# Lowest Landau level theory of the bosonic Jain states

Hart Goldman and T. Senthil  Department of Physics, Massachusetts Institute of
Technology, Cambridge, MA 02139, USA

###### Abstract

Quantum Hall systems offer the most familiar setting where strong inter-
particle interactions combine with the topology of single particle states to
yield novel phenomena. Despite our mature understanding of these systems, an
open challenge has been to to develop a microscopic theory capturing both
their universal and non-universal properties, when the Hamiltonian is
restricted to the non-commutative space of the lowest Landau level. Here we
develop such a theory for the Jain sequence of bosonic fractional quantum Hall
states at fillings $\nu={p\over p+1}$. Building on a lowest Landau level
description of a parent composite fermi liquid at $\nu=1$, we describe how to
dope the system to reach the Jain states. Upon doping, the composite fermions
fill non-commutative generalizations of Landau levels, and the Jain states
correspond to integer composite fermion filling. Using this approach, we
obtain an approximate expression for the bosonic Jain sequence gaps with no
reference to any long-wavelength approximation. Furthermore, we show that the
universal properties, such as Hall conductivity, are encoded in an effective
non-commutative Chern-Simons theory, which is obtained on integrating out the
composite fermions. This theory has the same topological content as the
familiar Abelian Chern-Simons theory on commutative space.

###### Contents

  1. 1 Introduction
  2. 2 Recap: The LLL composite fermion theory of bosons at $\nu=1$
    1. 2.1 Basics of the LLL and mean field theory of the metallic state at $\nu=1$
    2. 2.2 Non-commutative composite fermion field theory and the Seiberg-Witten map
  3. 3 Doping away from $\nu=1$
    1. 3.1 A physical picture: quantum mechanics of charge-vortex dipoles
    2. 3.2 Mean field theory of the composite fermion Landau problem
      1. 3.2.1 Mean field Hamiltonian and Landau level spectrum
      2. 3.2.2 Some formalism: Covariant coordinates
      3. 3.2.3 Bosonic Jain sequence and gaps
  4. 4 The bosonic Jain sequence: Universal features
    1. 4.1 Fluctuations and Hall response
    2. 4.2 Non-commutative Chern-Simons theory
    3. 4.3 Toward a non-commutative mutual Chern-Simons theory
  5. 5 Discussion
  6. A Landau level degeneracy on the non-commutative torus
  7. B Chern-Simons effective action

##  1 Introduction

Two dimensional many-particle systems in a strong magnetic field display some
of the most famous examples of quantum correlated phenomena. At special,
partial fillings $\nu$ of a Landau level, an incompressible phase is formed
with Hall conductivity, $\sigma_{xy}=\nu e^{2}/h$, in what is known as the
fractional quantum Hall effect (FQHE). Since Laughlin’s original explanation
of the basic physics of the FQHE [1], a number of distinct theoretical
approaches have been developed which provide a deeper and more versatile
understanding of the FQHE. A prominent and successful approach is built on the
framework of flux attachment [2], which trades the original theory of charged
particles in a magnetic field for a theory of new entities, dubbed composite
bosons or composite fermions, interacting with a Chern-Simons gauge field. In
particular, the composite fermion construction provides a simple unifying
explanation of the vast majority of observed FQH phases as integer quantum
Hall (IQH) states of composite fermions [3, 4, 5]. It also provides the
foundation to understand the metallic states found in electronic systems near
even denominator filling fractions [6, 7, 8, 9, 10, 11], and enables parent
descriptions of non-Abelian quantum Hall states [12, 13].

For the gapped quantum Hall states, the universal long distance properties are
captured by Chern-Simons topological quantum field theories, enriched by a
global $U(1)$ symmetry that corresponds to particle number conservation [14,
15]. Within the flux attachment framework, these Chern-Simons theories can be
usefully viewed as arising from integrating out Landau levels of composite
fermions. This description encapsulates the topological order and symmetry
fractionalization data of the quantum Hall state. However, it is not suitable
if we are interested in estimating non-universal, microscopic quantities, such
as the magnitude of the gaps, given a microscopic Hamiltonian. Especially
salient in this regard are Hamiltonians defined by projecting to the lowest
Landau level (LLL). Then the electron kinetic energy is quenched and all
energy scales are determined by the interaction strength. The interactions
must therefore be treated completely non-perturbatively, and the problem is
typically only amenable to study through a variety of numerical methods, such
as variational wavefunction calculations, exact diagonalization, and the
density matrix renormalization group. An interesting analytical approach, due
to Murthy and Shankar (for a review, see Ref. [16]), used a Hamiltonian
description of composite fermions in the LLL to yield good estimates of gaps
and other non-universal quantities for a variety of filling fractions.
Nevertheless, the universal properties of both the composite Fermi liquid and
the Jain states seem harder to directly extract from this formalism.

An important open question in quantum Hall physics is to obtain a unified
analytic framework that captures both universal and non-universal aspects of
the physics. This task is particularly challenging when the many-particle
Hilbert space is restricted to a single Landau level. Recently, inspired by
the earlier efforts of Pasquier, Haldane, and Read [17, 18], one of us,
together with Zhihuan Dong, made progress on this problem by developing a
composite fermion approach to the metallic state of bosons at $\nu=1$ that is
strictly in the LLL [19, 20]. This theory, like earlier approaches, involves
composite fermions coupled to a fluctuating $U(1)$ gauge field, although this
time the fields live on the non-commutative space of the LLL. Despite this
progress for the metallic state, there continues to be no construction of
gapped quantum Hall states that allows for a derivation of the long wavelength
Chern-Simons field theory but also provides estimates of the gaps and other
non-universal features.

Figure 1: A schematic of the emergence of the bosonic Jain states. Our
starting point is a theory of bosons in a magnetic field with interaction
energy, $V_{\mathrm{int}}$, and cyclotron frequency, $\omega_{c}$. The bosons
are projected to the lowest Landau level (LLL) when
$V_{\mathrm{int}}<<\omega_{c}$ (we use units $\hbar=c=1$). By changing
variables to composite fermions, $c$, coupled to a fluctuating gauge field,
$a_{\mu}$, we consider a composite Fermi liquid (CFL) theory on the non-
commutative space of the LLL, which is valid below the interaction scale. We
access the Jain states by turning on a background vector potential, $\delta
A_{i}$, such that the composite fermions fill $p$ Landau levels with a gap,
$\Delta_{p}$. Integrating out the composite fermions leads to an effective
_non-commutative_ Chern-Simons theory, which captures the topological data of
the Jain state.

The purpose of this work is to provide such a theory for the specific case of
Jain states of bosons at fillings $\nu=\frac{p}{p+1}$, with $p$ a positive
integer, which approach the metallic state of bosons at $\nu=1$ as
$p\rightarrow\infty$. An outline of our construction is given in Figure 1.
Starting with the description of the metallic composite Fermi liquid ground
state of bosons at $\nu=1$ in terms of non-commutative composite fermion field
theory, we show that the proximate Jain states can be obtained by filling
Landau levels of composite fermions, in a manner directly analogous to the
derivation using ordinary composite fermions. In the process, we obtain an
approximate mean field expression for the bosonic Jain sequence gaps,

$\displaystyle\Delta_{p}$
$\displaystyle=\frac{2\pi\bar{\rho}}{m_{\star}}\frac{1}{1+p}\,,$ (1.1)

where $\bar{\rho}$ is the charge density, $m_{*}$ is the composite fermion
effective mass, and $p=2\pi\bar{\rho}/\delta B$ is the Jain sequence index.
This expression is corrected compared to the usual composite fermion mean
field result, in which the gap is proportional to the effective magnetic field
felt by the composite fermions, $m_{*}\Delta=\delta B=2\pi\bar{\rho}/p$. In
our result, working on the non-commutative space of the LLL has corrected
$p\rightarrow p+1$. Using the Hartree-Fock approximation of the effective
mass, our result returns values for the gap that are compatible with exact
diagonalization estimates [21, 22]. In principle, other non-universal data,
such as collective modes and structure factors, can also be sought using our
mean field framework.

Furthermore, we show that integrating out the composite fermions yields a non-
commutative Chern-Simons (NCCS) field theory for these Jain states, which
describes the correct topological order. Many years ago, based on hydrodynamic
considerations, Susskind and Polychronakos [23, 24] proposed the NCCS field
theory and its matrix model regularization to describe quantum Hall states at
short distances, and it has been found to correctly capture ground state
topological features and wave functions [25, 26, 27]. More recently, the
quantum Hall matrix models have been generalized to the Jain sequences [28]
and to non-Abelian quantum Hall states [29, 30]. Their universal geometric
response has also been studied [31, 32]. However, the connection of the NCCS
theory to realistic microscopic Hamiltonians and to composite fermion field
theory has been opaque. These relationships are explained in our approach. The
NCCS theory appears as an effective description at scales
$\omega<<\Delta_{p}$, and its non-commutativity is set by the charge density,
rather than the total magnetic field, in agreement with the hydrodynamic
approach of Susskind [23].

Since the universal properties of fractional quantum Hall phases are well
described by the usual commutatice Chern-Simons field theory, one may question
what is gained by using the non-commutative version. It is therefore
interesting that in our work, which is an approximate treatment of microscopic
Hamiltonians in the LLL, the non-commutative theory emerges rather naturally,
arising as a bridge between a fully microscopic theory and the usual long
wavelength topological quantum field theory.

We finally comment that although there is an extensive literature on non-
commutative field theories (for reviews, see Refs. [33, 34]), the structure of
the theories we encounter differs from what is most commonly discussed. In
particular, our focus is on non-commutative theories of fermions at a non-zero
density of global $U(1)$ charge and in a background magnetic field. Our work
contributes to the general understanding of this class of problems.
Nevertheless, we will make contact with and use some of the results of the
extensive existing literature, as appropriate, throughout the paper.

We proceed as follows. In Section 2, we review the non-commutative composite
fermion approach the problem of bosons at $\nu=1$ engineered in Refs. [19]. We
then proceed to develop our mean field approach to the bosonic Jain sequence
in Section 3, in the process obtaining our result for the gaps. In Section 4,
we integrate out the composite fermions to obtain a NCCS field theory
describing the universal aspects of the Jain sequence states. We conclude in
Section 5.

##  2 Recap: The LLL composite fermion theory of bosons at $\nu=1$

###  2.1 Basics of the LLL and mean field theory of the metallic state at
$\nu=1$

We start by considering a 2d system of (bosonic or fermionic) charged
particles with density $\bar{\rho}$ in a strong magnetic field, $B$, such that
the filling, $\nu=2\pi\bar{\rho}/B$, is less than or equal to one. The
particles then form highly degenerate Landau levels, and they can each fill
states in the lowest Landau level. In the limit of large magnetic field, we
may therefore project the Hilbert space to states in the LLL. On restricting
to the LLL, the particles are only described by their guiding center
coordinates,

$\displaystyle R_{i}$
$\displaystyle=x_{i}-\ell_{B}^{2}\varepsilon_{ij}\pi_{j}\,,$ (2.1)

where $\ell_{B}=|B|^{-1/2}$ is the magnetic length and $\pi_{i}$ is the gauge
invariant momentum. Assuming canonical commutation relations, these
coordinates do not commute,

$\displaystyle[R_{i},R_{j}]$
$\displaystyle=-i\ell_{B}^{2}\varepsilon_{ij}\equiv
i\Theta\varepsilon_{ij}\,.$ (2.2)

Therefore, the geometry of the LLL is non-commutative, with non-commutativity
parameter $\Theta=-\ell_{B}^{2}$.

