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arXiv:cond-mat/9404094 · International Journal of Modern Physics B

ON EXACT BOSONIZATION OF THE CALOGERO-SUTHERLAND MODEL

Dmitri V. Khveshchenko

Published Jun 30, 1995• 12 citations
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Abstract

We elaborate on an algebraic method of a momentum space bosonization of exactly solvable one-dimensional models in the framework of the Calogero-Sutherland model. The approach provides an exact representation of the local field operator of the Calogero-Sutherland model in terms of quasiparticle eigenstate’s creation-annihilation operators and furnishes a simple method to compute exact correlation functions.

DOI
10.1142/s0217979295000707

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We elaborate on an algebraic method of a momentum space bosonization of exactly solvable one-dimensional models in the framework of the Calogero-Sutherland model. The approach provides an exact representation of the local field operator of the Calogero-Sutherland model in terms of quasiparticle eigenstate’s creation-annihilation operators and furnishes a simple method to compute exact correlation functions.
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 propose an approach to treating (1 + 1)-dimensional fermionic systems based on the idea of algebraic bosonization. This amounts to decomposing the elementary low-lying excitations around the Fermi surface in terms of basic building blocks carrying a representation of the [Formula: see text] algebra, which is the dynamical symmetry of the Fermi quantum incompressible fluid. This symmetry simply expresses the local particle number current conservation at the Fermi surface. The general approach is illustrated in detail through two examples: the Heisenberg and Calogero–Sutherland models, which allow comparison with the exact Bethe ansatz solution.
Using the collective field technique, we give the description of the spin Calogero-Sutherland Model (CSM) in terms of free bosons. This approach can be applicable for arbitrary coupling constant and provides the bosonized Hamiltonian of the spin CSM. The boson Fock space can be identified with the Hilbert space of the spin CSM in the large N limit. We show that the eigenstates corresponding to the Young diagram with a single row or column are represented by the vertex operators. We also derive a dual description of the Hamiltonian and comment on the construction of the general eigenstates. JSPS fellow
We establish a one-to-one correspondence between the “composite particles” with N particles and the Young tableaux with at most N rows. We apply this correspondence to the models of Calogero–Sutherland and Ruijsenaars–Schneider and we obtain a momentum space representation of the “composite particles” in terms of creation operators attached to the Young tableaux. Using the technique of bosonization, we obtain a position space representation of the “composite particles” in terms of products of vertex operators. In the special case where the “composite particles” are bosons and if we add one extra quasiparticle or quasihole, we construct the ground state wave functions corresponding to the Jain series ν=p/(2np±1) of the fractional quantum Hall effect.
Using the collective field technique, we give the description of the spin Calogero - Sutherland model (CSM) in terms of free bosons. This approach can be applicable for arbitrary coupling constant and provides the bosonized Hamiltonian of the spin CSM. The boson Fock space can be identified with the Hilbert space of the spin CSM in the large-N limit. We show that the eigenstates corresponding to the Young diagram with a single row or column are represented by the vertex operators. We also derive a dual description of the Hamiltonian and comment on the construction of the general eigenstates.
Matrix models and related Spin-Calogero-Sutherland models are of major relevance in a variety of subjects, ranging from condensed matter physics to QCD and low dimensional string theory. They are characterized by integrability and exact solvability. Their continuum, field theoretic representations are likewise of definite interest. In this paper we describe various continuum, field theoretic representations of these models based on bosonization and collective field theory techniques. We compare various known representations and describe some nontrivial applications.