}$. Note that
the CS term has the correct coefficient, $\tfrac{1}{8\pi}$. Also, in the
linear NC correction $S^{(1)}$ the Dirac momentum density couples to the
external electric field as expected from the Galilean invariance [4, 18]. On
the other hand, the presence of a coupling to the internal electric field
$\hat{a}_{\mu}$ is quite natural and expected given the influence of other
particles on a selected one. Also, we find that the presence of internal and
external (i.e., departure from the uniform) magnetic field induces a change in
the coefficient of the kinetic terms $\hat{c}^{\dagger}\partial_{\tau}\hat{c}$
and $\hat{d}^{\dagger}\partial_{\tau}\hat{d}$. Thus, we can conclude that the
NC formulation, up to the first order in $\theta$, recovers known results but
also systematically adds terms that we may expect on physical grounds.

Although we reproduced the version of the Son’s theory from a conjectured NC
field theory that is partially based on the microscopic approach, we may
wonder whether there is a formulation of an NC field theory that can
reproduce, via the Seiberg-Witten map, the original version of the Son’s
theory. In that case, there would be no terms that are not invariant under the
PH transformation, like the CS term for gauge field $a_{\mu}$, and we would
not have to assume that their effect will be erased or canceled by a
particular way of regularization.

With the experience of the previous derivation that resulted in Eqs. (78) and
(79) for covariant derivatives [and action 80], we may ask ourselves how Eqs.
(76) and (77) can be modified in a way that they still express the microscopic
constraints, but represent valid covariant derivatives. A way to do that is by
including a symmetry between physical and unphysical sectors, by assuming that
the background field $A_{\mu}$ couples to both sectors.

Thus, we consider

$\displaystyle D_{\mu}c$
$\displaystyle=\partial_{\mu}c-i(a_{\mu}^{(1)}+A_{\mu})\star
c-ic\star(a_{\mu}^{(2)}-A_{\mu}),$ (86) $\displaystyle D_{\mu}d$
$\displaystyle=\partial_{\mu}d-i(a_{\mu}^{(2)}+A_{\mu})\star
d-id\star(a_{\mu}^{(1)}-A_{\mu}).$ (87)

Note that, as before, in the course of the implementing microscopic
constraints, (69) and (70), where each one constrains densities both in $R$
and $L$ sector, the implied gauge fields connect i.e., transform, at the same
time, in the $R$ and $L$ sectors of fields $c$ and $d$. We warn the reader
that, as it stands, in the proposed covariant derivatives, only fields
$b^{(1)}_{\mu,\pm}\equiv a_{\mu}^{(1)}\pm A_{\mu}$ and
$b^{(2)}_{\mu,\pm}\equiv a_{\mu}^{(2)}\pm A_{\mu}$ are assumed to transform
canonically (see Appendix B for details), and fields $a_{\mu}^{(1)}$,
$a_{\mu}^{(2)}$, and $A_{\mu}$, that enter their decompositions (as they refer
both to $R$ and $L$ sectors) do not. The complete set-up of the new NC theory
is given in terms of $b^{(1)}_{\mu,\pm}$ and $b^{(2)}_{\mu,\pm}$ only. In the
Appendix B we explain why necessarily $b^{(1)}_{\mu,\pm}=b^{(2)}_{\mu,\pm}$ =
$b_{\mu,\pm}$, i.e., $a_{\mu}^{(1)}=a_{\mu}^{(2)}$, and we have an emerging
symmetry between the unphysical and physical sectors. The fields
$a_{\mu}=b_{\mu,+}+b_{\mu,-}$ and $A_{\mu}=(b_{\mu,+}-b_{\mu,-})/2$ will
assume their expected $U(1)$ gauge field roles in the commutative limit of the
new theory in which we get the original version of the Son’s theory that
incorporates the PH symmetry (CP or CT in [3]). The resulting NC action up to
first order is given by

$\displaystyle S_{NC}$ $\displaystyle=\int d\tau
d^{2}\boldsymbol{x}\Big{(}\hat{c}^{\dagger}D_{\tau}\hat{c}+\hat{d}^{\dagger}D_{\tau}\hat{d}+\hat{c}^{\dagger}(iD_{x}+D_{y})\hat{d}+\hat{d}^{\dagger}(iD_{x}-D_{y})\hat{c}+i\hat{a}_{0}\bar{\rho}_{e}\Big{)}+\frac{i}{4\pi}\varepsilon^{\alpha\beta\gamma}\int
d\tau d^{2}\boldsymbol{x}\;A_{\alpha}\partial_{\beta}a_{\gamma}$
$\displaystyle+\theta\int d\tau
d^{2}\boldsymbol{x}\left(\hat{c}^{\dagger}\hat{F}_{12}D_{\tau}\hat{c}-\hat{c}^{\dagger}\hat{F}_{10}D_{y}\hat{c}+\hat{c}^{\dagger}\hat{F}_{20}D_{x}\hat{c}+\hat{d}^{\dagger}\hat{F}_{12}D_{\tau}\hat{d}-\hat{d}^{\dagger}\hat{F}_{10}D_{y}\hat{d}+\hat{d}^{\dagger}\hat{F}_{20}D_{x}\hat{d}\right).$
(88)

Thus, we obtained an effective (long-wavelength) NC field theory that
consistently reproduces the Son’s theory in the commutative limit. The
introduced formalism and NC setup can be used to systematically generate
corrections in small parameter $\theta$. The achieved NC descriptions need
further understanding and analysis especially concerning the questions of
regularization and scaling of gauge fields [3, 17, 19]. At first glance, it
seems that our approach does not have the scaling problem.