On LLL projection, the kinetic energy is quenched, and the Hamiltonian may be
expressed in terms of interactions of the projected density operator111The
projection of the density operator will also include a form factor
$e^{-q^{2}l_{B}^{2}/4}$. We will follow the common practice of incorporating
this form factor into the interaction so that the Hamiltonian is expressed in
terms of the operator $\rho_{L}$ which satisfies the GMP algebra. , which we
denote $\rho_{L}(\mathbf{q})=\sum_{n}e^{i\mathbf{q}\cdot\mathbf{R}_{n}}$,
where boldface denotes spatial vectors and $n$ is a particle index. Using Eq.
(2.2), the density operator can be seen to satisfy the famous Girvin-
MacDonald-Platzman (GMP) algebra [35],

$\displaystyle[\rho_{L}(\mathbf{q}),\rho_{L}(\mathbf{p})]$
$\displaystyle=2i\sin\left(\frac{\ell_{B}^{2}(\mathbf{q}\times\mathbf{p})}{2}\right)\rho_{L}(\mathbf{q}+\mathbf{p})\,.$
(2.3)

The GMP algebra contains information on the geometric symmetries of the LLL:
$\rho_{L}$ can be expanded in terms of the generators of area-preserving
diffeomorphisms of space, which in turn satisfy the $\mathcal{W}_{\infty}$
algebra [36, 37, 38]. While these are not symmetries of the microscopic
Hamiltonian they encode the structure of the non-commutative space of the LLL.
Our interest is in projected Hamiltonians of the form,

$\displaystyle H$
$\displaystyle=\frac{1}{2}\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}\tilde{U}(\mathbf{q})\rho_{L}(-\mathbf{q})\rho_{L}(\mathbf{q})\,.$
(2.4)

where $\tilde{U}(\mathbf{q})=U_{0}\exp{-\frac{q^{2}l_{B}^{2}}{2}}$ is the
Fourier transform of a contact interaction augmented with a form factor. We
first review the solution of this problem developed in Ref. [19] for the
particular case of bosons at $\nu=1$.

The physical underpinnings of the lowest Landau level description is a view of
the composite fermions as charge-vortex dipoles [39]. Unlike in ordinary flux
attachment, these composite fermions are understood to be neutral, because the
vortices deplete a unit of charge at their cores. They possess a dipole moment
proportional to their momentum,

$\displaystyle\mathbf{d}=\ell_{B}^{2}\mathbf{k}\times\hat{z}$ (2.5)

The composite fermions thus naturally acquire a quadratic dispersion from
their dipole energy, $\epsilon(\mathbf{k})\sim|\mathbf{d}|^{2}$. We will refer
back to this basic physical picture at many points in this work.

We begin with a formal representation of the many body bosonic Hilbert space
and the density operator introduced by Pasquier and Haldane [17], and
developed further by Read [18]. For $N$ particles, any state in the many-body
Hilbert space of of bosons at $\nu=1$ can be written as

$\displaystyle|\Psi\rangle=\sum_{\\{m_{i}\\}}a_{m_{1}\dots
m_{N}}|m_{1},\dots,m_{N}\rangle\,,$ (2.6)

where $m_{i}=1,\dots,N$ is an orbital index for the $N$ single particle states
in the Landau level (thus the filling $\nu=1$), $|m_{1},\dots,m_{N}\rangle$ is
a product of single particle orbitals, and $a_{m_{1}\dots m_{N}}$ are
constants that are symmetric under exchange of indices. One may represent the
basis states in terms of composite fermion creation (annihilation) operators,
$c^{\dagger}_{mn}$ ($c_{mn}$), where $m$ and $n$ are again orbital indices,

$\displaystyle|m_{1},\dots m_{N}\rangle$
$\displaystyle=\varepsilon^{n_{1}\dots n_{N}}c^{\dagger}_{m_{1}n_{1}}\dots
c^{\dagger}_{m_{N}n_{N}}|0\rangle\,,$ (2.7)

where $|0\rangle$ is the vacuum state annihilated by $c_{mn}$. We may
interpret the left and right indices, respectively, as being associated with
(bosonic) charges and vortices, which bind to form the composite fermion. In
terms of the fermion operators, the projected density operator is

$\displaystyle(\rho_{L})_{mm^{\prime}}=c^{\dagger}_{mn}c_{m^{\prime}n}$ (2.8)

and is the generator of the physical electromagnetic symmetry.

This fermionic description is redundant. The physical states are invariant
under $SU(N)$ rotations of the $n_{i}$ indices, meaning that the theory has a
$SU(N)$ gauge symmetry. The presence of this gauge symmetry implies a
constraint, in that the physical states satisfy

$\displaystyle(\rho_{R})_{nn^{\prime}}|\Psi\rangle=c^{\dagger}_{mn}c_{mn^{\prime}}|\Psi\rangle=\delta_{nn^{\prime}}|\Psi\rangle\,.$
(2.9)

Here $\rho_{R}$ is the $SU(N)$ charge density, and we may interpret it as the
density of vortices in the LLL. Note that its diagonal part is shared with
that of the charge density, $\rho_{L}$, and represents the physical EM charge,
$\operatorname{Tr}[\rho_{L}]=\operatorname{Tr}[\rho_{R}]=N$.

We may construct momentum-space representations of the fermion operators,
$c(\mathbf{q})$, $c^{\dagger}(\mathbf{q})$ using the orbital matrix elements
of the magnetic translation operator,
$\tau_{\mathbf{q}}=e^{i\mathbf{q}\cdot\mathbf{R}}$. These operators satisfy
the standard anticommutation relations

$\displaystyle\\{c(\mathbf{q}),c^{\dagger}(\mathbf{q}^{\prime})\\}=(2\pi)^{2}\delta^{(2)}(\mathbf{q}-\mathbf{q}^{\prime})\,.$
(2.10)

In terms of these operators, we may construct $\rho_{L}$ and $\rho_{R}$ as
follows,

$\displaystyle\rho_{L}(\mathbf{q})=\int\frac{d^{2}\mathbf{k}}{(2\pi)^{2}}c^{\dagger}(\mathbf{k}-\mathbf{q})c(\mathbf{q})e^{-i\Theta\mathbf{k}\times\mathbf{q}/2}\,$
$\displaystyle,\,\rho_{R}(\mathbf{q})=\int\frac{d^{2}\mathbf{k}}{(2\pi)^{2}}c^{\dagger}(\mathbf{k}-\mathbf{q})c(\mathbf{q})e^{i\Theta\mathbf{k}\times\mathbf{q}/2}\,.$
(2.11)

where we recall $\Theta=-\ell_{B}^{2}=-1/(2\pi\bar{\rho})$ at $\nu=1$. In this
representation, the constraint in Eq. (2.9) becomes

$\displaystyle\rho_{R}(\mathbf{q})$
$\displaystyle=(2\pi)^{2}\bar{\rho}\,\delta^{(2)}(\mathbf{q})\,.$ (2.12)

Using the anticommutation relation in Eq. (2.10), it is immediate that both of
these operators furnish representations of the GMP algebra: $\rho_{L}$
satisfies Eq. (2.3) while $\rho_{R}$ satisfies Eq. (2.3) but with a minus sign
on the right hand side. Using the constraint in Eq. (2.12) and expanding
$\rho_{L}$ in powers of $\mathbf{q}$, it is straightforward to see that the
composite fermions possess the dipole moment222Each term in the expansion of
$\rho_{L}$ corresponds to a generator of diffeomorphisms in the LLL, which
satisfy the $\mathcal{W}_{\infty}$ algebra. The dependence of the dipole
moment on momentum is natural from this point of view: it is the generator of
translations. in Eq. (2.5),

$\displaystyle d_{i}=\ell_{B}^{2}\,\varepsilon_{ij}\mathcal{P}^{j}\,,$ (2.13)

where $\mathcal{P}$ is the composite fermion momentum operator.

Equipped with the fermionic representation of the density operator, we may
revisit the Hamiltonian in Eq. (2.4), which is now a four-fermion interaction
plus the constraint (2.12). In Refs. [18, 19, 20] this problem was attacked
using a Hartree-Fock mean field approach, in which the authors sought a ground
state of fermions at finite density, consistent with the expectation of a
composite Fermi liquid at $\nu=1$,

$\displaystyle\langle c^{\dagger}(\mathbf{q})c(\mathbf{q})\rangle$
$\displaystyle\neq 0\,.$ (2.14)

This leads to a Hartree-Fock Hamiltonian of the form

$\displaystyle H_{\mathrm{HF}}$
$\displaystyle=\int\frac{d^{2}\mathbf{q}}{(2\pi)^{2}}\epsilon(\mathbf{q})c^{\dagger}(\mathbf{q})c(\mathbf{q})\,,$
(2.15)

which describes a metallic state of composite fermions, with effective mass of
order the interaction strength. The quadratic dispersion,
$\epsilon(\mathbf{q})\sim|\mathbf{q}|^{2}$ at small $\mathbf{q}$, is naturally
understood as the dipole energy of the composite fermion.

The assumption of a non-vanishing expectation value for the Hartree-Fock order
parameter, $c^{\dagger}(\mathbf{q})c(\mathbf{q})$, spontaneously breaks the
$SU(N)$ gauge symmetry: it only commutes with $\rho_{R}(\mathbf{q})$ at
$\mathbf{q}=0$. The important fluctuations about the Hartree-Fock ground state
are therefore those near $\mathbf{q}=0$, which can be described by coupling to
a $U(1)$ gauge field, $a_{\mu}$. However, this is not an ordinary $U(1)$ gauge
symmetry, since the fields under consideration live on the non-commutative
space of the LLL. Therefore, one must consider a non-commutative composite
fermion field theory [19, 20]. This theory will anchor our analysis through
the rest of this work, and we are now prepared to introduce it.

###  2.2 Non-commutative composite fermion field theory and the Seiberg-Witten
map

To build a non-commutative field theory, it is necessary to understand how to
multiply functions on non-commutative space. These requires the introduction
of the Moyal star product. For functions of non-commutative coordinates,
$x_{i}$, the Moyal star product is

$\displaystyle f(x)\star g(x)$ $\displaystyle=\lim_{x^{\prime}\rightarrow
x}\exp\left(i\frac{\Theta}{2}\varepsilon^{ij}\frac{\partial}{\partial
x^{\prime\,i}}\frac{\partial}{\partial x^{j}}\right)f(x^{\prime})g(x)\,.$
(2.16)

Forming products in this way, the effective composite fermion action[19, 20],
including gauge field fluctuations, is

$\displaystyle S$ $\displaystyle=\int d\tau
d^{2}\mathbf{x}\left[c^{\dagger}\star
D_{\tau}c+\frac{1}{2m_{*}}(D_{i}c)^{\dagger}\star(D^{i}c)+ia_{0}\bar{\rho}\right]\,,$
(2.17) $\displaystyle D_{\mu}c$ $\displaystyle=\partial_{\mu}c-i\,\delta
A_{\mu}\star c-ic\star a_{\mu}\,.$ (2.18)

Here $\delta A_{\mu}$ is the background _probe_ electromagnetic vector
potential, such as, for instance, that needed to described the deviation of
the theory from $\nu=1$. On the other hand, $a_{\mu}$ is an emergent,
fluctuating $U(1)$ gauge field. The equation of motion of $a_{\tau}$ enforces
the constraint,

$\displaystyle c^{\dagger}\star c=\bar{\rho}=\frac{B}{2\pi}\,,$ (2.19)

where $B$ is the magnetic field felt by the underlying bosonic charges, which
are at filling $\nu=1$ when $\delta A=0$. The theory in Eq. (2.17) therefore
describes a metallic state of composite fermions at density set by the
background magnetic field.

Unlike in ordinary commutative field theory, on non-commutative space even
$U(1)$ gauge symmetries have non-Abelian representations. The physical
electromagnetic $U(1)$ symmetry acts to the left on the non-commutative
composite fermions,

$\displaystyle c$ $\displaystyle\rightarrow U_{L}\star c\,,$ (2.20)
$\displaystyle\delta A$ $\displaystyle\rightarrow U_{L}\star\delta A\star
U_{L}^{\dagger}-i\,\partial U_{L}\star U_{L}^{\dagger}\,,$ (2.21)
$\displaystyle a$ $\displaystyle\rightarrow a\,,$ (2.22)

while the emergent $U(1)$ gauge symmetry associated with the constraint acts
to the right,

$\displaystyle c$ $\displaystyle\rightarrow c\star U_{R}\,,$ (2.23)
$\displaystyle\delta A$ $\displaystyle\rightarrow\delta A\,,$ (2.24)
$\displaystyle a$ $\displaystyle\rightarrow U_{R}^{\dagger}\star a\star
U_{R}-i\,U_{R}^{\dagger}\star\partial U_{R}\,.$ (2.25)

Respectively, these gauge symmetries correspond to the conserved charge
densities,

$\displaystyle\rho_{L}=-c\star c^{\dagger}$
$\displaystyle,\rho_{R}=c^{\dagger}\star c\,,$ (2.26)

On quantization, each of these density operators can be seen to satisfy their
own GMP algebra, Eq. (2.3), as they correspond to the right and left densities
introduced in the previous subsection.

