ON THE DOMAIN WALL PARTITION FUNCTIONS OF LEVEL-1 AFFINE so(n) VERTEX MODELS

Abstract

In [1], Korepin introduced domain wall (DW) boundary conditions in the context of the six vertex, spin-1/2 or level-1 A (1) 1 model on a finite lattice, and obtained recursion relations that determine the partition function in that case. In [2], Izergin solved Korepin’s recursion relations and obtained a determinant expression for the level-1 A (1) 1 DW partition function. In [3], determinant expressions were obtained for the spin-k/2, or level-k A (1) 1 DW partition functions, k ∈ N, using the fact that these models can be obtained from the level-1 model using fusion [4,5]. Fusion was necessary in proving the levelk A (1) 1 result because an Izergin type proof, based on Lagrange interpolation, fails to extend for general k. The reason is that the DW spin-k/2 partition function is a polynomial of degree k(L − 1) in each (multiplicative) rapidity variable (where ‘degree’is defined as the difference of the highest and lowest powers in the variable). Thus for Lagrange interpolation to work, one needs to impose k(L− 1)+1 conditions on the partition function, which are not available for k > 2 because the DW boundary conditions offer exactly 2L such conditions (L from the upper left corner and L from the upper right corner, while the lower corners offer no new conditions due to the symmetries of the lattice and the vertex weights). For k = 2, there is an ‘extended’ Izergin type proof, in addition to the fusion proof. This proof was used, in [3], as a check, in the special k = 2 case, on the general k fusion proof. However, strictly speaking, this proof was unnecessary and the extra L conditions could have remained unused. This is intriguing, since exactly solvable models are notoriously parsimonious. In this work we find an application for them. Since the 19 vertex, spin-1, or level-2 A (1) 1 model coincides with the level-1 affine so(3) vertex model (as can be seen by comparing the vertex weights), and because the degree of the vertex weights of level-1 affine so(n) models, as polynomials in the rapidities, is independent of n [6], it is natural to expect that the DW partition functions of these models have the same form as that of the level-2 A (1) 1 model. In this work, we propose a determinant expression for the DW partition functions of level-1 affine so(n) vertex models, for n ≥ 4, based on the result of [3] for the level-2 A (1) 1 case.

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Cite this paper

@inproceedings{Dow2006ONTD, title={ON THE DOMAIN WALL PARTITION FUNCTIONS OF LEVEL-1 AFFINE so(n) VERTEX MODELS}, author={Alan Dow and O FODA}, year={2006} }