Exact dynamical correlation functions of Calogero-Sutherland model and one-dimensional fractional statistics.

One-dimensional (1D) model of non-relativistic particles with inverse-square interaction potential known as Calogero-Sutherland Model (CSM) is shown to possess fractional statistics. Using the theory of Jack symmetric polynomial the exact dynamical density-density correlation function and the one-particle Green’s function (hole propagator) at any rational interaction coupling constant λ = p/q are obtained and used to show clear evidences of the fractional statistics. Motifs representing the eigenstates of the model are also constructed and used to reveal the fractional exclusion statistics (in the sense of Haldane’s “Generalized Pauli Exclusion Principle”). This model is also endowed with a natural exchange statistics (1D analog of 2D braiding statistics) compatible with the exclusion statistics. (Submitted to PRL on April 18, 1994) PACS 05.30-d, 71.10.+x Typeset using REVTEX 1 The quantum particles obeying fractional statistics known as anyons have been a subject of intense study since the discovery of the fractional quantum Hall effect and the High Tc super-conductors [1]. The subject, however, is far from complete. In fact, even the simplest model of anyons, namely the ideal gas, has not yet been fully understood. To add to the confusion, there are now two seemingly non-equivalent definitions of fractional statistics. While the popular definition of anyons is based on the quantum phase arising from the exchange of particles [1], Haldane’s definition [2] is based on so called the “Generalized Pauli Exclusion Principle.” The main difference in the two approach is that while in the former the statistics is assigned to the Newtonian point particles, in the latter it is obeyed by the elementary excitations of condensed matter system. There are no fully solvable two-dimensional models where the ideas of fractional statistics can be rigorously tested. In one-dimension, however, there is such a model known as Calogero-Sutherland Model (CSM) [3]. I show in this Letter that the two definitions of fractional statistics can coexist in CSM without any inconsistency. Following Ref. [4], I construct the motifs for all the excited states and explicitly demonstrate that the quasi-particles and quasi-holes obey the Haldane’s exclusion statistics. I also solve the exact ground state dynamical density-density correlation function (DDDCF) and the hole propagator part of the one-particle Green’s function (HPOGF) and show that the contributing intermediate states involve only a finite number of quasi-particles (holes) consistent with the ideal “anyon” gas structure of this model. In calculating the correlation function and propagator a new mathematical technique based on the theory of Jack symmetric orthogonal polynomial [5] is used. The CSM Hamiltonian, which describes a system of N non-relativistic particles interacting with inverse-square exchange, is given by H = − N ∑