粒子数不守恒量子可积模型的本征值和本征态
发布时间：2018-11-12 10:08
【摘要】：本论文的研究对象是量子可积模型,一类在数学及物理领域均起着重要作用的模型。在文中为了求解量子可积模型的本征值和反演Bethe态,我们介绍和利用了几种最常用的方法:坐标Bethe Ansatz方法,代数Bethe Ansatz方法,Baxter提出的T-Q关系,分离变量法以及非对角Bethe Ansatz方法。文章的第一部分中我们对可积性,Yang-Baxter方程,反射方程,量子可积模型以及几种经典的方法做了简单的介绍。第二部分我们分别研究了反周期XXZ自旋链,开边界XXX自旋链与开边界XXZ自旋链,并且给出了一套基于非齐次T-Q关系和SoV基反演系统Bethe态的方法。反演系统Bethe态的具体思路是:首先我们利用非对角Bethe Ansatz方法构建系统的非齐次T-Q关系式并且给出相应的Bethe Ansatz方程;其次我们利用SoV方法构建系统Hilbert空间的一组完备基,这组基是某个算符X(u)的本征态或者赝本征态;接着我们求出这组完备基与转移矩阵本征态的内积,这组内积可以确定转移矩阵本征态;最后我们利用算符{X(uj)}和一个合适的参考态构建系统的Bethe态并利用上一步求出内积证明其是转移矩阵本征态。构建的反周期XXZ自旋链Bethe态中的参考态是个高度纠缠的迭加态,对应的算符X(uj)是单值矩阵的非对角元。开边界XXX自旋链和开边界XXX自旋链的Bethe态有着相似的形式,我们引入两组或者两套变换分别找到了构建Bethe态的算符和参考态。最后的结果显示三角化K-矩阵给出参考态,对角化K+矩阵给出产生算符。第三部分我们分别给出了具有非平行边界场的一维超对称t-J模型以及具有非对角边界的AdS/CFT自旋链的严格解。利用坐标Bethe Ansatz或者代数Bethe Ansatz方法,我们将这两种模型的本征值问题转换成具有非平行边界场的自旋链模型的本征值问题,而这一模型的严格解已经由非对角Bethe Ansatz方法给出。根据非对角Bethe Ansatz方法的结果,我们首次给出这两种非平凡模型的严格解。
[Abstract]:The object of this paper is quantum integrable model, which plays an important role in mathematics and physics. In order to solve the eigenvalue of quantum integrable model and inverse Bethe state, we introduce and utilize several most commonly used methods: coordinate Bethe Ansatz method, algebraic Bethe Ansatz method, T-Q relation proposed by Baxter. The method of separating variables and the method of non-diagonal Bethe Ansatz. In the first part of this paper, we briefly introduce integrability, Yang-Baxter equation, reflection equation, quantum integrable model and several classical methods. In the second part, we study counterperiodic XXZ spin chain, open boundary XXX spin chain and open boundary XXZ spin chain, and give a set of methods based on nonhomogeneous T-Q relation and Bethe state inversion system based on SoV basis. The concrete idea of inversion system Bethe states is as follows: firstly, we use the non-diagonal Bethe Ansatz method to construct the non-homogeneous T-Q relation of the system and give the corresponding Bethe Ansatz equation; Secondly, we use the SoV method to construct a set of complete bases in the system Hilbert space, which are the eigenstates or pseudo-eigenstates of an operator X (u). Then we obtain the inner product of the complete basis and the eigenstates of the transition matrix, which can be used to determine the eigenstates of the transition matrix. Finally, we construct the Bethe state of the system by using the operator {X (uj)} and a suitable reference state, and prove that it is the eigenstate of the transfer matrix by using the inner product of the previous step. The reference state in the Bethe state of the counter-periodic XXZ spin chain is a highly entangled superposition state, and the corresponding operator X (uj) is a non-diagonal element of a single-valued matrix. The Bethe states of open boundary XXX spin chains and open boundary XXX spin chains have similar forms. We introduce two sets of transformations to find the operators and reference states to construct Bethe states respectively. The results show that the triangulated K-matrix gives the reference state and the diagonalized K-matrix gives the production operator. In the third part, we give the one-dimensional supersymmetric t-J model with non-parallel boundary field and the strict solution of the AdS/CFT spin chain with non-diagonal boundary, respectively. By using coordinate Bethe Ansatz or algebraic Bethe Ansatz method, we transform the eigenvalue problem of these two models into the eigenvalue problem of spin chain model with nonparallel boundary field, and the strict solution of this model has been given by the non-diagonal Bethe Ansatz method. Based on the results of the non-diagonal Bethe Ansatz method, we obtain the strict solutions of these two nontrivial models for the first time.