Non-commutative field theories like Eq. (2.17) are exceptionally challenging
to work with, especially in the presence of a constraint like Eq. (2.19). A
common approach to studying them is to map them to a corresponding field
theory on commutative space, using a mapping developed by Seiberg and Witten
[40]. This map relates the fields $c,a,\delta A$ in Eq. (2.17) to new fields,
$\psi,\hat{a},\delta\hat{A}$, defined on commutative space333Frequently in the
literature on non-commutative field theory, the hat notation actually denotes
the _non-commutative_ fields. For consistency with Refs. [19, 20], we continue
to use the inverted notation., in an expansion in powers of the non-
commutative parameter, $\Theta=-\ell_{B}^{2}$. Remarkably, applying the
Seiberg-Witten map to the non-commutative CFL theory in Eq. (2.17), one
obtains[19] the Halperin-Lee-Read (HLR) theory [6] for bosons at $\nu=1$, but
with additional short-ranged corrections,

$\hat{\mathcal{L}}=\psi^{\dagger}(\partial_{\tau}-i\hat{a}_{\tau}-i\delta\hat{A}_{\tau})\psi+\frac{1}{2m_{*}}|(\partial_{i}-i\hat{a}_{i}-i\delta\hat{A}_{i})\psi|^{2}-i\frac{1}{4\pi}\hat{a}d\hat{a}+\Theta\mathcal{L}_{\mathrm{corr}}\,.$
(2.27)

Via the Seiberg-Witten map, the constraint term, $ia_{\tau}\bar{\rho}$, in the
non-commutative CFL theory becomes a Chern-Simons term in the commutative
theory, since $2\pi\bar{\rho}\Theta=-\nu=-1$. Flux attachment in the HLR
theory can therefore be thought of as “emerging” from the constraint on
$\rho_{R}$, Eq. (2.19), in the non-commutative CFL theory! The terms in
$\Theta\mathcal{L}_{\mathrm{corr}}$ are fixed by the Seiberg-Witten map but
are sub-leading in powers of $\Theta$, meaning that they are irrelevant at
long wavelengths. For more details on this calculation, see Ref. [19].

The Seiberg-Witten map thus exchanges a theory of composite fermions on non-
commutative space, whose density has a non-trivial form factor, for a theory
of ordinary composite fermions augmented by additional short-ranged terms.
This theory is therefore easily amenable to doping away from $\nu=1$,
resulting in the usual Jain sequence of bosonic FQH states, in for which the
composite fermions fill an integer number of Landau levels,

$\displaystyle\nu=\frac{p}{p+1}\,.$ (2.28)

Because this theory corresponds to a non-commutative CFL theory in the LLL,
one can also calculate estimates for LLL dynamical features at large $p$ (weak
$\delta\hat{B}\equiv\delta\hat{F}_{xy}$). For example, using this theory, one
can calculate the gaps for the Jain sequence states. One finds444We note that
Ref. [19] contains a sign error in its final expression for the gaps. We
correct it here.

$\displaystyle\Delta(\delta\hat{B})$
$\displaystyle=\frac{|\delta\hat{B}|}{m_{*}}\left(1+\Theta\delta\hat{B}\right)+\mathcal{O}\left((\Theta\delta\hat{B})^{2}\right)=\frac{2\pi\bar{\rho}}{m_{*}}\left(\frac{1}{p}-\frac{1}{p^{2}}\right)+\mathcal{O}(p^{-3})\,,$
(2.29)

where we have used the fact that in the Jain state the filling fraction of the
composite fermions is $\nu_{\mathrm{CF}}=2\pi\bar{\rho}/\delta\hat{B}=p$. In
this work we will go much further by doping the non-commutative CFL theory,
Eq. (2.17), directly. This will enable us to compute dynamical properties of
the LLL bosonic Jain states, such as their gaps, in a mean field approximation
valid to all orders in $\Theta$. We further show that our results match the
long wavelength estimate of Eq. (2.29). In addition, we will obtain the low
energy effective theory for gauge fluctuations about the mean field. This will
enable us to provide a microscopic derivation of the non-commutative Chern-
Simons theory description of quantum Hall states and make contact with prior
work that proposed such a description based on hydrodynamic arguments [23,
24].

##  3 Doping away from $\nu=1$

###  3.1 A physical picture: quantum mechanics of charge-vortex dipoles

Before turning on a non-commutative magnetic field in the composite fermion
field theory of Eq. (2.17), a clear physical picture can be formed by
considering the single particle problem of charges and vortices in the LLL,
with the composite fermions viewed as charge-vortex dipoles. The Jain states
are then obtained by filling Landau levels of the composite fermions.
Remarkably, we will show later on that this framework corresponds precisely to
the single particle limit of the non-commutative composite fermion field
theory.

Technically, our analysis of the charge-vortex dipole problem will rely on the
classic work of Nair and Polychronakos [41] which solved the Landau level
problem for particles moving in a non-commutative space. This work has since
seen considerable followup, e.g. in Refs. [42, 43, 44, 45, 46, 47, 48, 49].
Although physical quantum Hall systems and composite fermion theory are
occasionally mentioned in this literature, the precise connection to this
physics has not been made before as far as we know. We explain this connection
below.

Consider bosonic charges and vortices in the $\nu=1$ system, respectively
characterized by guiding center coordinates, $R_{i}$ and $R^{v}_{i}$. Because
the system has a finite charge density, the vortices see the same magnetic
field but with opposite sign to the charges. Hence their coordinates satisfy
an algebra with opposite sign,

$\displaystyle[R_{i},R_{j}]=-i\ell_{B}^{2}\,\varepsilon_{ij}\,,\,[R^{v}_{i},R_{j}^{v}]=+i\ell_{B}^{2}\,\varepsilon_{ij}\,,[R_{i},R^{v}_{j}]=0\,.$
(3.1)

We also define “composite fermion” coordinates,
$r^{\mathrm{CF}}_{i}=(R_{i}+R_{i}^{v})/2$, which commute,

$\displaystyle[r^{\mathrm{CF}}_{i},r^{\mathrm{CF}}_{j}]$ $\displaystyle=0\,.$
(3.2)

As expected, the composite fermion feels no magnetic field. Since the
composite fermion is a dipole of a charge and a vortex, we may define a dipole
moment, $d_{i}=R^{v}_{i}-R_{i}$, which is proportional to the canonical
momentum of the composite fermion,

$\displaystyle
p^{\mathrm{CF}}_{i}=\frac{1}{\ell_{B}^{2}}\varepsilon_{ij}d^{j}\,,\,[r^{\mathrm{CF}}_{i},p^{\mathrm{CF}}_{j}]=i\delta_{ij}\,,\,[p^{\mathrm{CF}}_{i},p^{\mathrm{CF}}_{j}]=0\,.$
(3.3)

It is thus natural to take the effective Hamiltonian to be the dipole energy,

$\displaystyle H$ $\displaystyle=\frac{1}{2m_{*}}(p^{\mathrm{CF}})^{2}\,.$
(3.4)

Indeed, this expectation is borne out in the Hartree-Fock analysis of Ref.
[19].

We now investigate how this picture changes on varying the background magnetic
field, $B\rightarrow B+\delta B$, while keeping the charge density fixed so
that $\nu_{\mathrm{CF}}=2\pi\frac{\bar{\rho}}{\delta B}=\frac{B}{\delta B}=p$,
where $p$ is an integer. The filling fraction of the bosonic charges is
therefore

$\displaystyle\nu=2\pi\frac{\bar{\rho}}{B+\delta B}=\frac{p}{1+p}\,,$ (3.5)

which is the bosonic Jain sequence. Tuning the magnetic field alters the
guiding center algebra of the charges while preserving that of the vortices,

$\displaystyle[R_{i},R_{j}]=-i\ell_{B+\delta
B}^{2}\,\varepsilon_{ij}\,,\,[R^{v}_{i},R_{j}^{v}]=+i\ell_{B}^{2}\,\varepsilon_{ij}\,,[R_{i},R^{v}_{j}]=0\,,$
(3.6)

where $\ell_{B+\delta B}^{2}=1/(B+\delta B)$. Now the composite fermions see a
magnetic field, and their coordinates no longer commute,

$\displaystyle[r^{\mathrm{CF}}_{i},r^{\mathrm{CF}}_{j}]$
$\displaystyle=i\frac{1}{4}(\ell_{B}^{2}-\ell_{B+\delta
B}^{2})\,\varepsilon_{ij}\equiv i\theta\,\varepsilon_{ij}\,.$ (3.7)

The composite fermion momentum defined in Eq. (3.3) now satisfies

$\displaystyle[r^{\mathrm{CF}}_{i},p^{\mathrm{CF}}_{j}]=\frac{\ell_{B+\delta
B}^{2}+\ell_{B}^{2}}{2\ell_{B}^{2}}\,i\delta_{ij}\,,\,[p^{\mathrm{CF}}_{i},p^{\mathrm{CF}}_{j}]=i\frac{\ell_{B}^{2}-\ell_{B+\delta
B}^{2}}{\ell_{B}^{4}}\varepsilon_{ij}=i\frac{\delta B}{1+\delta
B/B}\varepsilon_{ij}\,.$ (3.8)

Notice that the momentum commutator no longer vanishes, corresponding to the
fact that the composite fermions are experiencing a magnetic field. This
commutation algebra, together with the Hamiltonian in Eq. (3.4), is precisely
the non-commutative Landau level problem studied by Nair and Polychronakos
[50] (after a trivial rescaling of the momentum, which we will perform below,
to make it canonically conjugate to the composite fermion coordinate).

As in the ordinary Landau problem, we may diagonalize the Hamiltonian in Eq.
(3.4) by constructing creation and annihilation operators out of the momentum,
proportional to $p_{x}^{\mathrm{CF}}\pm ip_{y}^{\mathrm{CF}}$. This allows one
to easily compute the gaps, which are

$\displaystyle\Delta_{p}$ $\displaystyle=\frac{\delta
B}{m_{*}}\frac{1}{1+B/\delta B}=\frac{2\pi\bar{\rho}}{m_{*}}\frac{1}{1+p}\,.$
(3.9)

Amazingly, we will find that analysis and the resulting gaps precisely match
that of the non-commutative composite fermion mean field theory we will
develop in the next subsection.

Implicit in the above analysis is the identification of $\delta B$ with the
variation of the physical magnetic field. Hence we expect the density of
states of the non-commutative Landau level to be $\delta B$. This can be
checked by constructing an operator, $\hat{\mathcal{D}}_{i}$, which commutes
with the Hamiltonian and generates magnetic translations of $r^{\mathrm{CF}}$,
the composite fermion guiding center coordinate [50]. To do this, we start by
defining a new momentum that satisfies canonical commutation relations with
$r^{\mathrm{CF}}$,

$\displaystyle\tilde{p}^{\mathrm{CF}}$
$\displaystyle=\frac{2\ell_{B}^{2}}{\ell^{2}_{B+\delta
B}+\ell_{B}^{2}}\,p^{\mathrm{CF}}\,,$ (3.10)

Using $\tilde{p}^{\mathrm{CF}}$, the theory is now defined by the algebra,

$\displaystyle[r^{\mathrm{CF}}_{i},r^{\mathrm{CF}}_{j}]=i\theta\,\varepsilon_{ij}\,,[r^{\mathrm{CF}}_{i},\tilde{p}^{\mathrm{CF}}_{j}]=i\delta_{ij}\,,\,[\tilde{p}^{\mathrm{CF}}_{i},\tilde{p}^{\mathrm{CF}}_{j}]=4i\frac{(\ell_{B}^{2}-\ell_{B+\delta
B}^{2})}{(\ell^{2}_{B+\delta B}+\ell_{B}^{2})^{2}}\varepsilon_{ij}\equiv
i\delta B_{\mathrm{CF}}\,\varepsilon_{ij}\,,$ (3.11)

It is now easy to see that the operators,
$\mathcal{D}_{i}=(\tilde{p}_{i}^{\mathrm{CF}}-\delta
B_{\mathrm{CF}}\varepsilon_{ij}r^{\mathrm{CF}}_{j})/(1-\theta\delta
B_{\mathrm{CF}})$, commute with the Hamiltonian. These operators satisfy the
algebra,

$\displaystyle[\mathcal{D}_{x},\mathcal{D}_{y}]=-i\frac{\delta
B_{\mathrm{CF}}}{1-\theta\delta B_{\mathrm{CF}}}=-i\delta B\,.$ (3.12)

This is a strong indication that the density of states of the Landau level is
$\delta B$. Further discussion of the degeneracy of Landau levels on the non-
commutative torus can be found in Refs. [51, 52, 53, 43], and we sketch the
derivation in Appendix A. Furthermore, we will observe below that the change
in the physical magnetic field here, $\delta B$, can be identified with the
the magnetic field obtained by mapping a non-commutative background field to
an ordinary one via the Seiberg-Witten map.

