# Emergence of Dirac Composite Fermions: Dipole Picture

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

###### Abstract

Composite fermions (CFs) are the particles underlying the novel phenomena
observed in partially filled Landau levels. Both microscopic wave functions
and semi-classical dynamics suggest that a CF is a dipole consisting of an
electron and a double $2h/e$ quantum vortex, and its motion is subject to a
Berry curvature that is uniformly distributed in the momentum space. Based on
the picture, we study the electromagnetic response of composite fermions. We
find that the response in the long-wavelength limit has a form identical to
that of the Dirac CF theory. To obtain the result, we show that the Berry
curvature contributes a half-quantized Hall conductance which, notably, is
independent of the filling factor of a Landau level and not altered by the
presence of impurities. The latter is because CFs undergo no side-jumps when
scattered by quenched impurities in a Landau-level with the particle-hole
symmetry. The remainder of the response is from an effective system that has
the same Fermi wavevector, effective density, Berry phase, and therefore long-
wavelength response to electromagnetic fields as a Dirac CF system. By
interpreting the half-quantized Hall conductance as a contribution from a
redefined vacuum, we can explicitly show the emergence of a Dirac CF effective
description from the dipole picture. We further determine corrections due to
electric quadrupoles and magnetic moments of CFs and show deviations from the
Dirac CF theory when moving away from the long wavelength limit.

##  I Introduction

The fractional quantum Hall (FQH) effect, a phenomenon discovered nearly four
decades ago [1], remains unique as the only topological effect driven by
electron correlations and observed in laboratories. In two dimensions,
electrons subjected to a strong magnetic field are confined in a Landau level
(LL). In such a system, the kinetic energy is completely quenched, and the
electron-electron interaction dominates its behavior. Surprisingly, according
to the theory of composite fermions [2], the system can be interpreted as a
non-interacting or weakly interacting system consisting of fictitious
particles called composite fermions (CFs). It is hypothesized that these
particles are bound states of electrons and quantum vortices, reside in a
hidden Hilbert space, and form simple quantum states [3]. The theory of CFs
prescribes an ansatz for mapping a state in the hidden Hilbert space to a
quantum state of a partially filled LL in the real world. The theory is very
successful: by interpreting FQH states as integer quantum Hall states of free
CFs, wave functions constructed from the ansatz reliably achieve high overlaps
with those determined by exact diagonalizations. The theory also prompts
experimental searches for CFs. In various geometric resonance experiments and
ballistic transport experiments [4, 5, 6, 7], CFs do exhibit behaviors
expected for genuine particles. One could treat CFs as if they are elementary
particles in the hidden Hilbert space, just like electrons are to the real
world.

To treat CFs as the elementary particles of the hidden Hilbert space, one
needs an effective theory to specify their dynamics. For a long time, the de
facto standard effective theory for CFs is the one proposed by Halperin, Lee,
and Read (HLR) [8]. The theory underlies the designs and interpretations of
most CF experiments [4, 5]. In the HLR theory, CFs feel an effective magnetic
field and behave just like ordinary particles by following the ordinary
Newtonian dynamics. Such a picture naturally emerges from the Chern-Simons
(CS) field theory of CFs which tries to justify the CF picture by interpreting
the electron-vortex binding as a singular CS gauge transformation [9, 10, 11].
However, the CS theory disregards the fact that the physics occurs essentially
in the projected Hilbert space of a partially filled LL. It contains artifacts
which have to be eliminated in the HLR theory. For instance, to remove the
unwelcome presence of the bare band mass of electrons without introducing
inconsistencies, one has to assume phenomenologically that a mass
renormalization is accompanied by a corresponding renormalization to the
effective interaction between CFs [12, 13]. Another obvious flaw, i.e., the
absence of a half-quantized CF Hall conductance expected for a half-filled LL
with the particle-hole symmetry [14], is remedied only recently by Wang et
al., who show that spatial fluctuations of the effective magnetic field can
induce CF scattering with side jumps and gives rise to a half-quantized CF
Hall conductance [15].

These difficulties prompt Son to propose the Dirac CF theory [16]. Besides the
aforementioned issues, Son also notices that the HLR theory does not exhibit
explicitly the particle-hole symmetry expected for a LL in the zero-mass
limit. As an effort to address the issue, Son conjectures that CFs are
massless Dirac particles. The interpretation of CFs in the Dirac CF theory
differs in many ways from that in the HLR theory. Most notably, a Dirac CF
traversing a Fermi circle acquires a $\pi$-Berry phase, which is absent in the
HLR theory. It leads to the predictions of suppression of back-scattering of
CFs [17] and an extra phase in SdH oscillations [18]. Moreover, the theory
shows the swap of the notions of density and magnetic field, i.e., the density
of Dirac CFs is set by the magnetic field, while the effective magnetic field
felt by CFs is set by the electron density. This is because Dirac CFs are
actually particles dual to electrons in a charge-neutral massless Dirac cone
[19]. This is distinct from the HLR theory in which CFs remain electron-like,
having the same density, Fermi wavevector and dynamics.

One of manifest distinctions between the two effective theories is in how the
conductivity tensor of electrons $\sigma(\omega)$ is related to that of CFs
$\tilde{\sigma}(\omega)$. In the HLR theory, the two conductivity tensors are
related by a simple resistivity-shifting rule [8, 2]

$\sigma^{-1}(\omega)=\tilde{\sigma}^{-1}(\omega)+\sigma_{\mathrm{CS}}^{-1},$
(1)

with

$\sigma_{\textrm{CS}}=\frac{e^{2}}{2h}\left(\begin{array}[]{cc}0&1\\\
-1&0\end{array}\right),$ (2)

whereas in the Dirac CF theory, they are related by [16]

$\sigma(\omega)=-\sigma_{\textrm{CS}}\tilde{\sigma}^{-1}(\omega)\sigma_{\textrm{CS}}+\sigma_{\textrm{CS}}\beta(\omega)$
(3)

with

$\beta(\omega)=\left(\begin{array}[]{cc}1&-\mathrm{i}\omega/\tilde{\omega}_{B}\\\
\mathrm{i}\omega/\tilde{\omega}_{B}&1\end{array}\right),$ (4)

where $\tilde{\omega}_{B}\equiv eB/m^{\ast}$ is a characteristic frequency set
by the external magnetic field $\bm{B}=-B\hat{z}$. The relation is established
in the long wavelength limit, and depends on a CF effective mass parameter
$m^{\ast}$ which is phenomenologically introduced to recover the Galilean
invariance for the Dirac CF theory [16, 20]. As a result of the distinction,
the two theories will predict different electromagnetic (EM) responses of CF
systems.

An alternative effective description parallel to these theories is the dipole
picture proposed by Read [21]. In the picture, a CF is not a point particle
but a dipole consisting of an electron and a double $2h/e$ quantum vortex
which is spatially separated from the electron. Intriguingly, the momentum of
a CF, which is proportional to its velocity, can be interpreted as the
transverse of its dipole vector. The picture emerges naturally from the
Rezayi-Read wave function $\Psi(\\{z_{a}\\})\propto
J(\\{z_{a}+\mathrm{i}k_{a}l_{\mathrm{B}}^{2}\\})\exp(\mathrm{i}\sum_{a}\bar{k}_{a}z_{a}/2)$
which is obtained by applying the CF ansatz
$\Psi(\\{z_{a}\\}=\hat{P}_{\mathrm{LLL}}J(\\{z_{a}\\})\psi_{\mathrm{CF}}(\\{\bm{r}_{a}\\})$
to a CF wave function
$\psi_{\mathrm{CF}}(\\{\bm{r}_{a}\\})=\exp(\mathrm{i}\sum_{a}\bm{k}_{a}\cdot\bm{r}_{a})$
for a set of free CFs with wavevectors $\\{\bm{k}_{a}\\}$ in the hidden
Hilbert space [22], where we denote $z_{a}=x_{a}+\mathrm{i}y_{a}$ for
$\bm{r}_{a}\equiv(x_{a},y_{a})$, $k_{a}=k_{ax}+\mathrm{i}k_{ay}$ for
$\bm{k}_{a}\equiv(k_{ax},k_{ay})$ and $\bar{k}_{a}$ the complex conjugate of
$k_{a}$, $J(\\{z_{a}\\})\propto\prod_{a<b}(z_{a}-z_{b})^{2}$ is the Bijl-
Jastrow factor, $l_{B}=\sqrt{\hbar/eB}$ is the magnetic length, and
$\hat{P}_{\mathrm{LLL}}$ is the projection to the lowest LL [2, 3]. By
interpreting zeros in the Bijl-Jastrow factor as quantum vortices, one
observes that a double-vortex is displaced from the electron to which it is
bound by $z_{a}\rightarrow z_{a}+\mathrm{i}k_{a}l_{B}^{2}$. Microscopic wave
functions prescribed by the theory of CFs are well tested and widely accepted
as precise descriptions of many-body states of fractionally filled LLs [23].
The fact that the dipole picture is directly inferred from a microscopic wave
function distinguishes it from the HLR theory and the Dirac CF theory, both of
which are based on conjectured effective field theories and cannot be directly
associated with actual microscopic wave functions except for a few special
cases [24, 25, 26, 27].

For a long time, the dynamics of CFs as dipoles is not explicitly specified.
There is even a misunderstanding that the dipole picture is just a complement
to the HLR theory and has the same dynamics. To this end, Shi and Ji derive
the semi-classical dynamics of CFs in a CF Wigner crystal that is basically a
set of CF wave packets in the hidden Hilbert space [28]. From the semi-
classical dynamics, they obtain a dipole picture similar to that suggested by
Read. Specifically, in the picture, CS-like fields emerge as a result of the
correlations embodied in the Bijl-Jastrow factor and couple only to vortices,
while external EM fields couple only to electrons. By adopting the same
interpretation of the momentum as in Read’s theory, they obtain a dynamics in
which CFs are subject to a uniform Berry curvature $\Omega_{z}=1/eB$ in the
momentum space [29, 30]. The presence of the Berry curvature is a clear
indication that a dipole is not identical to the CF envisioned in the HLR
theory.

Notably, even though the dipole picture and the Dirac CF theory are very
different microscopically, their CFs share some basic properties (see Sec.
II.2): (1) a CF traversing a Fermi circle acquires a $\pi$-Berry phase; (2)
the Fermi wavevector $k_{\mathrm{F}}=\sqrt{eB/\hbar}$ is set only by the
magnetic field and independent of the electron density. These properties
underly various effects predicted for the Dirac CF theory [17, 18, 31]. On the
other hand, different from the Dirac CF theory, the dipole picture does not
exhibit explicitly the particle-hole symmetry. It is due to the lack of the
particle-hole symmetry of the vacuum assumed in the picture: the vacuum is an
empty LL for electrons and a filled LL for holes. The latter has a nonzero
Chern number $C_{\mathrm{vac}}=1$. The difference is superficial and may not
have a physical consequence (see Sec. VI.2). Therefore, it is reasonable to
believe that although the two pictures are distinct microscopically, they may
share predictions for macroscopic effects, e.g., the EM response.

In this paper, we determine the EM response of CFs in the dipole picture. We
show that, in the long-wavelength limit, the response has a form identical to
Eq. (3) of the Dirac CF theory, and is different from Eq. (1) even though the
dipole picture is closer microscopically to the HLR theory. The Dirac CF-like
EM response arises because the Berry curvature contributes a half-quantized
intrinsic Hall conductance which, notably, is independent of the filling
factor. We show that CFs undergo no side jumps when scattered in a LL with the
particle-hole symmetry. Therefore, the intrinsic contribution due to the Berry
curvature is not altered by extrinsic impurity scattering. The remainder of
the response is from an effective system with a Fermi wavevector
$k_{\mathrm{F}}=\sqrt{eB/\hbar}$ and a $\pi$-Berry phase, indistinguishable
from a Dirac CF system in the long-wavelength limit. By redefining the vacuum
of the dipole picture and interpreting the intrinsic response as a
contribution from the vacuum, we can explicitly show the emergence of a Dirac
CF effective description. On the other hand, although we can reproduce the
result by Wang et al. at half filling [15], the dipole theory and the HLR
theory are in general not equivalent when the filling factor of a LL deviates
from $1/2$. Finally, we show that deviations from the Dirac CF theory arise
when higher order corrections due to electric quadrupoles and magnetic moments
of CFs are considered.

The remainder of the paper is organized as follows. In Sec. II, we introduce
the dipole picture of CFs, and discuss its dynamics as well as a constraint
imposed by the particle-hole symmetry. In Sec. III, we determine the intrinsic
response of CFs, and show that it contributes a half quantized Hall
conductance independent of the filling factor. In Sec. IV, we show that CFs in
a particle-hole symmetric LL undergo no side-jumps when scattered. Therefore,
the intrinsic response is not altered by the scattering. In Sec. V, we solve
the Boltzmann equation and determine the extrinsic response of CFs. In Sec.
VI, we apply CS self-consistent conditions and determine the total response of
CFs to EM fields, and show the emergence of the Dirac CF effective theory in
the dipole picture. In Sec. VII, we determine corrections due to electric
quadrupoles and magnetic moments of CFs. In Sec. VIII, we test our theory by
fitting to experimental data. Finally, we summarize and discuss our results in
Sec. IX.