【学位授予单位】：中国科学院大学(中国科学院物理研究所)
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O41
本文编号：2326802
[Abstract]:The object of this paper is quantum integrable model, which plays an important role in mathematics and physics. In order to solve the eigenvalue of quantum integrable model and inverse Bethe state, we introduce and utilize several most commonly used methods: coordinate Bethe Ansatz method, algebraic Bethe Ansatz method, T-Q relation proposed by Baxter. The method of separating variables and the method of non-diagonal Bethe Ansatz. In the first part of this paper, we briefly introduce integrability, Yang-Baxter equation, reflection equation, quantum integrable model and several classical methods. In the second part, we study counterperiodic XXZ spin chain, open boundary XXX spin chain and open boundary XXZ spin chain, and give a set of methods based on nonhomogeneous T-Q relation and Bethe state inversion system based on SoV basis. The concrete idea of inversion system Bethe states is as follows: firstly, we use the non-diagonal Bethe Ansatz method to construct the non-homogeneous T-Q relation of the system and give the corresponding Bethe Ansatz equation; Secondly, we use the SoV method to construct a set of complete bases in the system Hilbert space, which are the eigenstates or pseudo-eigenstates of an operator X (u). Then we obtain the inner product of the complete basis and the eigenstates of the transition matrix, which can be used to determine the eigenstates of the transition matrix. Finally, we construct the Bethe state of the system by using the operator {X (uj)} and a suitable reference state, and prove that it is the eigenstate of the transfer matrix by using the inner product of the previous step. The reference state in the Bethe state of the counter-periodic XXZ spin chain is a highly entangled superposition state, and the corresponding operator X (uj) is a non-diagonal element of a single-valued matrix. The Bethe states of open boundary XXX spin chains and open boundary XXX spin chains have similar forms. We introduce two sets of transformations to find the operators and reference states to construct Bethe states respectively. The results show that the triangulated K-matrix gives the reference state and the diagonalized K-matrix gives the production operator. In the third part, we give the one-dimensional supersymmetric t-J model with non-parallel boundary field and the strict solution of the AdS/CFT spin chain with non-diagonal boundary, respectively. By using coordinate Bethe Ansatz or algebraic Bethe Ansatz method, we transform the eigenvalue problem of these two models into the eigenvalue problem of spin chain model with nonparallel boundary field, and the strict solution of this model has been given by the non-diagonal Bethe Ansatz method. Based on the results of the non-diagonal Bethe Ansatz method, we obtain the strict solutions of these two nontrivial models for the first time.
【学位授予单位】：中国科学院大学(中国科学院物理研究所)
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O41
【相似文献】
相关期刊论文 前10条
1 范洪义;产生标符a~+本征态恒等于零吗?[J];大学物理;1985年07期
2 范嗣林;互易厄密矩阵共同本征态计算的“组合法”[J];四川师范学院学报(自然科学版);1997年03期
3 梁麦林;α~(2M)本征态的偶奇表示形式[J];量子电子学报;1998年03期
4 吴式玉,周子舫;边界条件对无序体系本征态的影响[J];物理学报;1984年12期
5 欧发;光子场的相位本征态的表示及有关问题[J];量子电子学;1985年02期
6 时维春,常健斌,韩士杰;构造α~N的各正交本征态的方法[J];物理学报;1993年03期
7 张炎勋;马爱群;时维春;;α~5的正交归一本征态的某些非经典特性研究[J];哈尔滨师范大学自然科学学报;1993年03期
8 蔡维理,范洪义;对于“相干态是a~(-1)的本征态吗”一文的答复[J];大学物理;1998年05期
9 夏道澄;一维晶格中电子本征态特征与动力学行为[J];菏泽师专学报;2001年04期
10 孙金祚,王传奎,王继锁;Aubry模型的Anderson转变区与能量的关系[J];物理学报;1991年11期
相关会议论文 前1条
1 施冬梅;谢俊磊;杜仕国;赵文轸;;Ni-Co-P镀覆本征态聚苯胺的电磁和吸波性能[A];2009中国功能材料科技与产业高层论坛论文集[C];2009年
相关博士学位论文 前1条
1 张鑫;粒子数不守恒量子可积模型的本征值和本征态[D];中国科学院大学(中国科学院物理研究所);2017年
相关硕士学位论文 前3条
1 王晓华;高次幂湮灭算符本征态和量子力学非定域性检验研究[D];西南交通大学;2011年
2 张义勇;量子力学中几何相位的研究[D];东北师范大学;2006年
3 来yN敏;光学态的非经典特性及N粒子相容可观测量共同本征态的研究[D];宁波大学;2009年
，本文编号：2326802
本文链接：https://www.wllwen.com/shoufeilunwen/jckxbs/2326802.html
