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

具有无限维对称性的可积有限尺寸量子系统中的能级交叉以及纳米或介观系统中的可解模型

基本信息

批准号：
17540351
负责人：
DEGUCHI Tetsuo
金额：
$ 2.43万
依托单位：
Ochanomizu University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2005
资助国家：
日本
起止时间：
2005 至 2007
项目状态：
已结题

项目摘要

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.

spin1/2 XXZ 自旋链是可积量子自旋系统中最基本的模型之一，当 q 是单位根时，XXZ 哈密顿量的对称性被放大，并且包含 sl(2) 循环代数，这是一个无限维李代数这里参数 q 由 XXZ 耦合 Δ 定义为 Δ =(q+1/q)/2。我们已经在某些领域严格证明了每个正则Bethe ansatz 特征向量是 sl(2) 循环代数的最高权向量为了分析谱简并性，我们已经证明了有限维最高权表示的标准是不可约的。此外，我们还制定了一种构造方法。所有最高可能的权重表示具有相同的给定最高权重因此，我们构建了一个基本算法，用于导出与 sl(2) 循环代数相关的谱简并性我们还展示了扭曲 XXZ 自旋链的 sl(2) 环代数对称性。我们已经证明，在手性 Potts 模型的超可积点，与给定正则 Bethe 态相关的类 Ising 谱由多项式表征，我们将其称为SCP 多项式我们还证明了 SCP 多项式与与 Bethe 态相关的简并本征空间的 Drinfeld 多项式相同。自旋 1/2 XXZ 链的 sl(2) 环代数对称性在手性 Potts 模型的传递矩阵的特征谱中也起着重要作用，该模型给出了 2D Ising 模型的推广。