A Proof of Selection Rules for Critical Dense Polymers

TL;DR

This paper proves a conjecture about the eigenvalues and degeneracies of the transfer matrix in the critical dense polymers model, connecting loop models with the XXZ spin chain through algebraic homomorphisms.

Contribution

It provides a rigorous proof of the eigenvalue degeneracy conjecture for the critical dense polymers model using algebraic homomorphism techniques.

Findings

Confirmed the eigenvalue degeneracy conjecture for the model

Established a link between loop models and the XXZ spin chain

Enhanced understanding of algebraic structures in critical dense polymers

Abstract

Among the lattice loop models defined by Pearce, Rasmussen and Zuber (2006), the model corresponding to critical dense polymers ($\beta = 0$) is the only one for which an inversion relation for the transfer matrix $D_N(u)$ was found by Pearce and Rasmussen (2007). From this result, they identified the set of possible eigenvalues for $D_N(u)$ and gave a conjecture for the degeneracies of its relevant eigenvalues in the link representation, in the sector with $d$ defects. In this paper, we set out to prove this conjecture, using the homomorphism of the $TL_N (\beta)$ algebra between the loop model link representation and that of the XXZ model for $\beta = -(q+q^{-1})$.