# Particle-Hole Symmetry in the Fermion-Chern-Simons and Dirac Descriptions of a Half-Filled Landau Level

###### Abstract

It is well known that there is a particle-hole symmetry for spin-polarized electrons with two-body interactions in a partially filled Landau level, which becomes exact in the limit where the cyclotron energy is large compared to the interaction strength, so one can ignore mixing between Landau levels. This symmetry is explicit in the description of a half-filled Landau level recently introduced by D. T. Son, using Dirac fermions, but it was thought to be absent in the older fermion-Chern-Simons approach, developed by Halperin, Lee, and Read and subsequent authors. We show here, however, that when properly evaluated, the Halperin, Lee, Read (HLR) theory gives results for long-wavelength low-energy physical properties, including the Hall conductance in the presence of impurities and the positions of minima in the magnetoroton spectra for fractional quantized Hall states close to half-filling, that are identical to predictions of the Dirac formulation. In fact, the HLR theory predicts an emergent particle-hole symmetry near half filling, even when the cyclotron energy is finite.

January 21, 2021

###### Contents

- I Introduction
- II Review of the HLR approach
- III DC transport at
- IV Commensurability Oscillations
- V Comparison with the Dirac theory
- VI Conclusions

## I Introduction

A series of recent developments have focused renewed attention on the problem of a two-dimensional system of interacting electrons at, or close to, a half-filled Landau level. In particular, in a highly original work, D. T. SonSon (2015) has proposed a description of the half-filled Landau level that employs a collection of relativistic Dirac fermions, interacting with an emergent gauge field with no Chern-Simons term. This description stands in contrast to the more traditional description in terms of non-relativistic “composite fermions” interacting with a Chern-Simons gauge field, developed by Halperin, Lee and Read (HLR)Halperin *et al.* (1993) and others, some twenty years ago. (See, e.g., Refs. Jain (1989); Kalmeyer and Zhang (1992); Lopez and Fradkin (1991); Greiter and Wilczek (1992); Rejaei and Beenakker (1992); Kim *et al.* (1994); Altshuler *et al.* (1994); Simon and Halperin (1993); Stern and Halperin (1995)).

The Son-Dirac description has led to a number of valuable insights into the conventional problem of two-dimensional electrons in a strong magnetic fieldMetlitski and Vishwanath (2015); Wang and Senthil (2016); Geraedts *et al.* (2016); Murthy and Shankar (2016); Wang and Senthil (2016a), and it has also served to elucidate connections to other physical problems, such as exotic electronic states that could arise at the surface of a three-dimensional topological insulatorMross *et al.* (2015); Wang and Senthil (2015); Metlitski and Vishwanath (2015), time-reversal invariant quantum spin liquids in three dimensionsWang and Senthil (2016b); Metlitski and Vishwanath (2015); Metlitski (2015), and a class of field theory dualities in dimensionsWang and Senthil (2015); Metlitski and Vishwanath (2015); Mross *et al.* (2016); Seiberg *et al.* (2016); Karch and Tong (2016); Murugan and Nastase (2016); Kachru *et al.* (2016).

The Dirac picture seems to have
some significant advantages compared with the HLR description for the conventional two-dimensional electron system, in particular with respect to particle-hole (PH) symmetry.
It is well known that a partially-filled Landau level of spin-polarized electrons with two-body interactions should have an exact PH symmetry about half-filling, in the limit where the electron-electron interaction is weak compared to the cyclotron energy, so one can neglect mixing between Landau levelsGirvin (1984). Numerical calculations, either through trial wave functions motivated by the composite fermi liquid pictureRezayi and Haldane (2000); Balram and Jain (2016), or through unbiased energetic calculationsRezayi and Haldane (2000); Geraedts *et al.* (2016), seem to confirm that this symmetry is unbroken in the incompressible phase. This symmetry is made manifest in the Dirac model by setting a single parameter equal to zero, the Dirac mass .

By contrast, the HLR approach is not explicitly PH symmetric, and in fact it has been questioned whether the approach is even compatible with PH symmetryKivelson *et al.* (1997); Barkeshli *et al.* (2015).
It has been suggested that the Dirac theory and the HLR theory actually represent different fixed points and that there might necessarily be some kind of discontinuous phase transition separating these fixed pointsKivelson *et al.* (1997); Son (2015); Barkeshli *et al.* (2015); Potter *et al.* (2016).
These suggestions have been based on analyses of several key physical properties, in which it appeared that predictions of HLR were contradictory to PH symmetry.

In this paper, we reexamine several of these properties, and we find that when properly analyzed, the HLR theory gives identical results to the Dirac theory, in the limit of long wavelengths and low energies, near half filling. Some of the confusion about these points has arisen simply because the predictions of HLR theory were not previously analyzed with sufficient care. Despite our limitations to long-wavelengths and low-energies, we believe that our analysis casts strong doubt on the possibility that there is any regime of parameters in which the Dirac description and the HLR description correspond to two different phases of matter. Specifically, we have carried out detailed studies of two types of properties where it has been suggested that there are irreconcilable differences between the HLR and Son-Dirac descriptions – the Hall conductance of a half-filled Landau level in the presence of disorder, and the momentum values of the minima in the magnetoroton spectra of fractional quantized Hall states that are symmetrically displaced from .