##  II Dipole picture of CFs

###  II.1 Lagrangian

In the dipole picture, a CF is the bound state of an electron and a double
quantum vortex. The electron and the vortex are separated spatially, and
bounded together by an attractive force due to the void created by the vortex
and the repulsive Coulomb interaction between electrons [21]. CS-like fields
emerge and couple to vortices. Meanwhile, electrons are coupled to external EM
fields including the strong magnetic field which confines electrons in a LL.
This is the dipole picture explicated in Ref. [28]. Our study will be based on
this particular dipole picture instead of other variants which can be found in
literatures [32, 33, 34, 35, 36].

The picture of CFs is described by the action in (2+1) dimensions

$S=\int\mathrm{d}^{3}x\left(-A\cdot j^{\mathrm{e}}-a\cdot
j+\frac{e^{2}}{4h}\epsilon^{\mu\nu\gamma}a_{\mu}\partial_{\nu}a_{\gamma}-\varepsilon\right),$
(5)

where
$\left(j^{\mathrm{e}}\right)^{\mu}\equiv(\rho^{\mathrm{e}},\bm{j}^{\mathrm{e}})$
and $j^{\mu}\equiv(\rho,\bm{j})$ are charge-current densities of electrons and
vortices, $A^{\mu}\equiv(\Phi,\bm{A})$ and $a^{\mu}\equiv(\phi,\bm{a})$ denote
the external EM fields and the emergent CS fields, respectively, and
$\varepsilon$ denotes the binding energy between electrons and quantum
vortices. The action assumes that both electrons and vortices have zero mass
and are subject to the external magnetic field and the CS magnetic field,
respectively. As a result, the motion of the electrons (vortices) is confined
in a LL created by the external (CS) magnetic field.

Under the mean-field approximation, differentiating the action with respect to
the CS fields $a$ gives rise to the self-consistent conditions

$\displaystyle\rho$ $\displaystyle=-\frac{e^{2}}{2h}b,$ (6)
$\displaystyle\bm{j}$ $\displaystyle=-\sigma_{\mathrm{cs}}\bm{e},$ (7)

where $\bm{e}=-\partial_{t}\bm{a}-\bm{\nabla}\phi$ and
$\bm{b}=\bm{\nabla}\times\bm{a}\equiv b\hat{z}$ are CS electric and magnetic
fields, respectively. While the HLR theory has the same set of self-consistent
conditions, the dipole picture differs in that (a) the CS fields only couple
to vortices and are not equivalent in effects to the EM fields; (b) the
conditions as well as the CS term in Eq. (5) should be regarded as an
approximation only for the long-wavelength limit. Corrections are expected
when the wavelength of the EM fields is comparable to the length scale of the
dipoles, i.e., the magnetic length $l_{\mathrm{B}}$ [28].

To proceed, we relate the charge-current densities of electrons
$j^{\mathrm{e}}$ to their counterparts for vortices $j$. The charge-current
densities of electrons are defined by

$\displaystyle\rho^{\mathrm{e}}\left(t,\bm{x}\right)$
$\displaystyle=-e\sum_{a}\delta\left(\bm{x}-\bm{x}_{a}^{\mathrm{e}}(t)\right),$
(8) $\displaystyle\bm{j}^{\mathrm{e}}\left(t,\bm{x}\right)$
$\displaystyle=-e\sum_{a}\dot{\bm{x}}_{a}^{\mathrm{e}}\delta\left(\bm{x}-\bm{x}_{a}^{\mathrm{e}}(t)\right),$
(9)

where $\\{\bm{x}_{a}^{\mathrm{e}}(t)\\}$ denotes the set of electron
coordinates. The charge-current densities for vortices $(\rho,\bm{j})$ can be
similarly defined by using the set of vortex coordinates
$\\{\bm{x}_{a}(t)\\}$. We introduce the dipole vector

$\bm{d}_{a}\equiv\bm{x}_{a}^{\textrm{e}}-\bm{x}_{a}.$ (10)

After applying multipole expansions [37], we obtain

$\displaystyle\rho^{\mathrm{e}}$
$\displaystyle=\rho-\bm{\nabla}\cdot\bm{P}+\bm{\nabla}\bm{\nabla}:\mathbb{Q}\dots\,,$
(11) $\displaystyle\bm{j}^{\textrm{e}}$
$\displaystyle=\bm{j}+\partial_{t}\left(\bm{P}-\bm{\nabla}\cdot\mathbb{Q}\right)+\bm{\nabla}\times\bm{M}+\dots\,,$
(12)

where we keep dipole and quadrupole corrections for the charge density, and
displacement current and magnetic dipole corrections for the current density.
Because CFs have a length scale $l_{B}$ and an energy scale
$\hbar\tilde{\omega}_{B}$, the expansions are in rising orders of $ql_{B}$ and
$\omega/\tilde{\omega}_{B}$, where $\omega$ and $q$ are the frequency and
wavenumber of probing EM fields, respectively. The dipole density $\bm{P}$,
the quadrupole density tensor $\mathbb{Q}$, and the magnetization density
$\bm{M}$ are defined, respectively, by

$\displaystyle\bm{P}\left(t,\bm{x}\right)$
$\displaystyle=-e\sum_{a}\bm{d}_{a}\delta\left(\bm{x}-\bm{x}_{a}\right),$ (13)
$\displaystyle\mathbb{Q}\left(t,\bm{x}\right)$
$\displaystyle=-\frac{e}{2}\sum_{n}\bm{d}_{a}\bm{d}_{a}\delta(\bm{x}-\bm{x}_{a}),$
(14) $\displaystyle\bm{M}\left(t,\bm{x}\right)$
$\displaystyle=-e\sum_{a}\bm{d}_{a}\times\left(\dot{\bm{x}}_{a}+\frac{1}{2}\dot{\bm{d}}_{a}\right)\delta\left(\bm{x}-\bm{x}_{a}\right).$
(15)

Substituting the expansions into Eq. (5) and applying integrals by parts, we
obtain

$S\approx\int\mathrm{d}^{2}\bm{x}\mathrm{d}t\left[-\left(a+A\right)\cdot
j+\frac{e^{2}}{4h}\epsilon^{\mu\nu\gamma}a_{\mu}\partial_{\nu}a_{\gamma}\right.\\\
\left.+\bm{M}\cdot\bm{B}-\left(\varepsilon-\bm{E}\cdot\bm{P}-\mathbb{Q}:\bm{\nabla}\bm{E}\right)\right].$
(16)

We then follow Read to define the momentum of a CF as [21]

$\bm{p}_{a}=eB\hat{z}\times\bm{d}_{a}.$ (17)

We find

$\bm{M}\cdot\bm{B}=\sum_{a}\bm{p}_{a}\cdot\dot{\bm{x}}_{a}+\frac{1}{2eB}\left(\bm{p}_{a}\times\dot{\bm{p}}_{a}\right)\cdot\hat{z}.$
(18)

Substituting it into Eq. (16) and interpreting $\bm{x}_{a}$ as the coordinate
of a CF, we obtain an action in which a CF is governed by the Lagrangian

$L_{a}=\bm{p}_{a}\cdot\dot{\bm{x}}_{a}+\frac{1}{2eB}\left(\bm{p}_{a}\times\dot{\bm{p}}_{a}\right)\cdot\hat{z}-e\dot{\bm{x}}_{a}\cdot\tilde{\bm{a}}(t,\bm{x}_{a})\\\
+e\tilde{\phi}(t,\bm{x}_{a})-\mathcal{E}_{a}.$ (19)

From the Lagrangian, we see that the CF is subject to an effective field
$\tilde{a}\equiv(\tilde{\phi},\tilde{\bm{a}})=a+A$, a Berry curvature
$\Omega_{z}=1/eB$, and an energy dispersion with electric dipole and
quadrupole corrections

$\mathcal{E}_{a}=\varepsilon\left(\bm{p}_{a}\right)-\frac{1}{B}\left(\bm{E}\times\hat{z}\right)\cdot\bm{p}_{a}\\\
+\frac{1}{2eB^{2}}\left(\bm{p}_{a}\times\hat{z}\right)\cdot\bm{\nabla}\bm{E}\cdot\left(\bm{p}_{a}\times\hat{z}\right),$
(20)

where $\varepsilon\left(\bm{p}_{a}\right)$ denotes the binding energy of a CF,
and $\varepsilon=\sum_{a}\varepsilon\left(\bm{p}_{a}\right)$.

We model the binding energy as a harmonic potential
$\varepsilon\left(\bm{p}_{a}\right)\propto|\bm{d}_{a}|^{2}$. After
substituting Eq. (17), the binding energy becomes the dispersion of CFs, and
can be written as

$\varepsilon\left(\bm{p}_{a}\right)=\frac{b}{B}\frac{p_{a}^{2}}{2m^{*}},$ (21)

where we interpret $m^{*}$ as the effective mass of CFs. For a reason which
will be clarified in Sec. II.3, we append a $b/B$ factor to the dispersion.
The extra factor becomes 1 at half filling.

###  II.2 Dynamics

From the Lagrangian Eq. (19), we can obtain equations of motion of CFs. For
the moment, we focus on the long-wavelength limit and ignore all spatial
gradients of $\bm{E}$, $\bm{B}$ and $\bm{b}$. The time dependence of $\bm{B}$
is also ignored since $\partial_{t}\bm{B}=-\bm{\nabla}\times\bm{E}$ is of the
same order of the gradient of $\bm{E}$. The equations read:

$\displaystyle\dot{\bm{x}}$
$\displaystyle=\frac{b}{B}\frac{\bm{p}}{m^{\ast}}-\frac{1}{B}\bm{E}\times\hat{z}-\frac{1}{eB}\dot{\bm{p}}\times\hat{z},$
(22) $\displaystyle\dot{\bm{p}}$
$\displaystyle=-e\tilde{\bm{e}}-e\tilde{b}\dot{\bm{x}}\times\hat{z},$ (23)

where for the simplicity of notations we drop the subscripts indexing
particles, $\tilde{\bm{e}}=\bm{e}+\bm{E}$ and $\tilde{b}=b-B$ are the
effective electric and magnetic fields experienced by CFs, respectively. The
equations should be regarded as a zeroth order approximation. Corrections due
to gradients of the EM and CS fields will be determined in Sec. VII.

The dynamics of CFs in the dipole picture is different from that assumed in
the HLR theory. Two differences are notable: (a) CFs are subject to a Berry
curvature $\Omega_{z}=1/eB$; (b) The external electric field $\bm{E}$
introduces a correction to the group velocity due to the dipole correction to
the energy. In the HLR theory, EM fields can be completely absorbed into the
effective fields. Therefore, they are equivalent to the CS fields in driving
CFs. This is not true anymore in the dipole picture. Besides being a part of
the effective fields, they also introduce the Berry curvature and the dipole
correction. As we will see, these corrections change fundamentally how a CF
system responses to EM fields.

The presence of the Berry curvature has important physical consequences. It
introduces a $\pi$-Berry phase for an electron traversing a Fermi circle. The
$\pi$-Berry phase is regarded as a feature of the Dirac CF theory, and its
effects has been extensively explored [17, 18]. Another consequence of the
Berry curvature is a correction to the phase-space density-of-states [38]. In
ordinary two-dimensional systems, a quantum state always occupies a $h^{2}$
phase-space volume. In the presences of both the Berry curvature and the
(effective) magnetic field, however, the phase space volume element
$\mathrm{d}^{2}x\mathrm{d}^{2}p/h^{2}$ is modified to
$D\mathrm{d}^{2}x\mathrm{d}^{2}p/h^{2}$ by the density-of-state correction
factor

$D=1+e\tilde{b}\Omega_{z}=\frac{b}{B}.$ (24)

For a homogeneous system, we have $D=2\nu$, where
$\nu\equiv-\rho^{\mathrm{e}}h/e^{2}B$ is the filling factor of the system. The
factor modifies the Fermi wavevector from its usual value
$k_{\mathrm{F}}^{0}=\sqrt{4\pi n}$, where $n$ is the density of CFs, to

$k_{\mathrm{F}}=k_{\mathrm{F}}^{0}/\sqrt{D}=\sqrt{\frac{eB}{\hbar}}.$ (25)

We see that, even though the CF density in the dipole picture is the same as
the electron density, the Fermi wavevector is set only by the magnetic field
$B$ and independent of the density. The prediction differs from the HLR theory
and agrees with the Dirac CF theory but has a different interpretation:
$k_{\mathrm{F}}$ is modified due not to a change of the CF density but the
phase-space density-of-state correction.

