Hierarchies of Spin Models related to Calogero-Moser Models

TL;DR

This paper explores the structure of integrable hierarchies of spin exchange models related to Calogero-Moser systems, demonstrating their properties through the example of the Heisenberg spin chain and explicitly calculating spectra for certain cases.

Contribution

It reveals the general structure of spin model hierarchies associated with Calogero-Moser models and provides explicit spectral results for models based on $A_r$ root systems.

Findings

Higher degeneracy in spectra compared to original models

Explicit spectra for models with $A_3$, $A_4$, $A_5$ root systems

Identification of integrable hierarchy features

Abstract

The universal formulation of spin exchange models related to Calogero-Moser models implies the existence of integrable hierarchies, which have not been explored. We show the general structures and features of the spin exchange model hierarchies by taking as examples the well-known Heisenberg spin chain with the nearest neighbour interactions. The energy spectra of the second member of the hierarchy belonging to the models based on the $A_r$ root systems $(r=3,4,5)$ are explicitly and {\em exactly} evaluated. They show many many interesting features and in particular, much higher degree of degeneracy than the original Heisenberg model, as expected from the integrability.