Using this single particle picture of charge-vortex dipoles to physically
ground us, we will now derive these same results using the non-commutative
composite fermion theory in Eq. (2.17), which we dope using a non-commutative
vector potential. In doing so, we will demonstrate how the introduction of a
uniform non-commutative magnetic field in the field theory picture corresponds
to the deformation of the non-commutativity of the charge degrees of freedom
presented above.

###  3.2 Mean field theory of the composite fermion Landau problem

####  3.2.1 Mean field Hamiltonian and Landau level spectrum

We now construct the bosonic Jain sequence by doping the non-commutative field
theory, Eq. (2.17) and studying the resulting problem in mean field theory.
While it is not possible to alter the density of the composite fermions given
the constraint, Eq. (2.19), one can still turn on a background magnetic field
associated with $\delta A_{i}$. Indeed, if we neglect fluctuations of $a_{i}$,
the non-commutative composite fermion theory in a uniform magnetic field is
remarkably similar to the analogous problem of electrons on an ordinary space.
The composite fermions continue to form Landau levels, with gaps set by the
cyclotron frequency. Furthering the analogy, when small fluctuations of
$a_{i}$ are introduced, we will show in Section 4 that integrating out the
filled Landau levels leads to a _non-commutative_ Chern-Simons theory. This
will ultimately lead us to the bosonic Jain sequence states.

We start by passing to a Hamiltonian formulation, fixing the density via the
constraint, Eq. (2.19). Neglecting fluctuations of $a_{i}$, i.e. $\langle
a_{i}\rangle=0$, the mean field Hamiltonian can be written as

$\displaystyle\mathcal{H}$ $\displaystyle=\int
d^{2}\mathbf{x}\,\frac{1}{2m_{*}}(D_{i}c)^{\dagger}\star
D^{i}c\,,D_{i}c=\partial_{i}c-i\delta A_{i}\star c\,.$ (3.13)

The equation of motion of this theory leads to the single particle Schrödinger
equation,

$\displaystyle Hc=-\frac{1}{2m_{*}}D_{i}D^{i}c=Ec\,.$ (3.14)

Rather than fixing to a specific gauge and computing the spectrum by solving
the resulting differential equation, it is more conceptually straightforward
in this non-commutative context to solve the theory simply by considering its
algebraic properties, as in Ref. [50] and the analysis of the previous
subsection. The commutator of the $D_{i}$ operators is the non-commutative
field strength,

$\displaystyle\delta F_{xy}=i[D_{x},D_{y}]_{\star}=\partial_{x}(\delta
A_{y})-\partial_{y}(\delta A_{x})-i[A_{x},A_{y}]_{\star}\,,$ (3.15)

where we have defined the star commutator, $[A,B]_{\star}=A\star B-B\star A$.
To solve the Schrödinger equation, Eq. (3.14), for uniform $\delta F_{xy}$, we
can again construct creation and annihilation operators using $D_{x}\pm
iD_{y}$, leading to Landau levels with energy set by $\delta F_{xy}$,

$\displaystyle\epsilon_{n}=\omega_{c}\left(n+\frac{1}{2}\right)\,,$ (3.16)

where

$\displaystyle\omega_{c}(\delta F_{xy})=\frac{\delta F_{xy}}{m_{*}}$ (3.17)

is the Landau level gap. This expression superficially differs from the result
for Landau level gaps in Eq. (2.29). However, we stress that the non-
commutative field strength, $\delta F_{xy}$, is not equal to the physical
magnetic field, which sets composite fermion Landau level degeneracy and
corresponds to the physical magnetic field, $\delta B$, in Section 3.1. We now
describe how to relate these two quantities.

####  3.2.2 Some formalism: Covariant coordinates

To define the filling fraction for the composite fermion Landau levels, it is
necessary to determine their degeneracy. While naïvely one may expect that the
degeneracy is given by $\delta F_{xy}\times\mathrm{Area}$, this is in fact not
the case. Indeed, in the analysis of the dipole model in Section 3.1, the
field setting the energy gap was found to be differ from the physical magnetic
field. A hint of this can be seen in the fact that the commutator of the
ordinary coordinate operators, $x_{i}$, with the gauge covariant momenta,
$D_{i}$, is not gauge invariant: $D_{i}$ transforms as an adjoint while
$x_{i}$ does not transform. Non-commutative gauge invariance therefore
dictates that we work with coordinate operators that transform under gauge
transformations. These are the so-called covariant coordinates,

$\displaystyle Y^{i}[\delta A]$
$\displaystyle=x^{i}+\Theta\,\varepsilon^{ij}\delta A_{j}\,,$ (3.18)

which transform as adjoints under the left-acting gauge transformations, Eq.
(2.21),

$\displaystyle Y^{i}\rightarrow U_{L}\star Y\star U_{L}^{\dagger}\,,$ (3.19)

ensuring gauge invariant commutation relations with $D_{i}$.

The transformation law follows from the close relationship between non-
commutative gauge transformations and area-preserving diffeomorphisms (APDs)
of the non-commutative space. For a more detailed review of this topic, see
Ref. [33]. Consider a gauge transformation, $U_{L}=e^{i\lambda}$. Then for any
function, $f(\mathbf{x})$,

$\displaystyle U_{L}\star f(\mathbf{x})\star U_{L}^{\dagger}$
$\displaystyle=f(\mathbf{x})-\Theta\,\varepsilon^{ij}\partial_{i}\lambda\,\partial_{j}f(\mathbf{x})+\mathcal{O}(\Theta^{2})=f(x^{i}+\Theta\varepsilon^{ij}\partial_{j}\lambda+\dots)\,.$
(3.20)

This means that a combination of left and right gauge transformations is
infinitesimally equivalent to a translation by a vector,
$\Theta\,\varepsilon^{ij}\partial_{j}\lambda$, which leaves the area element
invariant. Therefore, under a left-acting gauge transformation, the covariant
coordinate, $Y^{i}$, transforms as an adjoint,

$\displaystyle Y^{i}\rightarrow x^{i}+\Theta\,\varepsilon^{ij}U_{L}\star\delta
A_{j}\star
U_{L}^{\dagger}+\Theta\,\varepsilon^{ij}\,\partial_{j}\lambda+\mathcal{O}(\Theta^{2})=U_{L}\star
Y^{i}\star U_{L}^{\dagger}\,.$ (3.21)

A useful consequence of this transformation law is that functions of covariant
coordinates also transform as adjoints,

$\displaystyle f[Y(\mathbf{x})]\rightarrow f\left[U_{L}(\mathbf{x})\star
Y(\mathbf{x})\star U_{L}^{\dagger}(\mathbf{x})\right]=U_{L}(\mathbf{x})\star
f[Y(\mathbf{x})]\star U_{L}^{\dagger}(\mathbf{x})\,.$ (3.22)

This property follows immediately from the equivalence of adjoint gauge
transformations with APDs, Eq. (3.20), and it will figure heavily in Section
4.

####  3.2.3 Bosonic Jain sequence and gaps

We are now prepared to extract the Landau level degeneracy and physical
magnetic field, as well as make contact with the dipole quantum mechanics
problem in Section 3.1. We start by replacing operators that act with a Moyal
product to the right with operators defined to act with left multiplication,
such that all operators only act to the left on states [50]. We may then
define new coordinate operators, which we (suggestively) name $R^{i}$ and
$R^{v}_{i}$, such that

$\displaystyle R_{i}c=Y_{i}\star c\,$ $\displaystyle,\,R^{v}_{i}c=c\star
x_{i}\,.$ (3.23)

These operators furnish two different, mutually commuting guiding center
algebras, which may be naturally expressed in terms of a new magnetic field,

$\displaystyle\delta\hat{B}$ $\displaystyle=\frac{\delta
F_{xy}}{1+\Theta\delta F_{xy}}\,,$ (3.24)

such that

$\displaystyle[R_{i},R_{j}]=i\Theta(1+\Theta\delta F_{xy})\varepsilon_{ij}$
$\displaystyle=-\frac{i}{B+\delta\hat{B}}\varepsilon_{ij}=-i\ell_{B+\delta\hat{B}}^{2}\varepsilon_{ij}\,,$
(3.25) $\displaystyle[R^{v}_{i},R^{v}_{j}]=i\ell_{B}^{2}\,\varepsilon_{ij}$
$\displaystyle,[R_{i},R^{v}_{j}]=0\,,$

where we recall $\Theta=-\ell_{B}^{2}$. This is the same guiding center
algebra as in the dipole quantum mechanics model, Eq. (3.1), with
$\delta\hat{B}$ identified with the physical magnetic field, which we had
earlier denoted $\delta B$.

The mean field Hamiltonian, Eq. (3.14), can also be expressed in terms of
$R_{i}$ and $R_{i}^{v}$, since in non-commutative field theory commutators
with $x^{i}$ are derivatives,

$\displaystyle[x^{i},c(\mathbf{x})]_{\star}=i\Theta\,\varepsilon^{ij}\partial_{j}c(\mathbf{x})\,.$
(3.26)

Therefore, the covariant derivative can be written as

$\displaystyle D^{i}c=\frac{i}{\Theta}\,\varepsilon^{ij}(R_{j}-R^{v}_{j})c\,,$
(3.27)

meaning that it is proportional to the composite fermion dipole moment, as
anticipated in Section 3.1. In mean field theory, the single particle
Hamiltonian of the composite fermions is therefore identical to the
Hamiltonian in Eq. (3.4),

$\displaystyle
H=\frac{1}{2m_{*}}\frac{1}{\ell_{B}^{4}}(R_{i}-R^{v}_{i})^{2}\,,$ (3.28)

The mean field composite fermion theory and the charge-vortex dipole problem
are equivalent! We can therefore immediately apply the result [50] for the
Landau level degeneracy in Section 3.1 to find that it is indeed set by
$\delta\hat{B}$,

$\displaystyle d_{LL}=\frac{\delta\hat{B}}{2\pi}\times\mathrm{Area}\,.$ (3.29)

See Refs. [51, 52, 53, 43] and Appendix A for a more detailed account of how
one arrives at this result for the case of the theory on a torus. We note that
the field strength, $\delta\hat{B}$, is also the value of the field strength
obtained from the Seiberg-Witten map, meaning that this result is consistent
with the requirement of flux quantization in the commutative approximation of
the theory [54, 55].

To summarize, by using the proper gauge covariant coordinates, we have shown
that the introduction of the non-commutative vector potential in the composite
fermion mean field theory corresponds precisely to the deformation of the non-
commutative parameter of the charges in the single-dipole model, with Eq.
(3.24) as the physical magnetic field. With this result, we can define the
composite fermion filling fraction,

$\displaystyle\nu_{\mathrm{CF}}$
$\displaystyle=2\pi\frac{\langle\rho_{L}\rangle}{\delta\hat{B}}=2\pi\frac{\bar{\rho}}{\delta\hat{B}}\,.$
(3.30)

The bosonic Jain states occur when the composite fermions form integer quantum
Hall states, with $\nu_{\mathrm{CF}}=p$, $p$ an integer.

Because $\delta\hat{B}$ sets the degeneracy of the composite fermion Landau
level, we interpret it as shift in the magnetic field felt by the underlying
bosonic charges from $B=2\pi\bar{\rho}$. Therefore, one is naturally led to
the conclusion that the filling fraction of the physical bosons is

$\displaystyle\nu=2\pi\frac{\bar{\rho}}{B+\delta\hat{B}}=\frac{p}{1+p}\,.$
(3.31)

This is precisely the expected form of the bosonic Jain sequence [21, 22]! In
terms of the density $\bar{\rho}$ and composite fermion Landau level index
$p$, one finds that the gap for each Jain sequence state is thus,

$\displaystyle\Delta_{p}=\frac{2\pi\bar{\rho}}{m_{\star}}\frac{1}{1+p}\approx\frac{2\pi\bar{\rho}}{m_{*}}\left(\frac{1}{p}-\frac{1}{p^{2}}\right)+\mathcal{O}(p^{-3})\,.$
(3.32)

This expression, for which we have not invoked any long wavelength
approximation, is one of the main results of this work. It matches the result
from the Seiberg-Witten map, Eq. (2.29), but involves no long wavelength
approximation and is thus valid even at small values of $p$.

