Abstract
We study the energy spectrum of two anyons on the sphere in a constant magnetic field. Making use of rotational invariance we reduce the energy eigenvalue equation to a system of linear differential equations for functions of a single variable, a reduction analogous to separating center of mass and relative coordinates on the plane. We solve these equations by a generalization of the Frobenius method and derive numerical results for the energies of non-analytically derivable states.
Highlights
In spite of the substantial research effort on various aspects of anyon systems, their analytical treatment remains essentially an open problem
In a recent companion paper [6] we studied the problem of anyons on the sphere with a constant magnetic field and identified a set of analytic energy eigenfunctions
The non separability of center of mass and relative coordinates on the sphere, unlike the plane, is a technical impediment, but it appears that the spherical geometry and topology create additional tension and complications for anyons
Summary
We will work in the so-called “singular gauge,” in which the Hamiltonian and all other operators are identical in form to those of free particles but the wavefunction is multivalued, acquiring a phase eiπα upon exchanging the two anyons on a counterclockwise path. The new wavefunction ψ remains multivalued and anyonic, like φ. Since the above Hamiltonian and angular momentum operators are the sum of two single-particle contributions it would be trivial to find their eigenstates, if the anyons were not effectively coupled through their nontrivial braiding properties. As it stands, the energy eigenvalue problem involves four coupled coordinates. Our task will be to reduce this problem to one variable by taking advantage of the angular momentum symmetry
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