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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.

自旋1/2 XXZ自旋链是可积量子自旋系统中最基本的模型之一。当q是单位的n次方根时，XXZ哈密顿量的对称性被扩大，它包含了sl(2)循环代数，这是一个无限维李代数，参数q由XXZ耦合Δ定义为Δ =(q+1/q)/2。我们已经严格地证明了在某些扇区中，每一个正则贝氏特征向量都是sl(2)循环代数的最高权向量。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准则。此外，我们还构造了一种具有相同给定最高权值的所有可能的最高权值表示的方法。因此，我们构造了一个推导与sl(2)环代数相关的谱简并度的基本算法。我们还证明了扭曲XXZ自旋链的sl(2)环代数对称性。我们证明了在手性波茨模型的超积点上，与给定正则贝特态相关的类伊辛谱是由一个多项式表征的，我们称之为SCP多项式。我们还证明了SCP多项式与与贝特态相关的简并特征空间的德林菲尔德多项式是相同的。因此，自旋1/2 XXZ链的sl(2)环代数对称性在手性Potts模型传递矩阵的特征谱中也起着重要作用，这是二维Ising模型的推广。