Dimensional Reduction on a Sphere
Abstract
The question of the dimensional reduction of twodimensional (2d) quantum models on a sphere to onedimensional (1d) models on a circle is addressed. A possible application is to look at a relation between the 2d anyon model and the 1d CalogeroSutherland model, which would allow for a better understanding of the connection between 2d anyon exchange statistics and Haldane exclusion statistics. The latter is realized microscopically in the 2d LLL anyon model and in the 1d Calogero model. In a harmonic well of strength ω or on a circle of radius R — both parameters ω and R have to be viewed as long distance regulators — the Calogero spectrum is discrete. It is well known that by confining the anyon model in a 2d harmonic well and projecting it on a particular basis of the harmonic well eigenstates, one obtains the CalogeroMoser model. It is then natural to consider the anyon model on a sphere of radius R and look for a possible dimensional reduction to the CalogeroSutherland model on a circle of radius R. First, the free onebody case is considered, where a mapping from the 2d sphere to the 1d chiral circle is established by projection on a special class of spherical harmonics. Second, the Nbody interacting anyon model is considered: it happens that the standard anyon model on the sphere is not adequate for dimensional reduction. One is thus led to define a new spherical anyonlike model deduced from the AharonovBohm problem on the sphere where each flux line pierces the sphere at one point and exits it at its antipode.
 Publication:

International Journal of Modern Physics B
 Pub Date:
 2006
 DOI:
 10.1142/S0217979206035503
 arXiv:
 arXiv:condmat/0509580
 Bibcode:
 2006IJMPB..20.3533M
 Keywords:

 Anyons;
 Calogero model;
 CalogeroSutherland model;
 coherent states;
 dimensional reduction;
 Condensed Matter  Mesoscopic Systems and Quantum Hall Effect
 EPrint:
 10 pages, 1 figure