本文選題：Temperley-Lieb代數(shù) + 海森堡XXZ模型�。� 參考：《東北師范大學(xué)》2016年碩士論文

【摘要】：1971年,H.N.V.Temperley和E.H.Lieb共同提出了Temperley-Lieb代數(shù)。Temperley-Lieb代數(shù)是為了研究二維統(tǒng)計(jì)力學(xué)而被引入的,它可以解決二維統(tǒng)計(jì)力學(xué)中的晶格模型問(wèn)題。Temperley-Lieb代數(shù)還與扭結(jié)不變量有著密切的聯(lián)系,因此,我們可以借助Temperley-Lieb代數(shù)構(gòu)造扭結(jié)不變量。最近的研究表明,發(fā)現(xiàn)Temperley-Lieb代數(shù)在量子信息領(lǐng)域扮演重要的角色。例如,在兩個(gè)qubit系統(tǒng)中,由Temperley-Lieb代數(shù)生成的辮子代數(shù)可以用來(lái)描述量子信息領(lǐng)域一組重要的量子態(tài)—Bell態(tài)。此外,Temperley-Lieb代數(shù)在研究自旋鏈模型方面也有重要的價(jià)值,我們可以借助Temperley-Lieb代數(shù)來(lái)構(gòu)造拓?fù)浠?進(jìn)而研究模型的求解,以及自旋鏈的性質(zhì)。此外,Temperley-Lieb代數(shù)在拓?fù)淞孔佑?jì)算方面也有著重要的作用,可以借助Temperley-Lieb代數(shù)來(lái)描述準(zhǔn)粒子的統(tǒng)計(jì)行為。本篇論文通過(guò)研究Temperley-Lieb代數(shù)的矩陣表示,構(gòu)造相應(yīng)的拓?fù)浠碚?并借助拓?fù)浠碚撗芯亢Ｉぷ孕溎Ｐ偷南嚓P(guān)性質(zhì)。已有的研究表明,一類(lèi)自旋鏈和Temperley-Lieb代數(shù)的生成元有著緊密的聯(lián)系,例如,海森堡自旋XXX模型,我們可以將這類(lèi)自旋鏈模型和Temperley-Lieb代數(shù)生成元建立聯(lián)系,接著研究Temperley-Lieb代數(shù)的作用空間,Temperley-Lieb代數(shù)的作用空間其實(shí)就是所謂的拓?fù)浠臻g,進(jìn)而我們可以研究Temperley-Lieb代數(shù)和Hamiltonian在拓?fù)淇臻g中的表示。本文借助含有q的Temperley-Lieb代數(shù)生成元,構(gòu)造了具有開(kāi)放邊界條件的海森堡XXZ自旋鏈模型。從而研究Temperley-Lieb代數(shù)和海森堡自旋鏈模型的聯(lián)系,最后研究了Temperley-Lieb代數(shù)的生成元和XXZ模型的哈密頓量之間的聯(lián)系,并借助拓?fù)浠碚?研究了具有開(kāi)放邊界條件的自旋XXZ模型。此類(lèi)模型可以看成是自旋XXX模型的推廣,具有量子代數(shù)U_q(su(2))的對(duì)稱(chēng)性。通過(guò)研究表明,拓?fù)淇臻g實(shí)際上就是q變形的單態(tài)空間。本文以四個(gè)粒子的自旋XXZ模型為例,借助拓?fù)浠碚撗芯苛四Ｐ偷那蠼?以及深入研究了該模型的基態(tài)和拓?fù)浠年P(guān)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.
【學(xué)位授予單位】：東北師范大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2016
【分類(lèi)號(hào)】：O413;O152.5

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本文編號(hào)：1987001