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arXiv:math-ph/0406021 · Journal of Statistical Mechanics Theory and Experiment

General boundary conditions for the and open spin chains

D. Arnaudon, Jean Avan, Nicolas Crampé, Anastasia Doikou, L. Frappat, E. Ragoucy

Published Aug 24, 2004• 79 citations
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Abstract

Two types of boundary conditions (‘soliton preserving’ and ‘soliton non-preserving’) are investigated for the and open spin chains. The appropriate reflection equations are formulated and the corresponding solutions are classified. The symmetry and the Bethe ansatz equations are derived for each case. The general treatment for non-diagonal reflection matrices associated with the ‘soliton preserving’ case is worked out. The connection between the ‘soliton non-preserving’ boundary conditions and the twisted (super-) Yangians is also discussed.

DOI
10.1088/1742-5468/2004/08/p08005

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Two types of boundary conditions (‘soliton preserving’ and ‘soliton non-preserving’) are investigated for the and open spin chains. The appropriate reflection equations are formulated and the corresponding solutions are classified. The symmetry and the Bethe ansatz equations are derived for each case. The general treatment for non-diagonal reflection matrices associated with the ‘soliton preserving’ case is worked out. The connection between the ‘soliton non-preserving’ boundary conditions and the twisted (super-) Yangians is also discussed.
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It is well known that integrable models associated to rational R matrices give rise to certain non-Abelian symmetries known as Yangians. Analogously boundary symmetries arise when general but still integrable boundary conditions are implemented, as originally argued by Delius, Mackay, and Short from the field theory point of view, in the context of the principal chiral model on the half-line. In the present study we deal with a discrete quantum mechanical system with boundaries, that is the N site gl(n) open quantum spin chain. In particular, the open spin chain with two distinct types of boundary condition known as soliton preserving and soliton nonpreserving is considered. For both types of boundaries we present a unified framework for deriving the corresponding boundary nonlocal charges directly at the quantum level. The nonlocal charges are simply coproduct realizations of particular boundary quantum algebras called boundary or twisted Yangians, depending on the choice of boundary conditions. Finally, with the help of linear intertwining relations between the solutions of the reflection equation and the generators of the boundary or twisted Yangians we are able to exhibit the exact symmetry of the open spin chain, namely we show that a number of the boundary nonlocal charges are in fact conserved quantities.
A quantum spin chain with non-conventional boundary conditions is studied. The distinct nature of these boundary conditions arises from the conversion of a soliton to an anti-soliton after being reflected by the boundary, hence the appellation soliton non-preserving boundary conditions. We focus on the simplest non-trivial case of this class of models based on the twisted Yangian quadratic algebra. Our computations are performed through the Bethe ansatz equations in the thermodynamic limit. We formulate a suitable quantization condition describing the scattering process and proceed with explicitly determining the bulk and boundary scattering amplitudes. The energy and quantum numbers of the low lying excitations are also derived.
We present an "algebraic treatment" of the analytical Bethe ansatz for open spin chains with soliton nonpreserving (SNP) boundary conditions. For this purpose, we introduce abstract monodromy and transfer matrices which provide an algebraic framework for the analytical Bethe ansatz. It allows us to deal with a generic [Formula: see text] open SNP spin chain possessing on each site an arbitrary representation. As a result, we obtain the Bethe equations in their full generality. The classification of finite dimensional irreducible representations for the twisted Yangians are directly linked to the calculation of the transfer matrix eigenvalues.