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arXiv:0902.1335 · Journal of Physics A Mathematical and Theoretical

A three-magnon problem for exactly rung-dimerized spin ladders: from a general outlook to the Bethe ansatz

P N Bibikov

Published Jul 16, 2009• 7 citations

Abstract

A three-magnon problem for an exactly rung-dimerized spin ladder is brought up separately in all total spin sectors. At first, a special duality transformation of the Schrödinger equation is found within the general outlook. Then the problem is treated within the coordinate Bethe ansatz. A straightforward approach is developed to obtain pure scattering states. At values S = 0 and S = 3 of total spin, the Schrödinger equation has a form inherent in the XXZ chain. At S = 1, 2, solvability holds only in five previously found completely integrable cases. Nevertheless, even in a general non-integrable case, there are some special Bethe solutions in both S = 1 and S = 2 sectors. Pure scattering states in all total spin sectors are presented explicitly.

DOI: 10.1088/1751-8113/42/31/315212

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A three-magnon problem for an exactly rung-dimerized spin ladder is brought up separately in all total spin sectors. At first, a special duality transformation of the Schrödinger equation is found within the general outlook. Then the problem is treated within the coordinate Bethe ansatz. A straightforward approach is developed to obtain pure scattering states. At values S = 0 and S = 3 of total spin, the Schrödinger equation has a form inherent in the XXZ chain. At S = 1, 2, solvability holds only in five previously found completely integrable cases. Nevertheless, even in a general non-integrable case, there are some special Bethe solutions in both S = 1 and S = 2 sectors. Pure scattering states in all total spin sectors are presented explicitly.

Three magnon bound states in all total spin sectors of general nonintegrable exactly rung-dimerized spin ladder are obtained by Bethe Ansatz. Basing on this result a dispersion law for

m

-magnon (

m>3

) bound states is conjectured. It is suggested that (depending on correlations between coupling constants) a behavior of the gas parameter (density of rung-triplets) on the boundary of the rung-dimerized phase may either be smooth or have a leap.

A detailed study of an S = ½ spin ladder model is given. The ladder consists of plaquettes formed by nearest-neighbor rungs with all possible SU (2)-invariant interactions. For properly chosen coupling constants, the model is shown to be integrable in the sense that the quantum Yang-Baxter equation holds and one has an infinite number of conserved quantities. The R -matrix and L -operator associated with the model Hamiltonian are given in a limiting case. It is shown that after a simple transformation, the model can be solved via a Bethe ansatz. The phase diagram of the ground state is exactly derived using the Bethe ansatz equation.