The bilinear-biquadratic model on the complete graph

TL;DR

This paper analyzes the spin-1 bilinear-biquadratic model on a complete graph, explicitly diagonalizing the Hamiltonian and mapping the ground-state phase diagram using group theory, providing exact energy spectra for any system size.

Contribution

It provides an exact diagonalization and phase diagram of the complete graph bilinear-biquadratic model using group representation theory, a novel analytical approach.

Findings

Explicit ground-state phase diagram mapped out.

Complete energy spectrum obtained analytically.

Degeneracies of energy levels characterized.

Abstract

We study the spin-1 bilinear-biquadratic model on the complete graph of N sites, i.e., when each spin is interacting with every other spin with the same strength. Because of its complete permutation invariance, this Hamiltonian can be rewritten as the linear combination of the quadratic Casimir operators of su(3) and su(2). Using group representation theory, we explicitly diagonalize the Hamiltonian and map out the ground-state phase diagram of the model. Furthermore, the complete energy spectrum, with degeneracies, is obtained analytically for any number of sites.