In the presence of a disorder potential that is statistically PH symmetric, symmetry dictates that the Hall conductance should be exactly , in the absence of Landau level mixing. Since 1997, it has been widely believed that HLR is incompatible with this requirement, and that HLR implies deviations in the Hall conductance proportional to the inverse square of the mean-free path of the composite fermions. We show below, however, that when properly evaluated, these deviations are absent in the HLR theory, at least in the case of weak, long-wavelength, disorder potentials.

For a system where the electronic filling factor is close to one half, oscillations in the conductivity at finite wave vector and frequency have been predicted, and in some cases observed, as a function of the deviations from half filling. These oscillations involve excitation or modulation at a non-zero wave vector , where maxima or minima in some characteristic of the response are predicted to occur at a series of values of , approximately given by

(1) |

where is the -th zero of the Bessel function, is the deviation of the magnetic field from the field at half filling, and is the Fermi wave vector of the composite fermions. PH symmetry requires that if the electron density is varied, while the magnetic field is held fixed, the wave vectors should be precisely independent of the sign of . In the Son-Dirac theory, Eq. (1) directly obeys this PH symmetry, because the value of is a constant, determined by the magnetic field, independent of the electron density. In HLR, however, is determined by the electron density, which will be slightly different for positive and negative values of . Therefore, if one were to treat Eq. (1) as an exact equality, using the definition of in HLR, one would find that PH symmetry is obeyed to first order in , but is violated at second order.

We show below that a careful evaluation of the locations of minima in the magnetoroton excitation spectrum in fractional quantum Hall states close to , originally discussed by Simon and Halperin (SH)Simon and Halperin (1993), using the HLR approach, gives predictions that are PH symmetric, at least to order . We show that these predictions coincide with the predictions of the Son-Dirac theory. The SH formulas actually contain corrections to Eq. (1), which vanish in the limit but are non-zero at order and which precisely eliminate the PH asymmetry at this order.

We note that the results described above were both obtained by careful evaluation of the HLR theory at the RPA level, and did not require any explicit assumption of particle hole symmetry, or any apparent assumption about the ratio between the electron interaction strength and the bare electron cyclotron energy. These results suggest that even when this ratio is finite, so the electrons are not projected into a single Landau level, there may be an emergent PH symmetry, which becomes asymptotically exact in the limit of low-frequency, long-wavelength and small deviation from half-filling. Our results show that for the properties we have analyzed, this is true at least to some nontrivial orders in frequency, momentum and deviation from half-filling.

Within the context of HLR theory, we find that a similar degree of PH symmetry should emerge in the vicinity of other fractions of the form , such as 1/4, 1/6, etc. As a practical matter, this is only of interest for small values of , since at least for the case of Coulomb interactions between the electrons, the ground state for values of appears to be a Wigner crystal of electrons, rather than a liquid of composite fermions. Nevertheless, an emergent PH symmetry at or would be noteworthy, since there is no exact particle hole symmetry about fractions other than 1/2, even for electrons confined to a single Landau level.

The structure of the paper is the following. In the next Section we review the HLR approach to the half-filled Landau level. In Section III we address the issue of transport at in the presence of disorder, and show how the HLR approach yields results which are consistent with the requirements of particle-hole symmetry. In Section IV we address “commensurability oscillations”, which occur at fillings slightly away from , with a focus on the locations of minima in the dispersion curves for the lowest-energy magnetoroton excitations in fractional quantized Hall states near half filling. We show how an analysis within the HLR approach yields results that are consistent with the requirements of particle-hole symmetry. In Section V, we review the Son-Dirac approach, and make a comparison between results of that approach and our analyses based on HLR. We conclude with a Summary section.

## Ii Review of the HLR approach

### ii.1 Definition of the Problem

We consider a two-dimensional system of interacting electrons in a strong magnetic field, with a Landau level filling fraction that is equal to or close to . We assume that the electrons are fully spin polarized, so we may neglect the spin degree of freedom. The Hamiltonian of the system may be written in the form

(2) |

where is a two-body interaction of the form

(3) |

and is the vector potential due to a uniform magnetic field in the -direction. In the case where is a long-range potential, the Hamiltonian must include interactions between the electrons and a uniform neutralizing background, which we include in . In the presence of impurities, we shall add a one-body potential which depends on position; for the present, however, we shall consider a system without impurities, so we take . Except where otherwise stated, we use units where the electron charge is positive and equal to unity, and .

The system under consideration has several important properties. First, it is Galilean invariant, so that it must obey Kohn’s theorem, which states that the response to a uniform time-varying electric field should be the same as for a system of non-interacting electrons in the given magnetic field. Second, as mentioned in the Introduction, in the limit where the electron mass is taken to zero, so that the cyclotron energy becomes infinite while the electron-electron interaction is held fixed, the system should manifest an exact PH symmetry about Landau-level filling fraction . We shall see to what extent these properties are preserved by approximations that have been proposed for treating the system.

### ii.2 The HLR hypothesis

The fermion-Chern-Simons approach employed in HLR began with an exact unitary transformation, a singular gauge transformation, where the many-body electron wave function is multiplied by a phase factor that depends on the positions of all the electrons, such that the transformed Hamiltonian acquires a Chern-Simons gauge field , with flux quanta attached to every electron. The transformed problem may be expressed in Lagrangian form by the following Lagrangian density:

(4) |

(5) |

(6) |

Taking the variation of the Lagrangian with respect to , we obtain the constraint

(7) |

In these equations, is the Grassmann field for a set of transformed “composite fermions” (CFs), whose density is identical to the electron density .

