Solution of Some Integrable One-Dimensional Quantum Systems

TL;DR

This paper demonstrates the integrability of certain one-dimensional multi-component quantum systems, determines their spectra and thermodynamics, and connects these findings to known models, providing explicit solutions and explanations for observed phenomena.

Contribution

It introduces a new class of integrable multi-component quantum systems and fully solves their spectra and thermodynamics, including strong interaction limits and connections to existing models.

Findings

Systems are proven to be integrable.

Explicit spectrum and thermodynamics are derived.

Solution reproduces numerical results for related lattice models.

Abstract

In this paper, we investigate a family of one-dimensional multi-component quantum many-body systems. The interaction is an exchange interaction based on the familiar family of integrable systems which includes the inverse square potential. We show these systems to be integrable, and exploit this integrability to completely determine the spectrum including degeneracy, and thus the thermodynamics. The periodic inverse square case is worked out explicitly. Next, we show that in the limit of strong interaction the "spin" degrees of freedom decouple. Taking this limit for our example, we obtain a complete solution to a lattice system introduced recently by Shastry, and Haldane; our solution reproduces the numerical results. Finally, we emphasize the simple explanation for the high multiplicities found in this model.