The Wave Functions for the Free-Fermion Part of the Spectrum of the $S{U}_q(N)$ Quantum Spin Models

Abstract

We conjecture that the free-fermion part of the eigenspectrum observed recently for the $S{U}_q(N)$ 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 $S{U}_q(N-1)$ 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 ($q=\exp (2i\pi /3)$), 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 $S{U}_q(2)$ case at $q=\exp (i2\pi /3)$ our numerical and analytical studies induce some conjectures for special rates of correlation functions.

arXiv comment25 pages and no figures

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• Phase diagram and spectral properties of a new exactly integrable spin one quantum chain
• A_k Generalization of the O(1) Loop Model on a Cylinder: Affine Hecke Algebra, q-KZ Equation and the Sum Rule