Universal amplitude ratios and Coxeter geometry in the dilute A model

TL;DR

This paper explores the spectrum and universal amplitude ratios of the dilute A model, revealing connections to Coxeter geometry and Lie algebras, and providing insights into integrable quantum field theories.

Contribution

It establishes a link between the dilute A model's excitation spectrum and Coxeter geometry, and computes universal amplitude ratios for various cases.

Findings

Spectrum satisfies functional equations similar to S-matrices.

Coxeter geometry relates Bethe roots to Lie algebras for specific L values.

Universal amplitude ratios are calculated for the model's universality class.

Abstract

The leading excitations of the dilute $A_L$ model in regime 2 are considered using analytic arguments. The model can be identified with the integrable $\phi_{1,2}$ perturbation of the unitary minimal series $M_{L,L+1}$. It is demonstrated that the excitation spectrum of the transfer matrix satisfies the same functional equations in terms of elliptic functions as the exact S-matrices of the $\phi_{1,2}$ perturbation do in terms of trigonometric functions. In particular, the bootstrap equation corresponding to a self-fusing process is recovered. For the special cases $L=3,4,6$ corresponding to the Ising model in a magnetic field, and the leading thermal perturbations of the tricritical Ising and three-state Potts model, as well as for the unrestricted model, $L=\infty$, we relate the structure of the Bethe roots to the Lie algebras $E_{8,7,6}$ and $D_4$ using Coxeter geometry. In these cases Coxeter geometry also allows for a single formula in generic Lie algebraic terms describing all four cases. For general $L$ we calculate the spectral gaps associated with the leading excitation which allows us to compute universal amplitude ratios characteristic of the universality class. The ratios are of field theoretic importance as they enter the bulk vacuum expectation value of the energy momentum tensor associated with the corresponding integrable quantum field theories.