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arXiv:math-ph/0406061 · Journal of Mathematical Physics

Remarkable identities related to the (quantum) elliptic Calogero-Sutherland model

Edwin Langmann A

Published Feb 1, 2006• 17 citations
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Abstract

We present remarkable functional identities related to the elliptic Calogero-Sutherland (eCS) system. We derive them from a second quantization of the eCS model within a quantum field theory model of anyons on a circle and at finite temperature. The identities involve two eCS Hamiltonians with arbitrary and, in general, different particle numbers N and M, and a particular function of N+M variables arising as anyon correlation function of N particles and M antiparticles. In addition to identities obtained from anyons with the same statistics parameter λ, we also obtain “dual” relations involving “mixed” correlation functions of anyons with two different statistics parameters λ and 1∕λ. We also give alternative, elementary proofs of these identities by direct computations.

DOI
10.1063/1.2167807

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We present remarkable functional identities related to the elliptic Calogero-Sutherland (eCS) system. We derive them from a second quantization of the eCS model within a quantum field theory model of anyons on a circle and at finite temperature. The identities involve two eCS Hamiltonians with arbitrary and, in general, different particle numbers N and M, and a particular function of N+M variables arising as anyon correlation function of N particles and M antiparticles. In addition to identities obtained from anyons with the same statistics parameter λ, we also obtain “dual” relations involving “mixed” correlation functions of anyons with two different statistics parameters λ and 1∕λ. We also give alternative, elementary proofs of these identities by direct computations.
We present further remarkable functional identities related to the elliptic Calogero-Sutherland (eCS) system. We derive them from a second quantization of the eCS model within a quantum field theory model of anyons on a circle and at finite temperature. The identities involve two eCS Hamiltonians with arbitrary and, in general, different particle numbers N and M, and a particular function of N + M variables arising as anyon correlation function of N particles and M anti-particles. In addition to identities obtained from anyons with the same statistics parameter λ, we also obtain “dual ” relations involving “mixed” correlation functions of anyons with two different statistics parameters λ and 1/λ. We also give alternative, elementary proofs of these identities by direct computations.
This thesis deals with a conformal field theoretical treatment of abelian anyons in (1 + 1)-dimensions and their relation to the integrable Calogero-Sutherland models. We generalize previous work relating anyons to the Calogero-Sutherland model by showing that the correlation function of the anyon field operators corresponds to the eigenfunctions of the deformed Calogero-Sutherland model. Our results suggest a physical application of the deformed Calogero-Sutherland model in the context of the fractional quantum Hall effect (FQHE). A key aspect for this work is the introduction of the dual anyon field operators, which obey a natural generalization of the canonical anti-commutation relation.
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
INTRODUCTION The Calogero-Sutherland model has recently attracted some attention mainly because, in spite of its simplicity, it yields nontrivial results which contribute to shape our understanding of fractional statistics [1]. Its most remarkable property is that its ground state is given by a Jastrow wave-function which is the one dimensional analogue of the Laughlin wave-function. The excitations are described by quasiparticles which carry a fraction of the quantum numbers of the fundamental particles. The wave functions are simple enough to allow the computation of physical quantities such as the dynamical correlation functions. Recently, it has become clear that the spin version of this model is closely related to the known spin chains such as the XXX chain. The wave functions of the spin models can be obtained as simply as those of the Calogero-Sutherland model by diagonalising simple differential operators known as the Dunkl operators [2]. In these lectures, I restrict to the