For the convenience of later applications, the equations of motion Eqs. (22,
23) can be solved for $\dot{\bm{x}}$ and $\dot{\bm{p}}$:

$\displaystyle\dot{\bm{x}}=$
$\displaystyle\frac{\bm{p}}{m^{*}}-\frac{1}{b}\hat{z}\times\bm{e},$ (26)
$\displaystyle\dot{\bm{p}}=$ $\displaystyle
e\tilde{b}\hat{z}\times\frac{\bm{p}}{m^{*}}-\frac{1}{D}e\bm{e}-e\bm{E}.$ (27)

###  II.3 Particle-hole symmetry and CF dispersion

The particle-hole symmetry imposes a constraint on how the CF dispersion
depends on the filling factor. To see that, we assume that CFs have the
dispersion $\varepsilon_{\bm{p}}=\bm{p}^{2}/2m^{\prime}$ as in the HLR theory.
Away from half-filling, CFs experience an effective magnetic field
$\tilde{b}$, and form $\Lambda$-levels [2]. In our case, the gap between
neighboring $\Lambda$-levels is [39, 38]

$\Delta=\frac{\hbar e|\tilde{b}|}{Dm^{\prime}}.$ (28)

The presence of $D$ in the denominator is notable. The particle-hole symmetry
requires that the gap at fillings $\nu$ and $1-\nu$ should be equal when one
fixes the external magnetic field and changes the density [2]. However, if
$m^{\prime}(1-\nu)=m^{\prime}(\nu)$ is assumed as in the HLR theory, the gap
predicted by Eq. (28) will be asymmetric for particles and holes since
$D=b/B=2\nu$ is proportional to the filling factor.

To compensate, we explicitly include a factor $D$ to the CF dispersion in Eq.
(21), and interpret $m^{\ast}$ as the effective mass of CFs. With the factor,
the particle-hole symmetry requires

$m^{*}(\nu)=m^{*}(1-\nu).$ (29)

We note that $m^{\ast}$ also depends on $B$, as shown in Ref. [8].

##  III Intrinsic response

###  III.1 Intrinsic Hall conductance

We first determine the intrinsic response, which is the EM response when the
system is in a local equilibrium state. In our case, the distribution function
of the local equilibrium state is

$f_{0}\left(\bm{x},\bm{p},t\right)=n_{\mathrm{F}}\left(\mathcal{E}(\bm{p})-\mu\right),$
(30)

where $n_{\mathrm{F}}$ is the Fermi-Dirac distribution function, $\mu$ is the
chemical potential, $\mathcal{E}(\bm{p})$ is the energy dispersion shown in
Eq. (20). The intrinsic contribution to the current is

$\bm{j}_{0}=-e\int\left[\mathrm{d}\bm{p}\right]D\dot{\bm{x}}f_{0}\left(\bm{p}\right)$
(31)

with $[\mathrm{d}\bm{p}]\equiv\mathrm{d}^{2}p/h^{2}$.

To evaluate the intrinsic current, we note that the electric field is to shift
the center of the CF dispersion, i.e.,
$\mathcal{E}(\bm{p})\approx\varepsilon\left(\bm{p}-m^{\ast}\bm{E}\times\hat{z}/b\right)$
to the linear order of $\bm{E}$. Substituting Eq. (26) and the dispersion into
Eq. (31), and making the substitution
$\bm{p}\rightarrow\bm{p}+m^{\ast}\bm{E}\times\hat{z}/b$, we have

$\bm{j}_{0}=-e\int[d\bm{p}]Dn_{\mathrm{F}}\left(\varepsilon(\bm{p})-\mu\right)\left(\frac{\bm{p}}{m^{*}}-\frac{\hat{z}\times\tilde{\bm{e}}}{b}\right).$
(32)

The first term of the integral vanishes, and the second term can be determined
by using the identity
$-e\int[d\bm{p}]Dn_{\mathrm{F}}\left(\varepsilon(\bm{p})-\mu\right)=\rho$ and
the self-consistent condition Eq. (6). We obtain:

$\displaystyle\bm{j}_{0}=$ $\displaystyle-\sigma_{\mathrm{CS}}\tilde{\bm{e}}.$
(33)

The intrinsic response contributes a Hall conductance
$\tilde{\sigma}_{xy}^{0}=-e^{2}/2h$ for the effective electric field. It is
notable that the Hall conductance is independent of the density or the filling
factor.

###  III.2 Definition of the CF coordinate

In the last subsection, we obtain a half-quantized intrinsic Hall conductance
for CFs. It seems to provide a solution to the issue raised by Kivelson et
al., i.e., CFs must have a half-quantized Hall conductance in a particle-hole
symmetric LL [14]. However, the issue is not fully resolved due to an
ambiguity in defining the coordinate of a CF. The interpretation of the
dynamics, in particular the presence of a Berry curvature, depends on the
definition we adopt for the CF coordinate, while the intrinsic Hall
conductance is directly related to the Berry curvature.

To see that, we examine the Lagrangian Eq. (19) to see how the definition of
the CF coordinate affects its interpretation. At half-filling, the Lagrangian
for a CF has the form
$L=\bm{p}\cdot\dot{\bm{x}}+\left(\bm{p}\times\dot{\bm{p}}\right)\cdot\hat{z}/2eB+e\tilde{\phi}(\bm{x})-\mathcal{E}$,
where $\bm{x}$ is the coordinate of the vortex in the CF. We implicitly define
the coordinate of the CF as the coordinate of its vortex and interpret the
second term as a Berry curvature $\Omega_{z}=1/eB$. However, the
interpretation changes with an alternative definition of the CF coordinate.
For instance, if we define the CF coordinate as the center coordinate
$\bm{x}^{\mathrm{c}}=\bm{x}+\bm{d}/2=\bm{x}+\bm{p}\times\hat{z}/2eB$, the
Lagrangian becomes
$L=\bm{p}\cdot\dot{\bm{x}}^{\mathrm{c}}+e\tilde{\phi}(\bm{x})-\mathcal{E}$,
and the Berry curvature vanishes. Similarly, if we define the CF coordinate as
the coordinate of its electron, the Berry curvature will reverse its sign.
Then, what would be the physical interpretation?

The answer lies in the local equilibrium state that the system adapts to. When
external fields are applied, the system rapidly relaxes to a local equilibrium
state with its momentum distribution determined by momentum relaxation
processes. By assuming the local equilibrium distribution Eq. (30), we
implicitly assume that the momentum relaxation processes do not change
$\bm{x}$, i.e., the position of the vortex, otherwise the effective potential
$\tilde{\phi}(\bm{x})$ would affect the energy conservation of relaxation
processes and therefore the detailed balance condition for determining the
local equilibrium distribution. It is the momentum relaxation process instead
of an arbitrary choice of the definition of the CF coordinate that determines
the intrinsic response.

To elucidate the point, we try an alternative definition of the CF coordinate,
say, the center of the dipole $\bm{x}^{\mathrm{c}}$. In this case, multipole
expansions lead to a dispersion
$\mathcal{E}^{\prime}(\bm{p})=\varepsilon(\bm{p})-[(\bm{E}-\bm{e})\times\hat{z}]\cdot\bm{p}/2B$.
We still assume that a momentum relaxation process does not change the
position of the vortex. It implies that a process inducing a momentum change
$\Delta\bm{p}$ also induces a change of the CF coordinate (side jump)
$\Delta\bm{x}^{\mathrm{c}}=\Delta\bm{p}\times\hat{z}/2eB$, and a change of
energy
$\Delta\mathcal{E}^{\prime}(\bm{p})+e(\bm{e}+\bm{E})\cdot\Delta\bm{x}^{\mathrm{c}}=\Delta\varepsilon(\bm{p})-(\bm{E}\times\hat{z})\cdot\Delta\bm{p}/B\equiv\Delta\mathcal{E}(\bm{p})$.
The principle of detailed balance will lead to the same local equilibrium
distribution Eq. (30).

In the next section, we will establish that the momentum relaxation processes
in a LL with the particle-hole symmetry do keep $\bm{x}$ unchanged. It makes
$\bm{x}$ a convenient choice as the CF coordinate since one needs not to deal
with the complexities associated with side jumps [15]. When the filling factor
deviates from $1/2$, there is another reason for choosing $\bm{x}$ as the CF
coordinate. In this case, the particular choice provides the simplest form of
the equations of motions. With other choices, non-diagonal Berry curvature
components will appear, making the equations more complicated and their
interpretation more difficult. The latter point is discussed in Ref. [28].

##  IV Scattering of CFs

In this section, we investigate the momentum relaxation processes of CFs. At
low temperatures, the processes are dominated by elastic scattering induced by
quenched impurities. For CF systems, there are two possible ways that
impurities can affect CFs: (a) impurities induce a random potential coupling
to the electrons in dipoles; (b) the random impurity potential induces a
spatial modulation of the CF density, and in turn induces a spatially
fluctuating CS magnetic field. We will show that CFs undergo no side-jumps
when scattered by both the fields as long as CFs adopt a dispersion consistent
to the particle-hole symmetry requirement Eq (29). It justifies our choice of
defining the coordinate of a CF as its vortex position.

We will first consider scattering by the impurity potential only. The more
complete consideration including the fluctuating CS field will be presented in
Sec. IV.2. Our consideration is similar to Wang et al.’s consideration for the
HLR theory [15]. The correspondence between the two considerations will be
discussed in Sec. IV.3. Our result also coincides with numerical observations
by Geraedts et al. [40]. It will be discussed in Sec. IV.4.

###  IV.1 Scattering by the impurity potential

At low temperatures, the momentum relaxation is dominated by elastic
scattering induced by impurities ubiquitously presented in real materials.
Impurities introduce a random potential into the system. As an electrostatic
potential, it only couples to the electrons in dipoles. Electrons feel an
effective random potential $V_{\textrm{eff}}(\bm{x}^{\mathrm{e}})$ which
consists of the random potential imposed directly by impurities and a
screening potential from other electrons. It is important to note that the
potential depends on $\bm{x}^{\mathrm{e}}$ instead of $\bm{x}$.

We first present a heuristic picture for a CF colliding with an impurity. We
assume that the scattering is instantaneous, i.e., the impurity exerts an
impulsive force to the electron with a total impulse $\Delta\bm{I}$. The
motion of an electron confined in a LL is governed by the equation
$eB\hat{z}\times\dot{\bm{x}}^{\textrm{e}}=\bm{F}$, where $\bm{F}$ is the total
force acting on the electron, including the binding force from the vortex and
the impulsive force exerted by the impurity. Since the binding force is
negligible in the infinitesimal time period of the collision, the equation
predicts that the electron will have a coordinate shift
$\Delta\bm{x}^{\textrm{e}}=\Delta\bm{I}\times\hat{z}/eB$ after the collision.
Meanwhile, the position of the vortex does not change. The process of the
collision is illustrated in Fig. 1. According to the dipole picture and Eq.
(17), it corresponds to a CF momentum change
$\Delta\bm{p}=eB\hat{z}\times\Delta\bm{x}^{\mathrm{e}}=\Delta\bm{I}$ and no
change to the CF coordinate. We therefore conclude that the collision does not
induce a side jump. Note that the conclusion is valid only when we define the
coordinate of the CF as its vortex position. Had we chosen
$\bm{x}^{\mathrm{c}}$ as the CF coordinate, for instance, we would conclude
that the collision induces a side jump
$\Delta\bm{x}^{\mathrm{c}}=\Delta\bm{I}\times\hat{z}/2eB=\Delta\bm{p}\times\hat{z}/2eB$.

Figure 1:  Collision between a CF and an impurity. Left: the impurity exerts
an impulse $\Delta\bm{I}$ to the electron in the CF. Right: after the
collision, the electron coordinate as well as the center coordinate
$\bm{x}^{\mathrm{c}}$ are shifted. Depending on the definition of the CF
coordinate as the center coordinate $\bm{x}^{\mathrm{c}}$ or the vortex
coordinate $\bm{x}$, the scattering can be interpreted as with or without a
side jump, respectively.

To treat the problem more formally, we follow the approach developed by
Sinitsyn et al. [41]. For a canonical quantum system, it is shown that the
side jump induced by elastic scattering is

$\Delta\tilde{\bm{x}}_{\tilde{\bm{p}}^{\prime},\tilde{\bm{p}}}=-\hbar(\partial_{\tilde{\bm{p}}}+\partial_{\tilde{\bm{p}}^{\prime}})\textrm{arg}[V_{\tilde{\bm{p}}^{\prime},\tilde{\bm{p}}}],$
(34)

where
$V_{\tilde{\bm{p}}^{\prime},\tilde{\bm{p}}}=\braket{\tilde{\bm{p}}^{\prime}}{\hat{V}}{\tilde{\bm{p}}}$
is the matrix element of a scattering potential $\hat{V}$ in the momentum
eigenstates $\ket{\tilde{\bm{p}}}$ and $\bra{\tilde{\bm{p}}^{\prime}}$, and
$\textrm{arg}[V_{\tilde{\bm{p}}^{\prime},\tilde{\bm{p}}}]$ denotes its phase
angle. Note that the formula is valid for a canonical system, i.e., its
classical Lagrangian should have the form of
$L=\tilde{\bm{p}}\cdot\dot{\tilde{\bm{x}}}-H(\tilde{\bm{x}},\tilde{\bm{p}})$,
where $\tilde{\bm{x}}$ and $\tilde{\bm{p}}$ are the canonical coordinate and
momentum, respectively.