Notably, for the $\nu=1/2$ Laughlin state, we find

$\displaystyle\Delta_{1}$
$\displaystyle=\frac{\pi\bar{\rho}}{m_{\star}}\approx(0.65)\,\pi\,U_{0}\bar{\rho}\,.$
(3.33)

This result uses the Hartree-Fock effective mass from Ref. [19] for the case
of a local contact potential and relies on mean field theory, $\langle
a\rangle=0$, but nevertheless incorporates the non-commutativity of the LLL
exactly. It is comparable to the value obtained with exact diagonalization,
which is $[0.095(5)]\,2\pi U_{0}\bar{\rho}/\nu\approx(0.38)\pi
U_{0}\bar{\rho}$. [21, 22]. We note, however, that the effective mass can in
principle depend on the magnetic field, $m_{*}(\delta\hat{B})$, which will
lead to corrections to our result, particularly for states far away from the
compressible state (small $p$) and very close to the compressible state (large
$p$), where gauge fluctuations likely play an important role. In Section 5, we
will comment further on the physics leading to such a magnetic field dependent
mass and how they could be handled theoretically to improve our result.

We now briefly comment on the states with negative composite fermion filling,
$p<0$, which correspond to $\nu>1$. For $p=-1$, which in HLR theory
corresponds to a superfluid, $\delta F_{xy}$ diverges, and so too does the gap
in Eq. (3.31). This indicates a singularity of the non-commutative field
theory, which presupposes $|\Theta|$ as the minimum uncertainty of the $x$ and
$y$ coordinates [50, 56]. In other words, all field strengths are cut off by
the non-commutativity of space at scales of $1/|\Theta|$. The non-commutative
field strength, $|\delta F_{xy}|$, continues to exceed $1/|\Theta|$ until
$p=-2$, which in HLR corresponds to the $\nu=2$ bosonic integer quantum Hall
state [57], and so the non-commutative composite fermion field theory is
problematic for $-2<\nu_{\mathrm{CF}}\leq 0$.

Before concluding this section, we note that it would have been quite
challenging to directly confirm that the magnetic field felt by the underlying
bosonic charges is indeed $B+\delta\hat{B}$ using the Pasquier-Haldane-Read
formalism, as the composite fermion operators become rectangular matrices when
the filling deviates from $\nu=1$. However, the correspondence with the simple
charge-vortex dipole model makes it clear that this is the only valid option.
Moreover, we will demonstrate that the topological orders associated with the
states at filling $\nu_{\mathrm{CF}}=p$ correspond precisely to that of the
$p^{\mathrm{th}}$ bosonic Jain state. This is the topic we now turn to.

##  4 The bosonic Jain sequence: Universal features

###  4.1 Fluctuations and Hall response

Having concluded that doping the non-commutative composite fermion field
theory, Eq. (2.17), leads to the bosonic Jain sequence of fractional quantum
Hall states, we are now prepared to assess their universal properties, which
for ordinary composite fermions are encoded in a Chern-Simons effective field
theory for the gauge fluctuations. In the original work of Lopez and Fradkin
[4] (based on the usual flux attachment transformation to composite fermions
without restricting to the LLL), the Chern-Simons effective action was
directly calculated by integrating out the composite fermions and expanding
the resulting functional determinant. We follow the same logic for the
composite fermions obtained in the present LLL construction. In this section,
we present a physically transparent derivation, in which we consider linear
response starting from the first quantized dipole Hamiltonian, Eq. (3.28), and
match the result to an effective _non-commutative_ Chern-Simons theory. In
Appendix B, we present a more formal derivation by calculating the
polarization tensor that determines the quadratic part of the effective action
for the gauge fields when the composite fermions are integrated out. We
perform our calculations without fixing to a particular gauge, so non-
commutative gauge invariance is manifest throughout.

We begin by introducing fluctuations into the Hamiltonian in Eq. (3.28). This
amounts to replacing the vortex guiding center coordinates, $R^{v}_{i}$, with
their covariant counterparts,

$\displaystyle R^{v}_{i}c=c\star X_{i}[a]\,,$ (4.1)

where

$\displaystyle X^{i}[a]=x^{i}-\Theta\,\varepsilon^{ij}a_{j}$ (4.2)

are the covariant coordinates for the fluctuating gauge field. Under a right-
acting gauge transformation, Eq. (2.25), they transform as

$\displaystyle X^{i}\rightarrow U_{R}^{\dagger}\star X^{i}\star U_{R}\,,$
(4.3)

and their commutator is

$\displaystyle[X_{i},X_{j}]_{\star}=i\Theta(\varepsilon_{ij}-\Theta
f_{ij})\,,$ (4.4)

although we will be primarily interested in situations where
$f_{ij}=\partial_{i}a_{j}-\partial_{j}a_{i}+i[a_{i},a_{j}]_{\star}=0$. With
this new definition for $R^{v}_{i}$, the covariant derivative continues to
take the form of Eq. (3.27). Therefore, the single particle Hamiltonian
retains the form of Eq. (3.28), but now it is understood to transform in the
adjoint representation under both left and right-acting gauge transformations.

We now study the response of the charges and vortices to the physical and
emergent electric fields, using the first quantized description. If $V_{b}[R]$
and $V_{v}[R^{v}]$ are the scalar potentials felt by the physical charges and
the vortices respectively, the Hamiltonian is

$\displaystyle H$
$\displaystyle=\frac{1}{2m_{*}}\frac{1}{\ell_{B}^{4}}(R_{i}-R_{i}^{v})^{2}+V_{b}[R_{i}]+V_{v}[R^{v}_{i}]\,.$
(4.5)

Because both $R_{i}$ and $R^{v}_{i}$ are defined in terms of the covariant
coordinates, the scalar potentials are each adjoints under gauge
transformations, as in Eq. (3.22). In terms of the field theory
representation, we may therefore consider them as background values of $\delta
A_{0}$ and $a_{0}$. We emphasize that the inclusion of scalar potentials for
both the charges and vortices is essential, since we will see that in order to
satisfy the constraints implemented by the gauge fluctuations, as in Eq.
(2.19), the vortices and the charges will simultaneously exhibit a Hall
effect. The same phenomenon occurs in ordinary flux attachment: within the FQH
state, establishing an electric field for the physical charges leads to an
average electric field for the emergent (statistical) gauge field.

The introduction of the scalar potentials leads to currents of boson charges
and vortices. In the Heisenberg picture, the world-line of a boson (vortex) is
$R_{i}(t)$ ($R^{v}_{i}(t)$), so we define the current densities,

$\displaystyle
j^{b}_{i}=\bar{\rho}\,\left\langle\frac{dR_{i}}{dt}\right\rangle$
$\displaystyle,\,\,j^{v}_{i}=\bar{\rho}\,\left\langle\frac{dR^{v}_{i}}{dt}\right\rangle\,.$
(4.6)

In particular, we consider $j_{i}^{b}$ (which is an adjoint under left-acting
gauge transformations), to be the physical current density. We also define the
physical and emergent electric fields as

$\displaystyle E^{i}$
$\displaystyle=-\frac{i}{\ell_{B+\delta\hat{B}}^{2}}\,\left\langle\varepsilon^{ij}\left[R_{j},V_{b}[R]\right]\right\rangle\equiv-\left\langle\partial_{R_{i}}V_{b}[R]\right\rangle\,,$
(4.7) $\displaystyle e^{i}$
$\displaystyle=\frac{i}{\ell_{B}^{2}}\left\langle\varepsilon^{ij}\left[R_{j}^{v},V_{v}[R^{v}]\right]\right\rangle\equiv-\left\langle\partial_{R_{i}^{v}}V_{v}[R^{v}]\right\rangle\,,$
(4.8)

For constant magnetic fields (as in the case of interest), these are simply
the gradients of the scalar potentials. One can confirm the physically
intuitive conclusion that $E$ is the physical electric field by noticing that,
if we write $V_{b}[Y]=\delta A_{0}(\mathbf{x})$, where $Y(\mathbf{x})$ is the
covariant coordinate and $\delta A_{0}(\mathbf{x})$ is a static scalar
potential, then we can use the star commutator with $Y$ to relate

$\displaystyle-\ell_{B+\delta\hat{B}}^{2}\,\partial_{Y_{i}}V_{b}[Y]=\ell_{B}^{2}\,\delta
F_{0i}\,,$ (4.9)

since $\delta A_{i}$ is time-independent. For a uniform electric field, this
leads to

$\displaystyle\delta F_{0i}=\frac{1}{1+\delta\hat{B}/B}E_{i}\,.$ (4.10)

One can check using the formulas in e.g. Ref. [19] that this is the relation
between the non-commutative electric field, $\delta F_{0i}$, and the ordinary
Abelian electric field of the Seiberg-Witten map. Since we have already
commented on how the Seiberg-Witten gauge field satisfies proper flux
quantization, this means that $E$ can be considered the physical electric
field.

We compute the DC response to uniform electric fields by solving the
Heisenberg equations of motion. To make the equations more compact, we
reintroduce the notation,
$p^{\mathrm{CF}}_{i}=\frac{1}{\ell_{B}^{2}}\varepsilon^{ij}(R^{v}_{j}-R_{j})$,
and we drop terms which vanish for $\langle f_{ij}\rangle=0$. Then we obtain

$\displaystyle\frac{dp^{\mathrm{CF}}_{i}}{dt}$
$\displaystyle=\frac{1}{i}\,[p^{\mathrm{CF}}_{i},H]=\delta
F_{xy}\,\varepsilon_{ij}\frac{p^{\mathrm{CF}}_{j}}{m_{*}}-\frac{1}{\ell_{B}^{2}}\left(\ell^{2}_{B+\delta\hat{B}}\,\partial_{R_{i}}V_{b}[R]+\ell_{B}^{2}\,\partial_{R^{v}_{i}}V_{v}[R^{v}]\right)\,,$
(4.11) $\displaystyle\frac{dR_{i}}{dt}$
$\displaystyle=\frac{1}{i}[R_{i},H]=\frac{\ell_{B+\delta\hat{B}}^{2}}{\ell_{B}^{2}}\frac{p^{\mathrm{CF}}_{i}}{m_{*}}-\ell_{B+\delta\hat{B}}^{2}\,\varepsilon_{ij}\partial_{R_{j}}V_{b}[R]\,,$
(4.12) $\displaystyle\frac{dR^{v}_{i}}{dt}$
$\displaystyle=\frac{1}{i}[R^{v}_{i},H]=\frac{p_{i}^{\mathrm{CF}}}{m_{*}}+\ell_{B}^{2}\,\varepsilon_{ij}\partial_{R^{v}_{i}}V_{v}[R^{v}]\,.$
(4.13)

The first equation determines the composite fermion drift velocity,
$p^{\mathrm{CF}}/m_{*}$, while the latter two equations determine the
individual charge and vortex responses.

In addition to the equations of motion, the theory also has the constraint in
Eq. (2.19), along with the equation of motion for $a_{i}$, which in the field
theory sets $\langle J_{R}\rangle=\frac{i}{2m}\langle c^{\dagger}\star
Dc-(Dc)^{\dagger}\star c\rangle=0$. Physically, we can understand these
constraints as the requirement that the vortices are fixed to have filling
$\nu=-1$. In the Pasquier-Haldane-Read language, this is the requirement that
the number of vortex orbitals is fixed to the number of physical bosons, even
on tuning the physical filling away from $\nu=1$. The constraint can therefore
be recast as the requirement that the vortices have unit Hall conductivity,

$\displaystyle j_{v}^{i}=-\frac{1}{2\pi}\varepsilon^{ij}e_{j}\,.$ (4.14)

Plugging this back into Eq. (4.13), we see that this is equivalent to the
statement,

$\displaystyle\langle p^{\mathrm{CF}}\rangle$ $\displaystyle=0\,.$ (4.15)

Because $p^{\mathrm{CF}}$ is proportional to the (gauge covariant) composite
fermion dipole moment, the boson and the vortex coordinates sit on top of each
other. With this constraint, Eq. (4.11) leads to a relation between the
physical and emergent electric fields,

$\displaystyle
E_{i}=-\frac{\ell_{B}^{2}}{\ell_{B+\delta\hat{B}}^{2}}e_{i}=-\frac{p+1}{p}\,e_{i}\,.$
(4.16)

Looking to Eq. (4.12), we immediately obtain the physical Hall conductivity,

$\displaystyle j_{b}^{i}$
$\displaystyle=\frac{1}{2\pi}\bar{\rho}\,\ell_{B+\delta\hat{B}}^{2}\,\varepsilon^{ij}E_{j}=\frac{1}{2\pi}\frac{p}{p+1}\varepsilon^{ij}E_{j}\,.$
(4.17)

Introducing units, the Hall conductivity is

$\displaystyle\sigma_{xy}=\nu\,\frac{e^{2}}{h}=\frac{p}{p+1}\frac{e^{2}}{h}\,.$
(4.18)

Hence the non-commutative composite fermion theory indeed leads to the correct
Hall conductivity for the bosonic Jain sequence states!

