本論文提出了在張量網路計算中可以透過有限系統的標度律分析，
使得我們可以得到系統發生量子相變的時候的臨界指數。在文中我們
以 AKLT 模型為例，在二維的方晶格中藉由將波函數變形我們可以調
整系統至臨界點。此相變為 AKLT-鐵磁相變。藉由計算臨界指數我們
得出此相變屬於二維易辛普適類 (2d Ising universality class)。
接下來我們討論考慮邊界效應的時候其邊界不尋常的表現。我們計算邊界
的異常維度 (anomalous dimension) 並且發現其數值與塊 (bulk) 不同，
但邊界兩方向與塊的異常維度能滿足特定的標度律法則。最後我們探討在
其他晶格結構下的臨界指數，除了方晶格以外我們探討了蜂窩狀晶
格的臨界指數，並且我們推測在 AKLT-鐵磁相變時兩者屬於同一類相
變。

In this thesis we study the finite size scaling analysis on 2d AKLT model.
With the help of tensor network technique, we can deform the AKLT state and the system would undergo the quantum phase transition. From previous results we have known that this system have an AKLT-Ferro-magnetic(FM) transition. In our analysis, we further recognize that this transition belongs 2d Ising universality class.
Next, we study the surface critical behavior of this system. We calculate the anomalous dimension η on both parallel and
perpendicular directions,which we call it η∥ and η⊥ respectively. Although η∥,η⊥ and η at bulk cases are all different. We check indeed they satisfy certain scaling rule.
Finally we study the different lattice structure. Besides square lattice, we study the honeycomb lattice and calculate the critical exponents. From our results, we recognize both systems belong to the same universality.

Contents
摘要 iii
Abstract v
誌謝 vii
1 Introduction 1
2 Model and methods 9
3 Results 21
4 Conclusion and future work 33
Bibliography 35

[1] Nishimori Hidetoshi,Ortiz Gerardo, “Elements of Phase Transitions and Critical Phenomena”, Oxford University Press, (2011)
[2] E. S. Raja Gopal, Critical Opalescence, Resonance, vol. 5, no. 4, pp. 37-45, (2000)
[3] Subir Sachdev, “Quantum Phase Transitions”, Cambridge University Press, (2011)
[4] G. G. Batrouni,R. T. Scalettar, ““ Quantum phase transitions” in Ultracold Gases and Quantum Information”, Oxford University Press, (2011)
[5] D. Bitko and T. F. Rosenbaum, Quantum Critical Behavior for a Model Magnet, Physical Review Letters, vol. 77, no. 5, pp. 940-943, (1996)
[6] R. Coldea,D. A. Tennant,E. M. Wheeler,E. Wawrzynska,D. Prabhakaran, M. Telling,K.Habicht,P. Smeibidl, K. Kiefer, Quantum Criticality in an Ising Chain: Experimental Evidence for Emergent E8 Symmetry, Science, vol. 327, no. 5962, pp. 177-180, (2010)
[7] P. Coleman, “Introduction to Many-Body Physics”, Cambridge University Press,(2015)
[8] Rom´an Or´us, A Practical Introduction to Tensor Networks:Matrix Product States and Projected Entangled Pair States, Annals of Physics, vol. 349, pp 117-158, (2014)
[9] Henk W. J. Blo¨te,Youjin Deng, Cluster Monte Carlo simulation of the transverse Ising model, Physical Review E, vol. 66, 066110, (2002)
[10] J. Jordan,R. Orús,G. Vidal,F. Verstraete,J. I. Cirac, Classical simulation of infinitesize quantum lattice systems in two spatial dimensions, Physical Review Letters, vol. 101, 250602, (2008)
[11] Ian Affleck,Tom Kennedy,Elliott H.Lieb,Hal Tasaki, Valence Bond Ground States in Isotropic Quantum Antiferromagnets, Communications in Mathematical Physics, vol. 115, no. 3, pp. 477-528, (1988)
[12] Frank Pollmann, “Symmetry Protected Topological Phases in One-Dimensional Systems”, Oxford University Press, (2015)
[13] Frank Pollmann,Erez Berg, Ari M. Turner, and Masaki Oshikawa, Symmetry protection of topological phases in one-dimensional quantum spin systems, Physical Review B, vol. 85, 075125, (2012)
[14] Chung-Yu Mou, “Theory of angular momentum and rotations”, Available: http://www.phys.nthu.edu.tw/ mou/teach.html,
[15] Paul Fendley, “Basic quantum statistical mechanics of spin systems.” Manuscript in preparation., Available: http://users.ox.ac.uk/ phys1116/book.html,
[16] Ulrich Schollw¨ock, The density-matrix renormalization group in the age of matrix product states, Annals of Physics , vol. 326, no. 1, (2010)
[17] Nikko Pomata,Ching-Yu Huang, and Tzu-Chieh Wei, Phase transitions of a 2D deformed-AKLT model, Physical Review B, vol. 98, 014432, (2018)
[18] Ching-Yu Huang, Maximilian Anton Wagner, and Tzu-Chieh Wei, Emergence of XY-like phase in deformed spin- 3/2 AKLT systems, Physical Review B, vol. 94, 165130, (2016)
[19] Z. Y. Xie,J. Chen,M. P. Qin,J. W. Zhu,L. P. Yang,and T. Xiang, Coarse-graining renormalization by higher-order singular value decomposition, Physical Review B, vol. 86, 045139, (2012)
[20] Michael Levin,Cody P. Nave, Tensor renormalization group approach to 2D classical lattice models, Physical Review Letters, vol. 99, 120601, (2007)
[21] Anders W. Sandvik, Computational Studies of Quantum Spin Systems, AIP Conf. Proc., vol. 1297, no. 1, pp. 135-338, (2010)
[22] Satoshi Morita, Naoki Kawashima, Calculation of higher-order moments by higherorder tensor renormalization group, Computer Physics Communications, vol. 236,pp. 65-71, (2019)
[23] Zhang Long,Wang Fa, Unconventional Surface Critical Behavior Induced by a Quantum Phase Transition from the Two-Dimensional Affleck-Kennedy-Lieb-Tasaki Phase to a Néel-Ordered Phase, Physical Review Letters, vol. 118, 087201,(2017)
[24] Tomita Yusuke,Okabe Yutaka, Finite-size scaling of correlation ratio and generalized scheme for the probability-changing cluster algorithm, Physical Review B, vol. 66, 180401, (2002)
[25] John L.Cardy, Conformal invariance and surface critical behavior, Nuclear Physics B, vol. 240, no. 4, pp. 514-532, (1984)
[26] F. Verstraete, J. I. Cirac, Renormalization algorithms for Quantum-Many Body Systems in two and higher dimensions, https://arxiv.org/pdf/cond-mat/0407066.pdf,(2004)