##  IX Inclusion of pairing

We have eliminated unphysical degrees of freedom in a way of constraint (47),
although in the case of the half-filled Landau level we also had to impose
additional bosonic correlations of the unphysical degrees of freedom to fix a
unique subspace of physical states - we called it spin-singlet sector of the
$SU(N)$ gauge symmetry of unphysical degrees of freedom. [Subsequently, we
also had to impose (48) - to eliminate hole degrees of freedom.]

We may search for another such state for unphysical degrees of freedom by
imposing other constraint(s), such as

$\rho_{nn^{\prime}}^{R(c)}=\rho_{n^{\prime}n}^{L(d)}.$ (89)

In the inverse space this corresponds to

$\rho^{R(c)}({\boldsymbol{q}})-\rho^{L(d)}({\boldsymbol{q}})=0.$ (90)

This choice seems natural as a requirement that will equalize and uniformly
distribute the electric charge of the unphysical degrees of freedom (similarly
to the special bilayer system).

The Hamiltonian that we can consider now is

$\displaystyle{\cal H}_{p}$ $\displaystyle={\cal H}-\frac{1}{2}\int
d^{2}{\boldsymbol{q}}\;V(|{\boldsymbol{q}}|)$
$\displaystyle\times\frac{\rho^{R(c)}({\boldsymbol{q}})-\rho^{L(d)}({\boldsymbol{q}})}{2}\frac{\rho^{R(c)}(-{\boldsymbol{q}})-\rho^{L(d)}(-{\boldsymbol{q}})}{2},$

which contains the same relevant two-body part for the Dirac physics as in the
previous inclusion of constraints (VI), but now we have the diagonal terms in
the $c$ and $d$-sector, like

$\displaystyle\sim\int
d^{2}{\boldsymbol{q}}\left[\rho^{L(c)}({\boldsymbol{q}})\rho^{L(c)}(-{\boldsymbol{q}})-\rho^{R(c)}({\boldsymbol{q}})\rho^{R(c)}(-{\boldsymbol{q}})\right]V(|{\boldsymbol{q}}|)$
$\displaystyle\sim\int d^{2}{\boldsymbol{q}}\int d^{2}{\boldsymbol{k}}_{1}\int
d^{2}{\boldsymbol{k}}_{2}\;c_{{{\boldsymbol{k}}_{1}}-{\boldsymbol{q}}}^{\dagger}c_{{\boldsymbol{k}}_{1}}c_{{{\boldsymbol{k}}_{2}}+{\boldsymbol{q}}}^{\dagger}c_{{\boldsymbol{k}}_{2}}$
(92) $\displaystyle\hskip 128.0374pt\times
i\left[({\boldsymbol{k}}_{1}-{\boldsymbol{k}}_{2})\times{\boldsymbol{q}}\right]V(|{\boldsymbol{q}}|),$

in the $c$-sector, that can lead to $p$-wave (Pfaffian in the $c$-sector and
anti-Pfaffian in the $d$-sector) instabilities.

We will assume a PH symmetry breaking and an effective Hartree-Fock-BCS
Hamiltonian of the following form,

$\displaystyle{\cal H}_{BCS}$ $\displaystyle=$
$\displaystyle\int\frac{d^{2}{\boldsymbol{k}}}{(2\pi)^{2}}\left(\Delta_{\boldsymbol{k}}^{*}d_{\boldsymbol{k}}^{\dagger}c_{\boldsymbol{k}}+\Delta_{\boldsymbol{k}}c_{\boldsymbol{k}}^{\dagger}d_{\boldsymbol{k}}\right)$
$\displaystyle+$
$\displaystyle\int\frac{d^{2}{\boldsymbol{k}}}{(2\pi)^{2}}\left({\tilde{\Delta}}_{\boldsymbol{k}}^{*}c_{-\boldsymbol{k}}c_{\boldsymbol{k}}+{\tilde{\Delta}}_{\boldsymbol{k}}c_{\boldsymbol{k}}^{\dagger}c_{-\boldsymbol{k}}^{\dagger}\right).$
(93)

Here, $\Delta_{\boldsymbol{k}}$ is defined in (57) and

${\tilde{\Delta}}_{\boldsymbol{k}}^{*}=|{\boldsymbol{k}}|\int
d^{2}{\boldsymbol{q}}\;\frac{V(|{\boldsymbol{q}}|)}{4(2\pi)^{2}}(i\hat{\boldsymbol{k}}\times{\boldsymbol{q}})\langle
c_{{\boldsymbol{k}}-{\boldsymbol{q}}}^{\dagger}c_{-{\boldsymbol{k}}+{\boldsymbol{q}}}^{\dagger}\rangle.$
(94)

We project to the $\alpha$ sector by taking

$c_{\boldsymbol{k}}\rightarrow\frac{1}{\sqrt{2}}\;\alpha_{\boldsymbol{k}},$
(95)

and

$d_{\boldsymbol{k}}\rightarrow\frac{1}{\sqrt{2}}e^{i\phi_{\boldsymbol{k}}}\alpha_{\boldsymbol{k}}.$
(96)

Thus,

${\cal
H}_{BCS}^{\alpha}=\int\frac{d^{2}{\boldsymbol{k}}}{(2\pi)^{2}}\left[|\Delta_{\boldsymbol{k}}|\alpha_{\boldsymbol{k}}^{\dagger}\alpha_{\boldsymbol{k}}+\left(\frac{{\tilde{\Delta}}_{\boldsymbol{k}}^{*}}{2}\alpha_{-\boldsymbol{k}}\alpha_{\boldsymbol{k}}+h.c.\right)\right].$
(97)

