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arXiv:1308.4446 · Journal of Physics A Mathematical and Theoretical

Algebraic Bethe ansatz for the six vertex model with upper triangularK-matrices

Rodrigo A. Pimenta, A Lima-Santos

Published Oct 24, 2013• 24 citations
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Abstract

We consider a formulation of the algebraic Bethe ansatz for the six vertex model with non-diagonal open boundaries. Specifically, we study the case where both left and right K -matrices have an upper triangular form. We show that the main difficulty entailed by those forms of the K -matrices is the construction of the excited states. However, it is possible to treat this problem with the aid of an auxiliary transfer matrix and by means of a generalized creation operator.

DOI
10.1088/1751-8113/46/45/455002

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We consider a formulation of the algebraic Bethe ansatz for the six vertex model with non-diagonal open boundaries. Specifically, we study the case where both left and right K -matrices have an upper triangular form. We show that the main difficulty entailed by those forms of the K -matrices is the construction of the excited states. However, it is possible to treat this problem with the aid of an auxiliary transfer matrix and by means of a generalized creation operator.
We study in this article the algebraic Bethe Ansatz from a combinatorial perspective.The commutation relations are represented by combinatorial diagrams so that the action of the transfer matrix over the nth excited state can be found through the analysis of some combinatorial trees.This allows us to compute the eigenvalues and eigenstates of the transfer matrix, as well as the respective Bethe Ansatz equations, in a straightforward way.As examples, we consider the six-vertex model with both periodic and non-periodic boundary conditions.In parallel, we also present a comprehensive introduction to the usual algebraic Bethe Ansatz (for both the periodic and the non-periodic cases).
We connect two alternative concepts of solving integrable models, Baxter's method of auxiliary matrices (or Q -operators) and the algebraic Bethe ansatz. The main steps of the calculation are performed in a general setting and a formula for the Bethe eigenvalues of the Q -operator is derived. A proof is given for states which contain up to three Bethe roots. Further evidence is provided by relating the findings to the six-vertex fusion hierarchy. For the XXZ spin-chain we analyse the cases when the deformation parameter of the underlying quantum group is evaluated both at and away from a root of unity.
We implement fully the algebraic Bethe ansatz for the XXX Heisenberg spin chain in the case when both boundary matrices can be brought to the upper-triangular form. We define the Bethe vectors which yield the strikingly simple expression for the off shell action of the transfer matrix, deriving the spectrum and the relevant Bethe equations. We explore further these results by obtaining the off shell action of the generating function of the Gaudin Hamiltonians on the corresponding Bethe vectors through the so-called quasi-classical limit. Moreover, this action is as simple as it could possibly be, yielding the spectrum and the Bethe equations of the Gaudin model.