本論文使用蒙地卡羅模擬對三維二聚體化自旋 1/2 反鐵磁海森堡模型做研究計算，其中使用非常有效率的隨機數列展開演算法，並且從適當的物理量和分析方法來計算倪耳溫度 $T_N$、交錯磁化密度 $M_s$ 和 $T^{\star}$。在不同的非無序的三維二聚體化自旋 1/2 海森堡模型中，先前文獻上的理論計算發現了 $T_N$ 和基態的 $M_s$ 存在著 3 個普適性比尺關係，有些也和 $\textrm{TlCuCl}_3$ 的實驗結果相符。這篇論文研究已發現的普適性比尺關係在無序模型上是否成立，並且考慮其他不同的非無序二聚體化海森堡模型。我們的計算結果不但確認普適性比尺關係在無序系統依然成立，還發現其中 2 個普適性比尺關係可以根據晶格點上的自旋與周圍自旋有較強自旋耦合的總數來做分類。

In this thesis, we use quantum Monte Carlo method to study three-dimensional (3D) spin-1/2 antiferromagnetic Heisenberg models. By employing every efficient algorithm, namely the stochastic series expansion (SSE), we calculate the Néel temperature $T_N$, the staggered magnetization density $M_s$, the spinwave velocity $c$, and $T^{\star}$ of these systems. It is established theoretically that there are three universal scaling relations between $T_N$ and $M_s$ for 3D clean spin models. Particularly, some of the predictions are consistent with the experimental results. Motivated by this, we have simulated 3D quantum spin models with certain kinds of quenched disorder and have found that these three universal scaling relations are valid for disordered systems. Finally, by simulating several clean 3D models, we also show that two of these scaling relations can be classified by the number of strong antiferromagnetic couplings touching a spin.

1. Greven, M., Birgeneau, R. J., Endoh, Y., Kastner, M. A., Matsuda, M., Shirane, G. Neutron scattering study of the two-dimensional spinS=1/2 square-lattice Heisenberg antiferromagnet Sr2CuO2Cl2. Zeitschrift für Physik B Condensed Matter, 96(4):465-477, Dec 1995.
2. Chakravarty, Sudip and Halperin, Bertrand I. and Nelson, David R. Low-temperature behavior of two-dimensional quantum antiferromagnets. Phys. Rev. Lett., 60:1057—1060, Mar 1988.
3. Haldane, F. D. M. O(3) Nonlinear σ Model and the Topological Distinction between Integer- and Half-Integer-Spin Antiferromagnets in Two Dimensions. Phys. Rev. Lett., 61:1029—1032, Aug 1988.
4. Reger, J. D. and Young, A. P. Monte Carlo simulations of the spin-1/2 Heisenberg antiferromagnet on a square lattice. Phys. Rev. B, 37:5978—5981, Apr 1988.
5. Chakravarty, Sudip and Halperin, Bertrand I. and Nelson, David R. Two-dimensional quantum Heisenberg antiferromagnet at low temperatures. Phys. Rev. B, 39:2344—2371, Feb 1989.
6. Singh, Rajiv R. P. Thermodynamic parameters of the T=0, spin-1/2 square-lattice Heisenberg antiferromagnet. Phys. Rev. B, 39:9760—9763, May 1989.
7. Singh, Rajiv R. P. Quantum renormalizations in the spin-1 Heisenberg antiferromagnet on the square lattice. Phys. Rev. B, 41:4873—4876, May 1990.
8. Zheng Weihong and Oitmaa, J. and Hamer, C. J. Square-lattice Heisenberg antiferromagnet at T=0. Phys. Rev. B, 43: 8321—8330, Apr 1991.
9. Chubukov, Andrey V. and Sachdev, Subir and Ye, Jinwu. Theory of two-dimensional quantum Heisenberg antiferromagnets with a nearly critical ground state. Phys. Rev. B, 49:11919—11961, May 1994.
10. Troyer, Matthias and Kontani, Hiroshi and Ueda, Kazuo. Phase Diagram of Depleted Heisenberg Model for CaVO9. Phys. Rev. Lett., 76:3822—3825, May 1996.
