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arXiv:1107.5627 · Nuclear Physics B

Determinant formula for the partition function of the six-vertex model with a non-diagonal reflecting end

Wen‐Li Yang, Xi Chen, Jun Min Feng, Kun Hao, Bo-Yu Hou, Kangjie Shi, Yao-Zhong Zhang

Published Nov 12, 2010• 8 citations
arXivHTMLPDF
DOI
10.1016/j.nuclphysb.2010.11.008

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Abstract We consider the six-vertex model with reflecting end boundary condition. We compute analytically boundary correlation functions, such as the boundary polarization and the emptiness formation probability. In order to do that, we use the Sklyanin’s reflection algebra to derive recursion relations for the partition function of the model as well as for the boundary correlations in terms of the partition function. Thanks to the Tsuchiya determinant formula, these recursion relations allow the boundary correlations to be also efficiently written in determinant form.
We review the (algebraic-)functional method devised by Galleas and further developed by Galleas and the author.We first explain the method using the simplest example: the computation of the partition function for the six-vertex model with domainwall boundary conditions.At the heart of the method lies a linear functional equation for the partition function.After deriving this equation we outline its analysis.The result is a closed expression in the form of a symmetrized sum -or, equivalently, multiple-integral formula -that can be rewritten to recover Izergin's determinant.Special attention is paid to the relation with other approaches.In particular we show that the Korepin-Izergin approach can be recovered within the functional method.We comment on the functional method's range of applicability, and review how it is adapted to the technically more involved example of the elliptic solid-on-solid model with domain walls and a reflecting end.We present a new formula for the partition function of the latter, which was expressed as a determinant by Tsuchiya-Filali-Kitanine.Our result takes the form of a 'crossing-symmetrized' sum with 2 L terms featuring the elliptic domain-wall partition function, which appears to be new also in the limiting case of the six-vertex model.Further taking the rational limit we recover the expression obtained by Frassek using the boundary perimeter Bethe ansatz.