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Modern Physics Letters A

GROUND STATE PERTURBATION OF N-ANYON IN A MAGNETIC FIELD

Yun Soo Myung, J.M. CHOI, M.J. UM, Changkeun Jue

Published Feb 10, 1993• 1 citation
Publisher

Abstract

We study N-anyon of the α-statistics in a uniform magnetic field, to investigate certain properties of the ground state of a fractional statistics. Using the improved bosonic end-perturbation theory, we obtain the first order perturbative energy shift of the ground state energy. It is realized that there exists a second order perturbative energy with Landau levels.

DOI
10.1142/s0217732393000350

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We study N-anyon of the α-statistics in a uniform magnetic field, to investigate certain properties of the ground state of a fractional statistics. Using the improved bosonic end-perturbation theory, we obtain the first order perturbative energy shift of the ground state energy. It is realized that there exists a second order perturbative energy with Landau levels.
We study the ground state of a large number N of 2D anyons in an external magnetic field. We consider a scaling limit where the statistics parameter α tends to zero when N tends to infinity which allows the statistics to be seen as a "perturbation from the bosonic end". Our model is that of bosons in a magnetic field and interacting through long-range magnetic potential generated by magnetic charges carried by each particle, smeared over discs of radius R. Our method allows to take R tends to not too fast at the same time as N tends to infinity. We use the information theoretic version of the de Finetti theorem of Brandão and Harrow to justify the so-called "average field approximation": the particles behave like independent, identically distributed bosons interacting via a self-consistent magnetic field.
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.
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.
The ground-state energy and wave function of many anyons in a harmonic oscillator potential is studied using perturbation theory from the fermionic and bosonic ends. As the number of anyons increases to infinity, the ground-state energy scales with the power 3/2. As the statistics parameter changes, there are repeated level crossings between states having different angular momenta. Hence the ground state is only piecewise continuous, and the number of pieces increases with the number of anyons. This is to be contrasted with the problem of interpolating statistics in one dimension where there are no level crossings in the ground state for any number of particles.
We study the ground state of a large number N of 2D extended anyons in an external magnetic field. We consider a scaling limit where the statistics parameter α is proportional to N−1 when N → ∞, which allows the statistics to be seen as a “perturbation around the bosonic end.” Our model is that of bosons in a magnetic field interacting through long-range magnetic potential generated by magnetic charges carried by each particle, smeared over discs of radius R. Our method allows us to take R → 0 not too fast at the same time as N → ∞ : R = N−1/4+ɛ. We use the information theoretic version of the de Finetti theorem of Brandão and Harrow to justify the so-called “average field approximation”: the particles behave like independent, identically distributed bosons interacting via a self-consistent magnetic field.