11. Beard, B. B. and Wiese, U.-J. Simulations of Discrete Quantum Systems in Continuous Euclidean Time. Phys. Rev. Lett., 77:5130—5133, Dec 1996.
12. Sandvik, Anders W. Finite-size scaling of the ground-state parameters of the two-dimensional Heisenberg model. Phys. Rev. B, 56:11678—11690, Nov 1997.
13. Rao, C. N. R. (Chintamani Nagesa Ramachandra) and Rao, K. J. Phase transitions in solids. New York : McGraw-Hill, 1997.
14. Newman, M and Barkema, G. Monte carlo methods in statistical physics. Oxford University Press: New York, USA, 1999.
15. Wenzel, Sandro and Janke, Wolfhard. Comprehensive quantum Monte Carlo study of the quantum critical points in planar dimerized/quadrumerized Heisenberg models. Phys. Rev. B, 79: 014410, Jan 2009.
16. Campostrini, Massimo and Hasenbusch, Martin and Pelissetto, Andrea and Rossi, Paolo and Vicari, Ettore. Critical exponents and equation of state of the three-dimensional Heisenberg universality class. Phys. Rev. B, 65:144520, Apr 2002.
17. Wang, Ling and Beach, K. S. D. and Sandvik, Anders W. High-precision finite-size scaling analysis of the quantum-critical point of S=1/2 Heisenberg antiferromagnetic bilayers. Phys. Rev. B, 73:014431, Jan 2006.
18. Albuquerque, A. Fabricio and Troyer, Matthias and Oitmaa, Jaan. Quantum phase transition in a Heisenberg antiferromagnet on a square lattice with strong plaquette interactions. Phys. Rev. B, 78:132402, Oct 2008.
19. Fritz, L. and Doretto, R. L. and Wessel, S. and Wenzel, S. and Burdin, S. and Vojta, M. Cubic interactions and quantum criticality in dimerized antiferromagnets. Phys. Rev. B, 83:174416, May 2011.
20. Wenzel, Sandro and Bogacz, Leszek and Janke, Wolfhard. Evidence for an Unconventional Universality Class from a Two-Dimensional Dimerized Quantum Heisenberg Model. Phys. Rev. Lett., 101:127202, Sep 2008.
21. F-J Jiang and U Gerber. Subtlety of determining the critical exponent ν of the spin-1/2 Heisenberg model with a spatially staggered anisotropy on the honeycomb lattice. Journal of Statistical Mechanics: Theory and Experiment, 2009(09):P09016, Sep 2009.
22. Jiang, F.-J. Monte Carlo simulations of an unconventional phase transition for a two-dimensional dimerized quantum Heisenberg model. Phys. Rev. B, 85:014414, Jan 2012.
23. Cavadini, N. and Heigold, G. and Henggeler, W. and Furrer, A. and Güdel, H.-U. and Krämer, K. and Mutka, H. Magnetic excitations in the quantum spin system TlCuCl3. Phys. Rev. B, 63: 172414, Apr 2001.
24. Rüegg, Ch and Cavadini, N and Furrer, A and Güdel, H-U and Krämer, Karl and Mutka, H and Wildes, A and Habicht, K and Vorderwisch, P. Bose--Einstein condensation of the triplet states in the magnetic insulator TlCuCl3. Nature, 423(6935):62, 2003.
25. Rüegg, Ch. and Normand, B. and Matsumoto, M. and Furrer, A. and McMorrow, D. F. and Krämer, K. W. and Güdel, H. -U. and Gvasaliya, S. N. and Mutka, H. and Boehm, M. Quantum Magnets under Pressure: Controlling Elementary Excitations in TlCuCl3. Phys. Rev. Lett., 100:205701, May 2008.
26. Matsumoto, Munehisa and Yasuda, Chitoshi and Todo, Synge and Takayama, Hajime. Ground-state phase diagram of quantum Heisenberg antiferromagnets on the anisotropic dimerized square lattice. Phys. Rev. B, 65:014407, Nov 2001.
