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Physical review. B, Condensed matter

Anyons in a magnetic field

M. D. Johnson, Geoffrey Canright

Published Apr 1, 1990• 78 citations
Publisher

Abstract

We study the properties of particles with fractional statistics (anyons) in a uniform magnetic field, to investigate the interplay between the statistical and physical flux. By exact solution of the two-body case, both free and with a harmonic attraction, we find a very clear example of the way in which an external field can offset the frustration caused by the statistics. These solutions also give the second virial coefficient, and hence a thermodynamic signature of the same effect. Finally, the same approach is used to study the problem of many anyons in a magnetic field, in particular the ground state and excitation spectrum.

DOI
10.1103/physrevb.41.6870

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We study the properties of particles with fractional statistics (anyons) in a uniform magnetic field, to investigate the interplay between the statistical and physical flux. By exact solution of the two-body case, both free and with a harmonic attraction, we find a very clear example of the way in which an external field can offset the frustration caused by the statistics. These solutions also give the second virial coefficient, and hence a thermodynamic signature of the same effect. Finally, the same approach is used to study the problem of many anyons in a magnetic field, in particular the ground state and excitation spectrum.
We consider the external magnetic field effects on the two types of anyons with fractional statistical parameter p/q with coprimes p and q, one with a fractional charge e/q and the other with a fractional flux p ϕ0/q by exact ground state wave function. We also study the geometry in which a two-dimensional strip of anyons contains an island of anyons with different statistical parameters in their equilibrium. The equilibrium inside an island is shown to be periodic with respect to the magnetic flux through the island. The period for the fractional charge anyon equals to the integer multiple of the fundamental flux quantum. On the other hand, in the case of the fractional flux anyon, the period is found to be the fractional multiple of the fundamental flux quantum.
We have investigated the interplay of fractional statistics and magnetic field for anyons on a lattice, by exact calculation of ground-state energies and wave functions for small systems. We find that the two types of ``flux'' (magnetic and statistical) are to some extent interchangeable in determining the ground-state properties, and thus that mean-field theories, which replace statistical phase shifts by magnetic fields, may be an appropriate method for treating fractional statistics.
The statistical mechanics of an anyon gas in a magnetic field is addressed. A harmonic regulator is used to define a proper thermodynamic limit. When the magnetic field is sufficiently strong, only exact N-anyon ground states, where anyons occupy the lowest Landau level, contribute to the equation of state. Particular attention is paid to the interval of definition of the statistical parameter \ensuremath{\alpha}\ensuremath{\in}[-1,0] where a gap exists. Interestingly enough, one finds that at the critical filling \ensuremath{\nu}=-1/\ensuremath{\alpha} where the pressure diverges, the external magnetic field is entirely screened by the flux tubes carried by the anyons.
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
We develop a density functional treatment of noninteracting Abelian anyons, which is capable, in principle, of dealing with a system of a large number of anyons in an external potential. Comparison with exact results for a few particles shows that the model captures the behavior qualitatively and semiquantitatively, especially in the vicinity of the fermionic statistics. We then study anyons with statistics parameter 1+1/n1+1/n1+1/n, which are thought to condense into a superconducting state. An indication of the superconducting behavior is the mean-field result that, for uniform density systems, the ground-state energy increases under the application of an external magnetic field independent of its direction. Our density functional theory based analysis does not find that to be the case for finite systems of anyons, which can accommodate a weak external magnetic field through density transfer between the bulk and the boundary rather than through transitions across effective Landau levels, but the ``Meissner repulsion'' of the external magnetic field is recovered in the thermodynamic limit as the effect of the boundary becomes negligible. We also consider the quantum Hall effect of anyons, and show that its topological properties, such as the charge and statistics of the excitations and the quantized Hall conductance, arise in a self-consistent fashion.