For more general Lagrangians in which dynamic variables are not necessarily
canonical, the side jump can be determined by the following procedure: (a)
find a transformation from original dynamical variables $(\bm{x},\bm{p})$ to
canonical variables $(\tilde{\bm{x}},\tilde{\bm{p}})$; (b) determine the
scattering matrix element and the side jump
$\Delta\tilde{\bm{x}}_{\tilde{\bm{p}}^{\prime},\tilde{\bm{p}}}$ by using Eq.
(34); (c) determine the side jump in terms of $(\bm{x},\bm{p})$ by
transforming the canonical variables back to the original dynamic variables.

We can determine the side jump for the current case by applying the procedure.
At half filling, the Lagrangian Eq. (19) is reduced to
$L=\bm{p}\cdot\dot{\bm{x}}+\left(\bm{p}\times\dot{\bm{p}}\right)\cdot\hat{z}/2eB-\varepsilon-
V_{\mathrm{eff}}$. It is easy to find the canonical variables
$\tilde{\bm{p}}=\bm{p}$ and
$\tilde{\bm{x}}=\bm{x}^{\mathrm{c}}=\bm{x}+\bm{p}\times\hat{z}/2eB$. Using the
canonical variables, the scattering potential can be written as
$V_{\textrm{eff}}(\bm{x}^{\mathrm{e}})=V_{\textrm{eff}}(\tilde{\bm{x}}+\tilde{\bm{p}}\times\hat{z}/2eB)$.
The scattering matrix element can be determined straightforwardly:

$\displaystyle V_{\tilde{\bm{p}}^{\prime},\tilde{\bm{p}}}$
$\displaystyle=\tilde{V}_{\textrm{eff}}(\bm{q})\exp\left[\mathrm{i}\frac{(\tilde{\bm{p}}^{\prime}\times\tilde{\bm{p}})\cdot\hat{z}}{2eB\hbar}\right],$
(35)

where $\tilde{V}_{\textrm{eff}}(\bm{q})$ is the Fourier transform of the
effective impurity potential, and
$\hbar\bm{q}=\tilde{\bm{p}}^{\prime}-\tilde{\bm{p}}\equiv\Delta\bm{p}$ is the
change of the momentum induced by the scattering. By applying Eq. (34), we
obtain

$\Delta\tilde{\bm{x}}_{\tilde{\bm{p}}^{\prime},\tilde{\bm{p}}}=\frac{\hbar\bm{q}\times\hat{z}}{2eB}.$
(36)

The result is identical to $\Delta\bm{x}^{\mathrm{c}}$ determined in the
heuristic argument. Transforming back to the original variables, we have

$\Delta\bm{x}_{\bm{p}^{\prime},\bm{p}}=\Delta\tilde{\bm{x}}_{\bm{p}^{\prime},\bm{p}}-\Delta\left[\frac{\bm{p}\times\hat{z}}{2eB}\right]=0.$
(37)

It confirms our conclusion from the heuristic argument that scattering of CFs
by impurities does not induce side jumps.

###  IV.2 Effect of the fluctuating CS magnetic field

In the last subsection, we analyze the scattering of CFs by considering only
the effective impurity potential. However, the potential can also induce a
density modulation of CFs. According to Eq. (6), the density modulation will
induce a spatially fluctuating CS magnetic field. Wang et al. analyze the
effect of the fluctuating CS magnetic field in the HLR theory, and find that
it induces side jumps and gives rise to a half-quantized CF Hall conductance
[15]. Here, we consider the same effect in the dipole picture.

We first determine the modification to the Lagrangian by the fluctuating CS
magnetic field. Following Wang et al.’s consideration, we can determine the
fluctuating CS magnetic field $\delta
b(\bm{x})=-(2m^{*}/e\hbar)V_{\textrm{eff}}(\bm{x})$ [15]. It gives rise to a
CS vector potential $\delta\bm{a}(\bm{x})$. The Lagrangian is modified to

$L=\bm{p}\cdot\dot{\bm{x}}+\frac{1}{2eB}\left(\bm{p}\times\dot{\bm{p}}\right)\cdot\hat{z}-e\delta\bm{a}(\bm{x})\cdot\dot{\bm{x}}-\varepsilon-
V_{\mathrm{eff}}\left(\bm{x}^{\mathrm{e}}\right).$ (38)

Following the procedure outlined in the last subsection, we proceed to find a
set of canonical variables $(\tilde{\bm{x}},\tilde{\bm{p}})$ for the modified
Lagrangian. Unlike the previous case, the presence of the CS vector potential
makes both $\bm{x}$ and $\bm{p}$ non-canonical. As in Wang et al.’s
consideration, we assume that the impurity potential is weak and slowly
varying in space. It thus suffices to find a transformation accurate to the
linear order of $\delta\bm{a}$ and its first spatial gradient. We have [42]:

$\displaystyle\bm{x}$
$\displaystyle=\tilde{\bm{x}}+\frac{1}{2eB}\hat{z}\times\left[\bm{p}+\frac{1}{2B}(\hat{z}\times\bm{p})\cdot\bm{\nabla}\delta\bm{a}(\tilde{\bm{x}})\right],$
(39) $\displaystyle\bm{p}$ $\displaystyle=\left[1-\frac{\delta
b(\tilde{\bm{x}})}{2B}\right]\tilde{\bm{p}}+e\delta\bm{a}(\tilde{\bm{x}}).$
(40)

It is straightforward to verify that
$L=\tilde{\bm{p}}\cdot\dot{\tilde{\bm{x}}}-\varepsilon-
V_{\mathrm{eff}}-\mathrm{d}\left[\delta\bm{a}(\bm{x})\cdot\left(\hat{z}\times\bm{p}\right)/2B\right]/\mathrm{d}t+O[(\delta
a)^{2},\nabla^{2}a]$, i.e., $\tilde{\bm{x}}$ and $\tilde{\bm{p}}$ are indeed
canonical.

We can then determine the perturbation to the hamiltonian of the canonical
system. Besides the direct perturbation from
$\hat{V}_{\mathrm{eff}}\left(\hat{\bm{x}}_{\mathrm{e}}\right)$, there is a
perturbation to the kinetic energy
$\varepsilon\left(\hat{p}\right)=D\hat{p}^{2}/2m^{\ast}$ induced by the CS
magnetic field. The perturbation can arise from: (a) the substitution of the
momentum operator Eq. (40); (b) the modulation of $D(\bm{x})\approx 1+\delta
b(\tilde{\bm{x}})/B$ induced by the CS magnetic field modulation; (c) the
dependence of $m^{\ast}$ on the filling factor $\nu$ which is also modulated
with the local density. For a particle-hole symmetric LL, the contribution
from (c) disappears in the linear order because $m^{\ast}$ is symmetric about
$\nu=1/2$. Moreover, the extra factor before $\tilde{\bm{p}}$ in Eq. (40) just
cancels the contribution from (b). Therefore, the total perturbation is:

$\hat{V}=\frac{e}{2m^{\ast}}\left[\hat{\tilde{\bm{p}}}\cdot\delta\bm{a}\left(\tilde{\bm{x}}\right)+\delta\bm{a}\left(\tilde{\bm{x}}\right)\cdot\hat{\tilde{\bm{p}}}\right]+\hat{V}_{\mathrm{eff}}\left(\hat{\bm{x}}_{\mathrm{e}}\right).$
(41)

We proceed to determine the scattering matrix element. The Fourier transform
of $\delta\bm{a}(\tilde{\bm{x}})$ is
$\delta\tilde{\bm{a}}(\bm{q})=-\mathrm{i}\delta\tilde{b}(\bm{q})(\hat{z}\times\bm{q}/q^{2})=\mathrm{i}(2m^{\ast}/e\hbar
q^{2})(\hat{z}\times\bm{q})\tilde{V}_{\mathrm{eff}}(\bm{q})$. Thus, the total
scattering matrix element is

$V_{\tilde{\bm{p}}^{\prime},\tilde{\bm{p}}}=\tilde{V}_{\textrm{eff}}(\bm{q})\left[e^{\mathrm{i}\frac{(\tilde{\bm{p}}^{\prime}\times\tilde{\bm{p}})\cdot\hat{z}}{2eB\hbar}}+\frac{2\mathrm{i}(\tilde{\bm{p}}^{\prime}\times\tilde{\bm{p}})\cdot\hat{z}}{\hbar^{2}q^{2}}\right].$
(42)

Following Wang et al.’s assumption that $\tilde{V}_{\textrm{eff}}(\bm{q})/q$
does not diverge at $q\rightarrow 0$, we determine that the phase angle of the
scattering matrix element is
$\mathrm{arg}(V_{\tilde{\bm{p}}^{\prime},\tilde{\bm{p}}})\approx-\hbar^{2}q^{2}/2[(\tilde{\bm{p}}^{\prime}\times\tilde{\bm{p}})\cdot\hat{z}]$
to the linear order of $ql_{B}$.

Applying Eq. (34), we determine the side jump in the canonical system:

$\Delta\tilde{\bm{x}}_{\tilde{\bm{p}}^{\prime},\tilde{\bm{p}}}=\frac{\bm{q}\times\hat{z}}{2k_{\mathrm{F}}^{2}},$
(43)

for CFs on the Fermi circle. Substituting $k_{\mathrm{F}}=\sqrt{eB/\hbar}$
into the relation, we find that it is identical to Eq. (36) determined for a
scalar potential.

Finally, we determine the side-jump in terms of $\bm{x}$ and $\bm{p}$. Note
that the side jump is determined by comparing particle trajectories in
asymptotic regions far from an impurity [41]. In these regions, the CS field
$\delta\bm{a}(\bm{x})$ induced by the impurity is negligible. Therefore, the
relation between $(\Delta\bm{x},\Delta\bm{p})$ and
$(\Delta\tilde{\bm{x}},\Delta\tilde{\bm{p}})$ is the same as Eq. (37). We thus
have

$\displaystyle\Delta\bm{x}_{\bm{p}^{\prime},\bm{p}}$ $\displaystyle=0.$ (44)

It indicates that when both the impurity potential and the fluctuating CS
magnetic field are considered, the scattering of CFs does not induce side
jumps in a LL with the particle-hole symmetry.

###  IV.3 Compared to Wang et al.’s result

The discussion in the last subsection closely follows Wang et al.’s model on
how impurities affect a CF system. An obvious difference between the two
considerations is in the choice of the CF coordinate: Wang et al. adopt
$\bm{x}^{\mathrm{c}}$ (or $\tilde{\bm{x}}$) as the coordinate of a CF, while
we use $\bm{x}$. The results obtained in the two considerations are identical:
Eq. (43) is exactly the side jump obtained by Wang et al. [15].

In the microscopic level, the presence of the Berry curvature and the internal
structure of CFs in the dipole picture do introduce differences. In our
theory, the impurity potential are coupled to electrons. It leads to a phase
factor in the first term of Eq. (42). It is inconsequential because its
contribution to the phase angle of the scattering matrix element is of the
higher order in $ql_{B}$. The more fundamental difference is in Eq. (40),
which is different from the usual Peierls substitution by a factor $\sim
1/\sqrt{D}$. It is also inconsequential because it is cancelled by the $D$
factor in the CF dispersion Eq. (21). The resulting perturbation Eq. (42) is
identical to that of Wang’s theory except for the aforementioned
inconsequential difference in the effective impurity potential.

###  IV.4 Compared to Geraedts et al.’s result

As an attempt to determine the Berry phase and Berry curvature of CFs,
Geraedts et al. numerically calculate the matrix element of the density
operator $\hat{\rho}_{-\bm{q}}=\sum_{a}\exp(\mathrm{i}\bm{q}\cdot\bm{r}_{a})$
between two Rezayi-Read states with different sets of wavevectors [40]. In Ref
[43], we argue that their calculation could be regarded as a “first-
principles” determination of the scattering matrix element from the
microscopic wave function. It is then interesting to see how the scattering
matrix element Eq. (42) determined from an effective picture is compared to
the “first-principles” result.

