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Journal of Physics A Mathematical and Theoretical

O(N)-matrix difference equations and a nested Bethe ansatz

H. Babujian, Angela Foerster, Michael Karowski

Published Jan 12, 2012• 14 citations
Publisher

Abstract

A system of O ( N )-matrix difference equations is solved by means of the off-shell version of the nested algebraic Bethe ansatz. In the nesting process, a new object, the Π-matrix, is introduced to overcome the complexities of the O ( N )-group structure. The highest weight property of the solutions is proved and some explicit examples are discussed.

DOI
10.1088/1751-8113/45/5/055207

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A system of O ( N )-matrix difference equations is solved by means of the off-shell version of the nested algebraic Bethe ansatz. In the nesting process, a new object, the Π-matrix, is introduced to overcome the complexities of the O ( N )-group structure. The highest weight property of the solutions is proved and some explicit examples are discussed.
A system of O(N)-matrix difference equations is solved by means of the off-shell version of the nested algebraic Bethe ansatz. In the nesting process a new object, the ΠΠΠ-matrix, is introduced to overcome the complexities of the O(N) group structure. The proof of the main theorem is presented in detail. In particular, the cancellation of all unwanted terms is shown explicitly. The highest weight property of the solutions is proved.
The system of SU(N)- and U(N)-matrix difference equations are solved by means of a nested version of a generalized Bethe ansatz, also the called `off shell' Bethe ansatz (Babujian H M 1990 Correlation functions in WZNW model as a Bethe wavefunction for the Gaudin magnets Proc. XXIVth Int. Symp. (Ahrenshoop, Zeuthen)). To solve this new Bethe ansatz in the algebraic language analogous to the conventional case, a new type of monodromy matrices is introduced. They fulfil a new type of Yang - Baxter equations which simplify the proofs. Using a similar approach as for the conventional nested Bethe ansatz the problem is solved iteratively. The vanishing of the `unwanted terms' of the first level ansatz is equivalent to a set of second level difference equations. The solutions are obtained as sums over `off-shell' Bethe vectors. These sums are `Jackson-type integrals'. The highest weight property of the solutions is proved. The solutions are calculated explicitly for several examples of SU(N)- and U(N)-representations.
We study the highest weight representations of the RTT-algebras for the R-matrix of sp(2n)sp(2n)sp(2n) type by the nested algebraic Bethe ansatz. For special representations these models were solved by Reshetikhin and by Martins and Ramos.