Equations [7] that follow and need to be solved self-consistently are

$|\Delta_{\boldsymbol{k}}|=|{\boldsymbol{k}}|\int
d^{2}{\boldsymbol{q}}\;\frac{V(|{\boldsymbol{q}}|)}{8(2\pi)^{2}}|{\boldsymbol{q}}|\sin^{2}(\phi_{\boldsymbol{q}})\left(1-\frac{|\Delta_{{\boldsymbol{k}}+{\boldsymbol{q}}}|-|\Delta_{q_{F}}|}{E_{{\boldsymbol{k}}+{\boldsymbol{q}}}}\right),$
(98)

and

$|{\tilde{\Delta}}_{\boldsymbol{k}}|=|{\boldsymbol{k}}|\int
d^{2}{\boldsymbol{q}}\;\frac{V(|{\boldsymbol{q}}|)}{16(2\pi)^{2}}|{\boldsymbol{q}}|\sin^{2}(\phi_{\boldsymbol{q}})\frac{|{\tilde{\Delta}}_{{\boldsymbol{k}}+{\boldsymbol{q}}}|}{E_{{\boldsymbol{k}}+{\boldsymbol{q}}}},$
(99)

where
$E_{\boldsymbol{q}}^{2}=(|\Delta_{{\boldsymbol{q}}}|-|\Delta_{q_{F}}|)^{2}+|{\tilde{\Delta}}_{\boldsymbol{q}}|^{2}$.
In this way, by specifying $V(|{\boldsymbol{q}}|)$ that will include factors
due to the projection to a fixed Landau level, we can find amplitudes in
${\tilde{\Delta}}_{\boldsymbol{k}}=|{\tilde{\Delta}}_{\boldsymbol{k}}|\exp\\{-i\phi_{\boldsymbol{k}}\\}$
and
${\Delta}_{\boldsymbol{k}}=|{\Delta}_{\boldsymbol{k}}|\exp\\{-i\phi_{\boldsymbol{k}}\\}$.

##  X Pairing solutions

The effective interaction in a fixed LL, $m=0,1,2,\ldots$, is given by the
following expression:

$V(|{\boldsymbol{q}}|)=V_{c}(|{\boldsymbol{q}}|)e^{-\frac{|{\boldsymbol{q}}|^{2}}{2}}\left[L_{m}\left(\frac{|{\boldsymbol{q}}|^{2}}{2}\right)\right]^{2},$
(100)

where

$V_{c}(|{\boldsymbol{q}}|)=\frac{V_{0}}{|{\boldsymbol{q}}|},$ (101)

represents the Coulomb interaction, $l_{B}=1$, and $L_{m}$ denotes the
Laguerre polynomial associated with a fixed LL with the quantum number $m$.

In the LLL, the Eqs. (98) and (99) lead to self-consistent solutions with the
following amplitudes, in the kinetic part,
$|{\Delta}_{\boldsymbol{k}}|=\Delta|\boldsymbol{k}|$ where $\Delta\approx
0.017086\;V_{0}$, and in the pairing part,
$|{\tilde{\Delta}}_{\boldsymbol{k}}|={\tilde{\Delta}}|\boldsymbol{k}|$ where
${\tilde{\Delta}}\approx 0.001414\;V_{0}$. The self-consistent solution is
numerically obtained using Mathematica with error estimation to $5\times
10^{-7}$. Obviously this represents a weak coupling case in which the kinetic
part dominates. Fig. 4 illustrates solutions of Eqs. (98) and (99) with the
corresponding self-consistent solution.

Figure 4: The red line and the blue line represent solutions of Eqs. (98) and
(99), respectively. The black dot marks the corresponding self-consistent
solution.

In the sLL, we found that Eqs. (98) and (99) do not support a coexistence of
(non-zero) kinetic and pairing amplitudes, and thus, if only pairing is
present it leads to a gapless (critical) $p$-wave state at this level of
approximation. We considered the question of coexistence when a cubic term is
generated in the expansion (VI), in the kinetic part. The resulting equations
are slightly modified Eqs. (98) and (99) ($V\rightarrow\frac{V}{3!}$, etc.),
and lead to a solution that describes a coexistence of pairing,
$|{\tilde{\Delta}}_{\boldsymbol{k}}|={\tilde{\Delta}}|\boldsymbol{k}|$ where
${\tilde{\Delta}}\approx 0.002337\;V_{0}$, and now
$|{\Delta}_{\boldsymbol{k}}|=\Delta|{\boldsymbol{k}}|^{3}$ where
$\Delta\approx 0.000591\;V_{0}\;(l_{B}=1)$. The numerically obtained solution
using Mathematica is shown in Fig. 5. The error is estimated to $1\times
10^{-6}$. Thus, this is a strong coupling, weak pairing case that can be
identified with the usual Pfaffian-state case in which all composite fermions
are paired in the same way of a $p$-wave.

Figure 5: The red line and the blue line illustrate solutions of modified Eqs.
(98) (with $V\rightarrow\frac{V}{3!}$, etc.) and (99), respectively. The black
dot shows the corresponding self-consistent solution with pairing
$|{\tilde{\Delta}}_{\boldsymbol{k}}|={\tilde{\Delta}}|\boldsymbol{k}|$, and
$|{\Delta}_{\boldsymbol{k}}|=\Delta|{\boldsymbol{k}}|^{3}$.

##  XI Conclusions

To conclude, we presented a microscopic derivation of the Dirac CF theory and
proposed an effective (long-wavelength) noncommutative field-theoretical
description. The developed formalism can serve as a base for a deeper
understanding of the strongly correlated physics of the half-filled LL. The
microscopic formulation is used to describe (in a mean-field approach) pairing
instabilities of the half-filled LL. The description is consistent with the
known experimental and numerical phenomenology and provides further insights
into the nature of pairing instabilities.

