Long-range to the Rescue of Yang-Baxter II

TL;DR

This paper extends the analysis of a spin chain model related to a deformed superconformal gauge theory, demonstrating long-range eigenstates, recursive structures, and Yang-Baxter relations for four-magnon solutions, building on previous three-magnon results.

Contribution

It constructs explicit four-magnon eigenvectors, reveals recursive relations with three-magnon states, and shows these solutions satisfy Yang-Baxter equations, advancing understanding of long-range integrability.

Findings

Four-magnon eigenstates expressed in terms of three-magnon solutions.

Validation through direct diagonalization confirms analytical results.

Four-magnon solutions satisfy an infinite tower of Yang-Baxter equations.

Abstract

We study the spin chain model capturing the one-loop spectral problem of the simplest $\mathcal{N}=2$ superconformal quiver gauge theory in four dimensions, obtained from a marginal deformation of the $\mathbb{Z}_2$ orbifold of $\mathcal{N}=4$ SYM. In Part I of this work \cite{Bozkurt:2024tpz}, we solved for the three-magnon eigenvector and found that it exhibits long-range behavior, despite the Hamiltonian being of nearest-neighbor type. In this paper, we extend the analysis to the four-magnon sector and construct explicit eigenvectors. These solutions are compatible with both untwisted and twisted periodic boundary conditions, and they allow for the computation of anomalous dimensions of single-trace operators of the gauge theory. We validate our results by direct comparison with brute-force diagonalization of the spin chain Hamiltonian. Additionally, we uncover a novel structural relation between eigenstates with different numbers of excitations. In particular, we show that the four-magnon eigenstates can be written in terms of the three-magnon solution, revealing a recursive pattern and hinting at a deeper underlying structure. Lastly, the four-magnon solution obeys an infinite tower of Yang-Baxter equations, as was the case for the three-magnon solution.