The Wave Functions for the Free-Fermion Part of the Spectrum of the $SU_q(N)$ Quantum Spin Models

TL;DR

This paper investigates the free-fermion spectrum of $SU_q(N)$ quantum spin models, proving a key conjecture for the SU(3) case and deriving eigenvector formulas, which advance understanding of their spectral properties and correlation functions.

Contribution

It proves the conjecture for the SU(3) spin chain, providing explicit eigenvector formulas and insights into the free-fermion spectrum of $SU_q(N)$ models.

Findings

Validated the conjecture for SU(3) case

Derived explicit eigenvector formulas for the auxiliary model

Suggested new conjectures for correlation functions

Abstract

We conjecture that the free-fermion part of the eigenspectrum observed recently for the $SU_q(N)$ Perk-Schultz spin chain Hamiltonian in a finite lattice with $q=\exp (i\pi (N-1)/N)$ is a consequence of the existence of a special simple eigenvalue for the transfer matrix of the auxiliary inhomogeneous $SU_q(N-1)$ vertex model which appears in the nested Bethe ansatz approach. We prove that this conjecture is valid for the case of the SU(3) spin chain with periodic boundary condition. In this case we obtain a formula for the components of the eigenvector of the auxiliary inhomogeneous 6-vertex model ($q=\exp (2 i \pi/3)$), which permit us to find one by one all components of this eigenvector and consequently to find the eigenvectors of the free-fermion part of the eigenspectrum of the SU(3) spin chain. Similarly as in the known case of the $SU_q(2)$ case at $q=\exp(i2\pi/3)$ our numerical and analytical studies induce some conjectures for special rates of correlation functions.