## References

  * [1] B. I. Halperin, P. A. Lee, and N. Read, Phys. Rev. B 47, 7312 (1993).
  * [2] G. Moore and N. Read, Nucl. Phys. B 360, 362 (1991).
  * [3] D. T. Son, Phys. Rev. X 5, 031027 (2015).
  * [4] V. Pasquier and F. D. M. Haldane, Nucl. Phys. B 516, 719 (1998).
  * [5] N. Read, Phys. Rev. B 58, 16262 (1998).
  * [6] Z. Dong and T. Senthil, Phys. Rev. B 102, 205126 (2020).
  * [7] N. Read and D. Green, Phys. Rev. B 61, 10267 (2000).
  * [8] G. Murthy and R. Shankar, Rev. Mod. Phys. 75, 1101 (2003).
  * [9] A. C. Balram, M. Barkeshli, and M. S. Rudner, Phys. Rev. B 98, 035127 (2018).
  * [10] R. V. Mishmash, D. F. Mross, J. Alicea, and O. I. Motrunich, Phys. Rev. B 98, 081107(R) (2018).
  * [11] M. Yutushui and D.F. Mross, Phys. Rev. B 102, 195153 (2020).
  * [12] L. Antonić, J. Vučičević and M. V. Milovanović, Phys. Rev. B 98, 115107 (2018).
  * [13] N. Read, Phys. Rev. Lett. 62, 86 (1989).
  * [14] M. V. Milovanović, S. Djurdjević, J. Vučičević and L. Antonić, Mod. Phys. Lett. B 34, 2030004 (2020).
  * [15] A. C. Potter, M. Serbyn and A. Vishwanath, Phys. Rev. X 6, 031026 (2016).
  * [16] C. Wang, N. R. Cooper, B. I. Halperin and A. Stern, Phys. Rev. X 7, 031029 (2017).
  * [17] N. Seiberg, T. Senthil, C. Wang, E. Witten, Ann. Phys. 374, 395 (2016).
  * [18] K. Prabhu, M. M. Roberts, arXiv: 1709.02814.
  * [19] D. Karabali, V. P. Nair, A. P. Polychronakos, Nucl. Phys. B627, 565 (2002).
  * [20] R. J. Szabo, Physics Reports, 378(4), 207-299 (2003).
  * [21] N. Seiberg and E. Witten, JHEP 09, 032 (1999).
  * [22] P. Aschieri, L. Castellani and M. Dimitrijević, Phys. Rev. D 87, 024017 (2013).
  * [23] P. Aschieri, M. Dimitrijević, P. Kulish, F. Lizzi, & J. Wess. Noncommutative spacetimes: symmetries in noncommutative geometry and field theory (Vol. 774). Springer Science & Business Media, (2009).
  * [24] B. Jurčo, S. Schraml, P. Schupp and J. Wess, Eur. Phys. J. C 17, 521 (2000).

##  Appendix A Basic structure of noncommutative gauge field theory

Noncommutative (NC) spaces are geometric structures for which the notion of a
point loses its meaning. They arise naturally in physics, the most famous
example being the “quantum” phase space of generalized coordinates and
momenta, understood as mutually non-commuting (incompatible) observables that
satisfy Heisenberg’s uncertainty relations
$[q^{i},p_{j}]=i\hbar\delta^{i}_{j}$. Another important example of NC space is
realized by a single particle moving in the LLL; the guiding center
coordinates of such a particle satisfy the following non-trivial commutation
relation, $[R_{x},R_{y}]=-il_{B}^{2}$. Finally, in the context of quantum
gravity, the idea of NC geometry is applied to the structure of space-time
itself. In this case, the degree to which space-time coordinates fail to
commute is usually taken to be proportional to the square of the Planck
length. For a more comprehensive review see, for example, [20].

Here we give a short review of the basic structure of a noncommutative gauge
field theory based on the Seiberg-Witten (SW) construction, first presented in
the context of String Theory [21]. An NC field theory is a field theory
defined on a noncommutative space-time, i.e., a space-time described by
mutually non-commuting coordinates, $[x^{\mu},x^{\nu}]\neq 0$. One way to
implement this NC structure is to deform the algebra of functions (fields) on
ordinary (commutative) space-time, by introducing an NC star-product, the
simplest of which is the Moyal-Weyl star-product,

$f(x)\star g(y)=e^{\frac{i}{2}\theta^{\mu\nu}\frac{\partial}{\partial
x^{\mu}}\frac{\partial}{\partial y^{\nu}}}f(x)g(y)|_{y\rightarrow x}.$ (102)

The deformation parameters $\theta^{\mu\nu}$ make a constant anti-symmetric
matrix (hence the term _constant noncommutativity_). Note that the first term
in the expansion (in powers of $\theta$) is just the ordinary commutative
product of functions; higher order terms represent NC corrections. When
applied to coordinate functions themselves, the above formula gives us

$[x^{\mu},x^{\nu}]=x^{\mu}\star x^{\nu}-x^{\nu}\star
x^{\mu}=i\theta^{\mu\nu},$ (103)

which is the simplest form of noncommutativity. Other types of
noncommutativity are related to different star-products, (see [22]), but we
will be working only with the Moyal-Weyl star-product. A more comprehensive
account on various aspects of NC geometry and NC gauge field theory can be
found in [23].

Let $\\{T_{A}\\}$ be a set of generators of a gauge group $\mathcal{G}$.
Infinitesimal SW gauge variation (NC gauge transformation) of an NC field
$\Phi$ transforming in the fundamental representation of the gauge group (e.g.
a matter field) is defined by

$\delta^{SW}_{\Lambda}\Phi=i\Lambda\star\Phi,$ (104)

where $\Lambda$ stands for an NC gauge parameter. This transformation rule is
analogous to the familiar one in ordinary, commutative gauge field theory,
except for the noncommutative Moyal-Weyl star-product.

