It is noted that the radial distribution function and the interaction energy per particle can be exactly computed for a class of composite fermion wave functions obtained at the mean-field level of the Chern-Simons theory for the fractional quantum Hall effect. These results can be quite instructive and useful to test different numerical methods and approximations used to study these phenomena.
It is noted that the radial distribution function and the interaction energy per particle can be exactly computed for a class of composite fermion wave functions obtained at the mean-field level of the Chern-Simons theory for the fractional quantum Hall effect. These results can be quite instructive and useful to test different numerical methods and approximations used to study these phenomena.
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.
The Fermi-hypernetted-chain (FHNC) theory and the effective hypernetted-chain method are applied to study the composite-fermion (CF) states of the fractional quantum Hall effect. Using this theory we compute, in the thermodynamic limit, the correlation energy, radial distribution function, and static structure factor for all unprojected CF wave functions. The unprojected excitation gaps for $\ensuremath{\nu}=1/3,1/5$ were obtained by adopting in the FHNC a scheme previously used to compute nuclear matter excitation spectra. The results obtained so far are consistent with Monte Carlo simulations and small-number exact diagonalizations.
We establish the quantum mechanics of composite fermions based on the dipole picture initially proposed by Read. It comprises three complimentary components: a wave equation for determining the wave functions of a composite fermion in ideal fractional quantum Hall states and when subjected to external perturbations, a wave-function for mapping a many-body wave function of composite fermions to a physical wave function of electrons, and a microscopic approach for determining the effective Hamiltonian of the composite fermion. The wave equation resembles the ordinary Schrödinger equation but has drift velocity corrections that are not present in the Halperin-Lee-Read theory. The wave-function constructs a physical wave function of electrons by projecting a state of composite fermions onto a half-filled bosonic Laughlin state of vortices. Remarkably, Jain's wave-function can be reinterpreted as the new in an alternative wave-function representation of composite fermions. The dipole picture and the effective Hamiltonian can be derived from the microscopic model of interacting electrons confined in a Landau level, with all parameters determined. In this framework, we can construct the physical wave function of a fractional quantum Hall state deductively by solving the wave equation and applying the wave-function , based on the effective Hamiltonian derived from first principles, rather than relying on intuition or educated guesses. For ideal fractional quantum Hall states in the lowest Landau level, the approach reproduces the well-established results of the standard theory of composite fermions. We further demonstrate that the reformulated theory of composite fermions can be easily generalized for flat Chern bands. Published by the American Physical Society 2024
The interaction energy of the two-dimensional electron system in the region of fractional quantum Hall effect is considered within the Chern–Simons composite fermion approach. In the limit when Coulomb interaction is very small comparing to the cyclotron energy the RPA results are obtained for the fillings ν=1/3, 1/5, 2/3, 2/5, 3/7 and compared with the exact diagonalization results for small systems (extrapolated for infinite systems). They show very poor agreement suggesting the need for looking for alternative approaches.