【摘要】：本論文的研究對(duì)象是量子可積模型,一類(lèi)在數(shù)學(xué)及物理領(lǐng)域均起著重要作用的模型。在文中為了求解量子可積模型的本征值和反演Bethe態(tài),我們介紹和利用了幾種最常用的方法:坐標(biāo)Bethe Ansatz方法,代數(shù)Bethe Ansatz方法,Baxter提出的T-Q關(guān)系,分離變量法以及非對(duì)角Bethe Ansatz方法。文章的第一部分中我們對(duì)可積性,Yang-Baxter方程,反射方程,量子可積模型以及幾種經(jīng)典的方法做了簡(jiǎn)單的介紹。第二部分我們分別研究了反周期XXZ自旋鏈,開(kāi)邊界XXX自旋鏈與開(kāi)邊界XXZ自旋鏈,并且給出了一套基于非齊次T-Q關(guān)系和SoV基反演系統(tǒng)Bethe態(tài)的方法。反演系統(tǒng)Bethe態(tài)的具體思路是:首先我們利用非對(duì)角Bethe Ansatz方法構(gòu)建系統(tǒng)的非齊次T-Q關(guān)系式并且給出相應(yīng)的Bethe Ansatz方程;其次我們利用SoV方法構(gòu)建系統(tǒng)Hilbert空間的一組完備基,這組基是某個(gè)算符X(u)的本征態(tài)或者贗本征態(tài);接著我們求出這組完備基與轉(zhuǎn)移矩陣本征態(tài)的內(nèi)積,這組內(nèi)積可以確定轉(zhuǎn)移矩陣本征態(tài);最后我們利用算符{X(uj)}和一個(gè)合適的參考態(tài)構(gòu)建系統(tǒng)的Bethe態(tài)并利用上一步求出內(nèi)積證明其是轉(zhuǎn)移矩陣本征態(tài)。構(gòu)建的反周期XXZ自旋鏈Bethe態(tài)中的參考態(tài)是個(gè)高度糾纏的迭加態(tài),對(duì)應(yīng)的算符X(uj)是單值矩陣的非對(duì)角元。開(kāi)邊界XXX自旋鏈和開(kāi)邊界XXX自旋鏈的Bethe態(tài)有著相似的形式,我們引入兩組或者兩套變換分別找到了構(gòu)建Bethe態(tài)的算符和參考態(tài)。最后的結(jié)果顯示三角化K-矩陣給出參考態(tài),對(duì)角化K+矩陣給出產(chǎn)生算符。第三部分我們分別給出了具有非平行邊界場(chǎng)的一維超對(duì)稱(chēng)t-J模型以及具有非對(duì)角邊界的AdS/CFT自旋鏈的嚴(yán)格解。利用坐標(biāo)Bethe Ansatz或者代數(shù)Bethe Ansatz方法,我們將這兩種模型的本征值問(wèn)題轉(zhuǎn)換成具有非平行邊界場(chǎng)的自旋鏈模型的本征值問(wèn)題,而這一模型的嚴(yán)格解已經(jīng)由非對(duì)角Bethe Ansatz方法給出。根據(jù)非對(duì)角Bethe Ansatz方法的結(jié)果,我們首次給出這兩種非平凡模型的嚴(yán)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.
【學(xué)位授予單位】：中國(guó)科學(xué)院大學(xué)(中國(guó)科學(xué)院物理研究所)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2017
【分類(lèi)號(hào)】：O41

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