At this stage, we have merely transformed one insoluble problem to another. However, the transformed problem admits a sensible mean-field approximation, whereas the original problem did not. In particular, if the Landau level is half full, so that there is one electron for each quantum of electromagnetic flux, the mean field problem describes a set of non-interacting fermions in zero magnetic field. To go beyond mean-field theory, one must include the effects of fluctuations in the gauge field and fluctuations in the two-body potential. The central hypothesis of HLR is that, in principle, one could obtain the correct properties of the system by starting from the mean field solution, treating the omitted fluctuation terms via perturbation theory. This assumes that the interacting ground state can be reached from the mean-field solution by turning on the perturbing terms adiabatically, without encountering any phase transition. Among the consequences of this assumption are that the ground state at should be compressible, and that there should be something like a Fermi surface, with a well-defined Fermi wave vector, Halperin *et al.* (1993); Altshuler *et al.* (1994); Kim *et al.* (1994); Stern and Halperin (1995).

Experimentally, in GaAs two-dimensional electron systems, it appears that the HLR hypothesis is correct for electrons in the lowest Landau level. However, it appears that the HLR hypothesis breaks down for electrons in the second Landau level, where one observes an incompressible fractional Hall state, with an energy gap, at half filling, in high quality samplesWillett *et al.* (1987). It is widely believed that this quantized Hall state may be understood as arising from an instability of the Fermi surface to formation of Cooper pairs in the second Landau levelMoore and Read (1991); Read and Green (2000); Levin *et al.* (2007); Lee *et al.* (2007). In still higher Landau levels, it appears that the Fermi surface is unstable with respect to the formation of charge density waves, which can lead to a large anisotropy in the measured electrical resistivity at low temperaturesLilly *et al.* (1999); Koulakov *et al.* (1996); Fogler and Koulakov (1997); Moessner and Chalker (1996).

If one is interested in dynamic properties, such as the response to a time-dependent and space-dependent electric field, the first level of approximation, beyond static mean field theory, is the random phase approximation (RPA), or time-dependent Hartree approximation. In this approximation, the composite fermions are treated as non-interacting fermions, with the bare mass , driven by an effective electromagnetic field which is the sum of the applied external electromagnetic field, the Hartree potential arising from the interaction , in the case where there are induced modulations in the self-consistent charge density, and induced electric and magnetic fields arising from modulations of the Chern Simons gauge field. These fields may be written as

(8) |

where is the electron current density at the point in question.

As we shall discuss further below, many properties of the system near are described properly by the RPA, including the response of the system to a uniform time-dependent electric field. However, use of the unrenormalized electron mass as assumed in the RPA, can lead to a serious error in the energy scale for various excitations. A proper low-energy description of the composite fermion liquid requires the use of an effective mass , which may be very different than the bare mass . In particular, one expects that the renormalized mass should be determined by the electron-electron interaction , and should be independent of , in the limit where and the cyclotron energy goes to infinity. The renormalized mass enters directly in the low temperature specific heat, and it also is manifest in the magnitudes of the energy gaps at fractional quantized Hall states of the form , where is a positive or negative integer, in the limit or Halperin *et al.* (1993); Stern and Halperin (1995).

A simple modification of the RPA, which we denote RPA*, would consist of replacing by in the RPA. Although this would correctly give the energy scale for the specific heat and energy gaps in the fractional quantized Hall states, this would change the response to a time-dependent uniform electric field, which was correctly given in RPA. Specifically, if we write , at frequency , then it is required by Kohn’s theorem that the resistivity tensor should be given by

(9) |

where is the unit antisymmetric tensor, . Using RPA*, one would find, incorrectly, that is replaced by in the formula for .

This defect in RPA* is familiar from the theory of ordinary Fermi liquids. In order to get the correct low-frequency response functions in the presence of a renormalized effective mass, it is necessary to include effects of the Landau interaction parameters . These may be defined by the energy cost to form a distortion of the Fermi surface. Specifically, a small distortion of the form

(10) |

will have an energy cost

(11) |

where . For a Galilean invariant system, we must have

(12) |

As noted in SH Simon and Halperin (1993), inclusion of these interaction parameters will also restore the correct response for the composite fermion system at . In the presence of a non-zero current, the parameters lead to an extra force on the electrons, which restores to in the resistivity tensor (9).

We remark that it is also necessary to take into account a Landau interaction parameter if one wishes to obtain the correct value for the electron compressibility. As in a normal Fermi liquid, we have

(13) |

where is the chemical potential (defined to exclude the contribution of the macroscopic electrostatic potential).

### ii.3 Infrared divergences

As was already observed in HLR, in the case of Coulomb interactions, which behave as for large separations , an analysis of contributions to the effective mass arising from long-wavelength fluctuations of the Chern-Simons gauge field predicts a logarithmic divergence in as one approaches the Fermi surface. A similar divergence is found in the Landau interaction parameters, however, so that Galilean invariance is preserved, and the compressibility remains finite. The decay rate for quasiparticles close to the Fermi energy is predicted to be small compared to the quasiparticle energy, in this case, so that the quasiparticle excitations remain well-defined, and the composite Fermion system may be described as a “marginal Fermi liquid.” Similar infrared divergences are found in the Son-Dirac theory of the half-filled Landau level.