There is a notorious problem concerning the closure axiom for SW gauge
transformations. Namely, assuming that NC gauge parameter is Lie algebra
valued, $\Lambda=\Lambda^{A}T_{A}$, the commutator of two SW transformations
is given by

$\displaystyle[\delta^{SW}_{1}\stackrel{{\scriptstyle\star}}{{,}}\delta^{SW}_{2}]\Phi=(\Lambda_{1}\star\Lambda_{2}-\Lambda_{2}\star\Lambda_{1})\star\Phi$
(105)
$\displaystyle=\frac{1}{2}\left([\Lambda^{A}_{1}\stackrel{{\scriptstyle\star}}{{,}}\Lambda_{2}^{B}]\\{T_{A},T_{B}\\}+\\{\Lambda^{A}_{1}\stackrel{{\scriptstyle\star}}{{,}}\Lambda_{2}^{B}\\}[T_{A},T_{B}]\right)\star\Phi.$

Due to the appearance of anticommutator $\\{T_{A},T_{B}\\}$, infinitesimal SW
transformations do not in general close in the Lie algebra of a gauge group. A
way to surmount this difficulty is to assume that NC gauge parameter $\Lambda$
belongs to the universal enveloping algebra (UEA), which is always infinite
dimensional [24]. This, however, leads to an infinite tower of new degrees of
freedom (new fields), which is not a preferable property. To see this,
consider a SW variation of the covariant derivative

$\delta^{SW}_{\Lambda}D_{\mu}\Phi=i\Lambda\star D_{\mu}\Phi,$ (106)

with $D_{\mu}\Phi=\partial_{\mu}\Phi-iV_{\mu}\star\Phi$ (here we only consider
“left” gauge field; the case of “right” gauge field is treated in a similar
manner). This implies the following transformation rule for the NC gauge
potential,

$\delta^{SW}_{\Lambda}V_{\mu}=\partial_{\mu}\Lambda+i[\Lambda\stackrel{{\scriptstyle\star}}{{,}}V_{\mu}],$
(107)

meaning that it is also an UEA-valued object, which leaves us with an infinite
number of new fields in the theory (one for each basis element of UEA).

SW map resolves this issue by demanding that NC fields can be expressed in
terms of the NC parameter $\theta^{\mu\nu}$, commutative gauge parameter
$\lambda=\lambda^{A}T_{A}$, commutative gauge potential
$v_{\mu}=v_{\mu}^{A}T_{A}$, and their derivatives,

$\displaystyle\Lambda$
$\displaystyle=\Lambda(\theta,\lambda,v_{\mu};\partial\lambda_{\mu},\partial
v_{\mu},\dots),$ (108) $\displaystyle V_{\mu}$
$\displaystyle=V_{\mu}(\theta,v_{\mu},\partial v_{\mu},\dots),$ (109)

where the ellipses stand for higher derivatives. In this way, NC theory is
defined by the corresponding commutative one. There are no new degrees of
freedom, just new interaction terms in the NC action.

NC gauge transformations are now induced by the corresponding commutative
ones,

$\displaystyle\delta^{SW}_{\lambda}\Lambda$
$\displaystyle=\Lambda(v_{\mu}+\delta_{\lambda}v_{\mu})-\Lambda(v_{\mu}),$
(110) $\displaystyle\delta^{SW}_{\lambda}V_{\mu}$
$\displaystyle=V_{\mu}(v_{\mu}+\delta_{\lambda}v_{\mu})-V_{\mu}(v_{\mu}),$
(111)

with $\delta_{\lambda}v_{\mu}=\partial_{\mu}\lambda+i[\lambda,v_{\mu}]$.

From (105) and (108) follows a consistency condition for NC gauge parameter,

$\Lambda_{1}\star\Lambda_{2}-\Lambda_{2}\star\Lambda_{1}+i(\delta^{SW}_{1}\Lambda_{2}-\delta^{SW}_{2}\Lambda_{1})=i\Lambda_{-i[\lambda_{1},\lambda_{2}]},$
(112)

which can be solved perturbatively (this makes sense because the star-product
is also defined perturbatively). To this end, we represent the NC gauge
parameter as an expansion in powers of the NC parameter $\theta$, with
coefficients built out of fields from the commutative theory,

$\Lambda=\lambda+\Lambda^{(1)}+\Lambda^{(2)}+\dots.$ (113)

The first term in the expansion (zeroth order in $\theta$) is the commutative
gauge parameter $\lambda=\lambda^{A}T_{A}$.

An NC gauge parameter, up to first order in $\theta^{\alpha\beta}$, is given
by

$\Lambda=\lambda-\frac{1}{2}\theta^{\alpha\beta}v_{\alpha}\partial_{\beta}\lambda+\mathcal{O}(\theta^{2}),$
(114)

where $\theta^{\alpha\beta}$ $(\alpha,\beta=0,1,2)$ is a constant
antisymmetric matrix of deformation parameters. In our case,
$\theta^{\alpha\beta}=\theta\epsilon^{\alpha\beta}$ where $\theta=-l_{B}^{2}$
and $\epsilon^{\alpha\beta}$ is defined by $\epsilon^{0i}=0$ $(i=1,2)$ and
$\epsilon^{12}=-\epsilon^{21}=1$. Note that we work in $2+1$ dimensions and
that the definition of $\epsilon^{\alpha\beta}$ implies that noncommutativity
is realized only between spatial coordinates; it does not involve the time
coordinate. Generalization to an arbitrary number of dimensions is
straightforward.

