Level crossings in integrable finite-size quantum systems with infinite-dimensional symmetry and solvable models in nanoscopic or mesoscopic systems

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The spin1/2 XXZ spin chain is one of the most fundamental models among integrable quantum spin systems. When q is an N-th root of unity, the symmetry of the XXZ Hamiltonian is enlarged and it contains the sl(2) loop algebra, which is an infinite-dimensional Lie algebra Here the parameter q is defined by the XXZ coupling Δ by Δ =(q+1/q)/2.We have shown rigorously in some sector that every regular Bethe ansatz eigenvector is a highest weight vector of the sl(2) loop algebra. In order to analyze the spectral degeneracy we have proven a criterion for a finite-dimensional highest weight representation to be irreducible. Furthermore, we have formulated a method for constructing all the possible highest weight representations with the same given highest weight Thus, we have constructed a fundamental algorithm for deriving the spectral degeneracy associated with the sl(2) loop algebra We have also shown the sl(2) loop algebra symmetry for the twisted XXZ spin chains.We have shown that at the superintegrable point of the chiral Potts model the Ising-like spectrum associated with a given regular Bethe state is characterized by a polynomial, which we call the SCP polynomial We have also shown that the SCP polynomial is identical to the Drinfeld polynomial of the degenerate eigenspace associated with the Bethe state. Thus, the sl(2) loop algebra symmetry of the spin-1/2 XXZ chain plays a significant role also in the eigenspectrum of the transfer matrix of the chiral Potts model, which gives a generalization of the 2D Ising model.