It is believed that these infrared divergences will be absent, and will remain finite, if one assumes an electron-electron interaction that falls off more slowly than , so that long-wavelength density fluctuations in the electron density are suppressed. Moreover, these divergences are irrelevant to the issues of PH symmetry which are the focus of the current investigation. Consequently, we shall assume, for the purposes of our discussion, that we are dealing with an electron-electron interaction that falls of more slowly than and that is finite.

We remark that for short-range electron-electron interactions, fluctuations in the gauge field lead to divergences that are stronger than logarithmic, and long-lived quasiparticles can no longer be defined at the Fermi surface. Nevertheless, it is believed that many predictions of the HLR theory remain valid in this caseKim *et al.* (1994); Altshuler *et al.* (1994).
We expect that the results of the present paper with regard to particle hole symmetry should also apply in the case of short-range interactions, but we have not investigated this case in detail.

### ii.4 Energy gaps at

According to the HLR picture, if there is a finite effective mass at , then for fractional quantized Hall states of the form , where is a positive or negative integer, the energy gaps, in the limit , should have the asymptotic form

(14) |

where the deviation from the magnetic field at for the given electron density, i.e.,

(15) |

Note that the allowed values of are symmetric about , assuming that the electron density is varied while is held fixed, since .

In the limit , PH symmetry requires that the energy gap should be the same for and , assuming that has been held fixed. Equation (14) will satisfy this requirement, at least to first order order in . Symmetry beyond first order depends on the choice of used in the formula. Although the HLR analysis specifies that should be evaluated under the condition of , there is still an ambiguity when , because one must decide whether to use the value appropriate for the given magnetic field or for the given electron density. These conditions are precisely equivalent to each other only when . If one employs in Eq. (14) the value of calculated at the given value of , then the formula will exhibit PH symmetry to all orders in . If one were to use the value of calculated at the given value of , however, there would be violations of PH symmetry at second order in .

In practice, the value of the renormalized mass cannot be calculated entirely within the HLR approach, so the value of to be used in the effective theory must be obtained from experiment or from some other microscopic calculation. Thus we can say that the HLR theory is compatible with PH symmetry in the fractional quantized Hall energy gaps, but it can only be deduced from the theory to first order in . We remark that the same situation occurs in the Son-Dirac theory. Precise PH symmetry in that case depends on a separate assumption that the renormalized value of the Dirac velocity should be determined by the magnetic field and not by the electron density.

## Iii DC transport at

PH symmetry, in the limit , implies that the Hall conductivity in response to a spatially uniform electric field should be precisely given by

(16) |

regardless of the applied frequency. This should be true even in the presence of impurities, provided that the disorder potential is PH symmetric in a statistical sense. This means that if one chooses the uniform background potential such that the average , then all odd moments of the disorder potential must vanish.

In the absence of impurities, we may use the result (9) for a Galilean invariant system to calculate the conductivity tensor

(17) |

If , this gives , which satisfies the condition for PH symmetry. As we have seen, the HLR theory will satisfy Galilean invariance if the interaction parameter is taken into account. However, if one were to use the renormalized mass without the interaction, one would find that is replaced by in Eq. (17), so that particle hole symmetry would not be satisfied for .

Of greater interest is the dc Hall conductivity in the presence of impurities. For many years, beginning with the work of Kivelson et al. in 1997Kivelson *et al.* (1997), it has been widely believed that the HLR approach must give
a result for the Hall conductivity that is inconsistent with PH symmetry, at least at the level of RPA and perhaps beyond, if the mean free path for composite fermions is finite. The reasoning goes as follows. Within the HLR approach, the electron resistivity tensor is related to the resistivity tensor of the composite fermions by

(18) |

where is the Chern-Simons resistivity tensor, given by

(19) |

One finds that in order to obtain the PH symmetric result for , if , it is necessary that However, it was argued that is necessarily equal to zero at . This is because, in the absence of impurities, the composite fermions see an average effective magnetic field equal to zero, which is effectively invariant under time reversal. The presence of impurities leads to non-uniformities in the electron density, which lead to local fluctuations in the effective magnetic field . These fluctuations, in turn, will be the dominant source of scattering of composite fermions, under conditions where the correlation length for the impurity potential is large compared to the Fermi wave length. If the impurity potential is statistically PH symmetric, then there will be equal probability to have a positive or negative value of at any point, so that the resulting perturbation to the composite fermions should again be invariant under time reversal in a statistical sense.

The fallacy we find here in this reasoning is that fluctuations in are correlated with fluctuations in the electrostatic potential, which though their effects are weak compared to the effects of , are sufficient to break the statistical time-reversal symmetry produced by the fluctuations alone. We shall see below that when these correlated fluctuations are taken into account we recover precisely the result required by PH symmetry.

