這份論文主要的目的是探究鐵磁性糾纏行為.在量子資訊科學裡，糾纏是一基本的概念,其與多體物理的關聯在近來倍受關注,尤其是在臨界的區域裡，一維量子自旋鏈的糾纏與連續性的共形場理論(CFT)之間的對應關係.我們闡釋這層概念是藉由自旋為1 的雙線性-雙二次(spin-1 BLBQ) 的模型來討論，並應用了SU(2)對稱性和周期邊界條件，接著設計一套方法去測量鐵磁性糾纏的中心電荷(central charge)大小。在這裡我們主要用的演算法，是應用了Steven R. White 創造出的密度矩陣重整化群(Density MatrixRenormalization Group)，這項發明可以解決希爾伯特空間(Hilbert space),隨指數成長的問題。此外，我們也使用了由Prof. Ian Peter McCulloch 所發展的Matrix Product Toolkit程式，來進行我們研究。

This thesis aims to explore the behavior of the ferromagnetic entanglement. Entanglement is an essential concept in quantum information science and the connection to many-body physics is recently enshrined. In particular the system of one-dimensional quantum critical chain and the corresponding conformal field theory (CFT) in the continuum limit. At first, we illustrate the idea by using spin-1 bi-linear bi-quadratic (BLBQ) model with SU(2) symmetry with periodic boundary condition. Then we design an approach to measure the effective central charge and the crossover length associated with the ferromagnetic entanglement. We applied the density matrix renormalization group (DMRG) method to obtain the ground state wave function and the entanglement. DMRG is invented by Prof. Steven R. White. It can address the problem of the exponential growth of the system size in the Hilbert space for the case of one-dimensional system. Finally we use the package called ``Matrix Product Toolkit’’ which was developed by Prof. Ian P. McCulloch to carry out the calculation.

1 Introduction 6
1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Entropy 9
2.1 Entanglement Entropy . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 R´enyi entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3 Symmetry 15
3.1 U(1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 SU(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3 SU(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4 Algorithm 22
4.1 iDMRG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.2 DMRG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5 Spin1 BLBQ 29
5.1 finite size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
5.2 infinite size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
6 Summary 59
A Singular value decomposition (SVD) 62
A.1 Right-canonical matrix product state . . . . . . . . . . . . . . . 62
A.2 Left-canonical matrix product state . . . . . . . . . . . . . . . 64
Reference 65

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