###  4.2 Non-commutative Chern-Simons theory

Equipped with the result for the Hall conductivity in Eq. (4.18) and the
relation between the electric fields in Eq. (4.16), we can construct an
effective Chern-Simons action at long wavelengths that reproduces them as the
equations of motion,

$\displaystyle S_{\mathrm{eff}}=-i\int
d^{2}\mathbf{x}d\tau\,\left[\frac{p+1}{4\pi}ada+\frac{p}{2\pi}adA^{\prime}+\frac{p}{4\pi}A^{\prime}dA^{\prime}+\mathcal{O}(\Theta)\right]\,,$
(4.19)

where we use the notation
$AdB=\varepsilon^{\mu\nu\lambda}A_{\mu}\partial_{\nu}B_{\lambda}$. To connect
with the discussion above, $a_{\mu}$ is simply the fluctuating gauge field,
while $A^{\prime}=(A^{\prime}_{0}(\mathbf{x}),0)$ is a probe field on top of
the background field $\delta A$ that gives rise to the (physical) electric
field, $E_{i}=-\partial_{i}A^{\prime}_{0}$. The equation of motion for $a_{i}$
reproduces the relation in Eq. (4.16), and integrating out $a$ altogether
returns the Hall response in Eq. (4.18).

Importantly, at no point have we actually invoked the Seiberg-Witten map to an
Abelian gauge theory. Indeed, the electric fields used in the analysis of
Section 4.1 transform as adjoints under the _non-commutative_ gauge
symmetries, Eqs. (2.21) and (2.25). Therefore, the true effective action
should display full non-commutative gauge invariance. Attempting to construct
such an action leads to significant complications: because the left and right-
acting gauge transformations are non-Abelian, there appears to be no gauge
invariant mutual Chern-Simons term that may be represented in terms of star
products of local operators. We will comment more on the pursuit of a non-
commutative mutual Chern-Simons term in Section 4.3.

For the purposes of diagnosing the topological order555Statements about
topological order here have the caveat that $a$ couples to a fermion field in
the fundamental representation. For ordinary commutative gauge theories, this
can be formally captured by viewing $a$ as a spinc connection rather than as
an ordinary $U(1)$ gauge field. For an explanation of this concept in a
condensed matter context, see Refs. [58, 59, 60, 13]. For the topological
order, this means that the quasiparticle statistics are shifted by $\pi$
compared to the usual formulas (equivalently, there is understood to be an
additional spin-$1/2$ Wilson line)., we may simply set $A^{\prime}=0$ and
construct a gauge invariant action for the fluctuating gauge field, $a$. The
only such action that can be represented in terms of star products of local
operators is the non-commutative Chern-Simons (NCCS) theory [23, 24],

$\displaystyle S_{\mathrm{NCCS}}$ $\displaystyle=-\int
d^{2}\mathbf{x}d\tau\,\frac{i(p+1)}{4\pi}\,\varepsilon^{\mu\nu\lambda}\left[a_{\mu}\star\partial_{\nu}a_{\lambda}+\frac{2i}{3}a_{\mu}\star
a_{\nu}\star a_{\lambda}\right]\,.$ (4.20)

As in non-Abelian gauge theories, gauge invariance dictates that this theory
has a cubic interaction term even though the gauge group is $U(1)$. In fact,
we can motivate the presence of the cubic term using the relation in Eq.
(4.16). If we turn off $E$, this equation becomes (dropping the brackets)

$\displaystyle 0$
$\displaystyle=-\partial_{R^{v}_{i}}V_{v}[R^{v}]=\frac{i}{\ell_{B}^{2}}\varepsilon^{ij}\left[R_{j}^{v},V_{v}[R^{v}]\right]\,.$
(4.21)

Now if we identify $V_{v}[X]$ with a fluctuation of $a_{0}(\mathbf{x})$ and
take $a_{i}(\mathbf{x})$ to be static, the commutator with the covariant
coordinate, $X_{i}$, gives the field strength,

$\displaystyle 0$
$\displaystyle=-\partial_{i}a_{0}+i[a_{0},a_{i}]_{\star}=f_{0i}\,.$ (4.22)

This matches the equation of motion of the NCCS theory, where the commutator
originates from differentiating the cubic term.

The NCCS theory, particularly in its representation of a matrix model [23,
24], has been extensively discussed as a short-wavelength description of
fractional quantum Hall phases. However, the connection of these models to
realistic microscopic Hamiltonians has been obscure. What is unique here is
that we have obtained this theory as a _long-wavelength_ effective field
theory of the bosonic Jain states that incorporates the non-commutativity of
the lowest Landau level. Our result thus explains the connection between the
non-commutative Chern-Simons theory and realistic microscopic models of
quantum Hall phases, which until now was poorly understood. Furthermore, we
have found that the non-commutativity of the Chern-Simons theory is set by the
charge density, $\bar{\rho}$ (since $\Theta=-1/2\pi\bar{\rho}$), rather than
the total magnetic field, $B+\delta\hat{B}$, in agreement with Susskind’s
original proposal [23].

The topological ground state properties, such as anyons and their braiding, of
NCCS theory are the same as the familiar Abelian Chern-Simons theory on
commutative space. Indeed, it has been shown that at the classical level the
NCCS action is equivalent to ordinary Chern-Simons action666The non-
commutative theory actually contains gauge transformations that are singular
in the corresponding commutative, Seiberg-Witten mapped theory, which lead to
quantization of the level even on the plane [41, 61, 56]. under the Seiberg-
Witten map [62, 56] (a similar result was derived for the corresponding Wess-
Zumino-Witten models [63]), and perturbative calculations have suggested that
this correspondence extends to the quantum level as well [64]. Furthermore,
the quantum Hall matrix models can be seen to reflect the correct topological
order: for example, for the Laughlin states, Polychronakos demonstrated the
existence of quasihole states with the correct fractional charge [24].

We note also that another argument for the emergence of NCCS theory was made
in Ref. [65], which obtained a cubic interaction with Moyal phase factors
using ordinary composite fermions (as in Ref. [4]) and proposed an emergent
non-commutative gauge symmetry. Our conclusion contrasts with the result in
Ref. [65], since the non-commutative gauge symmetry we consider is
incorporated _a priori_ into the parent microscopic theory. Indeed, we do not
believe there is any reason for non-commutative gauge symmetry to emerge
unless it is inhereted from a short distance, LLL theory.

There is also an interesting parallel between the first quantized composite
fermion Hamiltonian we considered in Eq. (4.5), which is stated in terms of
covariant world-line coordinates, and the quantum Hall matrix models. Indeed,
covariant worldline coordinates are also the basic variables in the matrix
model description of non-commutative Chern-Simons theory, in which the
definition of the covariant coordinates as non-commuting coordinates plus
gauge fields is implemented dynamically [23, 24]. However, unlike the usual
analysis of the matrix models, we did not introduce a regulator at long
distances in order to convert the covariant coordinate operators to finite-
dimensional matrices. It would be interesting in the future to explore what
can be learned from applying such an approach to our composite fermion
Hamiltonian.

Before moving on, we pause to make a technical comment regarding the
specification of the topological order described by Eq. (4.20). In discussing
ordinary non-Abelian Chern-Simons theory, one must be careful to specify a
regularization. Due to the cubic interaction, the choice of regularization at
short distances can lead to a one-loop exact shift in the non-Abelian Chern-
Simons level [66, 67, 68], which is matched by an analogous quantum shift in
in the corresponding Wess-Zumino-Witten model, where it appears in the
computation of the central charge. The NCCS theory is no different.
Perturbative calculations using a Maxwell regulator have found a level shit of
$k\rightarrow k+\operatorname{sgn}(k)$ in $U(1)$ NCCS theory with level $k$
[69], although this does not affect the topological order [70]. The same
shift777We note that Ref. [61] found the one-loop shift due to the Maxwell
regulator to be $k\rightarrow k+2\operatorname{sgn}(k)$, based on some
differences in normalization with Ref. [69]. This difference can be properly
settled by computing the free energy in the large-$k$ limit using the
background field formalism, as in Ref. [67]. We leave this for future work.
arises in the matrix models, where it comes from normal ordering a constraint
[24], and in this case it does affect the topological order (and shift the
filling fraction accordingly). For our purposes, the NCCS level in Eq. (4.20)
is meant to reflect the full quantum Chern-Simons level, i.e. we implicitly
choose a regulator in which no such shift appears.

###  4.3 Toward a non-commutative mutual Chern-Simons theory

We now revisit the question of how to construct an effective Chern-Simons
action for both the background ($\delta A$) and fluctuating ($a$) gauge
fields. As we commented above, because the left and right-handed non-
commutative $U(1)$ gauge symmetries are not Abelian, there is no _local_ gauge
invariant mutual Chern-Simons term. However, it should be possible to
construct a non-local mutual Chern-Simons term, one which leverages the
inherent non-locality of field theories on non-commutative space.
Unfortunately, despite much effort, particularly in Refs. [71, 72], this term
has proven elusive. While we will not completely solve this problem here, we
will propose an action (which is not necessarily the generalization of Eq.
(4.19)) that is at least gauge invariant to $\mathcal{O}(\Theta)$. We expect
that the intuition underlying this construction may prove useful in the
pursuit of a full solution to this problem.

The basic problem with constructing a mutual Chern-Simons theory on non-
commutative space is the same as in ordinary non-Abelian gauge theory: gauge
invariance would necessitate that both participating gauge fields transform
simultaneously, which does not appear possible by definition. However, with
covariant coordinates, it is possible to _induce_ left-acting gauge
transformations on the right-handed gauge field and _vice versa_. For example,
right-acting gauge transformations act as APDs on $\delta A[X]$,

$\displaystyle\delta A[X]$ $\displaystyle\rightarrow\delta
A\left[U(\mathbf{x})\star X\star
U^{\dagger}(\mathbf{x})\right]=U(\mathbf{x})\star\delta A[X]\star
U^{\dagger}(\mathbf{x})\,,$ (4.23)

and analogously for $a[Y]$.

Leveraging this property, we can define a non-commutative Chern-Simons action
that is gauge invariant to $\mathcal{O}(\Theta)$ under both left and right-
acting gauge transformations by using the combination $a[Y]+\delta A[X]$ ,

$\displaystyle S_{\mathrm{NCCS}}\left[a[Y]+\delta A[X]\right]\,,$ (4.24)

where $S_{\mathrm{NCCS}}$ is defined in Eq. (4.20). Notice that such an action
is fully gauge invariant under right (left) gauge transformations if $Y$ ($X$)
is replaced with the coordinate $\mathbf{x}$, at the cost of breaking
invariance under the other gauge group.