In the subsections below, we show how disorder leads to the desired result for . As there are some subtleties involved in these calculations, we present here two different derivations, which bring different insights to the problem and which may be applicable in somewhat different regimes. The first derivation employs a semi-classical analysis and uses the Kubo formula, which expresses the conductivity in terms of equilibrium correlation functions. The second derivation employs the Born Approximation and the Boltzmann Equation , and calculates the conductivity by analyzing the effect of the electric field on the particles’ dynamics. We also discuss consequences for thermoelectric transport at .

Our calculations are restricted to the case where the Fourier components of the disorder potential have wave vectors small compared to . Neither the HLR nor the Dirac theories, in their simplest forms, can describe quantitatively the effect of potential fluctuations with wave vectors comparable to or larger than . In either theory, the coupling between a short-wavelength potential fluctuation and the operators that scatter a composite fermion from one point to another on the Fermi surface will be affected by vertex corrections, whose value is determined by microscopic considerations and cannot be calculated within the low-energy theory itself.

It should be emphasized that while the effects discussed below may be important as a matter of principle, they are all sub-leading corrections to the transport in the presence of impurities. For small impurity concentrations, the CF Hall conductance is small compared to the diagonal CF conductance, , which is proportional to , where is the transport mean free path for composite fermions. Conversely, if one were to set , this would lead to a deviation of the electronic from the PH-symmetric value by an amount proportional to , which is small compared to as well as to , in the limit of large .

### iii.1 Disorder potential and fluctuations of the magnetic field

In general, density fluctuations produced by an external electrostatic potential such as will tend to screen the external potential and give rise to a combined self-consistent potential, which we denote . Within a mean-field approximation, for long-wavelength potential fluctuations, the induced density fluctuation should be related to by

(20) |

where = is the compressibility of noninteracting fermions. We assume here that the potential contains only Fourier components with wave vectors that are small compared to , which is appropriate for a remotely doped system, where the impurities are set back from the 2DES by a distance large compared to the Fermi wavelength.

Beyond the mean field approximation, we should replace by , and we should redefine the potential to include effects of the Landau parameter. The effective magnetic field produced by a fluctuation in the redefined is then given by

(21) |

Equivalently, we may describe this in terms of the induced vector potential , which may be written in Fourier space as

(22) |

Since the gauge fluctuation will couple to the momentum of a composite fermion with a term , we find that the total effect of the impurity potential is a term in the Hamiltonian whose matrix element between an initial state of momentum and a final state is given by

(23) |

where .

### iii.2 Semiclassical analysis using the Kubo formula

In this subsection, we employ a semiclassical analysis of the dynamics of CFs of mass in the presence of the (screened) impurity potential . We restore factors of and , and we consider a more general situation, where , where is an integer, not necessarily equal to 1. Then Eq (21) for the effective magnetic field should be replaced by

(24) |

The semiclassical equations of motion are then

(25) | |||||

(26) |

(We assume, here, and in the formulas below, that the product of the electron charge and the -component of the external magnetic field is positive. Results for the opposite case may be obtained by interchanging indices for the and axes.)

We shall consider to be a random function, symmetrically distributed around . Its correlation length is assumed large compared to with the Fermi momentum and the Fermi energy, as required for validity of the semiclassical approximation. Note that the Lorentz force (of order ) is then large compared to the force exerted by gradient of the potential (of order ) by a factor . The validity of the semiclassical analysis also requires that the typical scattering angle from this Lorentz force, , is large compared to the diffraction angle , i.e. .

It is convenient to separate into radial and angular co-ordinates, by writing

(27) |

For a particle of energy

(28) |

while the angle must be found by integrating

(29) |

along the trajectory of the particle.

We shall use the classical form of the Kubo formulas for the conductivity in terms of velocity-velocity correlation functions. To this end, we construct the correlator

(30) |

with the average taken over the distribution of particles in phase space. To represent the degenerate Fermi gas we shall consider the microcanonical distribution at the Fermi energy . The conductivities are then

(31) |

where the prefactor involves the compressibility. For fixed Fermi energy , large compared to , we use (28) expanded to first order in , to write

and then use (29) to replace for at both and , leading to

(33) |

The correlator

(34) |

depends on how the particles move in real space. Assuming that the composite mean free path is large compared to the correlations length for fluctuations in the potential , we may expect that each particle will explore phase space with the probability of the microcanonical distribution, . (The assumption is clearly valid in the limit where the magnitude of the potential fluctuations is small while is held fixed.) Integrating the microcanonical distribution over 2D momentum leads to a uniform real-space density distribution [since ]. Thus, each particle moves in such a way that its time-varying potential has the same probability distribution as . For example, from Eqn (29), vanishes under time-averaging. More specifically, since the distribution of is invariant under , so too is that of under , such that (34) is real. Hence, from (33)

(35) |

Inserting this in Eqn (31), and noting that the correlator (34) will vanish at for any disordered potential, we obtain the result

(36) |

For the case , where , we recover our desired result , in units where . More generally, the result (36) implies that the electron Hall conductivity at is precisely given by , even in the presence of impurities. Thus there seems to be a kind of emergent PH symmetry at fractions such as and .

