New Developments in the Eight Vertex Model II. Chains of odd length

Details New Developments in the Eight Vertex Model II. Chains of odd length New Developments in the Eight Vertex Model II. Chains of odd length

2004-10-05

arxiv.org/abs/cond-mat/0410113v3

Journal of Statistical Physics, 120, 37 (2005)

Physics

Condensed Matter

Statistical Mechanics

29 pages. A typo in the published version of table I has been corrected

Scientific paper

10.1007/s10955-005-4410-5

We study the transfer matrix of the 8 vertex model with an odd number of lattice sites $N.$ For systems at the root of unity points $\eta=mK/L$ with $m$ odd the transfer matrix is known to satisfy the famous ``$TQ$'' equation where ${\bf Q}(v)$ is a specifically known matrix. We demonstrate that the location of the zeroes of this ${\bf Q}(v)$ matrix is qualitatively different from the case of even $N$ and in particular they satisfy a previously unknown equation which is more general than what is often called ``Bethe's equation.'' For the case of even $m$ where no ${\bf Q}(v)$ matrix is known we demonstrate that there are many states which are not obtained from the formalism of the SOS model but which do satisfy the $TQ$ equation. The ground state for the particular case of $\eta=2K/3$ and $N$ odd is investigated in detail.

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