Yangian-invariant spin models and Fibonacci numbers

TL;DR

This paper investigates Yangian-invariant supersymmetric spin models, revealing their degeneracy patterns are linked to Fibonacci numbers and grow exponentially, with implications for understanding their spectral structure and connections to anyons.

Contribution

It provides a closed-form expression for the minimum average degeneracy of Yangian-invariant models and connects their spectral levels to generalized Fibonacci numbers and quasi-particle models.

Findings

Minimum average degeneracy grows exponentially with system size.

Degeneracy is significantly higher than in generic models.

Spectral levels relate to an effective quasi-particle model.

Abstract

We study a wide class of finite-dimensional su(m|n)-supersymmetric models closely related to the representations of the Yangian Y(sl(m|n)) labeled by border strips. We quantitatively analyze the degree of degeneracy of these models arising from their Yangian invariance, measured by the average degeneracy of the spectrum. We compute in closed form the minimum average degeneracy of any such model, and show that in the non-supersymmetric case it can be expressed in terms of generalized Fibonacci numbers. Using several properties of these numbers, we show that (except in the simpler su(1|1) case) the minimum average degeneracy grows exponentially with the number of spins. We apply our results to several well-known spin chains of Haldane-Shastry type, quantitatively showing that their degree of degeneracy is much higher than expected for a generic Yangian-invariant spin model. Finally, we show that the set of distinct levels of a Yangian-invariant spin model is described by an effective model of quasi-particles. We study this effective model, discussing its connections to one-dimensional anyons and properties of generalized Fibonacci numbers.