### iii.3 Calculation using the Born Approximation and Boltzmann Equation

It seems reasonable that we are justified in using a semiclassical approximation for our problem, because we are necessarily focused on potential fluctuations on a length scale that is large compared to . However, the requirement also that the classical scattering angle exceeds the diffraction angle, [i.e., the condition discussed above], leads to some subtleties in the applicability of the classical results for weak potentialsD’yakonov and Khaetskii (1991). It can be shown that the transport scattering cross section, (i.e., the integrated cross section weighted by the square of the momentum transfer) is correctly given by the semiclassical approximation in this case, and it agrees with a quantum mechanical calculation based on the Born approximation. However, the total scattering cross section, as well as the differential cross section at any particular angle, is generally not given correctly by a semiclassical analysis. Therefore, it seems useful to check that our semiclassical calculation of the off-diagonal part of the CF conductivity tensor can be duplicated in a more quantum mechanical calculation.

Here we follow closely the analysis used by Nozières and Lewiner (NL)Nozières and Lewiner (1973) for the anomalous Hall effect due to spin-orbit interactions in a spin-polarized semiconductor. In their analysis, NL employed a Boltzmann equation to study the evolution of the electron system in a uniform applied electric field, paying careful attention to the effects of spin orbit coupling on the collision integral in the presence of the field.

In our case, we wish to study carefully the scattering of a composite fermion by an impurity described by an effective Hamiltonian of the form (23). In order to use the NL analysis directly, we must impose the condition that the scattering matrix element due to a single impurity is zero in the limit . This means that the associated potential should vanish for faster than . In real space, this means that the space integral of the potential should vanish, as well as its first spatial moments. If individual impurities do not satisfy these conditions, the NL analysis may still be used if impurities can be grouped into small clusters that satisfy the conditions. In any case, the purpose of this subsection is to provide a check of the validity of the above-described semiclassical approximation as a matter of principle, rather than to check the validity in a realistic situation.

It is instructive to describe our calculation in two parts. In the first part we consider the scattering of a single composite fermion from momentum to momentum by the potential (23) in the absence of an electric field. We show - following NL - that this scattering involves a “side-jump” , i.e., a motion of the electron in the direction perpendicular to the momentum transferred from the disordered potential to the composite fermion. When averaged over all scattering processes from a momentum each scattering event involves a side-jump, which results in a net motion perpendicular to the direction of . In the presence of an electric field , the net flux of electrons that experience scattering by the potential is proportional to , where is the transport scattering time. As they scatter from impurities, the extra electrons acquire a velocity in the -direction given by where is the cumulative side jump during the time in which their direction of motion is randomized. Since is of order , this results in a current in the -direction of the order of , which gives rise to a non-zero contribution to that is independent of the mean free path.

In the second part we consider the effect of an applied electric field on the scattering. In the presence of that field the change in position associated with the side-jump implies that the scattering of the composite fermion involves also a change in its kinetic energy. As explained below, that change results in another contribution to the Hall current, equal in magnitude and sign to the first contribution. Throughout this subsection, we assume , and return to units where .

#### iii.3.1 Scattering rate of a single composite fermion

For the first part, suppose that a composite fermion, described by a Gaussian wave packet, centered at a momentum on the Fermi surface, is incident on the impurity. As discussed in Appendix B of NL, we may write the wave function of the CF as

(37) |

(38) |

where describes the incident wave:

(39) |

where is the energy of a fermion of wave vector , and is a normalization constant, and and are of order and respectively. (Note that the incident wave packet is centered at the origin at time .) In the limit of large positive times one finds that

(40) |

As noted in NL, the average position of the particle at time can be written as

(42) |

where

(43) |

There are two contributions to the shift of the average position. The first is seen when we consider a momentum in the scattered wave, with , so that . Then, to lowest order, may be replaced by , and the phase is equal to the phase of . Using (23) for , we find that has an extra argument, beyond the contribution from , arising from the complex value of . This extra argument has the form , and it leads to an extra displacement of the center of the scattered wave packet by an amount

(44) |

The second contribution to the average displacement comes from weight that has been asymmetrically removed from the incident part of the wave packet, where is close to . Here there is an interference between and . If one assumes that is vanishing for , then one finds that the contribution from this term is given by

(45) |

Summing the two contributions we find that the net displacement (“side jump”) associated with a particle that scatters from a direction into direction depends on the transferred momentum, and is given by

(46) |

This side jump contributes directly to the total current through a net charge displacement per unit time

(47) |

where is the occupation probability for a state of momentum , and is the transition probability [see Eq. (52) below]. We can express the side-jump contribution in terms of the current in the absence of that contribution, . Using Eq. (46) for the displacement, and noticing that the transport scattering rate is given by

(48) |

we can simplify Eq. (47) to

(49) |

Since, to leading order, , the term leads to a contribution to of the form

(50) |

#### iii.3.2 Scattering rate in a composite fermion liquid in the presence of an electric field

Eq. (50) is half of the amount we need for PH symmetry. The second half is a consequence of having a liquid of composite fermions, in which an applied electric field affects the occupation of momentum states. While the scattering rate from momentum to momentum is symmetric with respect to the sign of for a single composite fermion in the absence of an electric field, the situation is more complicated in the presence of both a liquid of composite fermions and an electric field. In that case, due to the electric field the side-jump is associated with a change of the composite fermion’s kinetic energy by an amount . The effect of this change of energy on the transport is best understood by means of the Boltzmann equation. For transport in the presence of impurities the equation reads

(51) |

where

(52) |

Here is the Fermi-Dirac distribution, is the force acting on the composite fermions, is the energy, and is the disordered potential. The -function expresses the change of the kinetic energy incurred by the scattered electron, a change which is our main focus here.

