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arXiv:1803.07767 · Physical review. B./Physical review. B

Composite fermion Hall conductivity and the half-filled Landau level

Prashant Kumar, S. Raghu, Michael Mulligan

Published Jun 7, 2019• 12 citations
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Abstract

We consider the Hall conductivity of composite fermions in the theory of Halperin, Lee, and Read (HLR) [B. I. Halperin, P. A. Lee, and N. Read, Phys. Rev. B 47, 7312 (1993)]. We present a fully quantum-mechanical numerical calculation that shows, under suitable conditions, the HLR theory exhibits a particle-hole symmetric dc electrical Hall response in the presence of quenched disorder. Remarkably, this response of the HLR theory remains robust even when the disorder range is of the order of the Fermi wavelength. We find that deviations from the particle-hole symmetric response can appear in the ac Hall conductivity at frequencies sufficiently large compared to the inverse system size. Our results agree with a recent semiclassical analysis by Wang et al. [C. Wang, N. R. Cooper, B. I. Halperin, and A. Stern, Phys. Rev. X 7, 031029 (2017)] and complement the arguments based on the fully quantum-mechanical model by Kumar et al. [P. Kumar, M. Mulligan, and S. Raghu, Phys. Rev. B 98, 115105 (2018)]. These results provide further evidence that the HLR theory is compatible with an emergent particle-hole symmetry.

