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 η = mK/L with m odd the transfer matrix is known to satisfy the famous "T Q" equation where Q(v) is a specifically known matrix. We demonstrate that the location of the zeroes of this 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 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 T Q equation. The ground state for the particular case of η = 2K/3 and N odd is investigated in
We propose an expression for the current form of the lowering operator of the sl 2 loop algebra symmetry of the six vertex model (XXZ spin chain) at roots of unity. This operator has poles which correspond to the evaluation parameters of representation theory which are given as the roots of the Drinfeld polynomial. We explicitly compute these polynomials in terms of the Bethe roots which characterize the highest weight states for all values of S z . From these polynomials we find that the Bethe roots satisfy sum rules for each value of S z .
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