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On the nature of composite fermions

Ajit Coimbatore Balram, Jainendra K. Jain

Published Apr 13, 2016• 0 citations
OpenAlex

Abstract

Motivated by the issue of particle-hole symmetry for the composite fermion Fermi sea at the half filled Landau level, Dam T. Son has made an intriguing proposal [Phys. Rev. X {\bf 5}, 031027 (2015)] that composite fermions are Dirac particles. We ask what features of the Dirac-composite fermion theory and its various consequences may be reconciled with the well established microscopic theory of the fractional quantum Hall effect and the 1/2 state, which is based on {\em non-relativistic} composite fermions. Starting from the microscopic theory, we derive the assertion of Son that the particle-hole transformation of electrons at filling factor $\nu=1/2$ corresponds to an effective time reversal transformation (i.e. $\{\vec{k}_j\}$$\rightarrow$$\{-\vec{k}_j\}$) for composite fermions, and discuss how this connects to the absence of $2k_{\rm F}$ backscattering in the presence of a particle-hole symmetric disorder. By considering bare holes in various composite-fermion $\Lambda$ levels (analogs of electronic Landau levels) we determine the $\Lambda$ level spacing and find it to be very nearly independent of the $\Lambda$ level index, consistent with a parabolic dispersion for the underlying composite fermions. Finally, we address the compatibility of the Chern-Simons theory with the lowest Landau level constraint, and find that the wave functions of the mean-field Chern-Simons theory, as well as a class of topologically similar wave functions, are surprisingly accurate when projected into the lowest Landau level. These considerations lead us to introduce a normal form for the unprojected wave functions of the $n/(2pn-1)$ states that correctly capture the topological properties even without lowest Landau level projection.