We find that the simple formula Eq. (42) captures all essential features
observed in Geraedts et al.’s calculation. To see that, we rewrite Eq. (42) in
the limit $\bm{q}\rightarrow 0$ as

$V_{\tilde{\bm{p}}^{\prime},\tilde{\bm{p}}}\approx-\mathrm{i}\frac{2\tilde{V}_{\textrm{eff}}(\bm{q})}{\hbar
q}\tilde{p}\sin\theta\exp\left(\mathrm{i}\frac{\hbar
q}{2\tilde{p}\sin\theta}\right)$ (45)

where $\theta$ denotes the polar angle from $\tilde{\bm{p}}$ to $\bm{q}$, and
$\tilde{V}_{\textrm{eff}}(\bm{q})$ in the current context is the effective
potential induced by a single-mode scalar potential
$V(\bm{r})=\exp\left(\mathrm{i}\bm{q}\cdot\bm{r}\right)$. The formula predicts
that (a) when $\bm{q}$ is parallel to $\tilde{\bm{p}}$, i.e., $\theta=0$, the
matrix element vanishes; (b) when $\bm{q}$ is perpendicular to
$\tilde{\bm{p}}$, i.e., $\theta=\pm\pi/2$, the matrix element has a phase
equal to $\pm(\hbar q/2\tilde{p}-\pi/2)$; (c) traversing anti-clockwise an
circular path around the origin of the momentum space and cumulating the first
part of the phase, one always obtains a total phase $\pi$, independent of the
radius of the path. These are exactly what are observed in Geraedts et al.’s
work.

The phase obtained from the scattering matrix element is in general not the
Berry phase [43]. Geraedts et al. interprets the lack of dependence of the
cumulated phase on the radius of the circular path as a manifestation of the
singular distribution of the Berry curvature in a massless Dirac cone.
However, we see that it is also a property of the scattering matrix element
Eq. (42) even though it is for a system with a uniform Berry curvature.

Nonetheless, Geraedts et al.’s result does provide a “first principles”
support for the model of impurity scattering shown in Sec. IV.2.

##  V Extrinsic response

###  V.1 Boltzmann equation

In this section, we determine the extrinsic response of CFs. It is the
contribution from the deviation of the non-equilibrium distribution of the
system from the local equilibrium distribution Eq. (30). The non-equilibrium
distribution function $f=f(t,\bm{x},\bm{p})$ is determined by the Boltzmann
equation:

$\displaystyle\partial_{t}f+\dot{\bm{x}}\cdot\left(\partial_{\bm{x}}f\right)+\dot{\bm{p}}\cdot\left(\partial_{\bm{p}}f\right)=\left.\frac{\partial
f}{\partial t}\right|_{\textrm{coll}},$ (46)

where $\left.\partial f/\partial t\right|_{\textrm{coll}}$ denotes the
collision contribution of impurity scattering.

The collision contribution can be explicitly specified. In the last section,
we show that CFs undergo no side jumps when scattered. It greatly simplifies
our consideration because we can model the collision term as usual for elastic
scattering [44]:

$\left.\frac{\partial f}{\partial
t}\right|_{\textrm{coll}}=-\int[d\bm{p}^{\prime}]DW_{\bm{p},\bm{p}^{\prime}}[f(t,\bm{x},\bm{p})-f(t,\bm{x},\bm{p}^{\prime})],$
(47)

where $W_{\bm{p},\bm{p}^{\prime}}$ denotes the transition probability from
state $\bm{p}^{\prime}$ to $\bm{p}$. For elastic scattering, the energy
conservation requires
$W_{\bm{p},\bm{p}^{\prime}}\propto\delta[\mathcal{E}(\bm{p})-\mathcal{E}(\bm{p}^{\prime})]$.

The solution of the Boltzmann equation can be decomposed into

$f(t,\bm{x},\bm{p})=f_{0}(t,\bm{x},\bm{p})+f_{1}(t,\bm{x},\bm{p}),$ (48)

where $f_{0}$ is the local equilibrium distribution specified in Eq. (30), and
$f_{1}$ is the deviation induced by driving fields. It is obvious that $f_{0}$
fulfills the condition of detailed balance, i.e., the collision term vanishes
for $f_{0}$.

The collision term can be further simplified. We assume that the scattering is
approximately isotropic,
$W_{\bm{p},\bm{p}^{\prime}}=W\delta[\mathcal{E}(\bm{p})-\mathcal{E}(\bm{p}^{\prime})]$.
By substituting it into the collision term, and noting that
$\mathcal{E}(\bm{p})$ and $\mathcal{E}(\bm{p}^{\prime})$ can be replaced with
$\varepsilon(\bm{p})$ and $\varepsilon(\bm{p}^{\prime})$ since $f_{1}$ has
already been in the linear order of driving fields, we have [45]

$\displaystyle\left.\frac{\partial f}{\partial
t}\right|_{\textrm{coll}}=-\frac{f_{1}-\bar{f}_{1}}{\tau},$ (49)

where
$1/\tau=\int[d\bm{p}^{\prime}]DW\delta(\varepsilon_{\bm{p}}-\varepsilon_{\bm{p}^{\prime}})$
is the reciprocal of the relaxation time, and

$\bar{f}_{1}=\frac{1}{2\pi}\int_{0}^{2\pi}f_{1}(\phi)d\phi$ (50)

is the average of $f_{1}$ over the polar angle $\phi$ of the momentum
$\bm{p}$. Strictly speaking, the transition probability for our case must be
anisotropic, as evident in Eq. (42). However, in the regime $ql_{B}\ll 1$ we
are interested in, the anisotropy will not introduce qualitative differences.
Therefore, we ignore the anisotropy for simplicity. The generalization for
anisotropic scattering is straightforward and can be found in Ref. [45].

To solve the Boltzmann equation to the linear order of driving fields, it
suffices to solve for single-mode fields
$[\bm{E}(t,\bm{x}),\,\bm{e}(t,\bm{x})]=(\bm{E},\,\bm{e})\exp(-\mathrm{i}\omega
t+\mathrm{i}\bm{q}\cdot\bm{x})$. The solution has the form
$f_{1}(t,\bm{x},\bm{p})=f_{1}\exp(-\mathrm{i}\omega
t+\mathrm{i}\bm{q}\cdot\bm{x})$. By substituting Eqs. (26, 27) into Eq. (46),
we have:

$\left[-1/\tau+\mathrm{i}\omega-\mathrm{i}\bm{v}\cdot\bm{q}-\omega_{\textrm{c}}^{*}(\hat{z}\times\bm{p})\cdot\partial_{\bm{p}}\right]f_{1}\\\
=e\bm{v}\cdot(\bm{e}+\beta\bm{E})\left(-\frac{\partial
f_{0}}{\partial\varepsilon}\right)-\frac{\bar{f}_{1}}{\tau},$ (51)

where $\bm{v}=\bm{p}/m^{\ast}$, $\beta$ is defined in Eq. (4), and
$\omega_{\textrm{c}}^{*}=e\tilde{b}/m^{*}$ is the effective CF cyclotron
frequency. The solutions of the equation for $\nu=1/2$ and $\nu\neq 1/2$ will
be discussed in the following subsections.

With the distribution function, the extrinsic contribution to the CF (vortex)
current density can be determined by

$\bm{j}_{1}=-e\int[d\bm{p}]Df_{1}\dot{\bm{x}}\approx-e\int[d\bm{p}]Df_{1}\frac{\bm{p}}{m}.$
(52)

###  V.2 $\nu=1/2$

We first consider the case of half-filling. In this case, the effective
magnetic field $\tilde{b}=0$, therefore $\omega_{\mathrm{c}}^{\ast}=0$.
Equation (51) can be readily solved. We have

$\displaystyle f_{1}$
$\displaystyle=\frac{e\bm{v}\cdot(\bm{e}+\beta\bm{E})\left(-\frac{\partial
f_{0}}{\partial\varepsilon}\right)-\bar{f}_{1}/\tau}{-1/\tau+\mathrm{i}\omega-\mathrm{i}\bm{q}\cdot\bm{v}}.$
(53)

By averaging both sides of Eq. (53) over the polar angle of the momentum, we
obtain a self-consistent equation for determining $\bar{f}_{1}$. After solving
the equation, we have

$\displaystyle\bar{f}_{1}=\tau
e\bar{\bm{v}}\cdot(\bm{e}+\beta\bm{E})\left(-\frac{\partial
f_{0}}{\partial\varepsilon}\right),$ (54)

where
$\bar{\bm{v}}=-iv_{\textrm{F}}\bm{q}l/\\{(1-i\omega\tau)(1+\gamma)[1-\gamma(1-i\omega\tau)]\\}$,
$v_{\textrm{F}}=\hbar k_{\textrm{F}}/m^{*}$ is the CF Fermi velocity,
$l=v_{\textrm{F}}\tau$ is the mean free path of CFs, and

$\gamma=\sqrt{1+\frac{(ql)^{2}}{(1-i\omega\tau)^{2}}}.$ (55)

The solution of the Boltzmann equation can be written as

$\displaystyle f_{1}$
$\displaystyle=\frac{e(\bm{v}-\bar{\bm{v}})\cdot(\bm{e}+\beta\bm{E})}{-1/\tau+\mathrm{i}\omega-\mathrm{i}\bm{q}\cdot\bm{v}}\left(-\frac{\partial
f_{0}}{\partial\varepsilon}\right).$ (56)

Substituting Eq. (56) into Eq. (52), we obtain

$\displaystyle\bm{j}_{1}$ $\displaystyle=\tilde{\sigma}(\bm{e}+\beta\bm{E}),$
(57) $\displaystyle\tilde{\sigma}_{11}$
$\displaystyle=\frac{2\sigma_{0}}{(1-\mathrm{i}\omega\tau)\left(1+\gamma\right)}\frac{\mathrm{i}\omega\tau}{1-\gamma(1-\mathrm{i}\omega\tau)},$
(58) $\displaystyle\tilde{\sigma}_{22}$
$\displaystyle=\frac{2\sigma_{0}}{(1-\mathrm{i}\omega\tau)(1+\gamma)},$ (59)

where $\sigma_{0}=e^{2}n\tau/m^{*}$ is the Drude conductivity, $n$ is the
density of CFs, and the subscripts $1$ and $2$ denote the longitudinal
($\parallel\bm{q}$) and transverse ($\parallel\hat{z}\times\bm{q}$)
directions, respectively. The non-diagonal elements of $\tilde{\sigma}$ vanish
at half-filling.

It is important to note that the extrinsic response is not driven by the
effective electric field $\tilde{\bm{e}}$ but by $\bm{e}+\beta\bm{E}$. As a
result, $\bm{E}$ is not equivalent to the CS electric field $\bm{e}$ in
driving the response. It manifests the fact that CFs in the dipole picture are
not point particles and subject to the dipole correction from which the
$\beta$ matrix arises. The dipole correction is derived from the internal
structure of CFs. This is in contrast with the Dirac CF theory, in which a CF
is regarded as a point particle and the dipole moment is introduced only as an
after-thought for fulfilling the requirement of the Galilean invariance [16,
46].

###  V.3 $\nu\neq 1/2$

Away from half filling, the effective magnetic field $\tilde{b}=b-B$ is
nonzero. Solving Eq. (51) becomes more complicated. To this end, we note that
Eq. (51) is identical to the Boltzmann equation for ordinary particles albeit
with a driving field $\bm{e}+\beta\bm{E}$. For ordinary systems, Mirlin and
W$\ddot{\textrm{o}}$lfle obtain a solution in Ref. [45]. Their solution can be
adapted to our case by noting the relation
$-f_{0}^{\prime}\left(\varepsilon(\bm{p})\right)=D^{-1}\delta\left((\bm{p}^{2}-p_{\mathrm{F}}^{2})/2m^{\ast}\right)$.
We obtain

$\displaystyle\tilde{\sigma}_{11}$
$\displaystyle=-2\tilde{\sigma}_{0}\frac{\mathrm{i}\omega\tau}{(ql)^{2}}\left(1-\mathrm{i}\omega\tau\frac{A_{00}}{A_{00}-\omega_{\textrm{c}}^{*}\tau}\right),$
(60) $\displaystyle\tilde{\sigma}_{22}$
$\displaystyle=\frac{2\tilde{\sigma}_{0}}{\omega_{\textrm{c}}^{*}\tau}\left(\frac{A_{0s}^{2}}{A_{00}-\omega_{\textrm{c}}^{*}\tau}+A_{ss}\right),$
(61) $\displaystyle\tilde{\sigma}_{12}$
$\displaystyle=-\tilde{\sigma}_{21}=-2\tilde{\sigma}_{0}\frac{\omega}{v_{\textrm{F}}q}\frac{A_{0s}}{A_{00}-\omega_{\textrm{c}}^{*}\tau},$
(62)

where $\tilde{\sigma}_{0}=\sigma_{0}/D\equiv\tilde{n}e^{2}\tau/m^{\ast}$is the
Drude conductivity for a system with the effective density
$\tilde{n}=n/D=eB/2h$, and the coefficients $A_{ij}$ are defined by:

$\displaystyle
A_{ij}=\frac{1}{2\pi}\int_{0}^{2\pi}d\phi\psi_{i}(\phi)\int_{\mp\infty}^{\phi}d\phi^{\prime}\psi_{j}(\phi^{\prime})\exp\\{K(\phi,\phi^{\prime})\\},$
$\displaystyle
K(\phi,\phi^{\prime})=-\frac{\phi-\phi^{\prime}}{\omega_{\textrm{c}}^{*}\tau}(1-\mathrm{i}\omega\tau)-\frac{\mathrm{i}qv_{\textrm{F}}}{\omega_{\textrm{c}}^{*}}(\sin\phi-\sin\phi^{\prime}),$
$\displaystyle\psi_{0}(\phi)=1,\quad\psi_{s}(\phi)=\sin\phi,$ (63)

where the lower bound of the integral over $\phi^{\prime}$ is $-\infty$
($\infty$) for $\tilde{b}>0$ ($\tilde{b}<0$).

