Radii of the E8 Gosset Circles as the Mass Excitations in the Ising Model

TL;DR

This paper links the mass spectrum of bound states in the 1D Ising model's E8 structure to the radii of Gosset's circles, revealing geometric and algebraic connections with Coxeter groups and polytopes.

Contribution

It establishes a novel geometric interpretation of E8 mass ratios through Gosset's circles and Coxeter projections, connecting algebraic, geometric, and physical aspects of the model.

Findings

Mass ratios match radii of Gosset's circles on the Coxeter plane.

Projections of root systems relate to observed bound state masses.

Numerical values closely align with experimental measurements.

Abstract

The Zamolodchikov's conjecture implying the exceptional Lie group E8 seems to be validated by an experiment on the quantum phase transitions of the 1D Ising model carried out by the Coldea et. al. The E8 model which follows from the affine Toda field theory predicts 8 bound states with the mass relations in the increasing order m1, m2= tau m1, m3, m4, m5, m6=tau m3, m7= tau m4, m8= tau m5, where tau= (1+\sqrt(5))/2 represents the golden ratio. Above relations follow from the fact that the Coxeter group W(H4) is a maximal subgroup of the Coxeter-Weyl group W(E8). These masses turn out to be proportional to the radii of the Gosset's circles on the Coxeter plane obtained by an orthogonal projection of the root system of E8 . We also note that the masses m1, m3, m4 and m5 correspond to the radii of the circles obtained by projecting the vertices of the 600-cell, a 4D polytope of the non-crystallographic Coxeter group W(H4). A special non-orthogonal projection of the simple roots on the Coxeter plane leads to exactly the numerical values of the masses of the bound states as 0.4745, 0.7678, 0.9438, 1.141, 1.403, 1.527, 1.846, and 2.270. We note the striking equality of the first two numerical values to the first two masses of the bound states determined by the Coldea et. al.