多體系統的研究對於了解材料的性質很重要，但經常有希爾伯特空間(Hilbert space)隨著系統的大小呈指數的成長的困難，特別是那些展現奇特的行為的強關聯系統。然而，在過去的二十年間已經有許多解決多體問題的演算法被發展出來，且這些演算法並不會受限於系統大小的成長。由Steven R. White 所開發的密度矩陣重整化群(Density Matrix Renormalization Group)是最重要的一個，而這個方法也是我們在這個論文裡主要討論的演算法。除此之外，我們在這個論文裡還要討論有關一維多體系統的纏結(entanglement)性質，這在最近數年間吸引了許多的注意。

The study of many-body systems is important for the properties of material and often has the difficulty in the exponential growth of Hilbert space with system size, especially for those strongly correlated systems which exhibit exotic behavior. However, in the past two decades have developed some useful algorithms of tackling many-body problems, which is not restricted by the growth of system size. The Density Matrix Renormalization Group (DMRG) invented by Steven R. White is the significant one and it is the major algorithm we discuss in this thesis. In addition, this thesis will show the entanglement properties of many-body systems in one dimension which attracts a lot of attention in recent years.

1. Introduction 1
1.1. The simulation of strongly correlated systems . . . . . . . . . . . . . . . 1
1.2. Entanglement in quantum systems . . . . . . . . . . . . . . . . . . . . . 2
1.3. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2. The entanglement measure 4
2.1. Entanglement of pure state . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2. entanglement entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3. The numerical methods 8
3.1. Matrix-product representation . . . . . . . . . . . . . . . . . . . . . . . . 8
3.1.1. Matrix-product diagram . . . . . . . . . . . . . . . . . . . . . . . 8
3.1.2. Matrix-product state . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.1.3. Matrix-product operator . . . . . . . . . . . . . . . . . . . . . . . 16
3.1.4. The operations of MPS and MPO . . . . . . . . . . . . . . . . . . 19
3.2. Density Matrix Renormalization Group . . . . . . . . . . . . . . . . . . . 20
3.2.1. Infinite-size algorithm (iDMRG) . . . . . . . . . . . . . . . . . . . 22
3.2.2. Finite-size algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3. Infinite Time-Evolving Block Decimation . . . . . . . . . . . . . . . . . . 26
3.3.1. The power method in ground state search . . . . . . . . . . . . . 26
3.3.2. iTEBD algorithm in ground state search . . . . . . . . . . . . . . 27
4. Examples of numerical simulation 32
4.1. The Heisenberg spin chain . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.1.1. The simulation and its result . . . . . . . . . . . . . . . . . . . . 33
4.2. Ising model with transverse field . . . . . . . . . . . . . . . . . . . . . . . 35
4.2.1. The simulation and its result . . . . . . . . . . . . . . . . . . . . 37
5. The ferromagnetic entanglement 40
5.1. Entanglement entropy and conformal field theory . . . . . . . . . . . . . 40
5.2. Entanglemet scaling of ferromagnetic system . . . . . . . . . . . . . . . . 41
5.2.1. The XXZ spin chain . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.2.2. The simulation results . . . . . . . . . . . . . . . . . . . . . . . . 43
5.2.3. Summary and discussion . . . . . . . . . . . . . . . . . . . . . . . 47
A. Useful Mathematical Tools 48
A.1. Singular Value Decomposition . . . . . . . . . . . . . . . . . . . . . . . . 48
A.2. Suzuki-Trotter expansion . . . . . . . . . . . . . . . . . . . . . . . . . . 49
B. Matrix Product Toolkit 51
B.1. Finite-size calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
B.2. Infinite-size calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
Bibliography 54

[1] J. Quintanilla and C. Hooley. Phys. World, 22:32, (2009).
[2] H. Q. Lin. Phys. Rev. B, 42:6561, (1990).
[3] J M Zhang and R X Dong. Eur. J. Phys., 31:591, (2010).
[4] Steven R. White. Phys. Rev. Lett., 69:2863, (1992).
[5] Steven R. White. Phys. Rev. B, 48:10345, (1993).
[6] Guifré Vidal. Phys. Rev. Lett., 91:147902, (2003).
[7] Guifré Vidal. Phys. Rev. Lett., 93:040502, (2004).
[8] Guifré Vidal. Phys. Rev. Lett., 98:070201, (2007).
[9] Luigi Amico, Rosario Fazio, Andreas Osterloh, and Vlatko Vedral. Rev. Mod. Phys., 80:517, (2008).
[10] Pasquale Calabrese and John Cardy. J. Stat. Mech.: Theory Exp., 2004:P06002, (2004).
[11] Artur Ekert and Peter L. Knight. Am. J. Phys., 63:415, (1995).
[12] J. Eisert, M. Cramer, and M. B. Plenio. Rev. Mod. Phys., 82:277, (2010).
[13] M B Hastings. J. Stat. Mech.: Theory Exp., 2007:P08024, (2007).
[14] Ulrich Schollwöck. Ann. Phys., 326:96, (2011).
[15] Stellan Östlund and Stefan Rommer. Phys. Rev. Lett., 75:3537, (1995).
[16] Stefan Rommer and Stellan Östlund. Phys. Rev. B, 55:2164, (1997).
[17] F. Verstraete, J. J. García-Ripoll, and J. I. Cirac. Phys. Rev. Lett., 93:207204, (2004).
[18] I. P. McCulloch. J. Stat. Mech.: Theory Exp., 2007:P10014, (2007).
[19] Gregory M. Crosswhite and Dave Bacon. Phys. Rev. A, 78:012356, (2008).
[20] Kenneth G. Wilson. Rev. Mod. Phys., 47:773, (1975).
[21] U. Schollwöck. Rev. Mod. Phys., 77:259, (2005).
[22] I. P. McCulloch. arXiv, 0804.2509.
[23] Steven R. White and David A. Huse. Phys. Rev. B, 48:3844, (1993).
[24] Pierre Pfeuty. Ann. Phys., 57:79, (1970).
[25] Mark Summerfield. Programming in Python 3: A Complete Introduction to the Python Language. Addison-Wesley Professional, 2 edition, (2009).
[26] Jonas A. Kjäll, Michael P. Zaletel, Roger S. K. Mong, Jens H. Bardarson, and Frank Pollmann. Phys. Rev. B, 87:235106, (2013).
[27] Vladislav Popkov and Mario Salerno. Phys. Rev. A, 71:012301, (2005).
[28] Olalla A. Castro-Alvaredo and Benjamin Doyon. Phys. Rev. Lett., 108:120401, (2012).
[29] Pochung Chen, Zhi long Xue, I P McCulloch, Ming-Chiang Chung, Miguel Cazalilla, and S-K Yip. J. Stat. Mech.: Theory Exp., 2013:P10007, (2013).
[30] Gilbert Strang. Introduction to Linear Algebra. Wellesley Cambridge Press, 4 edition, (2009).
[31] Masuo Suzuki. Commun. Math. Phys., 51:183, (1976).
[32] Naomichi Hatano and Masuo Suzuki. Lect. Notes Phys., 679:37, (2005).