Exact Dynamical Correlation Functions of Calogero-Sutherland Model and One-Dimensional Fractional Statistics

TL;DR

This paper derives exact dynamical correlation functions for the Calogero-Sutherland model, demonstrating fractional statistics through analytical results and eigenstate motifs, revealing both exclusion and exchange statistics in one dimension.

Contribution

It provides the first exact calculations of dynamical correlation functions for the CSM at rational coupling, linking fractional exclusion and exchange statistics.

Findings

Exact density-density correlation functions obtained for all rational couplings

Evidence of fractional exclusion statistics in eigenstate motifs

Demonstration of 1D exchange statistics compatible with exclusion statistics

Abstract

One-dimensional 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 $\lambda = p/q$ 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 {\it exclusion} statistics (in the sense of Haldane's ``Generalized Pauli Exclusion Principle''). This model is also endowed with a natural {\it exchange } statistics (1D analog of 2D braiding statistics) compatible with the {\it exclusion} statistics. (Submitted to PRL on April 18, 1994)