Spin dependent extension of Calogero-Sutherland model through anyon like representations of permutation operators

TL;DR

This paper extends the spin-dependent Calogero-Sutherland model by introducing anyon-like permutation representations, enabling exact solutions and revealing new nonlocal interaction variants with parameter-dependent degeneracies.

Contribution

It introduces a novel class of anyon-like permutation representations in the spin Calogero-Sutherland model, allowing exact solutions and new nonlocal interaction variants.

Findings

Exact eigenfunctions and energy eigenvalues derived

New permutation representations with phase factors constructed

Degeneracy factors depend on discrete parameters

Abstract

We consider a $A_{N-1}$ type of spin dependent Calogero-Sutherland model, containing an arbitrary representation of the permutation operators on the combined internal space of all particles, and find that such a model can be solved as easily as its standard $su(M)$ invariant counterpart through the diagonalisation of Dunkl operators. A class of novel representations of the permutation operator $P_{ij}$, which pick up nontrivial phase factors along with interchanging the spins of $i$-th and $j$-th particles, are subsequently constructed. These `anyon like' representations interestingly lead to different variants of spin Calogero-Sutherland model with highly nonlocal interactions. We also explicitly derive some exact eigenfunctions as well as energy eigenvalues of these models and observe that the related degeneracy factors crucially depend on the choice of a few discrete parameters which characterise such anyon like representations.