The reason the action in Eq. (4.24) is only invariant to $\mathcal{O}(\Theta)$
stems from the fact that it is non-local. Full gauge invariance under e.g.
right-acting gauge transformations would require

$\displaystyle\delta A[X]\rightarrow U^{\dagger}[Y]\star\delta A[X]\star
U[Y]=U^{\dagger}(\mathbf{x})\star\delta A[X]\star
U(\mathbf{x})+\mathcal{O}(\Theta^{2})\,,$ (4.25)

where the $\mathcal{O}(\Theta^{2})$ term is non-vanishing. Resolving this
issue would require introducing new Wilson line-like operators $V,W$ which
transform under both left and right gauge transformations as follows

$\displaystyle U(1)_{L}$ $\displaystyle:\,V\rightarrow U^{\dagger}[X]\star
V\star U(\mathbf{x})\,,\,W\rightarrow U[X]\star W\star U^{\dagger}[X]\,,$
(4.26) $\displaystyle U(1)_{R}$ $\displaystyle:\,V\rightarrow U[Y]\star W\star
U^{\dagger}[Y]\,,\,W\rightarrow U^{\dagger}[Y]\star W\star U(\mathbf{x})\,.$
(4.27)

The gauge invariant Chern-Simons action would then be

$\displaystyle S_{\mathrm{NCCS}}\left[V^{\dagger}\star a[Y]\star
V+W\star\delta A[X]\star W^{\dagger}\right]\,.$ (4.28)

Unfortunately, we have not been able to construct explicit expressions for the
operators, $V$ and $W$, and we leave this for future work. It is also not
clear to us how to construct more general $K$-matrices than $(p,p,p)$ using
this approach. We finally note that a discussion of non-commutative Chern-
Simons theories with such $K$-matrices can be found in Ref. [71]. However, the
authors of that work fiat the mixed transformation laws for each gauge field,
instead of attempting to induce them using covariant coordinates. The theories
discussed in Ref. [71] therefore cannot be obtained using the composite
fermion approach outlined in our work, in which left and right gauge
transformations do not mix.

##  5 Discussion

A major challenge in quantum Hall physics has been to develop a microscopic
theory that is defined in the lowest Landau level and is capable of capturing
both universal and non-universal physics. In this work, we have met this
challenge for the specific case of bosonic Jain sequences at fillings
$\nu=\frac{p}{p+1}$ using a composite fermion construction [17, 18, 19, 20]
that, unlike the standard flux attachment, explicitly lives in the lowest
Landau level. Previous work [18, 19] employed this construction to discuss the
metallic composite fermi liquid state for bosons at $\nu=1$. An effective
field theory description for this state consists of composite fermions coupled
to a $U(1)$ gauge field on the non-commutative space of the LLL. Starting from
this description, we doped the theory away from $\nu=1$ to access the Jain
sequence states. This is achieved by subjecting the composite fermions to a
background, non-commutative magnetic field while holding their density fixed.
Integrating out the composite fermions, we obtained a non-commutative Chern-
Simons field theory, which encodes the topological features of the ground
state. This conclusion significantly clarifies long-standing questions about
the role of non-commutative Chern-Simons theory in the study of the fractional
quantum Hall effect at short distances. It is a low-energy effective theory
arising from integrating out composite fermions in the LLL. It captures the
correct topological order of the ground state but does not contain any
dynamical information on its own. Its non-commutativity is set by the charge
density, as in the original proposal of Susskind [23].

Our microscopic approach incorporates both the universal and non-universal
data of quantum Hall states within a single theoretical framework. As an
important demonstration, we presented an elegant, closed-form expression for
the Jain sequence gaps, invoking only a mean field approximation. It should
also be possible to extract other dynamical features as well, such as the
dispersion of the the GMP mode or the momentum dependence of the static
structure factor. In approaching such calculations, it is important to note
that in non-commutative field theory the density and current operators are not
gauge invariant at finite momentum, meaning that some care will be necessary
to ensure that results are gauge invariant. Another related problem that will
be important to attack in the future is the structure of the mutual Chern-
Simons term in non-commutative field theory, which we expect to lack a
representation in terms of star products of local operators. Another problem
that could be tackled within our description is to study the evolution between
the Jain ststes and a bosonic superfluid state by turning on a periodic
potential in the LLL (along the lines of what was done at filling $\nu=1$ in
Ref. [20]).

The success of our approach for the bosonic Jain sequence invites the question
of how to extend our framework to the fermionic Jain sequences. This would
require a fully LLL theory of the composite Fermi liquid states at even
denominator fillings in fermionic systems. Constructing such a theory is of
great importance. For example, it would shed light on the emergence of a
particle-hole symmetric composite Fermi liquid theory at $\nu=1/2$, like the
Dirac theory proposed by Son [9] (an analogous “reflection symmetry” was
proposed by one of us for the states at $\nu=1/2n$ [10], but the status of
that symmetry on LLL projection in clean systems is an open question; for
alternate proposals in the LLL limit, see Ref. [73]). A non-commutative, LLL
field theory of the $\nu=1/2$ state was proposed recently [74], but further
work is needed in this direction.

We now comment on how our approximate result for the bosonic Jain sequence
gaps in Eq. (3.32) can be improved. In obtaining Eq. (3.32), we took the
composite fermion effective mass, $m_{*}$, to be given by the result of the
Hartree-Fock calculation at $\nu=1$. A better approximation would be to
calculate the effective mass directly at the filling of the Jain state, which
would reveal if the effective mass has a field dependence, $m_{*}=m_{*}(\delta
B)$. This could alter the dependence of the gaps, $\Delta_{p}$, on the Jain
state index, $p$, from the form in Eq. (3.32). Such a calculation of
$m_{*}(\delta B)$ can conceivably be performed within the Hamiltonian theory
of Murthy and Shankar [16]. Indeed, we may regard the Hartree-Fock calculation
within the Hamiltonian theory as providing an improved mean field ansatz on
top of which fluctuation effects can be included using the non-commutative
field theory.

Gauge fluctuations can also lead to field dependence of the effective mass. In
the composite Fermi liquid itself, these fluctuations lead to a diverging
effective mass. As emphasized by HLR in Ref. [6], on moving to proximate Jain
fractions, this divergence will be cut-off at an energy scale given by the
Jain gap, $\Delta_{p}$. This leads to a $\Delta_{p}$ that behaves (for small
$\delta B$ and short ranged interactions) as $\Delta_{p}\sim|\delta
B|^{\frac{3}{2}}\sim|p|^{-3/2}$. However, this asymptotic form is likely to
only be relevant for very small $|\delta B|$, i.e. very large $p$. In the
present problem of the bosonic Jain states, describing the region of large $p$
using the composite Fermi liquid will be additionally problematic, as the true
ground state at $\nu=1$ is the paired Pfaffian state. The large $p$ region
will then involve competition between pairing and Landau level formation of
composite fermions, and this will determine the details of the gap sizes and
other non-universal characteristics. Therefore, for $p$ that is not too large,
we expect that the mean field description used in this work will be adequate.

Finally, a rich subject that we have not yet touched on is the response of
quantum Hall systems to spatial curvature, which straddles universal and
dynamical data [75, 38, 76]. The Hall viscosity, or the parity-odd response to
shears, is not universal _a priori_ , but is determined by the Wen-Zee shift –
a universal quantity – in Galilean invariant quantum Hall states [75, 77, 78].
Indeed, the matrix model regularization of non-commutative Chern-Simons theory
has been shown to yield the correct Hall viscosity (up to an orbital
contribution) for the Laughlin states and some non-Abelian quantum Hall states
[31, 32]. These arguments indicate that the non-commutative Chern-Simons
theories we obtain at low energies will encode the Hall viscosity for the
bosonic Jain states. This can also be checked by adapting the response
calculations in Section 4.1 to finite wave vector, as the Hall viscosity
appears in the coefficient of the $\mathcal{O}(q^{2})$ contribution to the
Hall conductivity [79, 80].

A major open problem in this area is to obtain a microscopic derivation of the
coupling of composite fermions to geometry, the so-called “orbital spin” of
the composite fermion. A framework like ours that can bridge the gap between
short and long-wavelength physics is an ideal platform on which to solve this
problem. However, an obstacle to such a construction is inherent to non-
commutative gauge theories: because of the relationship between gauge symmetry
and area-preserving diffeomorphisms, it is generally not possible to construct
a gauge invariant (or covariant) stress tensor that satisfies a local
continuity equation [81] (a non-commutative equivalent to the Belinfante
procedure is not known to us), and the construction of a Wen-Zee term suffers
from similar challenges to the ordinary mutual Chern-Simons term. Resolving
these questions and developing a non-commutative composite fermion field
theory including a coupling to curvature will be an important direction for
future studies.

## Acknowledgements

We thank Zhihuan Dong, Eduardo Fradkin, Prashant Kumar, and Hong Liu for
discussions and comments on the manuscript. HG is supported by the Gordon and
Betty Moore Foundation EPiQS Initiative through Grant No. GBMF8684 at the
Massachusetts Institute of Technology. This work was supported by NSF grant
DMR-1911666, and partially through a Simons Investigator Award from the Simons
Foundation to Senthil Todadri. This work was also partly supported by the
Simons Collaboration on Ultra-Quantum Matter, which is a grant from the Simons
Foundation (651440, TS).

##  Appendix A Landau level degeneracy on the non-commutative torus

In this Appendix, we sketch the derivation of the degeneracy of Landau levels
on the non-commutative torus, which proceeds in analogy to the derivation on
the ordinary torus. For more details, see Refs. [51, 52, 53, 43], as well as
the extensive literature on field theory on the non-commutative torus and
$T$-duality, which is reviewed in Ref. [33].

Without loss of generality, we choose to work on the square torus and
identify,

$\displaystyle x_{i}\sim x_{i}+2\pi R\,,$ (A.1)

where $R$ is the compactification radius. We define the physical position
operators by exponentiation,

$\displaystyle V_{i}=e^{ix_{i}/R}\,,$ (A.2)

which are invariant under shifts of $2\pi R$. They satisfy the algebra

$\displaystyle V_{1}V_{2}=V_{2}V_{1}e^{-i\Theta/R^{2}}\,.$ (A.3)

We work with the single particle Hamiltonian in Eq. (3.13), but we now choose
to work in Landau gauge,

$\displaystyle(\delta A_{x},\delta A_{y})$ $\displaystyle=(0,\delta
F_{xy}x)\,,$ (A.4)

The covariant derivatives, $D_{i}=\partial_{i}-i\delta A_{i}$, satisfy

$\displaystyle[D_{i},D_{j}]=-i\delta F_{xy}\,.$ (A.5)

Because the covariant derivatives involve $x$ rather than $U_{2}=e^{ix/R}$,
they transform under shifts of $x$ as

$\displaystyle D_{y}\rightarrow D_{y}-i\delta F_{xy}(2\pi R)\,.$ (A.6)

As in the case of the ordinary torus, this shift can be eliminated by a
suitable gauge transformation, but now such gauge transformations are _non-
commutative_. The non-commutativity of the gauge group will ultimately be what
alters the degeneracy from the non-commutative gauge flux.

The covariant derivative transforms as an adjoint under non-commutative (left-
acting) gauge transformations. In Landau gauge, it changes as

$\displaystyle D_{y}\rightarrow U\star D_{y}\star
U^{\dagger}=\partial_{y}+U\star\partial_{y}U^{\dagger}-i\delta F_{xy}\,U\star
x\star U^{\dagger}\,.$ (A.7)

To determine the gauge transformation that cancels the shift in Eq. (A.6), let
$U=e^{-i\alpha y/R}$. Then

$\displaystyle D_{y}\rightarrow\partial_{y}-i\delta
F_{xy}x+i\left(1+\Theta\delta F_{xy}\right)\frac{\alpha}{R}\,.$ (A.8)

To cancel the shift in Eq. (A.6), we therefore must have

$\displaystyle\alpha=\frac{1}{2\pi}\frac{\delta F_{xy}}{1+\Theta\delta
F_{xy}}\times(2\pi R)^{2}=\frac{\delta\hat{B}}{2\pi}\times(\mathrm{Area})\,.$
(A.9)

$\alpha$ is therefore the flux of the “physical” magnetic field defined in Eq.
(3.24) through the torus! Requiring that the gauge transformation $U$ itself
be periodic on the torus therefore yields the flux quantization condition,

$\displaystyle\oint_{T^{2}}\frac{\delta\hat{B}}{2\pi}=\frac{1}{2\pi}\frac{\delta
F_{xy}}{1+\Theta\delta F_{xy}}\times(2\pi R)^{2}=n\in\mathbb{Z}\,.$ (A.10)

Now, under the transformation $U$, the wave function on non-commutative space,
$\Psi[x,y]$, transforms as $\Psi\rightarrow U\star\Psi$, so we seek a complete
set of wave functions with the following properties,

$\displaystyle\Psi[x,y+2\pi R]$ $\displaystyle=\Psi[x,y]$ (A.11)
$\displaystyle\Psi[x+2\pi R,y]$ $\displaystyle=U^{\dagger}\star\Psi[x,y]\,.$
(A.12)

The space of such wave functions constitutes the space of degenerate ground
states on the torus. In ordinary, commutative space, the ground states
correspond to the set of theta functions, and the dimension of the space of
ground states (the LL degeneracy) is given by the flux piercing the torus. In
this case as well, the space of ground states (the fundamental sections of the
non-commutative gauge theory on the torus) has dimension set by the number of
flux quanta, $|n|$ [51, 52, 53]. This completes the argument that the Landau
level degeneracy is set by the magnetic field in Eq. (3.24).

