Yangian-invariant spin models and Fibonacci numbers

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.

LaTex, 31 pages, 7 figures. Minor additions in Section 2, one reference also added

References in corpus (6)

• Physics and Mathematics of Calogero particles
• Exact partition function of SU(m|n) supersymmetric Haldane-Shastry spin chain
• Low energy properties of the SU(m|n) supersymmetric Haldane-Shastry spin chain
• The Berry-Tabor conjecture for spin chains of Haldane-Shastry type
• An exactly solvable supersymmetric spin chain of BC_N type
• A new perspective on the integrability of Inozemtsev's elliptic spin chain

Cited by in corpus (11)

• Resurrecting the partially isotropic Haldane-Shastry model
• Spin-Ruijsenaars, q-deformed Haldane-Shastry and Macdonald polynomials
• Integrable open spin chains related to infinite matrix product states
• How coordinate Bethe ansatz works for Inozemtsev model
• A novel class of translationally invariant spin chains with long-range interactions
• The open Haldane-Shastry chain: thermodynamics and criticality
• Rational quantum integrable systems of D_N type with polarized spin reversal operators
• Supersymmetric $t$-$J$ models with long-range interactions: partition function and spectrum
• Supersymmetric $t$-$J$ models with long-range interactions: thermodynamics and criticality
• Thermodynamics and criticality of su($m$) spin chains of Haldane-Shastry type
• On the Integrable Structure of Super Yang-Mills Scattering Amplitudes