Long-range to the Rescue of Yang-Baxter

TL;DR

This paper introduces a novel long-range solution to the three-magnon eigenvalue problem in a complex spin chain model related to an $ cal{N}=2$ SCFT, overcoming previous limitations due to broken permutation symmetry.

Contribution

It presents a new method that generalizes the coordinate Bethe Ansatz to solve for three magnons in a non-Yang-Baxter-satisfying spin chain.

Findings

Discovered a long-range solution satisfying an infinite tower of Yang-Baxter equations.

Extended Bethe Ansatz techniques to cases with broken permutation symmetry.

Provided insights into the spectral problem of a challenging four-dimensional SCFT model.

Abstract

We study the spin chain model which captures the one-loop spectral problem of a prototypical example of an $\mathcal{N}=2$ SCFT in four dimensions. Up to date, this spin chain model remains unfathomable; the coordinate Bethe Ansatz does not lead to a solution from three magnons on, as the Yang-Baxter equation is not satisfied by the two-magnon scattering coefficients. In this paper, we find a long-range solution to the eigenvalue problem for three magnons. Remarkably, the scattering coefficients of our solution together with the position-dependent corrections, obey an infinite tower of Yang-Baxter equations. Our method of solving the three-magnon problem is interesting in its own right and generalizes the coordinate Bethe Ansatz approach to cases where the permutation symmetry is broken.