本文選題：量子糾纏 + SPT態(tài)��； 參考：《清華大學(xué)》2016年博士論文

【摘要】：低維關(guān)聯(lián)量子物態(tài)是當(dāng)前凝聚態(tài)物理的研究熱點,其非平庸拓撲信息很大程度上隱藏在基態(tài)波函數(shù)的量子糾纏特性中。通過將體系一個子部分的約化密度矩陣取對數(shù)得到的糾纏哈密頓量,以及其本征值譜、即糾纏譜,則是定量描述糾纏的工具。通過左右劃分得到的邊緣糾纏譜的低能部分與拓撲態(tài)邊緣激發(fā)能譜一一對應(yīng),從而可以反映拓撲信息。本文集中于一維有對稱性的短程糾纏態(tài)、即對稱性保護拓撲態(tài)(SPT態(tài)),通過研究其共振價鍵固體(VBS)構(gòu)型的嚴格可解點基態(tài)波函數(shù)的量子糾纏特性,來刻畫SPT態(tài)的拓撲性質(zhì)。主要內(nèi)容如下。在第一部分我們首先介紹了反鐵磁整數(shù)自旋鏈的Haldane相,以及作為其嚴格可解點的AKLT模型。我們說明只有奇數(shù)自旋AKLT態(tài)才屬于非平庸的SPT態(tài),并以該模型為例證明,SPT態(tài)邊緣糾纏譜簡并度中,既有拓撲相關(guān)的部分、也有非普適的部分,而非平庸的前者可通過由對稱性決定的拓撲退糾纏算符除去。該工作厘清了SPT態(tài)邊緣糾纏譜的結(jié)構(gòu),對用糾纏譜刻畫拓撲態(tài)有指導(dǎo)作用。從第二部分起,我們研究如何通過量子糾纏,從VBS型基態(tài)波函數(shù)中讀取原SPT態(tài)與平庸相之間的量子臨界點的信息。該臨界點不能用朗道的經(jīng)典理論描述,其低能元激發(fā)是原VBS的邊緣自由度。我們對VBS波函數(shù)引入對稱的延展劃分:將每l個格點組合成一個子塊,以所有奇數(shù)子塊作為子系統(tǒng),得到體糾纏哈密頓量HE。我們說明,HE在l選取合適的情況下就描述退禁閉的邊緣自由度之間的滲透,從而給出量子臨界點的完整能譜。我們證明自旋S-AKLT態(tài)的HE是自旋S/2的海森堡模型,對于非平庸的奇數(shù)S并取l=偶數(shù),HE就描述Haldane相與平庸相之間的量子臨界點,低能有效理論是S U(2)1WZW場論。在第三部分,我們將延展劃分的方法運用到具有S O(5)對稱性的自旋2-VBS態(tài),并證明HE在l=奇數(shù)時就描述與原SPT態(tài)相連的量子臨界點,低能有效理論是S U(4)1WZW場論。在第四部分,我們用自旋構(gòu)造了具有S U(N)(N≥3)結(jié)構(gòu)的一維VBS態(tài):在總長度為偶數(shù)時,兩端邊緣互為共軛,破壞鏡像對稱性;而當(dāng)總長度為奇數(shù)時,兩端邊緣相同,屬于非平庸SPT態(tài),且其對應(yīng)的量子臨界點由取l=奇數(shù)的體糾纏哈密頓量描述,低能有效理論是S U(N)1WZW場論。通過延展劃分我們不僅可以得到量子臨界點低能有效理論,更能寫出對應(yīng)的哈密頓量,從而建立了體-邊緣-量子臨界點對應(yīng),深化了對拓撲態(tài)的理解。
[Abstract]:Low-dimensional correlated quantum state is a hot topic in condensed matter physics. Its non-mediocre topological information is largely hidden in the quantum entanglement properties of ground state wave function. The entangled Hamiltonian obtained by taking the reduced density matrix of a subpart of the system, and its eigenvalue spectrum, that is, the entanglement spectrum, is a tool for quantificational description of entanglement. The low-energy part of the edge-entangled spectrum obtained by the left and right partition corresponds to the excited energy spectrum of the topological state, which can reflect the topological information. In this paper, we focus on the one dimensional symmetric short range entangled state, which is the symmetric protected topological state. By studying the quantum entanglement properties of the strictly solvable ground state wave function of its resonant valence bond solid state, we characterize the topological properties of the SPT state. The main contents are as follows. In the first part we first introduce the Haldane phase of the antiferromagnetic integer spin chain and the AKLT model as its strictly solvable point. We show that only the odd-number spin AKLT states are non-mediocre SPT states, and take the model as an example to prove that the degeneracy of the edge entanglement spectrum of the AKLT states is not only topological dependent, but also non-universal. The non-mediocre former can be removed by the topological deentanglement operator determined by symmetry. This work clarifies the structure of the edge entanglement spectrum of SPT states and provides guidance for the characterization of topological states by entanglement spectra. From the second part, we study how to read the information of the quantum critical point between the original SPT state and the mediocre phase from the VBS type ground state wave function by quantum entanglement. This critical point cannot be described by Landau's classical theory, and its low energy element excitation is the edge degree of freedom of the original VBS. We introduce a symmetric extension partition for VBS wave functions: every l lattice points are combined into a subblock, and all odd sub-blocks are used as subsystems to obtain the volume entangled Hamiltonian. We show that the percolation between the edge degrees of freedom of decommissioning is described under the suitable condition of l, and the complete energy spectrum of the quantum critical point is obtained. We prove that the HE of the spin S-AKLT state is the Heisenberg model of spin S / 2. For the non-mediocre odd number S and the even number he, we describe the quantum critical point between the Haldane phase and the mediocre phase. The low energy efficient theory is S U(2)1WZW field theory. In the third part, we apply the method of extended partition to the spin 2-VBS states with S _ O _ (5) symmetry, and prove that he describes the quantum critical points connected with the original SPT states when l = odd. The low energy efficient theory is S U(4)1WZW 's field theory. In the fourth part, we construct a one-dimensional VBS state with S U(N)(N 鈮，

本文編號：1908969