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Motivated by the issue of particle-hole symmetry for the composite fermion Fermi sea at the half filled Landau level, Dam T. Son has made an intriguing proposal [Phys. Rev. X {\bf 5}, 031027 (2015)] that composite fermions are Dirac particles. We ask what features of the Dirac-composite fermion theory and its various consequences may be reconciled with the well established microscopic theory of the fractional quantum Hall effect and the 1/2 state, which is based on {\em non-relativistic} composite fermions. Starting from the microscopic theory, we derive the assertion of Son that the particle-hole transformation of electrons at filling factor ν=1/2\nu=1/2ν=1/2 corresponds to an effective time reversal transformation (i.e. \{\vec{k}_j\}$$\rightarrow$$\{-\vec{k}_j\}) for composite fermions, and discuss how this connects to the absence of 2kF2k_{\rm F}2kF​ backscattering in the presence of a particle-hole symmetric disorder. By considering bare holes in various composite-fermion Λ\LambdaΛ levels (analogs of electronic Landau levels) we determine the Λ\LambdaΛ level spacing and find it to be very nearly independent of the Λ\LambdaΛ level index, consistent with a parabolic dispersion for the underlying composite fermions. Finally, we address the compatibility of the Chern-Simons theory with the lowest Landau level constraint, and find that the wave functions of the mean-field Chern-Simons theory, as well as a class of topologically similar wave functions, are surprisingly accurate when projected into the lowest Landau level. These considerations lead us to introduce a normal form for the unprojected wave functions of the n/(2pn−1)n/(2pn-1)n/(2pn−1) states that correctly capture the topological properties even without lowest Landau level projection.
Motivated by the particle-hole symmetry at the half-filled lowest Landau level, many articles have taken the point of view, following D. T. Son, that composite fermions are Dirac particles and their Fermi sea is akin to that occurring at the surface of a three dimensional topological insulator. This presents a paradox, because the microscopic theory of composite fermions, which treats composite fermions as non-relativistic particles, has been confirmed in great detail. The authors discuss what features of the Dirac-composite fermion view can be reconciled with the microscopic theory. They also show how particle-hole symmetry can emerge in the Chern-Simons formulation of composite fermions.
We propose a particle-hole symmetric theory of the Fermi-liquid ground state of a half-filled Landau level. This theory should be applicable for a Dirac fermion in the magnetic field at charge neutrality, as well as for the v = 1/2 quantum Hall ground state of nonrelativistic fermions in the limit of negligible inter-Landau-level mixing. We argue that when particle-hole symmetry is exact, the composite fermion is a massless Dirac fermion, characterized by a Berry phase of π around the Fermi circle. We write down a tentative effective field theory of such a fermion and discuss the discrete symmetries, in particular, C P. The Dirac composite fermions interact through a gauge, but non-Chern-Simons, interaction. The particle-hole conjugate pair of Jain-sequence states at filling factors n/(2n + 1) and (n + 1)/(2n + 1), which in the conventional composite fermion picture corresponds to integer quantum Hall states with different filling factors, n and n + 1, is now mapped to the same half-integer filling factor n + 1/2 of the Dirac composite fermion. The Pfaffian and anti-Pfaffian states are interpreted as d -wave Bardeen-Cooper-Schrieffer paired states of the Dirac fermion with orbital angular momentum of opposite signs, while s -wave pairing would give rise to a particle-hole symmetric non-Abelian gapped phase. When particle-hole symmetry is not exact, the Dirac fermion has a C P -breaking mass. The conventional fermionic Chern-Simons theory is shown to emerge in the nonrelativistic limit of the massive theory.
The particle-hole (PH) symmetry at half-filled Landau level requires the relationship between the flux number N_phi and the particle number N on a sphere to be exactly N_phi - 2(N-1) = 1. The wave functions of composite fermions with 1/2 "orbital spin", which contributes to the shift "1" in the N_phi and N relationship, are proposed, shown to be PH symmetric, and validated with exact finite system results. It is shown the many-body composite electron and composite hole wave functions at half-filling can be formed from the two components of the same spinor wave function of a massless Dirac fermion at zero-magnetic field. It is further shown that away from half-filling, the many-body composite electron wave function at filling factor nu and its PH conjugated composite hole wave function at 1-nu can be formed from the two components of the very same spinor wave functions of a massless Dirac fermion at non-zero magnetic field. This relationship leads to the proposal of a very simple Dirac composite fermion effective field theory, where the two-component Dirac fermion field is a particle-hole spinor field coupled to the same emergent gauge field, with one field component describing the composite electrons and the other describing the PH conjugated composite holes. As such, the density of the Dirac spinor field is the density sum of the composite electron and hole field components, and therefore is equal to the degeneracy of the Lowest Landau level. On the other hand, the charge density coupled to the external magnetic field is the density difference between the composite electron and hole field components, and is therefore neutral at exactly half-filling. It is shown that the proposed particle-hole spinor effective field theory gives essentially the same electromagnetic responses as Son's Dirac composite fermion theory does.
Building on previous work [N. Read, Phys. Rev. B 58, 16262 (1998); Z. Dong and T. Senthil, Phys. Rev. B 102, 205126 (2020)] on the system of bosons at filling factor $\ensuremath{\nu}=1$, we derive the Dirac composite fermion theory for a half-filled Landau level from first principles and apply the Hartree-Fock approach in a preferred representation. On the basis of the microscopic formulation, in the long-wavelength limit, we propose a noncommutative field-theoretical description, which in a commutative limit reproduces the Son's theory, with additional terms that may be expected on physical grounds. The microscopic representation of the problem is also used to discuss pairing instabilities of composite fermions. We find that a presence of a particle-hole symmetry breaking leads to a weak (BCS) coupling ppp-wave pairing in the lowest Landau level, and strong coupling ppp-wave pairing in the second Landau level that occurs in a band with nearly flat dispersion, a third power function of momentum.
We construct explicit lowest-Landau-level wave functions for the composite-fermion Fermi sea and its low-energy excitations following a recently developed approach [Pu, Wu, and Jain, Phys. Rev. B 96, 195302 (2017)] and demonstrate them to be very accurate representations of the Coulomb eigenstates. We further ask how the Berry phase associated with a closed loop around the Fermi circle, predicted to be $\ensuremath{\pi}$ in a Dirac composite fermion theory satisfying particle-hole symmetry [D. T. Son, Phys. Rev. X 5, 031027 (2015)], is affected by Landau-level mixing. For this purpose, we consider a simple model wherein we determine the variational ground state as a function of Landau-level mixing within the space spanned by two basis functions: the lowest-Landau-level projected and the unprojected composite-fermion Fermi sea wave functions. We evaluate Berry phase for a path around the Fermi circle within this model following a recent prescription, and find that it rotates rapidly as a function of Landau-level mixing. We also consider the effect of a particle-hole symmetry-breaking three-body interaction on the Berry phase while confining the Hilbert space to the lowest Landau level. Our study deepens the connection between the $\ensuremath{\pi}$ Berry phase and the exact particle-hole symmetry in the lowest Landau level.
In a two-dimensional electron gas under a strong magnetic field, correlations generate emergent excitations distinct from electrons. It has been predicted that "composite fermions"--bound states of an electron with two magnetic flux quanta--can experience zero net magnetic field and form a Fermi sea. Using infinite-cylinder density matrix renormalization group numerical simulations, we verify the existence of this exotic Fermi sea, but find that the phase exhibits particle-hole symmetry. This is self-consistent only if composite fermions are massless Dirac particles, similar to the surface of a topological insulator. Exploiting this analogy, we observe the suppression of 2k(F) backscattering, a characteristic of Dirac particles. Thus, the phenomenology of Dirac fermions is also relevant to two-dimensional electron gases in the quantum Hall regime.