Lattice Ising model in a field: E$_8$ scattering theory

TL;DR

This paper demonstrates that the dilute A3 lattice model, near its critical point, exhibits an E8 structure in the scaling limit, linking it to the E8 scattering theory related to the Ising model in a magnetic field.

Contribution

The study shows that the dilute A3 model's thermodynamics in the scaling limit naturally lead to the E8 scattering theory, establishing a concrete lattice realization.

Findings

The dilute A3 model exhibits E8 symmetry in the scaling limit.

The model's thermodynamics align with the E8 scattering theory.

A direct relation between the lattice model and E8 structure is established.

Abstract

Zamolodchikov found an integrable field theory related to the Lie algebra E$_8$, which describes the scaling limit of the Ising model in a magnetic field. He conjectured that there also exist solvable lattice models based on E$_8$ in the universality class of the Ising model in a field. The dilute A$_3$ model is a solvable lattice model with a critical point in the Ising universality class. The parameter by which the model can be taken away from the critical point acts like a magnetic field by breaking the $\Integer_2$ symmetry between the states. The expected direct relation of the model with E$_8$ has not been found hitherto. In this letter we study the thermodynamics of the dilute A$_3$ model and show that in the scaling limit it exhibits an appropriate E$_8$ structure, which naturally leads to the E$_8$ scattering theory for massive excitations over the ground state.