The result has an important feature: it is identical to the response of a
system with an effective density $\tilde{n}=eB/2h$ and a Fermi wavevector
$k_{\mathrm{F}}=\sqrt{4\pi\tilde{n}}=\sqrt{eB/\hbar}$, which are exactly the
parameters of a Dirac CF system [16]. The effective density is set by the
magnetic field instead of the electron density, exactly the swap of notions
observed in the Dirac CF theory.

The formula hides a subtle difference from the HLR theory which could have an
experimental effect. At finite $q$, the formula predicts a magneto-
conductivity oscillation with respect to
$qv_{\mathrm{F}}/|\omega_{\mathrm{c}}^{\ast}|=qR_{\mathrm{c}}^{\ast}$, where
$R_{\mathrm{c}}^{\ast}=\hbar k_{\mathrm{F}}/e|\tilde{b}|$ is the effective
cyclotron radius of CFs [8, 45, 4] (see Sec. VIII). Because the Fermi
wavevector is now determined by the magnetic field, the magneto-conductivity
oscillation will show asymmetry about $\nu=1/2$ in the positions of
conductivity extrema [31], similar to that observed in recent geometric
resonance experiments [47, 48].

##  VI Electromagnetic response

###  VI.1 CF current

Combining the intrinsic response Eq. (33) and the extrinsic response Eq. (57)
of the CF current, we have

$\bm{j}=-\sigma_{\mathrm{CS}}(\bm{e}+\bm{E})+\tilde{\sigma}\left(\bm{e}+\beta\bm{E}\right).$
(64)

It is important to note that $\tilde{\sigma}$ is a Fermi surface property. As
we have shown in the last section, it is identical to that of an effective
Dirac CF system.

To determine the actual response to external EM fields, we apply the self-
consistent condition Eq. (7), and find

$\bm{e}=\left(\tilde{\sigma}^{-1}\sigma_{\mathrm{CS}}-\beta\right)\bm{E}.$
(65)

Substituting the relation into Eq. (7), we obtain

$\bm{j}=\left(-\sigma_{\mathrm{CS}}\tilde{\sigma}^{-1}\sigma_{\mathrm{CS}}+\sigma_{\mathrm{CS}}\beta\right)\bm{E}.$
(66)

It yields a conductivity tensor exactly the form of Eq. (3).

###  VI.2 Emergence of the Dirac CF theory

To elucidate how a Dirac CF-like EM response emerges in the dipole picture, we
identify the extrinsic contribution
$\bm{j}_{1}=\bm{j}-\bm{j}_{0}=\bm{j}+\sigma_{\mathrm{CS}}\tilde{\bm{e}}$ as
the “Dirac CF” current $\tilde{\bm{j}}$, and
$\bm{j}_{\mathrm{D}}=\bm{j}-\sigma_{\mathrm{CS}}\bm{E}$ as the “Dirac
electron” current [16]. By applying the equations of continuity for CF and
electron densities, we can generalize them to the densities
$\tilde{\rho}=\rho+(e^{2}/2h)\tilde{b}$ and
$\rho_{\mathrm{D}}=\rho+e^{2}B/2h$, respectively. We thus have
$j_{\mathrm{D}}^{\mu}=j^{\mu}+(e^{2}/2h)\epsilon^{\mu\nu\gamma}\partial_{\nu}A_{\gamma}$.
Substituting the CS self-consistent conditions Eqs. (6, 7), or
$j^{\mu}=(e^{2}/2h)\epsilon^{\mu\nu\gamma}\partial_{\nu}a_{\gamma}$ into the
relation, we obtain

$\displaystyle j_{\mathrm{D}}^{\mu}$
$\displaystyle=\frac{e^{2}}{2h}\epsilon^{\mu\nu\gamma}\partial_{\nu}\tilde{a}_{\nu}=j_{0}^{\mu}.$
(67)

We find that the “Dirac electron” current is nothing but the intrinsic current
determined in Eq. (33). Substituting the relation and the CS self-consistent
conditions into the identity $\tilde{j}=j-j_{0}$, We obtain

$\tilde{j}^{\mu}=-\frac{e^{2}}{2h}\epsilon^{\mu\nu\gamma}\partial_{\nu}A_{\gamma}.$
(68)

The two relations Eqs. (67, 68) are exactly the self-consistent conditions
yielded by the action of the Dirac CF theory [16].

We can rewrite the action Eq. (16) in terms of $\tilde{j}$ and the effective
fields $\tilde{a}$, we have

$S=\int\mathrm{d}^{2}\bm{x}\mathrm{d}t\left[\frac{e^{2}}{4h}\epsilon^{\mu\nu\gamma}A_{\mu}\partial_{\nu}A_{\gamma}-\frac{e^{2}}{2h}\epsilon^{\mu\nu\gamma}\tilde{a}_{\mu}\partial_{\nu}A_{\gamma}\right.\\\
\left.-\tilde{a}\cdot\tilde{j}+L_{\mathrm{Dipoles}}\right].$ (69)

It has the same gauge structure as the Dirac CF action, although its particles
are governed by a different Lagrangian
$L_{\mathrm{Dipoles}}=\bm{M}\cdot\bm{B}-\mathcal{E}-\mathcal{E}_{\mathrm{0}}$
with $\mathcal{E}_{\mathrm{0}}=\tilde{a}[j_{\mathrm{D}}]\cdot
j_{\mathrm{D}}/2$ and $\tilde{a}[j_{\mathrm{D}}]$ determined by solving Eq.
(67). Nonetheless, as far as Fermi-surface properties, e.g., the response of
$\tilde{j}$ to external fields, are concerned, the particles are
indistinguishable from Dirac CFs because they have the same effective density,
Fermi wavevector and Berry phase (see Sec. V.3). We thus find that a Dirac CF
effective theory emerges from the dipole picture.

Transforming the action Eq. (16) to Eq. (69) redefines the vacuum of the
effective theory from an empty (or filled) LL of electrons to a half-filled LL
of vortices. It is feasible because the intrinsic response Eq. (33)
contributes a half-quantized Hall conductance which is independent of the
filling factor and not altered by impurity scattering. We can interpret the
intrinsic response as the contribution from the redefined vacuum,
$\mathcal{E}_{0}$ as the polarization energy of the vacuum, and the remainder
of the response as from an effective Dirac CF system.

###  VI.3 Electron current

The electron current $\bm{j}^{\mathrm{e}}$ is the current actually measured in
experiments. It is related to the CF current $\bm{j}$ by Eq. (12). To
determine the electron current, we need to determine $\bm{P}$ and $\bm{M}$.

To determine the dipole density $\bm{P}$, we make use of Eq. (26). Applying
the average $-e\int[\mathrm{d}\bm{p}]Df$ to both sides of the equation, we
obtain

$\bm{j}=\tilde{\omega}_{B}\hat{z}\times\bm{P}-\sigma_{\mathrm{CS}}\bm{e}.$
(70)

Applying the self-consistent condition Eq. (7), we find

$\bm{P}=0.$ (71)

For the magnetization density $\bm{M}$, it is easy to see that $\bm{M}$ is
proportional to $\bm{B}$ and $\bm{b}$. Since we ignore all variations of
$\bm{B}$ and $\bm{b}$ for the moment, we conclude that $\bm{M}$ does not
contribute to the response of $\bm{j}^{\mathrm{e}}$.

We thus have

$\bm{j}^{\mathrm{e}}=\bm{j}$ (72)

when only the dipole correction is considered.

##  VII Quadrupole and magnetic corrections

In the previous sections, we ignore the quadrupole term in Eq. (20) as well as
the temporal and spatial dependences of the magnetic fields. In this section,
we will consider their effects on the EM response.

The corrections can be classified by orders of $ql_{B}$. The EM response
determined in the last section is accurate only to the zeroth order of
$ql_{B}$. By considering the electric quadrupole correction in Eq. (21), we
can determine the response accurate to the linear order of $ql_{B}$. Because
$\partial_{t}\bm{B}=-\bm{\nabla}\times\bm{E}$ is of the same order of the
quadrupole correction, it should be considered on an equal footing. On the
other hand, corrections due to the spatial gradient of $\bm{B}$ is of the
quadratic order in $ql_{B}$, and considering them requires a multipole
expansion of $\bm{j}^{\mathrm{e}}$ to at least the order of the magnetic
quadrupole [37]. It is beyond the scope of the current study.

Corrections due to the variation of $\bm{b}$ should be fully accounted for.
Even thought $\bm{b}$ looks like the counterpart of $\bm{B}$, it is actually
equivalent to the density because of the self-consistent condition Eq. (6).
Since it is not subject to the approximation of the multipole expansions,
considering its gradients will not introduce inconsistencies. As we will see,
its corrections are essential for obtaining a density response function of CFs
with a correct static limit.

For simplicity, we consider these corrections only for half filling.

###  VII.1 Quadrupole correction

By using Eq. (19) and the complete form of Eq. (20), we determine
modifications to the equations of motion Eqs. (22, 23) due to the quadrupole
correction to the energy and the time dependence of $B$. We have

$\displaystyle\dot{\bm{x}}$
$\displaystyle=\frac{b}{B}\frac{\bm{p}}{m^{\ast}}-\frac{1}{eB}\dot{\bm{p}}\times\hat{z}+\frac{1}{eB^{2}}\hat{z}\times\bm{\nabla}\bm{E}\cdot(\bm{p}\times\hat{z}),$
(73) $\displaystyle\dot{\bm{p}}$
$\displaystyle=-\frac{1}{B}\bm{\nabla}\bm{E}\cdot(\bm{p}\times\hat{z}),$ (74)

where we keep only terms related to $\bm{\nabla}\bm{E}$ since only they are
relevant to the quadrupole correction for the linear response to EM fields.
Substituting Eq. (74) into Eq. (73), we obtain $\dot{\bm{x}}=\bm{p}/m^{\ast}$,
i.e., the quadrupole correction does not change Eq. (26).

The Boltzmann equation can then be solved. We decompose the distribution
function into $f=f_{0}+f_{1}$, where $f_{0}$ still assumes the same form as
Eq. (30) but with the quadrupole term in Eq. (20) included. It is easy to see
that it introduces no correction to $\bm{j}_{0}$. The correction to $f_{1}$ is

$\displaystyle\Delta f_{1}$
$\displaystyle=\frac{e\left(\bm{v}^{\prime}-\bar{v}^{\prime}\right)\cdot\bm{E}}{-1/\tau+\mathrm{i}\omega-\mathrm{i}\bm{q}\cdot\bm{v}}\left(-\frac{\partial
f_{0}}{\partial\varepsilon}\right),$ (75)

where
$\bm{v}^{\prime}=-(\omega/2\tilde{\omega}_{B}^{2})\bm{q}\cdot(\hat{z}\times\bm{v})(\hat{z}\times\bm{v})$
and
$\bar{\bm{v}}^{\prime}=-v_{\textrm{F}}(\bm{q}l_{B})(\omega/\tilde{\omega}_{B})\gamma/[2(1+\gamma)(1-\gamma(1-i\omega\tau))]$.

The quadrupole correction to $\bm{j}_{1}$ can then be determined by using Eq.
(52). It is obvious from Eq. (20) that the quadrupole term has a similar
effect as the dipole term in making $\bm{E}$ nonequivalent to $\bm{e}$.
Therefore, the quadrupole correction can be characterized by a correction to
the $\beta$ matrix. We have

$\displaystyle\Delta\beta_{11}$
$\displaystyle=-\frac{\left(ql_{B}\right)^{2}}{4}\frac{1+\gamma(1-i\omega\tau)}{(1+\gamma)(1-i\omega\tau)}\approx
0,$ (76) $\displaystyle\Delta\beta_{22}$
$\displaystyle=-\frac{ql_{B}}{4}\frac{\mathrm{i}\omega\left(ql\right)}{\tilde{\omega}_{B}}\frac{1}{(1+\gamma)(1-i\omega\tau)},$
(77)

and non-diagonal elements vanish. The correction then propagates to Eq. (66)
for the total EM response.