Using the expansion (114) and the transformation rule (104) we readily obtain

$\Phi=\phi-\frac{\theta^{\alpha\beta}}{2}v_{\alpha}\partial_{\beta}\phi+\frac{i\theta^{\alpha\beta}}{4}v_{\alpha}v_{\beta}\phi+\mathcal{O}(\theta^{2}).$
(115)

Also, from (107) follows the transformation law for the NC gauge field,

$V_{\mu}=v_{\mu}-\frac{\theta^{\alpha\beta}}{2}v_{\alpha}\left(\partial_{\beta}v_{\mu}+F_{\beta\mu}\right).$
(116)

In connection to our model, with left (L) and right (R) NC $U(1)$ gauge
transformations acting on NC CF fields $c$ and $d$, it is convenient to
combine the two CFs into a two-component spinor,

$\Psi=\begin{pmatrix}c\\\ d\end{pmatrix}.$ (117)

Under an NC gauge transformation it changes as

$\Psi\rightarrow\Psi^{\prime}=U_{L}\star\Psi\star U_{R},$ (118)

where $U_{L/R}=e^{i\Lambda_{L/R}}$. The NC gauge parameters $\Lambda_{L/R}$
have a special form given by

$\Lambda_{L/R}=\begin{pmatrix}\Lambda_{L/R}^{c}&0\\\
0&\Lambda_{L/R}^{d}\end{pmatrix}.$ (119)

Infinitesimally, we get

$\delta\Psi=i\left(\Lambda_{L}\star\Psi+\Psi\star\Lambda_{R}\right).$ (120)

Covariant derivative of $\Psi$ is defined by

$\displaystyle D_{\mu}\Psi$ $\displaystyle=\partial_{\mu}\Psi-
iV^{L}_{\mu}\star\Psi-i\Psi\star V^{R}_{\mu}$ (121)

where we introduced left, $V^{L}_{\mu}$, and right, $V^{R}_{\mu}$, NC gauge
potential. One can show that $D_{\mu}\Psi$ transforms covariantly,

$(D_{\mu}\Psi)^{\prime}=U_{L}\star(D_{\mu}\Psi)\star U_{R},$ (122)

provided that NC gauge potentials $V_{\mu}^{L}$ and $V^{R}_{\mu}$ transform in
the following way:

$\displaystyle(V^{L}_{\mu})^{\prime}$ $\displaystyle=$ $\displaystyle
U_{L}\star V^{L}_{\mu}\star U_{L}^{\dagger}-i\partial_{\mu}U_{L}\star
U_{L}^{\dagger},$ $\displaystyle(V_{\mu}^{R})^{\prime}$ $\displaystyle=$
$\displaystyle U_{R}^{\dagger}\star V_{\mu}^{R}\star
U_{R}-iU_{R}^{\dagger}\star\partial_{\mu}U_{R}.$ (123)

The NC action (in Euclidean signature) reads as

$S_{NC}=\int d\tau
d^{2}\boldsymbol{x}\left(\Psi^{\dagger}\star\tau^{\mu}D_{\mu}\Psi+i(V^{L}_{0}+V_{0}^{R})\bar{\rho}_{e}\right),$
(124)

where we have

$\tau^{0}=\begin{pmatrix}1&0\\\ 0&1\end{pmatrix},\hskip
5.69046pt\tau^{1}=\begin{pmatrix}0&i\\\ i&0\end{pmatrix},\hskip
5.69046pt\tau^{2}=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}.$ (125)

In terms of components, the action is

$\displaystyle S_{NC}=\int d\tau d^{2}\boldsymbol{x}\Big{(}c^{\dagger}\star
D_{\tau}c+d^{\dagger}\star D_{\tau}d$ (126) $\displaystyle+$ $\displaystyle
c^{\dagger}\star(iD_{x}+D_{y})d+d^{\dagger}\star(iD_{x}-D_{y})c+i(V^{L}_{0}+V_{0}^{R})\bar{\rho}_{e}\Big{)}.$

The SW map allows us to represent NC fields in terms of ordinary fields from
the corresponding commutative theory; these commutative fields will be denoted
by a hat symbol. NC fields are organized as perturbation series in powers of
the deformation parameter, and they reduce to ordinary fields in the
$\theta=0$ limit.

SW expansions of NC gauge parameters $\Lambda_{L/R}$ (up to first order) are
(note the sign difference) given by

$\displaystyle\Lambda_{L}$
$\displaystyle=\hat{\Lambda}_{L}-\frac{\theta}{2}\epsilon^{\alpha\beta}\hat{V}^{L}_{\alpha}\partial_{\beta}\hat{\Lambda}_{L},$
$\displaystyle\Lambda_{R}$
$\displaystyle=\hat{\Lambda}_{R}+\frac{\theta}{2}\epsilon^{\alpha\beta}V_{\alpha}^{R}\partial_{\beta}\hat{\Lambda}_{R}.$
(127)

Likewise, SW expansions of L and R NC gauge potentials are (again, note the
sign difference)

$\displaystyle V_{\mu}^{L}$
$\displaystyle=\hat{V}^{L}_{\mu}-\frac{\theta}{2}\epsilon^{\alpha\beta}\hat{V}^{L}_{\alpha}\Big{(}\partial_{\beta}\hat{V}^{L}_{\mu}+\hat{F}^{L}_{\beta\mu}\Big{)},$
(128) $\displaystyle V_{\mu}^{R}$
$\displaystyle=\hat{V}^{R}_{\alpha}+\frac{\theta}{2}\epsilon^{\alpha\beta}\hat{V}^{R}_{\alpha}\Big{(}\partial_{\beta}\hat{V}^{R}_{\mu}+\hat{F}^{R}_{\beta\mu}\Big{)},$
(129)

with (commutative) gauge field strengths

$\displaystyle\hat{F}^{L}_{\alpha\beta}$
$\displaystyle=\partial_{\alpha}\hat{V}_{\beta}^{L}-\partial_{\beta}\hat{V}_{\alpha}^{L},$
$\displaystyle\hat{F}^{R}_{\alpha\beta}$
$\displaystyle=\partial_{\alpha}\hat{V}^{R}_{\beta}-\partial_{\beta}\hat{V}^{R}_{\alpha}.$
(130)

