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Reduction of Anyons to One Dimension and Calogero-Sutherland-type Models

Radhika Vathsan

Published Jan 30, 2013• 3 citations
OpenAlex

Abstract

The two-dimensional anyon system, when reduced to one dimension, yields models related to the Calogero-Sutherland model. One such reduction leads to a new model with a class of exact solutions. This model is one of a family of models obtained upon dimensional reduction of spherically symmetric models in arbitrary dimensions.

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REDUCTION OF ANYONS TO ONE DIMENSION AND CALOGERO–SUTHERLAND-TYPE MODELS

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We demonstrate that the exact form-factors of the Calogero-Sutherland model which were recently found in [10] in confirmation of the congectures earlier made by Haldane (Ref. [9]) can be reproduced in the framework of some bosonization procedure in momentum space. This observation implies a possibility of an exact bosonization of this model describing one-dimensional anyons. 1 The Calogero-Sutherland (CS) model [1] of one-dimensional (1D) spinless particles with long-ranged 1/x 2 interaction ( V (x) = π2 λ(λ−1) L 2 (sin πx L)2 in a finite size L system) provides a remarkable example of an exactly solvable system which can be viewed as a continuous deformation away from either ideal Fermi or Bose gas. At special values λ = 1/2, 1 and 2 the original continuous model describes statistics of eigenvalues of random matrices belonging to one of the three Dyson’s ensembles. Recently
We construct a new many-body Hamiltonian with two- and three-body interactions in two space dimensions and obtain its exact many-body ground state for an arbitrary number of particles. This ground state has a novel pairwise correlation. A class of exact solutions for the excited states is also found. These excited states display an energy spectrum similar to the Calogero-Sutherland model in one dimension. The model reduces to an analog of the well-known trigonometric Sutherland model when projected on to a circular ring.