##  Appendix B Chern-Simons effective action

In this Appendix, we show explicitly that integrating out $p$ non-commutative
composite fermion Landau levels leads to an effective Chern-Simons action with
level $p+1$ for the fluctuating gauge field, $a_{\mu}$, as in Eq. (4.19). We
do this by computing the polarization tensor, $\pi_{xy}(\omega)$, in the
uniform ($\mathbf{q}\rightarrow 0$) limit. The other components of the
polarization tensor are then fixed by gauge invariance. Additionally, while we
do not compute the cubic term in the non-commutative Chern-Simons action here
(it is only nonzero at finite wave vector), it is also required to appear by
gauge invariance.

For spatially uniform fluctuations, $a_{i}(\tau)$, the coupling to the
composite fermions is

$\displaystyle\Gamma_{ac^{\dagger}c}$ $\displaystyle=\int
d^{2}\mathbf{x}d\tau\,\frac{i}{2m_{*}}a_{i}(\tau)\left[c^{\dagger}\star
D^{(0)}_{i}c-(D^{(0)}_{i}c)^{\dagger}\star
c\right]\,,\,D^{(0)}_{i}c=\partial_{i}c-i\delta A_{i}\star c\,,$ (B.1)

plus a diamagnetic term, which will not play a role here. We may rewrite this
coupling in terms of the momentum operator,
$p^{\mathrm{CF}}_{i}=-iD_{i}^{(0)}$, which is the same as the operator we
introduced in Section 3.1. Then we may write

$\displaystyle\Gamma_{ac^{\dagger}c}$ $\displaystyle=-\int
d^{2}\mathbf{x}d\tau\,a_{i}(\tau)\,c^{\dagger}\,\frac{p^{\mathrm{CF}}_{i}}{m_{*}}\,c\,.$
(B.2)

We work with a complete basis of non-commutative Landau level eigenstates,
which we denote $\\{|M,\alpha\rangle\\}$, where $M$ is a Landau level index,
and $\alpha$ parameterizes the Landau level degeneracy. The fermion propagator
may then be written as

$\displaystyle G_{M}(\omega)$
$\displaystyle=\sum_{\alpha}\frac{|M,\alpha\rangle\langle M,\alpha|}{i\omega-
E_{M}}\,,$ (B.3)

where $E_{M}=\epsilon_{M}-\mu$, $\mu$ is a chemical potential that fixes the
density to $\bar{\rho}$, and $\epsilon_{M}$ are the Landau level energies, Eq.
(3.16). Using this form for the propagator, along with the vertex in Eq.
(B.2), the polarization tensor may be expressed as

$\displaystyle\pi_{xy}(\omega)$
$\displaystyle=\frac{1}{\mathrm{Area}}\int\frac{d\Omega}{2\pi}\sum_{M,N}\sum_{\alpha,\beta}\frac{\langle
M,\alpha|\frac{p^{\mathrm{CF}}_{x}}{m_{*}}|N,\beta\rangle\langle
N,\beta|\frac{p_{y}^{\mathrm{CF}}}{m_{*}}|M,\alpha\rangle}{[i(\omega+\Omega)-E_{M}][i\Omega-
E_{N}]}\,$ (B.4) $\displaystyle=\frac{i\omega}{\mathrm{Area}}\sum_{M\neq
N}\sum_{\alpha,\beta}\frac{\langle
M,\alpha|\frac{p^{\mathrm{CF}}_{x}}{m_{*}}|N,\beta\rangle\langle
N,\beta|\frac{p_{y}^{\mathrm{CF}}}{m_{*}}|M,\alpha\rangle}{(E_{M}-E_{N})^{2}}(\Theta(-E_{M})-\Theta(-E_{N}))+\mathcal{O}(\omega^{2})\,.$
(B.5)

We now use the fact

$\displaystyle\frac{p^{\mathrm{CF}}_{i}}{m_{\star}}$
$\displaystyle=\frac{1}{i}[R^{v}_{i},H]\,,$ (B.6)

where here $R^{v}_{i}$ is defined (in mean field theory) to act as
$R^{v}_{i}c=c\star x_{i}$. This allows us to rewrite,

$\displaystyle\pi_{xy}(\omega)$
$\displaystyle=\frac{i\omega}{\mathrm{Area}}\sum_{M\neq
N}\sum_{\alpha,\beta}\langle M,\alpha|R_{x}^{v}|N,\beta\rangle\langle
N,\beta|R^{v}_{y}|M,\alpha\rangle(\Theta(-E_{M})-\Theta(-E_{N}))\,.$ (B.7)

We now define the Landau level projection operators,

$\displaystyle\mathcal{P}_{M}=\sum_{\alpha}|M,\alpha\rangle\langle
M,\alpha|\,,$ (B.8)

such that

$\displaystyle\pi_{xy}(\omega)$
$\displaystyle=\frac{i\omega}{\mathrm{Area}}\sum_{M}\operatorname{Tr}_{\alpha}\left[\mathcal{P}_{M}R^{v}_{x}(1-\mathcal{P}_{M})R^{v}_{y}-\mathcal{P}_{M}R^{v}_{y}(1-\mathcal{P}_{M})R^{v}_{x}\right]\Theta(-E_{M})\,,$
(B.9)
$\displaystyle=\frac{i\omega}{\mathrm{Area}}\sum_{M}\operatorname{Tr}_{\alpha}\left[\mathcal{P}_{M}[R^{v}_{x},R^{v}_{y}]+\mathcal{P}_{M}R^{v}_{y}\mathcal{P}_{M}R^{v}_{x}-\mathcal{P}_{M}R^{v}_{x}\mathcal{P}_{M}R^{v}_{y}\right]\,.$
(B.10)

Here $\operatorname{Tr}_{\alpha}$ is the trace over the degenerate indices. To
compute the projected vortex coordinates,
$\mathcal{P}_{M}R_{i}^{v}\mathcal{P}_{M}$, we recall
$p^{\mathrm{CF}}_{i}=\frac{1}{\ell_{B}^{2}}\varepsilon_{ij}(R^{v}_{j}-R_{j})$,
and we introduce operators,

$\displaystyle r^{\mathrm{CF}}_{i}$
$\displaystyle=\frac{1}{2}(R^{v}_{i}+R_{i})\,,$ (B.11)
$\displaystyle\tilde{p}_{i}^{\mathrm{CF}}$
$\displaystyle=\frac{2\ell_{B}^{2}}{\ell_{B+\delta\hat{B}}^{2}+\ell_{B}^{2}}p_{i}^{\mathrm{CF}}\,,$
(B.12) $\displaystyle\mathcal{D}_{i}$ $\displaystyle=\frac{1}{1-\theta\delta
B_{\mathrm{CF}}}(\tilde{p}_{i}^{\mathrm{CF}}-\delta
B_{\mathrm{CF}}\varepsilon_{ij}r^{\mathrm{CF}}_{j})\,,$ (B.13)
$\displaystyle\theta$
$\displaystyle=\frac{1}{4}(\ell_{B}^{2}-\ell_{B+\delta\hat{B}}^{2})\,,\,\delta
B_{\mathrm{CF}}=4\frac{\ell_{B}^{2}-\ell^{2}_{B+\delta\hat{B}}}{(\ell^{2}_{B+\delta\hat{B}}+\ell_{B}^{2})^{2}}\,,$
(B.14)

which satisfy the same algebra as the corresponding operators in Section 3.1.
Importantly, $\mathcal{D}_{i}$ commutes with $p_{i}^{\mathrm{CF}}$ and
therefore the Hamiltonian, meaning that it survives Landau level projection.
As discussed in Section 3.1, it also satisfies the algebra,

$\displaystyle[\mathcal{D}_{x},\mathcal{D}_{y}]$
$\displaystyle=-i\delta\hat{B}\,.$ (B.15)

In terms of these operators, we can express $R^{v}_{i}$ as

$\displaystyle R^{v}_{i}=\varepsilon_{ij}\left[\frac{1-\theta\delta
B_{\mathrm{CF}}}{\delta B_{\mathrm{CF}}}\mathcal{D}_{j}-\frac{1}{\delta
B_{\mathrm{CF}}}\left(1+\frac{\ell^{2}_{B}-\ell_{B+\delta\hat{B}}^{2}}{\ell_{B}^{2}+\ell_{B+\delta\hat{B}}^{2}}\right)\tilde{p}_{j}^{\mathrm{CF}}\right]\,.$
(B.16)

Because the creation and annihilation operators are built out of
$p^{\mathrm{CF}}$, the second term vanishes on projection, i.e.

$\displaystyle\mathcal{P}_{M}R^{v}_{i}\mathcal{P}_{M}=\frac{1-\theta\delta
B_{\mathrm{CF}}}{\delta
B_{\mathrm{CF}}}\varepsilon_{ij}\mathcal{D}_{j}=\frac{1}{\delta\hat{B}}\,\varepsilon_{ij}\mathcal{D}_{j}\,.$
(B.17)

Thus, since $\mathcal{D}$ commutes with $p^{\mathrm{CF}}$ and the degeneracy
of each Landau level is $\frac{\delta\hat{B}\times\mathrm{Area}}{2\pi}$, if
$p$ Landau levels are filled,

$\displaystyle\pi_{xy}(\omega)$
$\displaystyle=\frac{i\omega}{\mathrm{Area}}\sum_{M}\operatorname{Tr}_{\alpha}\left[[R^{v}_{x},R^{v}_{y}]-\frac{1}{\delta\hat{B}^{2}}[\mathcal{D}_{x},\mathcal{D}_{y}]\right]\Theta(-E_{M})$
(B.18) $\displaystyle=-\frac{\omega}{\mathrm{Area}}\times
p\times\frac{\delta\hat{B}\times\mathrm{Area}}{2\pi}\times\left(\ell_{B}^{2}+\frac{1}{\delta\hat{B}}\right)\,.$
(B.19)

But, due to the constraint, the composite fermion density is fixed to
$\bar{\rho}=\frac{B}{2\pi}$, so $\delta\hat{B}\ell_{B}^{2}=1/p$, and this
result becomes

$\displaystyle\pi_{xy}(\omega)$ $\displaystyle=-\frac{\omega}{2\pi}(p+1)\,.$
(B.20)

The resulting Chern-Simons effective action is therefore

$\displaystyle S_{\mathrm{eff}}$ $\displaystyle=-\int
d^{2}\mathbf{x}d\tau\,\frac{i(p+1)}{4\pi}ada+\mathcal{O}(\Theta)\,,$ (B.21)

reflecting the correct topological order for the $\nu=p/(p+1)$ Jain state.
Notably, the constraint has played an essential role in generating a properly
quantized Chern-Simons level.

We now comment on the background and mutual Chern-Simons terms. For a
spatially uniform background probe field, $A_{i}^{\prime}(\tau)$, the coupling
to the composite fermions is the same as for $a_{i}(\tau)$, as in Eq. (B.2).
The calculation of these terms is therefore identical, and their coefficients
are also $p+1$. However, one should be careful in the interpretation of this
result, since $\partial_{0}A^{\prime}_{i}$ is not the exactly the physical
electric field, as discussed in Section 4.1. Indeed, $A^{\prime}$ does not
even transform covariantly under left-acting $U(1)$ gauge transformations.
Instead, it satisfies the modified infinitesimal transformation law,

$\displaystyle A^{\prime}_{i}\rightarrow
A^{\prime}_{i}+i\,\partial_{i}\lambda+i[\lambda,\delta
A_{i}+A^{\prime}_{i}]_{\star}\,.$ (B.22)

This means that the Seiberg-Witten map cannot be applied directly to the
probe, $A^{\prime}_{i}$, and it must be modified accordingly. Rather than
doing so here, particularly given the difficulties with constructing a gauge
invariant mutual Chern-Simons term, we leave this to future work. Instead, we
emphasize the physically transparent derivation of the Hall conductivity in
Section 4.1, which should be consistent with such an analysis.

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