Finally, the SW expansion of the NC matter field reads as

$\Psi=\hat{\Psi}-\frac{\theta}{2}\epsilon^{\alpha\beta}\left((\hat{V}^{L}_{\alpha}-\hat{V}^{R}_{\alpha})\partial_{\beta}\hat{\Psi}-i\hat{V}^{L}_{\alpha}\hat{V}^{R}_{\beta}\hat{\Psi}\right).$
(131)

In particular, if we take $V^{L}_{\mu}=A_{\mu}$ and
$V^{R}_{\mu}=a_{\mu}-A_{\mu}$ (both $V^{L/R}_{\mu}$ act as scalars), we get
covariant derivatives (78) and (79), and the previous equation becomes

$\Psi=\hat{\Psi}+\frac{\theta}{2}\epsilon^{\alpha\beta}\left((\hat{a}_{\alpha}-2\hat{A}_{\alpha})\partial_{\beta}\hat{\Psi}-i\hat{a}_{\alpha}\hat{A}_{\beta}\hat{\Psi}\right).$
(132)

In general, to compute a first order NC correction for a product of two NC
fields, we apply the following formula,

$(A\star
B)^{(1)}=A^{(1)}\hat{B}+\hat{A}B^{(1)}+\frac{i}{2}\theta^{\alpha\beta}\partial_{\alpha}\hat{A}\partial_{\beta}\hat{B},$
(133)

where $A^{(1)}$ and $B^{(1)}$ are first order NC corrections of $A$ and $B$,
respectively.

Thus, we can readily compute the NC covariant derivative,

$\displaystyle D_{\mu}\Psi=\partial_{\mu}\Psi-iV^{L}_{\mu}\star\Psi-i\Psi\star
V_{\mu}^{R}=D_{\mu}\hat{\Psi}+\frac{\theta}{2}\epsilon^{\alpha\beta}\left[(\hat{a}_{\alpha}-2\hat{A}_{\alpha})\partial_{\beta}D_{\mu}\hat{\Psi}-i\hat{a}_{\alpha}\hat{A}_{\beta}D_{\mu}\hat{\Psi}-(\hat{f}_{\alpha\mu}-2\hat{F}_{\alpha\mu})D_{\beta}\hat{\Psi}\right],$
(134)

with commutative field strengths
$\hat{f}_{\mu\nu}=\partial_{\mu}\hat{a}_{\nu}-\partial_{\nu}\hat{a}_{\mu}$ and
$\hat{F}_{\mu\nu}=\partial_{\mu}\hat{A}_{\nu}-\partial_{\nu}\hat{A}_{\mu}$.
Commutative covariant derivative is simply
$D_{\mu}\hat{\Psi}=\partial_{\mu}\hat{\Psi}-i\hat{a}_{\mu}\hat{\Psi}$ .

Inserting (128), (129), (132), and (134) into the NC action (124) and applying
the rule (133), we obtain the commutative action (82) and its first-order (in
$\theta$) NC correction (VIII).

In a similar fashion, if we consider covariant derivatives (86) and (87),
i.e.,

$\displaystyle D_{\mu}c$
$\displaystyle=\partial_{\mu}c-i(a_{\mu}^{(1)}+A_{\mu})\star
c-ic\star(a_{\mu}^{(2)}-A_{\mu}),$ (135) $\displaystyle D_{\mu}d$
$\displaystyle=\partial_{\mu}d-i(a_{\mu}^{(2)}+A_{\mu})\star
d-id\star(a_{\mu}^{(1)}-A_{\mu}).$ (136)

and require $a_{\mu}^{(1)}=a_{\mu}^{(2)}=\tfrac{1}{2}a_{\mu}$ (a full symmetry
between physical and unphysical sectors), by the same procedure we obtain
action (88) that reproduces the Son’s theory at the commutative level.

##  Appendix B The structure of NC covariant derivatives

Let us assume that the structure of NC covariant derivatives is given by Eqs.
(76) and (77), i.e.,

$\displaystyle D_{\mu}c$
$\displaystyle=\partial_{\mu}c-i(a_{\mu}^{(1)}+A_{\mu})\star c-ic\star
a_{\mu}^{(2)},$ (137) $\displaystyle D_{\mu}d$
$\displaystyle=\partial_{\mu}d-ia_{\mu}^{(2)}\star
d-id\star(a_{\mu}^{(1)}-A_{\mu}),$ (138)

as consistent with the microscopic constraints.