To determine the electron current $\bm{j}^{\mathrm{e}}$, we need to determine
the displacement current
$\bm{J}^{\mathrm{Q}}=-\partial\bm{\nabla}\cdot\mathbb{Q}/\partial t$. Note
that $\bm{P}=0$ is still true since Eq. (26) is not modified by the quadrupole
correction. One may evaluate $\bm{J}^{\mathrm{Q}}$ directly, or apply the
Onsager relation which yields in the current context

$\bm{J}^{\mathrm{Q}}=\left(\Delta\beta\tilde{\sigma}\right)\left(\bm{e}+\beta\bm{E}\right)=\Delta\beta\sigma_{\mathrm{CS}}\bm{E},$
(78)

where we apply Eq. (65) to get the last form. The total correction to the
electron conductivity tensor of the system is

$\Delta\sigma=\sigma_{\mathrm{CS}}\Delta\beta+\Delta\beta\sigma_{\mathrm{CS}},$
(79)

where the first term is from the $\Delta\beta$ correction to Eq. (66). To the
linear order of $ql_{B}$, it contributes a correction to the Hall conductivity

$\Delta\sigma_{xy}=-\frac{e^{2}}{h}\frac{ql_{B}}{8}\frac{\mathrm{i}\omega}{\tilde{\omega}_{B}}\frac{ql}{(1+\gamma)(1-i\omega\tau)}.$
(80)

###  VII.2 Magnetic correction

In this subsection, we determine corrections due to the variation of the CS
magnetic field $\delta b$. It has two effects. First, it modifies the
equations of motion, and introduces corrections to the responses of the CF
current. Second, it gives rise to a magnetization current because the CF
dispersion Eq. (21) implies that each CF carries a magnetic moment
$\bm{m}_{b}=-(p^{2}/2m^{\ast}B)\hat{z}$ with respect to the CS magnetic field.
Both the effects should be considered.

First, we determine the corrections to the CF responses. In this case, the
local equilibrium distribution is assumed to be

$f_{0}\left(\bm{x},\bm{p},t\right)=n_{\mathrm{F}}\left(\varepsilon(\bm{p})-\varepsilon(\bm{p}_{\mathrm{F}})\right),$
(81)

where we drop terms in the energy irrelevant to $\delta b$ and assume a local
chemical potential
$\mu(\bm{x})=\varepsilon(\bm{p}_{\mathrm{F}})=b(\bm{x})p_{\mathrm{F}}^{2}/2m^{\ast}B$
so that the distribution has a uniform density. Note that the choice of
$\mu(\bm{x})$ does not affect the steady-state solution of the Boltzmann
equation.

We can then determine the correction to the intrinsic current $\bm{j}_{0}$.
$\delta b$ introduces extra forces $-(p^{2}/2m^{\ast}B)\bm{\nabla}\delta
b-e\delta b\dot{\bm{x}}\times\hat{z}$ into the system. As a result, Eqs. (26,
27) are modified to

$\displaystyle\dot{\bm{x}}$
$\displaystyle=\frac{\bm{p}}{m^{*}}-\frac{1}{b}\hat{z}\times\bm{e}+\frac{1}{eb}\bm{m}_{b}\times\bm{\nabla}\delta
b,$ (82) $\displaystyle\dot{\bm{p}}$ $\displaystyle=e\delta
b\hat{z}\times\frac{\bm{p}}{m^{*}}-e\left(\frac{\bm{e}}{D}+\bm{E}\right)-\frac{\bm{\nabla}\delta
b}{b}\frac{p^{2}}{2m^{\ast}},$ (83)

The correction to $\bm{j}_{0}$ is

$\Delta\bm{j}_{0}=-\int[\mathrm{d}\bm{p}]Df_{0}\frac{\bm{m}_{b}\times\bm{\nabla}\delta
b}{b}=\frac{e^{2}}{16\pi m^{*}}\hat{z}\times\bm{\nabla}\delta b.$ (84)

The correction to the extrinsic current $\bm{j}_{1}$ can also be determined.
It is straightforward to determine that $\delta b$ introduces a correction to
$f_{1}$:

$\Delta
f_{1}=\frac{e(\bm{v}-\bar{\bm{v}})\cdot\left(\frac{\hbar}{2m^{\ast}}\bm{\nabla}\delta
b\right)}{-1/\tau+\mathrm{i}\omega-\mathrm{i}\bm{q}\cdot\bm{v}}\left(-\frac{\partial
f_{0}}{\partial\varepsilon}\right),$ (85)

i.e., the effect of $\delta b$ is equivalent to an effective electric field
$(\hbar/2m)\bm{\nabla}\delta b$. Therefore, the correction to $\bm{j}_{1}$ is

$\Delta\bm{j}_{1}=\tilde{\sigma}\frac{\hbar}{2m^{*}}\bm{\nabla}\delta b.$ (86)

Next, we determine the magnetization current. The CS magnetization is
determined by $\bm{M}_{b}=\int[d\bm{p}]Df\bm{m}_{b}.$ We decompose
$\bm{m}_{b}$ as
$\bm{m}_{b}(p_{\mathrm{F}})+[\bm{m}_{b}(p)-\bm{m}_{b}(p_{\mathrm{F}})]$, and
note that $f_{1}$ does not contribute to the expectation value of the second
term. We obtain

$\displaystyle\bm{M}_{b}$
$\displaystyle=\frac{\hbar\rho}{2m^{\ast}}\hat{z}-\int[d\bm{p}]Df_{0}\frac{p^{2}-p_{\mathrm{F}}^{2}}{2m^{\ast}B}\hat{z}=\frac{\hbar\rho}{4m^{\ast}}\hat{z}.$
(87)

After applying Eq. (6), we find that the magnetization current is

$\bm{j}^{\mathrm{m}}=\bm{\nabla}\times\bm{M}_{b}=\frac{e^{2}}{16\pi
m^{*}}\hat{z}\times\bm{\nabla}\delta b=\Delta\bm{j}_{0}.$ (88)

Summing all the corrections, we find that Eq. (64) is modified by substituting
$\bm{e}$ with $\bm{e}+(\hbar/2m^{\ast})\bm{\nabla}\delta b$. After applying
the self-consistent conditions Eqs. (6, 7), we obtain

$\bm{j}=-\sigma_{\mathrm{CS}}\tilde{\sigma}^{-1}\sigma_{\mathrm{CS}}\bm{E}^{\ast}+\sigma_{\mathrm{CS}}\left[\beta\bm{E}+\frac{1}{e^{2}\tilde{\chi}_{0}}\bm{\nabla}\rho\right],$
(89)

where $\tilde{\chi}_{0}=-m^{\ast}/2\pi\hbar^{2}$ is the static density
susceptibility of a free CF system at $\bm{q}=0$, and

$\bm{E}^{\ast}=\bm{E}+\frac{1}{e^{2}\tilde{\chi}_{0}}\bm{\nabla}\rho.$ (90)

Finally, we determine the electron current $\bm{j}^{\mathrm{e}}$. We need to
determine the polarization density $\bm{P}$ and the magnetization density
$\bm{M}$. The polarization density does not vanish due to the magnetic
correction in Eq. (82). Applying the average $-e\int[\mathrm{d}\bm{p}]Df$ to
both sides of Eq. (82) and noting
$-e\int[\mathrm{d}\bm{p}]D\dot{\bm{x}}f=\bm{j}-\bm{j}^{\mathrm{m}}$ in the
current context, we obtain

$\bm{P}=-\frac{1}{\tilde{\omega}_{B}}\frac{e^{2}}{8\pi
m^{\ast}}\bm{\nabla}\delta b.$ (91)

Next, we determine the magnetization density $\bm{M}$. Each CF carries a
magnetic moment with respect to $B$: $\bm{m}_{B}=\partial
L/\partial\bm{B}=(\bm{p}\times\dot{\bm{p}})/2eB^{2}-\hat{z}(b/B^{2})(p^{2}/2m^{\ast})+\bm{m}_{B}^{\prime}$,
where $\bm{m}_{B}^{\prime}$ denotes contributions from such as the
$B$-dependence of the effective mass or the zero-point energy of a LL as
discussed in the M2RPA theory [49, 13]. Substituting Eq. (83), we obtain
$\bm{m}_{B}=\bm{m}_{\bm{b}}+\bm{m}_{B}^{\prime}$. The terms related to driving
fields in Eq. (83) are ignored because their contributions are of higher
orders of $ql_{B}$.

The total electron current can then be obtained by applying Eq. (12). We note
that $\bm{j}^{\mathrm{m}}$ should be subtracted from $\bm{j}$ when applying
the relation. It cancels part of the contribution from $\bm{m}_{B}$. The net
magnetization correction to the electron current is thus contributed only by
$\bm{M}^{\prime}\equiv\int[\mathrm{d}\bm{p}]D\bm{m}_{B}^{\prime}f$. We have

$\bm{j}^{\mathrm{e}}=\left(-\sigma_{\mathrm{CS}}\tilde{\sigma}^{-1}\sigma_{\mathrm{CS}}+\sigma_{\mathrm{CS}}\beta\right)\bm{E}^{\ast}+\bm{\nabla}\times\bm{M}^{\prime}.$
(92)

Applying the relation
$\rho(\omega,\bm{q})\approx\rho^{\mathrm{e}}(\omega,\bm{q})=\bm{q}\cdot\bm{j}^{\mathrm{e}}(\omega,\bm{q})/\omega$,
we can rewrite the electron current as
$\bm{j}^{\mathrm{e}}=\sigma\bm{E}+\bm{\nabla}\times\bm{M}^{\prime}$ with the
conductivity tensor $\sigma$ determined by

$\sigma^{-1}=\left(\sigma^{\ast}\right)^{-1}-\frac{\mathrm{i}q^{2}}{\omega}\frac{1}{e^{2}\tilde{\chi}_{0}}\left(\begin{array}[]{cc}1&0\\\
0&0\end{array}\right),$ (93)

where
$\sigma^{\ast}=-\sigma_{\mathrm{CS}}\tilde{\sigma}^{-1}\sigma_{\mathrm{CS}}+\sigma_{\mathrm{CS}}\beta$
is the conductivity tensor with respect to $\bm{E}^{\ast}$.

###  VII.3 Density response function

From Eq. (93), we can determine the density response function $\chi$ of
electrons by using the relation
$\chi(\omega,\bm{q})=-\mathrm{i}q^{2}\sigma_{11}(\omega,\bm{q})/e^{2}\omega$.
We have

$\frac{1}{\chi}=\frac{1}{\tilde{\chi}_{0}}+\frac{\mathrm{i}\omega\tilde{\sigma}_{22}}{q^{2}}\frac{\left(2h/e\right)^{2}}{1+\frac{2h}{e^{2}}\frac{\mathrm{i}\omega}{\tilde{\omega}_{B}}\tilde{\sigma}_{22}},$
(94)

where $\tilde{\sigma}_{22}$ is the transverse component of $\tilde{\sigma}$
shown in Eq. (59).

It can be shown that $\chi$ is related to the magnetic susceptibility of CFs.
The first term in the right hand side of Eq. (94), which is contributed by the
magnetic correction, can be absorbed into the second term as a correction to
$\tilde{\sigma}_{22}$. In the low-frequency limit $\omega\rightarrow 0$, we
have
$\Delta\tilde{\sigma}_{22}\approx(e/2h)^{2}q^{2}/\mathrm{i}\omega\tilde{\chi}_{0}$.
Because the magnetic susceptibility of CFs is related to the transverse
component of the CF conductivity tensor by
$\tilde{\chi}^{\mathrm{m}}=\mathrm{i}\omega[\tilde{\sigma}_{22}+\Delta\tilde{\sigma}_{22}]/q^{2}$
[50], we have

$\frac{1}{\chi_{0}}=\left(\frac{2h}{e}\right)^{2}\tilde{\chi}_{0}^{\mathrm{m}},$
(95)

where $\chi_{0}$ ($\tilde{\chi}_{0}^{\mathrm{m}}$) denotes the static density
(magnetic) susceptibility of electrons (CFs).

Equation (95) implies a self-consistency condition for the model of CFs. In
Sec. IV.2, we assume the relation $\delta
b(\bm{x})=-(2m^{*}/e\hbar)V_{\textrm{eff}}(\bm{x})$ for determining the
magnitude of the fluctuating CS magnetic field. It leads to the conclusion of
no-side-jumps in Sec. IV.2, and our derivation of the extrinsic response is
based on the conclusion. The relation requires that the density response
function of electrons should have the long-wavelength static limit
$\chi_{0}=\tilde{\chi}_{0}$. Therefore, to make our considerations self-
consistent, the static density susceptibility $\tilde{\chi}_{0}$ of CFs should
be related to $\tilde{\chi}_{0}^{\mathrm{m}}$ by the same relation as Eq.
(95). It is easy to verify that the relation indeed holds for our
semiclassical model. However, our semi-classical approach cannot fully
determine the magnetic susceptibility without taking account of quantum
effects (e.g., Landau diamagnetism) and specifying an effective hamiltonian to
the quadratic order of $\delta b$ [51]. Therefore, we need to assume that a
full effective hamiltonian of CFs has these corrections canceled.