27. Kulik, Y. and Sushkov, O. P. Width of the longitudinal magnon in the vicinity of the O(3) quantum critical point. Phys. Rev. B, 84:134418, Oct 2011.
28. Oitmaa, J. and Kulik, Y. and Sushkov, O. P. Universal finite-temperature properties of a three-dimensional quantum antiferromagnet in the vicinity of a quantum critical point. Phys. Rev. B, 85:144431, Apr 2012.
29. Jin, Songbo and Sandvik, Anders W. Universal Néel temperature in three-dimensional quantum antiferromagnets. Phys. Rev. B, 85:020409, Jan 2012.
30. Qin, Yan Qi and Normand, B. and Sandvik, Anders W. and Meng, Zi Yang. Multiplicative logarithmic corrections to quantum criticality in three-dimensional dimerized antiferromagnets. Phys. Rev. B, 92:214401, Dec 2015.
31. Scammell, H. D. and Sushkov, O. P. Nonequilibrium quantum mechanics: A ``hot quantum soup'' of paramagnons. Phys. Rev. B, 95:024420, Jan 2017.
32. Scammell, H. D. and Sushkov, O. P. Multiple universalities in order-disorder magnetic phase transitions. Phys. Rev. B, 95:094410, Mar 2017.
33. Kao, MT and Jiang, FJ. Investigation of a universal behavior between Néel temperature and staggered magnetization density for a three-dimensional quantum antiferromagnet. The European Physical Journal B, 86(10):419, 2013.
34. Merchant, P and Normand, B and Krämer, KW and Boehm, M and McMorrow, DF and Rüegg, Ch. Quantum and classical criticality in a dimerized quantum antiferromagnet. Nature physics, 10(5):373, 2014.
35. Tan, Deng-Ruei and Jiang, Fu-Jiun. Scaling relations of three-dimensional random-exchange quantum antiferromagnets. The European Physical Journal B, 88(11):289, Nov 2015.
36. Tan, D.-R. and Jiang, F.-J. Universal scaling of Néel temperature, staggered magnetization density, and spin-wave velocity of three-dimensional disordered and clean quantum antiferromagnets. Phys. Rev. B, 95:054435, Feb 2017.
37. Tan, D.-R. and Li, C.-D. and Jiang, F.-J. Classification for the universal scaling of Néel temperature and staggered magnetization density of three-dimensional dimerized spin-1/2 antiferromagnets. Phys. Rev. B, 97:094405, Mar 2018.
38. Sandvik. Stochastic Series Expansion algorithm and program for the S=1/2 Heisenberg antiferromagnet. http://physics.bu.edu/~sandvik/programs/ssebasic/ssebasic.html
39. Huang, Yi-Zhen and Xi, Bin and Chen, Xi and Li, Wei and Wang, Zheng-Chuan and Su, Gang. Quantum phase transition, universality, and scaling behaviors in the spin-1/2 Heisenberg model with ferromagnetic and antiferromagnetic competing interactions on a honeycomb lattice. Phys. Rev. E, 93:062110, Jun 2016.
40. Nohadani, Omid and Wessel, Stefan and Haas, Stephan. Quantum phase transitions in coupled dimer compounds. Phys. Rev. B, 72;024440, Jul 2005.
41. Sandvik, Anders W. Classical percolation transition in the diluted two-dimensional S=1/2 Heisenberg antiferromagnet. Phys. Rev. B, 66:024418, Jul 2002.
42. R. Kenna. Finite size scaling for O(N) φ4-theory at the upper critical dimension. Nuclear Physics B, 691(3):292 – 304, 2004.
43. Jiang, F.-J. Method of calculating the spin-wave velocity of spin-1/2 antiferromagnets with O(N) symmetry in a Monte Carlo simulation. Phys. Rev. B, 83:024419, Jan 2011.
44. Sen, Arnab and Suwa, Hidemaro and Sandvik, Anders W. Velocity of excitations in ordered, disordered, and critical antiferromagnets. Phys. Rev. B, 92:195145, Nov 2015.