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

自旋1/2XXZ自旋链是可积量子自旋系统中最基本的模型之一。当q是N次单位根时，XXZ Hamilton算子的对称性得到了扩大，它包含了无限维李代数sl（2）loop代数。这里参数q由XXZ耦合Δ定义，Δ =（q+1/q）/2。为了分析谱简并性，我们证明了有限维最高权表示不可约的一个判据。此外，我们已经制定了一种方法，用于构造具有相同给定最高权重的所有可能的最高权重表示。我们构造了一个基本的算法，导出了与sl（2）loop代数有关的谱简并度。我们还证明了扭曲XXZ自旋链的sl（2）loop代数对称性。我们证明了在手征Potts模型的超可积点处，Ising-我们还证明了SCP多项式与Bethe态简并本征空间的Drinfeld多项式是相同的。因此，自旋为1/2的XXZ链的sl（2）圈代数对称性在手征Potts模型的转移矩阵的本征谱中也起着重要的作用，这给出了二维Ising模型的推广.

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.