As customary, linear response to is analyzed by setting to be on the left-hand side of (51) and by writing on the right-hand side. The transfer of energy affects the expansion of the distribution functions on the right hand side. Specifically we have,

We now make use of the definition of the transport scattering rate (48) to write the Boltzmann equation as

(53) |

which amounts to

(54) |

As this expression shows, in the limit of a small scattering rate the shift of the Fermi sea that results from the application of the electric field is primarily parallel to the electric field, but includes also a small term perpendicular to the field. This term contributes to the Hall conductivity.

The current is , with . The angular integral gives for both components of the current (each component from a different term), leading to , and . This contribution to the Hall conductivity adds to the side-jump contribution calculated in the previous subsection, with the sum of the two being .

### iii.4 Thermopower and thermal transport

#### iii.4.1 General considerations

In this subsection, we again restore and the electron charge . The formulas are correct for either sign of , provided that the product of the electron charge and the -component of the external magnetic field is positive. For , the and axes should be interchanged.

The thermoelectric and thermal responses for the CFs can be obtained from standard results for non-interacting fermions, based on interpreting the CF conductivity in terms of an energy-dependent conductivity through

(55) |

with the Fermi distribution. We explore the consequences, making use only of the fact that , independent of the Fermi energy, and hence that . Here we focus on the state with .

Although observations of thermal effects require that the temperature should not be too small, the calculations here also assume that the temperature should not be too high. In particular, we assume that the temperature is sufficiently low that the mean free path for inelastic scattering of composite fermions is large compared to the mean free path for elastic scattering by impurities. This restriction becomes more severe as the sample becomes more ideal.

#### iii.4.2 Thermopower

The heat current induced by a field applied to the CFs is described by a response function, , assuming that the temperature is a constant. For a non-interacting Fermi gas, at low temperatures, expanding around the Fermi level leads to the general result

(56) |

Since the Hall conductivity of the CFs is fixed to , requiring , then

(57) |

This (diagonal) result is of the form required by PH symmetry, as discussed in Potter *et al.* (2016), so that
and are each characterized by a
single non-universal quantity, and
.

To construct the thermoelectric response tensor for the electrons (not the CFs), one must take account of the fact that the electric field that couples to the electrons is

(58) |

where is the current of either electrons or CFs and

(59) |

The response tensors for the electrons are readily found to be

(60) | |||||

(61) |

With our specific forms of and , these become

(62) |

(63) |

In a thermopower experiment, one measures a voltage gradient induced when there is a heat current, but no electric current, flowing through the sample. Making use of an Onsager relationCooper *et al.* (1997), as well as the relations between CF and electron coefficients, one finds

(64) |

We see that the thermopower tensor
has non-zero off-diagonal elements, since is not diagonal. This contrasts with predictions based on a naive application of the HLR theory, pointed out by Potter *et al.* (2016), in which the off-diagonal thermopower vanishes. It recovers the central result of their PH symmetric theory.

#### iii.4.3 Thermal Transport

In a thermal transport experiment, one seeks to measure the heat current induced by a temperature gradient , under conditions where the electrical current is zero. As shown in Ref. Wang and Senthil (2016), the diagonal thermal conductivity at should be related by the Wiedemann-Franz law to the conductivity of the composite fermions, that is

(65) |

This result is obtained in both the HLR theory and the Dirac theory. Note that the thermal conductivity will become large as the mean free path becomes large, while the diagonal electrical conductivity approaches zero in this limit.

It was also suggested in Ref. Wang and Senthil (2016a) that for a system confined to the lowest Landau level, with a particle-hole symmetric distribution of impurities, the off-diagonal thermal conductivity should be given precisely by

(66) |

However, in an actual experiment in a strong magnetic field, one expects that thermal gradients and currents will be quite inhomogeneous, and a major part of the thermal Hall current will be associated with chiral heat flow near the sample boundaries, where particle-hole symmetry is strongly brokenCooper *et al.* (1997). Moreover, the transverse heat flow will be small compared to the longitudinal heat current, if the disorder scattering is weak. A proper analysis of the transverse heat flow is, therefore, a non-trivial problem, which we shall not address here.

## Iv Commensurability Oscillations

An important property investigated in HLR, which turns out to be sensitive to PH symmetry, was the wavevector dependent longitudinal conductivity, , for a wave vector in the -direction, in the limit of frequency . Precisely at , In the absence of impurities, it was found, using the RPA that

(67) |

independent of the renormalized mass or the bare mass. Subsequent analyses supported the idea that this result should be correct to all orders in perturbation theory, even in the case of short range electron-electron interactions or of interactions, where the effective mass is found to diverge at the Fermi energyKim *et al.* (1994). In the presence of disorder, it was predicted that Eq (67) should hold for ,
where is the transport mean free path for the composite fermions. For , the electrical conductivity approaches a constant, given by

(68) |

(This equation may be taken as a definition of ).

