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Laughlin type wave function for two-dimensional anyon fields in a KMS-state

Nevena Ilieva, W. Thirring

Published Jan 1, 2000• 5 citations
OpenAlex

Abstract

The correlation functions of two-dimensional anyon fields in a KMS-state are studied. The n-particle wave functions of noncanonical fermions of level , odd, are shown to be of Laughlin type of order . For the first noncanonical Fermi class (=3) this is simply the finite-temperature generalization of Laughlin wave function. This relates the rst and second quantized pictures of the fractional quantum Hall effect.

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Physical review. B, Condensed matter•Feb 15, 2000•1 citation
The correlation functions of two-dimensional anyon fields in a KMS-state are studied. The n-particle wave functions of noncanonical fermions of level , odd, are shown to be of Laughlin type of order . For the first noncanonical Fermi class (=3) this is simply the finite-temperature generalization of Laughlin wave function. This relates the rst and second quantized pictures of the fractional quantum Hall effect.
A two-dimensional system of electrons with repulsive two-particle interaction is considered in a homogeneous magnetic field. Assuming the limit of high magnetic field the motion of the electrons is restricted to the lowest Landau level. We present quantum-mechanical wave functions for the many-particle system of N spin-polarized fermions on a parallelogram with periodic boundary conditions and with ${\mathit{N}}_{\mathrm{\ensuremath{\Phi}}}$ quanta of magnetic flux flowing through its area. These wave functions are generalizations of the Laughlin wave function to more general rational values of the occupation fraction \ensuremath{\nu}=N/${\mathit{N}}_{\mathrm{\ensuremath{\Phi}}}$. We divide the N electrons into n groups and assume Laughlin-Jastrow-type correlations for electrons within the same group, while electrons in different groups are correlated with a second Jastrow function. The corresponding wave functions exhibit a special structure that depends on \ensuremath{\nu} but is not affected by the interaction between the particles. Furthermore, we consider excited states with quasielectrons and quasiholes. Our theory leads to a unified view of the composite fermion theory of Jain and the hierarchy theory of Haldane, Laughlin, and Halperin. We present a qualitative argument why the fractions of Jain are observed preferably in the experiments.
We study the behavior of an interacting two-dimensional charged particle system in a transverse magnetic field and give a simple picture of the features that lead to the fractional quantum Hall effect. In particular, we discuss how the interparticle interactions give rise to an energy-level substructure in the lowest Landau level and develop a relatively simple model that gives insight into the analytical form of the Laughlin wave function. We analyze the model by transforming to a reference frame that rotates at half the cyclotron angular frequency. In the rotating frame, the vector potential for the magnetic field is replaced by a quadratic scalar potential. It is then evident that the system of interacting particles is a bound state, even in the absence of a charge-neutralizing background. We solve exactly a many-particle system in the rotating frame using a harmonic model interaction. In addition to being a good example for demonstrating basic methods in quantum mechanics, we show explicitly that the Laughlin function is an eigenstate of this model system.
The complete set of ground state wave functions for N anyons in an external magnetic field on the torus is found. The cases when the filling factor is less than or equal to one are considered. The single valued description of anyons is employed through out by coupling bosons to a Chern-Simons field. At the end, the Chern-Simons interaction is removed by a singular gauge transformation as a result of which the wave functions become multi-component in agreement with other studies. USITP-93-07 1 Anyons [1], particles obeying fractional statistics, have been the subject of intense study in the past few years. Besides the purely theoretical interest in fractional statistics a new motivation has been provided by the Laughlin theory of the fractional quantum hall effect (FQHE)[2]. It was realized some time ago that certain excitations of the Laughlin state are anyonic [3]. Anyons are, therefore, characteristic of the elusive Laughlin state. An understanding of anyons in a magnetic field may help to provide, in turn, an understanding of the rich structure of the Laughlin state. This has been the motivation behind much of the study of anyons in
Fractional quantum Hall eect is a remarkable behaviour of correlated electrons, observed exclusively in two dimensions, at low temperatures, and in strong magnetic elds.The most prominent fractional quantum Hall state occurs at Landau level lling factor ν = 1/3 and it is described by the famous Laughlin wave function, which (apart from the trivial Gaussian factor) is an example of Jack polynomial.Fermionic Jack polynomials also describe another pair of candidate fractional quantum Hall states: MooreRead and ReadRezayi states, believed to form at the ν = 1/2 and 3/5 llings of the second Landau level, respectively.Bosonic Jacks on the other hand are candidates for certain correlated states of cold atoms.We examine here a continuous family of fermionic Jack polynomials whose special case is the Laughlin state as approximate wave functions for the 1/3 fractional quantum Hall eect.
Laughlin's quantum Hall wave function is obtained as an exact ground state of an N-particle Hamiltonian, in which the electrons themselves are coupled to the Chern-Simons field. The proof requires quantum field-theoretic methods in the Schr\"odinger picture, and clearly exhibits the relation to conformal field theory since the Knizhnik-Zamolodchikov connection shows up and the Laughlin ground state is recognized as a conformal block. A refined version of this approach can be applied for toroidal boundary conditions, i.e., the electrons are confined to the physical interior of the Hall sample. The result turns out to be a nontrivial modification of the p-fold degenerate Haldane-Rezayi wave function. Furthermore, it is shown that the degeneracy of the ground state is accounted for by the quantum group Uq(su2){U}_{q}{(su}_{2})Uq​(su2​) with deformation parameter $q=\mathrm{exp}(2\ensuremath{\pi}i/p)$ and p odd; put differently, the possible quantum numbers p of the quantum Hall effect, determining the filling fractions $\ensuremath{\nu}=1/p,$ can be explained by an underlying quantum symmetry.