An elliptic current operator for the 8 vertex model

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We compute the operator which creates the missing degenerate states in the algebraic Bethe ansatz of the 8 vertex model at roots of unity and relate it to the concept of an elliptic current operator. We find that in sharp contrast with the corresponding formalism in the six-vertex model at roots of unity the current operator is not nilpotent with the consequence that in the construction of degenerate eigenstates of the transfer matrix an arbitrary number of exact strings can be added to the set of regular Bethe roots. Thus the original set of free parameters s,t of an eigenvector of T is enlarged to become s,t,λc,₁,..., λc,ₙ} with arbitrary string centers λc,ⱼ and arbitrary n.