The non-trivial -dependence of results from an inverse -dependence of the transverse conductivity for composite fermions, which is non-local, because at , the composite fermions can travel in straight lines for distances of the order of , which can be very large compared to the inter-particle distance . For filling factors that differ slightly from , the composite fermions will no longer travel in straight lines, but rather should follow cyclotron orbits with an effective cyclotron radius given by

(69) |

One would expect, therefore, that the conductivity should become independent of for wavelengths large compared to , or Analysis at the RPA level, using a semiclassical description of the composite fermion trajectories, found that the value of the conductivity in this regime is essentially the same as the conductivity at . By contrast, in the regime , if , one finds that the longitudinal conductivity depends on and , and is a non-monotonic function of these variables. If either or is varied, one finds a series of maxima and minima, with the maxima occurring roughly at points which satisfy Eq. (1), or equivalently

(70) |

Since , with a high degree of accuracy, it is natural to describe the oscillatory dependence as a commensurability phenomenon, with maxima in where the diameter of the cyclotron orbit is approximately times the wavelength . The calculated peaks and valleys are generally broad if is of order unity, but the peaks are predicted to become sharp, and the positions of the maxima to become more precisely defined, in the limit of a clean sample and small .

Experimentally, the values of , at relatively low frequencies, have been extracted from accurate measurements of the propagation velocity of surface acoustic waves, as a function of acoustic wavelength and applied magnetic field, in a sample containing a two-dimensional electron gas, by Willett and coworkersWillett *et al.* (1993).
These surface acoustic wave experiments were, in fact, very important in establishing the validity of the HLR picture.

Another type of commensurability oscillation, commonly referred to as Weiss oscillations, may be observed by measuring the dc resistivity in the presence of a periodic electrostatic potential, which may be imposed by a periodic array of gates or etched defects on the surfaceKamburov *et al.* (2014); Kang *et al.* (1993); Smet *et al.* (1996, 1998, 1999); Willett *et al.* (1999); Zwerschke and Gerhardts (1999). In this case, theory predicts, and experiments have seen, maxima in the resistivity at magnetic fields where the wave vector of the array satisfies approximately Eq (1) or (70).

In the following subsections, we shall examine a third type of commensurability oscillation related to the existence of local minima in the spectrum of so-called magnetoroton excitations in a fractional quantized Hall state with close to 1/2. Magnetorotons may be understood as bound states of a quasiparticle in the lowest empty composite-Fermion Landau level and a quasihole in the highest filled level. As was discussed by Simon and HalperinSimon and Halperin (1993), the spectrum should have a series of minima, at wave vectors given approximately by Eq. (1), which become increasingly sharp for small values of . The frequencies are manifest as poles in the response function to an applied electric field at frequency and wave vector . For certain filling fractions the magnetoroton minima have been numerically calculated using composite fermion trial wave functionsScarola *et al.* (1999).

Although the magnetoroton spectrum may be difficult to measure experimentally in the region of interest to us^{1}^{1}1However, the magnetoroton spectrum has been successfully measured at filling fractions , and by Kukushkin et al.Kukushkin *et al.* (2009)., it has a big advantage from a theoretical point of view compared to predictions for the magnetoresistance in a periodic potential or the zero-frequency longitudinal conductance. The last two quantities are well defined only in the presence of a small but finite density of impurities. However, the behavior of a partially full composite-fermion Landau level in the presence of weak impurity scattering may be quite complicated, and is certainly not well understood.
By contrast, the magnetoroton spectrum may be studied in system without impurities, in a fractional quantized Hall state where there is an energy gap and where the magnetoroton may be precisely defined, as the lowest energy excitation for the given value of . We shall comment briefly on our understanding of the Weiss oscillations at the end of this section.

The requirements imposed by PH symmetry on the magnetoroton minima were stated in the Introduction. They are not satisfied in a naive application of the HLR approach. Below we show how they are satisfied by a more careful application of the HLR theory.

### iv.1 Magnetoroton spectrum at

We now look for the dispersion minima of the magneto-roton modes within the HLR composite fermion theory, at filling fraction , when is large. The magnetoroton frequencies will appear as poles in the current response matrix to an electric field at wave vector and frequency , defined by

(71) |

We shall take to lie along the -axis, so the indices and refer to longitudinal and transverse components respectively.

Our analysis will follow closely the work of SHSimon and Halperin (1993), and we shall first consider the response function using the RPA. Following Eqs. (27) and (28) of SH, we may write

(72) |

(73) |

where is the resistivity tensor of the composite fermions, and has matrix elements

(74) |

where is the two-body interaction, defined above.

According to SH, the composite fermion conductivity tensor, for a general value of , can be expressed in terms of an infinite sum of terms involving associated Laguerre polynomials. It the limit of large , one can employ a semiclassical approximation, where the sums can be carried out, and one can write the conductivity tensor in closed form in terms of Bessel functions. For the moment, we shall employ this semiclassical approximation, and shall comment later on the corrections that would be expected if one were to employ the full expressions for .

#### iv.1.1 Semiclassical calculation of

The semiclassical results of SH may be written (in units where ) as

(75) |

where and is the cyclotron radius of the composite fermion), , and is the Bessel function of the first kind. The full resistivity is given by the composition rule (

(76) |

We begin by looking for the poles of the physical conductivity tensor, which correspond to zeros of . To leading order in , these poles are located at the zeros of , which would yield dispersion minima at where is the ’th zero of the Bessel function . Here, however, we calculate the momenta () at these dispersion minima to next order in and address the question of their PH symmetry near half-filling.

To leading order in and , the tensor is given by