Three-magnon problem for exactly rung-dimerized spin ladders: from general outlook to Bethe Ansatze

TL;DR

This paper analyzes the three-magnon problem in rung-dimerized spin ladders, developing a Bethe Ansatz approach and revealing integrability properties across different total spin sectors.

Contribution

It introduces a duality transformation for the Schrödinger equation and constructs explicit pure scattering states for all total spin sectors.

Findings

Identifies integrable cases for S=1,2 sectors.

Develops a straightforward method for pure scattering states.

Reveals partial Bethe solutions in non-integrable models.

Abstract

Three-magnon problem for exactly rung-dimerized spin ladder is brought up separately at all total spin sectors. At first a special duality transformation of the $\rm Schr\ddot odinger$ equation is found within general outlook. Then the problem is treated within Coordinate Bethe Ansatze. A straightforward approach is developed to obtain pure scattering states. At values S=0 and S=3 of total spin the $\rm Schr\ddot odinger$ equation has the form inherent in the $XXZ$ chain. For $S=1,2$ solvability holds only in five previously found {\it completely integrable} cases. Nevertheless a partial S=1 Bethe solution always exists even for general non integrable model. Pure scattering states for all total spin sectors are presented explicitly.