##  VIII An experimental test

The longitudinal conductivity $\sigma_{11}(\omega,\bm{q})$ of a two
dimensional electron gas (2DEG) at finite $\bm{q}$ can be measured by using a
surface acoustic wave (SAW) that traverses the 2DEG. The SAW is attenuated by
the 2DEG and its velocity is altered [52, 53, 54]. It is shown that the shift
of the sound velocity $v_{\mathrm{s}}$ is related to
$\sigma_{11}(\omega,\bm{q})$ by [55, 52, 45]

$\frac{\Delta
v_{\mathrm{s}}}{v_{\mathrm{s}}}=\frac{\alpha^{2}}{2}\mathrm{Re}\frac{1}{1+\mathrm{i}\sigma_{11}(\omega,\bm{q})/\sigma_{\mathrm{m}}},$
(96)

where $\alpha^{2}/2=3.2\times 10^{-4}$ is the piezoelectric coupling constant
for GaAs, $\sigma_{\mathrm{m}}=2\epsilon_{\mathrm{eff}}v_{\mathrm{s}}$, and
$\epsilon_{\mathrm{eff}}$ is the effective dielectric constant for the 2DEG.

The dipole theory (as well as the Dirac CF theory) predicts a longitudinal
conductivity $\sigma_{11}(\omega,\bm{q})$ different from that of the HLR
theory. The most important difference is contributed by the off-diagonal
component of the $\beta$ matrix in Eq. (3) which originates from the dipole
correction (see Sec. V.2). We have:

$\sigma_{11}^{\mathrm{Dipole}}-\sigma_{11}^{\mathrm{HLR}}\approx\mathrm{i}\frac{e^{2}}{2h}\frac{\omega}{\tilde{\omega}_{\mathrm{B}}}.$
(97)

The magnitude of the correction ($\sim 3\times 10^{-7}\mathrm{S}$ for the
experiment considered below) turns out to be comparable to that of
$\sigma_{\mathrm{m}}$. Since the correction to the imaginary part of the
conductivity scales the velocity shift: $\Delta
v_{\mathrm{s}}/v_{\mathrm{s}}\approx(\alpha^{2}/2)\mathrm{[1-\mathrm{Im}(\sigma_{11}/\sigma_{\mathrm{m}})]/[\mathrm{Re}(\sigma_{11}/\sigma_{\mathrm{m}})]^{2}}$,
there is a good chance to detect its effect in a SAW experiment.

As a test of our theory and an effort of looking for the dipole correction, we
fit the data presented in Ref. [56]. The experiment is one of a series of SAW
experiments carried out by Willett et al. on CF systems [57, 54, 56, 6]. It
employs a high SAW frequency (10.7GHz) and thus has the best data quality. The
data is shown in Fig. 2.

Figure 2:  Experimental data of the SAW velocity shift adapted from Ref. [56] (black circles) and fits to the data by using the dipole theory (red thick lines) and the HLR theory (blue thin lines). Two different ways of fittings are shown: (I) fits the data as it is by treating $\sigma_{\mathrm{m}}$ as a fitting parameter; (II) fits the data after subtracting a constant background and fixes the value of $\sigma_{\mathrm{m}}$ to $6.8\times 10^{-4}\mathrm{S}$. For clarity, (II) is offset vertically by $-0.8\times 10^{-4}$. |  | $v_{\mathrm{s}}$ | $m^{\ast}$ | $\tau$ | $\sigma_{\mathrm{m}}$ | $\alpha^{2}/2$ | $(\Delta v_{\mathrm{s}}/v_{\mathrm{s}})_{0}$
---|---|---|---|---|---|---|---
($10^{3}\mathrm{m/s}$) | ($m_{\mathrm{e}}$) | (ps) | ($10^{-7}\mathrm{S}$) | ($10^{-4}$) | ($10^{-4}$)
I | Dipole | $3.1$ | $0.6$ | $14$ | $14$ | $\underline{3.2}$ | $\underline{0}$
HLR | $3.0$ | $\underline{0.3}$ | $7.3$ | $12$ | $\underline{3.2}$ | $\underline{0}$
II | Dipole | $3.1$ | $0.7$ | $20$ | $\underline{6.8}$ | $2.8$ | $0.89$
HLR | $3.1$ | $\underline{0.3}$ | $5.7$ | $\underline{6.8}$ | $4.4$ | $0.30$
Reference |  $3.0$ [58] |  $0.8$ [56] | – | $6.8$ | $3.2$ | –
Table 1:  The values of fitting parameters. Underlined numbers indicate values
fixed in fittings. $(\Delta v_{\mathrm{s}}/v_{\mathrm{s}})_{0}$ is the value
of the background subtracted from the data. The last line shows the reference
values of the parameters as well as their sources.

We first fit the data as it is. We find that the dipole theory fits the data
rather well. It reproduces the main features of the resonances and coincides
well with the data in the whole range of $\Delta B$, as evident in Fig. 2
(denoted as I). The parameters for the fit are shown in Table 1. All
parameters except $\sigma_{\mathrm{m}}$ are close to their respective
reference values. $\sigma_{\mathrm{m}}$, on the other hand, is approximately a
factor of two larger than the expected value. In comparison, the HLR theory
fits the data less well. It cannot fit the data near $\Delta B\sim 0$ and at
the two ends simultaneously, and produces sub-resonance features that are too
pronounced. Moreover, we cannot treat $m^{\ast}$ as a fitting parameter for
the fitting to the HLR theory because it converges to an unphysical negative
value. We therefore set $m^{\ast}=0.3m_{\mathrm{e}}$. Smaller values of
$m^{\ast}$ yield slightly better fits but not much different from the one
shown. The fitted value of $\sigma_{\mathrm{m}}$ is also significantly larger
than its expected value.

The disparity between the fitted values of $\sigma_{\mathrm{m}}$ and its
expected value is an issue. In literatures, the value of $\sigma_{\mathrm{m}}$
is usually quoted as $3.5\times 10^{-7}\mathrm{S}$ [57, 54, 56, 6]. The value
is obtained by assuming $\epsilon_{\mathrm{eff}}=(\epsilon_{0}+\epsilon)/2$,
where $\epsilon_{0}$ and $\epsilon$ are the dielectric constants of vacuum and
GaAs, respectively. The assumption is appropriate only when the 2DEG is on top
of a sample surface, while in reality 2DEG is always a finite depth $d$
underneath the surface. When $qd\gtrsim 1$, which is likely true for the
current case, one should use $\epsilon_{\mathrm{eff}}\approx\epsilon$ [58]. It
yields the reference value $\sigma_{\mathrm{m}}=6.8\times\times
10^{-7}\mathrm{S}$ shown in Table 1. Unfortunately, it is not easy to account
for the remaining disparity in the value of $\sigma_{\mathrm{m}}$. We are not
aware of another mechanism that could increase $\sigma_{\mathrm{m}}$.

We therefore try to fit the data by insisting
$\sigma_{\mathrm{m}}=6.8\times\times 10^{-7}\Omega^{-1}$. In this case, both
the theories cannot fit the data as they show sizable deviations and too
strong resonances. Actually, in the regime $ql\gg 1$ ($ql\approx 8$ in the
current case) necessary for the observation of strong resonances, we have

$\sigma_{11}\left(\Delta
B=0\right)\approx\frac{e^{2}}{4h}\left(\frac{q}{k_{\mathrm{F}}}\pm\mathrm{i}\frac{\omega}{\tilde{\omega}_{\mathrm{B}}}\right),$
(98)

where the $+$ and $-$ signs are for the dipole theory and the HLR theory,
respectively. The real (and dominant) part of the conductivity is set only by
external parameters (the wavenumber and the density). It predicts $\Delta
v_{\mathrm{s}}/v_{\mathrm{s}}\sim 0.4\times 10^{-4}$ at $\Delta B=0$ for
$\sigma_{\mathrm{m}}=6.8\times 10^{-7}\mathrm{S}$, much lower than the value
actually observed in the experiment. Moreover, because $\sigma_{11}$ is
determined by the transverse component of the CF conductivity
$\tilde{\sigma}_{22}$ whose real part is not renormalized by interaction in
the clean limit [59], the disparity cannot be eliminated by considering a
residual interaction between CFs.

We thus conjecture that there may exist extrinsic mechanisms which contribute
a background to the experimentally measured SAW velocity shifts. We treat the
value of the background as a fitting parameter. We also treat the
piezoelectric coupling constant $\alpha^{2}/2$ as a fitting parameter because
it is shown to be a function of the depth $d$ of the 2DEG layer [58]. The
fittings are shown in Fig. 2 with parameters shown in Table 1 (denoted as II).
It is evident that the dipole theory has the better fit. It yields parameters
close to their respective reference values or in the reasonable regime. On the
contrary, in the fitting to the HLR theory, we find sizable deviations and a
fitted value of $\alpha^{2}/2$ which is too large 111The constant has an
oscillatory dependence on $d$. Our own calculation shows the same qualitative
behavior as that shown in Ref [58] but with quantitative differences. It
indicates that the constant has a maximal value $3.7\times 10^{-4}$ at $d=0$
and a second peak $3.1\times 10^{-4}$ at $qd\sim 2.5$..

Although the dipole theory consistently provides better fits to the
experimental data, the support to it cannot be regarded conclusive before
either the value of $\sigma_{\mathrm{m}}$ or the conjectured background can be
clarified. We also apply the fitting to the set of data presented in Ref.
[57]. We find that both the theories can reproduce main features but with
sizable quantitative deviations. More experiments may be needed for clarifying
various factors that may affect the measurement. Our analysis, nevertheless,
does show that SAW experiments have the potential for measuring the dipole
correction and differentiating different theories of CFs.

Finally, we note that the magnetic correction shown in Eq. (93) only slightly
modifies $\sigma_{11}$, and the quadrupole correction, which only alters the
Hall conductivity, does not have an effect to the velocity shift. Therefore,
SAW experiments can not distinguish the dipole theory from the Dirac CF
theory.

##  IX Summary and discussion

In summary, we determine the EM response of CFs of the dipole picture. We show
how the Dirac CF-like EM response emerges in the dipole picture (Sec. VI.2).
We show that a CF system has an intrinsic Hall response which is independent
of the filling factor (Sec. III.1) and not altered by impurity scattering
(Sec. IV). When one ascribes the intrinsic response to a redefined vacuum, the
remainder of the EM response can be interpreted as from a Dirac CF system as
far as the the low energy and long wavelength response is concerned.

When we go beyond the zeroth order in $ql_{B}$, deviations from the Dirac CF
theory start to show up (Sec. VII). We see that the quadrupole correction
gives rise to a correction to the Hall conductivity, and the magnetic
correction modifies the density response function. Although it could be hard
to detect these corrections in experiments, they do manifest an important fact
of the dipole picture, i.e., a CF has an internal structure.

It is natural to ask whether or not the internal structure of a CF can be
probed in experiments. In the dipole picture, a CF has a length scale $l_{B}$,
comparable to its Fermi wavelength. It suggests that carefully designed
experiments that can measure the Fermi wavelength could potentially probe the
internal structure as well. To this end, we fit the SAW measurement data by
Willett et al., and show that the contribution from the dipole correction is
sizable and improves fittings. To distinguish the dipole model and the Dirac
CF theory, on the other hand, we need to detect quadrupoles or higher order
corrections. In Ref. [31], we propose such an experiment. We show that a
geometrical resonance experiment with a modulated magnetic field will give
rise to asymmetry opposite to that observed in a similar experiment but with a
modulated scalar potential (or the CS magnetic field).

Our theory suggests a couple of constraints on the effective model describing
the dipole, including the density-of-state factor appeared in the dispersion
Eq. (21) and the relation Eq. (95) between the magnetic and density
susceptibilities. In principle, one could infer an effective Hamiltonian
directly from microscopic wave-functions. It is necessary to develop a scheme
to do that and test the viability of our assumptions. This will be left as a
future investigation.

We do not consider the effects of residual interaction between CFs for a
reason. For both the HLR theory and the Dirac CF theory, the consideration is
essential because both the theories rely on renormalizations to eliminate the
unwelcome presences of bare parameters, i.e., the band mass for the HLR theory
and the velocity of the massless Dirac cone for the Dirac CF theory. For our
theory, on the other hand, the consideration is less important because our
model is inferred directly from a microscopic CF wave function defined in the
projected Hilbert space of a LL. Spurious dependences on bare parameters have
been eliminated from the beginning. The residual interaction between CFs,
which should also be derived directly from the microscopic CF wave function,
would renormalize model parameters. In contrast to the HLR theory and the
Dirac CF theory, the renormalization in our theory needs not relate the
parameters to their bare counterparts.

###### Acknowledgements.

This work is supported by the National Basic Research Program of China (973
Program) Grants No. 2018YFA0305603 and No. 2015CB921101 and the National
Science Foundation of China Grant No. 11325416.

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