Generalization of Laughlin's Theory for the Fractional Quantum Hall Effect

TL;DR

This paper extends Laughlin's theory for the fractional quantum Hall effect by introducing a quasiparticle operator that reproduces composite fermion wavefunctions, unifies different filling fractions, and predicts a quantum critical point at half filling.

Contribution

It proposes a modified quasiparticle operator and constructs ground state wavefunctions for various filling fractions as superpositions of coupled Laughlin condensates, unifying different theoretical descriptions.

Findings

Wavefunctions for general states at filling fractions _{n,m}^{\u00b1} are constructed as superpositions of coupled Laughlin condensates.

The wavefunctions for _{n,m}^{-} states match those of noninteracting composite fermions.

Half filling of the lowest Landau level is identified as a quantum critical point for phase transitions.

Abstract

Motivated by the quasiparticle wavefunction in the composite fermion (CF) theory for fractional quantum Hall filling factor $\nu = 1/m$, I consider a suitable quasiparticle operator in differential form, as a modified form of Laughlin's quasiparticle operator, that reproduces quasiparticle wave function as predicted in the CF theory. I further consider the conjugate of this operator as quasihole operator for obtaining a novel quasihole wave function for $1/m$ state. Each of these wave functions is interpreted as expelled electron into a different Hilbert subspace from the original Hilbert space of Laughlin condensate while still maintaining its correlation (although changed) with the electrons in the condensate such that the expelled electron behaves as a CF with respect to the electrons in the condensate. With this interpretation, I show that the ground state wavefunctions for general states at filling fractions $\nu_{n,m}^{\pm} = n/[n(m\mp 1)\pm 1]$, respectively, can be constructed as coherent superposition of $n$ coupled Laughlin condensates and their conjugates, formed at different Hilbert subspaces. The corresponding wave functions, specially surprising for $\nu_{n,m}^{-}$ sequence of states, are identical with those proposed in the theory of noninteracting CFs. The states which were considered as fractional quantum Hall effect of interacting CFs, can also be treated in the same footing as for the prominent sequences of states describing as the coupled condensates among which one is a non-Laughlin condensate in a different Hilbert subspace. Further, I predict that the half filling of the lowest Landau level is a quantum critical point for phase transitions between two topologically distinct phases each corresponding to a family of states.