If we want both (137) and (138) to transform covariantly, i.e.,

$\displaystyle D_{\mu}c$ $\displaystyle\rightarrow U^{c}_{L}\star
D_{\mu}c\star U_{R}^{c},$ (139) $\displaystyle D_{\mu}d$
$\displaystyle\rightarrow U^{d}_{L}\star D_{\mu}d\star U_{R}^{d},$ (140)

we get two separate transformation laws for the gauge field $a_{\mu}^{(2)}$
(in the $c$-case it acts from the right and in the $d$-case from the left):

$\displaystyle a_{\mu}^{(2)}$ $\displaystyle\rightarrow U^{c\dagger}_{R}\star
a_{\mu}^{(2)}\star U_{R}^{c}-iU_{R}^{c\dagger}\star\partial_{\mu}U_{R}^{c},$
(141) $\displaystyle a_{\mu}^{(2)}$ $\displaystyle\rightarrow U^{d}_{L}\star
a_{\mu}^{(2)}\star U_{L}^{d\dagger}-i\partial_{\mu}U_{L}^{d}\star
U_{L}^{d\dagger}.$ (142)

Consistency of the two demands the following constraints:

$\displaystyle U_{L}^{d}$ $\displaystyle=U_{R}^{c\dagger},$ (143)
$\displaystyle U_{R}^{c\dagger}\star\partial_{\mu}U_{R}^{c}$
$\displaystyle=U_{L}^{d}\star\partial_{\mu}U_{L}^{d\dagger}=\partial_{\mu}U_{L}^{d}\star
U_{L}^{d\dagger}.$ (144)

The second equation implies that $(\partial_{\mu}U_{L}^{d})\star
U_{L}^{d\dagger}-U_{L}^{d}\star(\partial_{\mu}U_{L}^{d\dagger})=0$, which is
not consistent with $U_{L}^{d}$ being unitary. Thus, we must change the
structure of the derivatives.

Now consider the case when two covariant derivatives have the same structure,

$\displaystyle D_{\mu}c$ $\displaystyle=\partial_{\mu}c-iA_{\mu}\star
c-ic\star b_{\mu},$ (145) $\displaystyle D_{\mu}d$
$\displaystyle=\partial_{\mu}d-iA_{\mu}\star d-id\star b_{\mu},$ (146)

with two gauge fields: the external background field $A_{\mu}$ (which is L
gauge field for both fields $c$ and $d$) and $b_{\mu}=a_{\mu}-A_{\mu}$ (which
is R gauge field for both fields $c$ and $d$). They transform under gauge
transformations in the following way:

$\displaystyle A_{\mu}\rightarrow$ $\displaystyle U_{L}\star A_{\mu}\star
U_{L}^{\dagger}-i(\partial_{\mu}U_{L})\star U^{\dagger}_{L},$ (147)
$\displaystyle b_{\mu}\rightarrow$ $\displaystyle U^{\dagger}_{R}\star
b_{\mu}\star U_{R}-iU^{\dagger}_{R}\star(\partial_{\mu}U_{R}).$ (148)

In the commutative limit, these become simple $U(1)$ gauge transformations,

$\displaystyle A_{\mu}$ $\displaystyle\rightarrow
A_{\mu}+\partial_{\mu}\varphi_{A},$ (149) $\displaystyle b_{\mu}$
$\displaystyle\rightarrow b_{\mu}+\partial_{\mu}\varphi_{b}.$ (150)

The relevant field $a_{\mu}=b_{\mu}+A_{\mu}$ represents an independent $U(1)$
gauge field with respect to $A_{\mu}$.

In the setup implied by (86) and (87), i.e.,

$\displaystyle D_{\mu}c$
$\displaystyle=\partial_{\mu}c-i(a_{\mu}^{(1)}+A_{\mu})\star
c-ic\star(a_{\mu}^{(2)}-A_{\mu}),$ (151) $\displaystyle D_{\mu}d$
$\displaystyle=\partial_{\mu}d-i(a_{\mu}^{(2)}+A_{\mu})\star
d-id\star(a_{\mu}^{(1)}-A_{\mu}),$ (152)

we have to consider four different gauge fields $b^{(1)}_{\mu,\pm}\equiv
a_{\mu}^{(1)}\pm A_{\mu}$ and $b^{(2)}_{\mu,\pm}\equiv a_{\mu}^{(2)}\pm
A_{\mu}$. Their transformation laws are given by

$\displaystyle b_{\mu+}^{(1)}$ $\displaystyle\longrightarrow U_{L}^{c}\star
b_{\mu+}^{(1)}\star U_{L}^{c\dagger}-i\partial_{\mu}U_{L}^{c}\star
U_{L}^{c\dagger},$ (153) $\displaystyle b_{\mu-}^{(1)}$
$\displaystyle\longrightarrow U_{R}^{d\dagger}\star b_{\mu-}^{(1)}\star
U_{R}^{d}-iU_{R}^{d\dagger}\star\partial_{\mu}U_{R}^{d},$ (154) $\displaystyle
b_{\mu+}^{(2)}$ $\displaystyle\longrightarrow U_{L}^{d}\star
b_{\mu+}^{(2)}\star U_{L}^{d\dagger}-i\partial_{\mu}U_{L}^{d}\star
U_{L}^{d\dagger},$ (155) $\displaystyle b_{\mu-}^{(2)}$
$\displaystyle\longrightarrow U_{R}^{c\dagger}\star b_{\mu-}^{(2)}\star
U_{R}^{c}-iU_{R}^{c\dagger}\star\partial_{\mu}U_{R}^{c}.$ (156)

However, these four fields are not mutually independent,
$b^{(1)}_{\mu,+}-b^{(1)}_{\mu,-}=b^{(2)}_{\mu,+}-b^{(2)}_{\mu,-}$, because
aside from the external field, the theory can have only two more independent
gauge fields, connected with two kinds of constraints. If we require that the
difference of gauge fields transforms as a gauge field, we will have
$U_{L}^{c}=U_{R}^{d\dagger}$ and $U_{L/R}^{c}=U_{L/R}^{d}$, i.e.,
possibilities for independent gauge transformations, independent gauge fields,
will be reduced to just one, but this would contradict our beginning set-up.
That is why we need to take $a_{\mu}^{(1)}=a_{\mu}^{(2)}$, and have two
independent gauge transformations and fields
$a_{\mu}=2a_{\mu}^{(1)}=2a_{\mu}^{(2)}$ and $A_{\mu}$, in the commutative
limit.