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arXiv:1003.3472

Three-magnon bound states in exactly rung-dimerized spin ladders

P. N. Bibikov

Published Mar 17, 2010• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.1003.3472

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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 mmm-magnon (m>3m>3m>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 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.
An effective model is suggested for an asymmetric spin ladder with dimerized rungs. Magnon mode truncation that originated from magnon decay (recently observed in the one compound $\mathrm{IPA}\text{\ensuremath{-}}\mathrm{Cu}{\mathrm{Cl}}_{3}$) is naturally described within this model. Using Bethe Ans\"atze, we studied a one-magnon sector and obtained relations between interaction constants of the model and experimentally observable quantities such as the gap and truncation energies, spin velocity, and truncation wave vector. It is also shown that the structure factor turns to zero at the truncation point.
An integrable spin-ladder model with nearest-neighbor exchanges and biquadratic interactions is proposed. With the Bethe ansatz solutions of the model Hamiltonian, it is found that there are three possible phases in the ground state, i.e., a rung-dimerized phase with a spin gap, and two massless phases. The possible fixed points of the system and the quantum critical behavior at the critical point J=J+c{J=J}_{+}^{c}J=J+c​ are discussed.
We present a family of spin-ladder models which can be solved exactly for the ground state and exhibit non-Haldane spin-liquid properties as predicted recently by Nersesyan and Tsvelik [Phys. Rev. Lett. 78, 3939 (1997)], and study their excitation spectra using a simple variational Ansatz. The elementary excitation is neither a magnon nor a spinon, but a pair of propagating triplet or singlet solitons connecting two spontaneously dimerized ground states. Second-order phase transitions separate this phase from the Haldane phase and the rung-dimer phase.