The 8V CSOS model and the \(\text{sl}_2\) loop algebra symmetry of the six-vertex model at roots of unity. (English) Zbl 1073.82535
Summary: We review an algebraic method for constructing degenerate eigenvectors of the transfer matrix of the eight-vertex cyclic solid-on-solid lattice model (8V CSOS model), where the degeneracy increases exponentially with respect to the system size. We consider the elliptic quantum group \(E_{\tau,\eta}(\text{sl}2)\) at the discrete coupling constants: \(2N\;\eta = m_1 + im_2 \tau\), where \(N\), \(m_1\) and \(m_2\) are integers. Then we show that degenerate eigenvectors of the transfer matrix of the six-vertex model at roots of unity in the sector \(S^Z \equiv 0\;(\mod N\)) are derived from those of the 8V CSOS model, through the trigonometric limit. They are associated with the complete N strings. From the result we see that the dimension of a given degenerate eigenspace in the sector \(S^Z \equiv 0\;(\mod N\)) of the six-vertex model at \(N\)th roots of unity is given by \(2^{2S^Z_{max}/N}\), where \(S^Z_{max}\) is the maximal value of the total spin operator \(S^Z\) in the degenerate eigenspace.
MSC:
82B23 | Exactly solvable models; Bethe ansatz |
82B20 | Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics |
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