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.
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.
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.
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 corresponds to an effective time reversal transformation (i.e. {kj}→{−kj}) for composite fermions, and discuss how this connects to the absence of 2kF backscattering in the presence of a particle-hole symmetric disorder. By considering bare holes in various composite-fermion Λ levels (analogs of electronic Landau levels) we determine the Λ level spacing and find it to be very nearly independent of the Λ 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.
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.
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.
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.
The particle-hole (PH) symmetry of electrons is an exact symmetry of the electronic Hamiltonian confined to a specific Landau level, and its interplay with the formation of composite fermions has attracted much attention of late. We investigate an emergent symmetry in the fractional quantum Hall effect, namely, the PH symmetry of composite fermions, which relates states at composite fermion filling factors ${\ensuremath{\nu}}^{*}=n+\overline{\ensuremath{\nu}}$ and ${\ensuremath{\nu}}^{*}=n+1\ensuremath{-}\overline{\ensuremath{\nu}}$, where the integer n is the $\mathrm{\ensuremath{\Lambda}}$-level index and $0\ensuremath{\le}\overline{\ensuremath{\nu}}\ensuremath{\le}1$. Detailed calculations using the microscopic theory of composite fermions demonstrate the following for low-lying $\mathrm{\ensuremath{\Lambda}}$ levels (small n): (i) The two-body interaction between composite-fermion particles is very similar, apart from a constant additive term and an overall scale factor, to that between composite-fermion holes in the same $\mathrm{\ensuremath{\Lambda}}$ level; and (ii) the three-body interaction for composite fermions is an order of magnitude smaller than the two-body interaction. Taken together, these results imply an approximate PH symmetry for composite fermions in low $\mathrm{\ensuremath{\Lambda}}$ levels, which is also supported by exact-diagonalization studies and available experiments. This symmetry, which relates states at electron filling factors $\ensuremath{\nu}=\frac{n+\overline{\ensuremath{\nu}}}{2(n+\overline{\ensuremath{\nu}})\ifmmode\pm\else\textpm\fi{}1}$ and $\ensuremath{\nu}=\frac{n+1\ensuremath{-}\overline{\ensuremath{\nu}}}{2(n+1\ensuremath{-}\overline{\ensuremath{\nu}})\ifmmode\pm\else\textpm\fi{}1}$, is not present in the original Hamiltonian and owes its existence entirely to the formation of composite fermions. With increasing $\mathrm{\ensuremath{\Lambda}}$-level index, the two-body and three-body pseudopotentials become comparable, but at the same time they both diminish in magnitude, indicating that the interaction between composite fermions becomes weak as we approach $\ensuremath{\nu}=1/2$.