DOI
10.1103/physrevb.99.235114

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We consider the Hall conductivity of composite fermions in the theory of Halperin, Lee, and Read (HLR) [B. I. Halperin, P. A. Lee, and N. Read, Phys. Rev. B 47, 7312 (1993)]. We present a fully quantum-mechanical numerical calculation that shows, under suitable conditions, the HLR theory exhibits a particle-hole symmetric dc electrical Hall response in the presence of quenched disorder. Remarkably, this response of the HLR theory remains robust even when the disorder range is of the order of the Fermi wavelength. We find that deviations from the particle-hole symmetric response can appear in the ac Hall conductivity at frequencies sufficiently large compared to the inverse system size. Our results agree with a recent semiclassical analysis by Wang et al. [C. Wang, N. R. Cooper, B. I. Halperin, and A. Stern, Phys. Rev. X 7, 031029 (2017)] and complement the arguments based on the fully quantum-mechanical model by Kumar et al. [P. Kumar, M. Mulligan, and S. Raghu, Phys. Rev. B 98, 115105 (2018)]. These results provide further evidence that the HLR theory is compatible with an emergent particle-hole symmetry.
An effective field theory of composite Dirac fermions was proposed by Son [Phys. Rev. X 5, 031027 (2015)] as a theory of the half-filled Landau level with explicit particle-hole symmetry. We compute the electromagnetic response of this Son-Dirac theory on the level of the random phase approximation (RPA), where we pay particular attention to the effect of an additional composite-fermion dipole term that is needed to restore Galilean invariance. We find that once this dipole correction is taken into account, spurious interband transitions and collective modes that are present in the response of the unmodified theory either cancel or are strongly suppressed. We demonstrate that this gives rise to a consistent theory of the half-filled Landau level valid at all frequencies, at least to leading order in the momentum. In addition, the dipole contribution modifies the Fermi-liquid response at small frequency and momentum, which is a prediction of the Son-Dirac theory within the RPA that distinguishes it from a separate description of the half-filled Landau level by Halperin, Lee, and Read within the RPA.
Recently, a Dirac (particle-hole symmetric) description of composite fermions in the half-filled Landau level (LL) was proposed [D. T. Son, Phys. Rev. X 5, 031027 (2015)], and we study its possible consequences on BCS (Cooper) pairing of composite fermions (CFs). One of the main consequences is the existence of anisotropic states in single-layer and bilayer systems, which was previously suggested in Jeong and Park [J. S. Jeong and K. Park, Phys. Rev. B 91, 195119 (2015)]. We argue that in the half-filled LL in the single-layer case the gapped states may sustain anisotropy, because isotropic pairings may coexist with anisotropic ones. Furthermore, anisotropic pairings with the addition of a particle-hole symmetry-breaking mass term may evolve into rotationally symmetric states, i.e., Pfaffian states of Halperin-Lee-Read (HLR) ordinary CFs. On the basis of the Dirac formalism, we argue that in the quantum Hall bilayer at total filling factor 1, with decreasing distance between the layers, weak pairing of ppp-wave paired CFs is gradually transformed from Dirac to ordinary, HLR-like, with a concomitant decrease in the CF number. Global characterization of low-energy spectra based on the Dirac CFs agrees well with previous calculations performed by exact diagonalization on a torus. Finally, we discuss features of the Dirac formalism when applied in this context.
We study fractional quantum Hall states at filling fractions in the Jain sequences using the framework of composite Dirac fermions. Synthesizing previous work, we write an effective field theory consistent with all symmetry requirements, including Galilean invariance and particle-hole symmetry. Employing a Fermi-liquid description, we demonstrate the appearance of the Girvin-Macdonald-Platzman algebra and compute the dispersion relation of neutral excitations and various response functions. Our results satisfy requirements of particle-hole symmetry. We show that while the dispersion relation obtained from the modified random-phase approximation (MRPA) of the Halperin-Lee-Read (HLR) theory is particle-hole symmetric, correlation functions obtained from this scheme are not. The results of the Dirac theory are shown to be consistent with the Haldane bound on the projected structure factor, while those of the MPRA of the HLR theory violate it.
Here, the authors investigate the possibility that particle-hole symmetry in the lowest Landau level at half filling arises as an emergent low-energy symmetry. Specifically, they start with the traditional approach to the half-filled Landau level: an effective field theory known as the composite Fermi liquid pioneered by Halperin, Lee, and Read. In this theory, composite fermions arise when flux is attached to electrons in a half-filled Landau level. However, an equivalent starting point is one where one starts with a filled Landau level, depletes electrons (or equivalently, adds holes) to the point of half filling. One then attaches flux to the holes, and the resulting composite hole description is another viable description of the half-filled Landau level. The authors construct a particle-hole symmetric theory by studying a system consisting of alternating quasi-one-dimensional strips of composite Fermi liquid (CFL) and composite hole liquid (CHL), both of which break particle-hole symmetry. Gauge invariance requires the existence of gapless edge modes at the interface between the CFL and CHL. Out of these modes, they construct a neutral Dirac fermion coupled to an emergent U(1) gauge field. Thus, at distances long compared to the strip width, they demonstrate that the system is described by a Dirac fermion coupled to an emergent gauge field, with an anti-unitary particle-hole symmetry, precisely the form conjectured recently by Son to be the low-energy description of a particle-hole symmetric half-filled Landau level.
The Universal Composite Fermion Hall Conductance at ν=1/2ν=1/2ν=1/2
We show that at electronic filling factor ν=1/2ν=1/2ν=1/2, the lowest Landau level constraint implies that at any temperature, and in the presence of {\it any} amount of particle-hole symmetric disorder, the {\it composite fermion} Hall conductivity is precisely −e2/2h-e^2/2h−e2/2h. (The electronic Hall conductivity is e2/2he^2/2he2/2h.) This is inconsistent with the response of a composite Fermi liquid in {\it zero} effective magnetic field. We also examine perturbatively the nature of the putative composite Fermi liquid in the case in which the bare particles are anyons with ``nearly'' Fermi statistics.
We study particle-hole symmetry at the integer quantum Hall plateau transition using composite fermion mean-field theory. Because this theory implicitly includes some electron-electron interactions, it also has applications to certain fractional quantum Hall plateau transitions. Previous work [P. Kumar et al., Phys. Rev. B 100, 235124 (2019)] using this approach showed that the diffusive quantum criticality of this transition is described by a nonlinear sigma model with topological $\ensuremath{\theta}=\ensuremath{\pi}$ term. This result, which holds for both the Dirac and Halperin, Lee, and Read composite fermion theories, signifies an emergent particle-hole (reflection) symmetry of the integer (fractional) quantum Hall transition. Here we consider the stability of this result to various particle-hole symmetry-violating perturbations. In the presence of quenched disorder that preserves particle-hole symmetry, we find that finite longitudinal conductivity at this transition requires the vanishing of a symmetry-violating composite fermion effective mass, which if present would generally lead to $\ensuremath{\theta}\ensuremath{\ne}\ensuremath{\pi}$ and a corresponding violation of particle-hole symmetric electrical transport ${\ensuremath{\sigma}}_{xy}\ensuremath{\ne}\frac{1}{2}\frac{{e}^{2}}{h}$. When the disorder does not preserve particle-hole symmetry, we find that $\ensuremath{\theta}$ can vary continuously within the diffusive regime. Our results call for further study of the universality of the quantum Hall plateau transition.