Scaling Limit of the Ising Model in a Field

TL;DR

This paper numerically investigates the dilute A_3 model, confirming its scaling limit aligns with E_8 mass ratios predicted by integrable quantum field theory, while also identifying a potential anomaly.

Contribution

It provides detailed numerical analysis of Bethe Ansatz solutions for the dilute A_3 model, supporting the E_8 structure in its scaling limit and exploring its critical spectrum.

Findings

Numerical results match E_8 mass ratios in the scaling limit.

Identifies a state that may violate Bethe Ansatz assumptions.

Analyzes the critical spectrum with massive excitations.

Abstract

The dilute A_3 model is a solvable IRF (interaction round a face) model with three local states and adjacency conditions encoded by the Dynkin diagram of the Lie algebra A_3. It can be regarded as a solvable version of an Ising model at the critical temperature in a magnetic field. One therefore expects the scaling limit to be governed by Zamolodchikov's integrable perturbation of the c=1/2 conformal field theory. Indeed, a recent thermodynamic Bethe Ansatz approach succeeded to unveil the corresponding E_8 structure under certain assumptions on the nature of the Bethe Ansatz solutions. In order to check these conjectures, we perform a detailed numerical investigation of the solutions of the Bethe Ansatz equations for the critical and off-critical model. Scaling functions for the ground-state corrections and for the lowest spectral gaps are obtained, which give very precise numerical results for the lowest mass ratios in the massive scaling limit. While these agree perfectly with the E_8 mass ratios, we observe one state which seems to violate the assumptions underlying the thermodynamic Bethe Ansatz calculation. We also analyze the critical spectrum of the dilute A_3 model, which exhibits massive excitations on top of the massless states of the Ising conformal field theory.