Temperley-Lieb代数在研究开放边界海森堡XXZ自旋链模型中的应用

发布时间：2018-06-06 14:45

本文选题：Temperley-Lieb代数 + 海森堡XXZ模型　；参考：《东北师范大学》2016年硕士论文

【摘要】：1971年,H.N.V.Temperley和E.H.Lieb共同提出了Temperley-Lieb代数。Temperley-Lieb代数是为了研究二维统计力学而被引入的,它可以解决二维统计力学中的晶格模型问题。Temperley-Lieb代数还与扭结不变量有着密切的联系,因此,我们可以借助Temperley-Lieb代数构造扭结不变量。最近的研究表明,发现Temperley-Lieb代数在量子信息领域扮演重要的角色。例如,在两个qubit系统中,由Temperley-Lieb代数生成的辫子代数可以用来描述量子信息领域一组重要的量子态—Bell态。此外,Temperley-Lieb代数在研究自旋链模型方面也有重要的价值,我们可以借助Temperley-Lieb代数来构造拓扑基,进而研究模型的求解,以及自旋链的性质。此外,Temperley-Lieb代数在拓扑量子计算方面也有着重要的作用,可以借助Temperley-Lieb代数来描述准粒子的统计行为。本篇论文通过研究Temperley-Lieb代数的矩阵表示,构造相应的拓扑基理论,并借助拓扑基理论研究海森堡自旋链模型的相关性质。已有的研究表明,一类自旋链和Temperley-Lieb代数的生成元有着紧密的联系,例如,海森堡自旋XXX模型,我们可以将这类自旋链模型和Temperley-Lieb代数生成元建立联系,接着研究Temperley-Lieb代数的作用空间,Temperley-Lieb代数的作用空间其实就是所谓的拓扑基空间,进而我们可以研究Temperley-Lieb代数和Hamiltonian在拓扑空间中的表示。本文借助含有q的Temperley-Lieb代数生成元,构造了具有开放边界条件的海森堡XXZ自旋链模型。从而研究Temperley-Lieb代数和海森堡自旋链模型的联系,最后研究了Temperley-Lieb代数的生成元和XXZ模型的哈密顿量之间的联系,并借助拓扑基理论,研究了具有开放边界条件的自旋XXZ模型。此类模型可以看成是自旋XXX模型的推广,具有量子代数U_q(su(2))的对称性。通过研究表明,拓扑空间实际上就是q变形的单态空间。本文以四个粒子的自旋XXZ模型为例,借助拓扑基理论研究了模型的求解,以及深入研究了该模型的基态和拓扑基的关系。
[Abstract]:In 1971, H.N.V. Temperley and E.H.Lieb jointly proposed Temperley-Lieb algebra. Temperley-Lieb algebra was introduced to study two-dimensional statistical mechanics. It can solve the lattice model problem in two-dimensional statistical mechanics. Temperley-Lieb algebra is also closely related to kink invariants. We can construct kink invariants with Temperley-Lieb algebra. Recent studies have shown that Temperley-Lieb algebras play an important role in the field of quantum information. For example, in two qubit systems, braided algebras generated by Temperley-Lieb algebras can be used to describe a group of important quantum states in the field of quantum information. In addition, Temperley-Lieb algebras are also of great value in the study of spin chain models. We can use Temperley-Lieb algebras to construct topological bases, and then to study the solution of the model and the properties of spin chains. In addition, Temperley-Lieb algebras also play an important role in topological quantum computation. Temperley-Lieb algebras can be used to describe the statistical behavior of quasiparticles. In this paper, the matrix representation of Temperley-Lieb algebra is studied, the corresponding topological basis theory is constructed, and the related properties of Heisenberg spin chain model are studied with the help of topological basis theory. Previous studies have shown that a class of spin chains are closely related to the generators of Temperley-Lieb algebras, such as the Heisenberg spin XXX model, which we can relate to the generators of Temperley-Lieb algebras. Then we study the action space of Temperley-Lieb algebra and the action space of Temperley-Lieb algebra is actually called topological base space, and we can study the representation of Temperley-Lieb algebra and Hamiltonian in topological space. In this paper, a Heisenberg XXZ spin chain model with open boundary conditions is constructed by means of the Temperley-Lieb algebraic generator with Q. The relation between Temperley-Lieb algebra and Heisenberg spin chain model is studied. Finally, the relation between the generator of Temperley-Lieb algebra and the Hamiltonian of XXZ model is studied. The spin XXZ model with open boundary conditions is studied with the help of topological basis theory. This kind of model can be regarded as a generalization of the spin XXX model with the symmetry of the quantum algebra U _ S _ Q _ (2). The results show that the topological space is actually the one-state space with q-deformation. In this paper, taking the spin XXZ model of four particles as an example, the solution of the model is studied with the help of the topological basis theory, and the relationship between the ground state and the topological basis of the model is studied in depth.
【学位授予单位】：东北师范大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O413;O152.5

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