二维正方晶格上含空位O（2）自旋模型相变的蒙特卡罗研究
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摘要
本文应用Monte Carlo方法和有限尺寸标度理论研究正方品格上含空位的O(2)自旋模型的相图和可能存在的三临界性质。我们发现在低空位权重下,系统会发生KT相变;在高空位权重下,系统存在一级相变线。我们对之间区域可能存在的三临界点位置及临界性质进行了初步研究。本文结构如下:
第一章,介绍了有关的背景知识。首先介绍O(n)模型和O(n)圈模型,以及O(n)圈模型的临界性质和三临界性质;然后介绍我们研究的动机;最后引入含空位的O(2)自旋模型,并介绍了KT相变的一些性质。
第二章,介绍了我们采用的蒙特卡罗方法:用Wolff集团算法结合Metropolis算法实现对含空位的O(2)自旋模型的较高效率的MC模拟。
第三章,详细描述了我们对含空位的O(2)自旋模型的研究,包括对KT相变区域的研究,一级相变区域的研究,以及中间区域可能存在的三临界相变行为的研究,并提出了下一步工作的重点。
第四章,对本文进行总结。
In this thesis,we study the phase diagram and the probable tricriticality of a dilute O(2) spin model on the square lattice by means of Monte Carlo simulations and finite size scaling methods.We find out that the KT behaviors are associated with low vacancy probability,while the first order transition behaviors happen in high vacancy probability area.Then we fbcus on the mid-area.Our results suggest that a tricritical point exists.The main contents are as fbllows:
In chapter one,we review relevant theoretical background,including the mapping from the O(n) spin model onto the O(n) loop model,the criticality and tricriticality of the O(n) loop model,the XY model and KT transition.The motivation of our research is also introduced.
In chapter two,we describe the Monte Carlo algorithms applied in our simulations. We use a hybrid algorithm in which Metropolis sweeps alternate with Wolff cluster steps.The cluster algorithm acts on the non-zero spin only.
In chapter three,we present the results of our numeric simulations of the dulite O(2) spin model.We find a KT transition line at low vacancy probability and a first order transition line at high vacancy probability.We preliminarily estimate the tricritical point which is located at v_(tri)=0.915±0.005,T_(tri)=0.46189±0.00001 and caculate the critical exponents.Furthermore we put forward the next work plan.
In chapter fbur,we conclude this thesis.
第一章,介绍了有关的背景知识。首先介绍O(n)模型和O(n)圈模型,以及O(n)圈模型的临界性质和三临界性质;然后介绍我们研究的动机;最后引入含空位的O(2)自旋模型,并介绍了KT相变的一些性质。
第二章,介绍了我们采用的蒙特卡罗方法:用Wolff集团算法结合Metropolis算法实现对含空位的O(2)自旋模型的较高效率的MC模拟。
第三章,详细描述了我们对含空位的O(2)自旋模型的研究,包括对KT相变区域的研究,一级相变区域的研究,以及中间区域可能存在的三临界相变行为的研究,并提出了下一步工作的重点。
第四章,对本文进行总结。
In this thesis,we study the phase diagram and the probable tricriticality of a dilute O(2) spin model on the square lattice by means of Monte Carlo simulations and finite size scaling methods.We find out that the KT behaviors are associated with low vacancy probability,while the first order transition behaviors happen in high vacancy probability area.Then we fbcus on the mid-area.Our results suggest that a tricritical point exists.The main contents are as fbllows:
In chapter one,we review relevant theoretical background,including the mapping from the O(n) spin model onto the O(n) loop model,the criticality and tricriticality of the O(n) loop model,the XY model and KT transition.The motivation of our research is also introduced.
In chapter two,we describe the Monte Carlo algorithms applied in our simulations. We use a hybrid algorithm in which Metropolis sweeps alternate with Wolff cluster steps.The cluster algorithm acts on the non-zero spin only.
In chapter three,we present the results of our numeric simulations of the dulite O(2) spin model.We find a KT transition line at low vacancy probability and a first order transition line at high vacancy probability.We preliminarily estimate the tricritical point which is located at v_(tri)=0.915±0.005,T_(tri)=0.46189±0.00001 and caculate the critical exponents.Furthermore we put forward the next work plan.
In chapter fbur,we conclude this thesis.
引文
[1] H. E. Stanley, Phys. Rev. Lett. 20, 589 (1968).
[2] B. Nienhuis, Phys. Rev. Lett. 49, 1062 (1982).
[3] B. Duplantier, H. Saleur, Phys. Rev. Lett. 59, 539 (1987).
[4] H. W. J. Blote, B. Nienhuis, J. Phys. A: Math. Gen. 22, 1415 (1989).
[5] W.-A. Guo, H. W. J. Blote, Y.-Y. Liu, Commun. Theor. Phys. 41, 911 (2004).
[6] B. Nienhuis, S. Warnaar, H. W. J. Blote, J. Phys. A: Math. Gen. 26, 477 (1993).
[7] B. Nienhuis, A. N. Berker, Eberhard K. Riedel, M. Schick, Phys. Rev. Lett. 43, 737 (1979).
[8] B. Nienhuis, Eberhard K. Riedel, M. Schick, J. Phys. A: Math. Gen. 13, L189 (1980).
[9] B. Nienhuis, J. Phys. A: Math. Gen. 15, L199 (1982).
[10] W.-A. Guo, B. Nienhuis, H. W. J. Blote, Phys. Rev. Lett. 96, 045704 (2006).
[11] J. M. Kosterlitz, D. J. Thouless, J. Phys. C: Solid State Phys. 5, L124 (1972).
[12] J. M. Kosterlitz, D. J. Thouless, J. Phys. C: Solid State Phys. 6, 1181 (1973).
[13] J. M. Kosterlitz, J. Phys. C: Solid State Phys. 7, 1046 (1974).
[14] K. Binder, D. W. Heermann, Monte Carlo Simulation in Statistical Physics (Springer, Berlin, Heidelberg, 2002).
[15] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Zeller, E. Zeller, J. Chem. Phys. 21, 1089 (1953).
[16] U. Wolff, Phys. Rev. Lett 62, 361 (1989).
[17] R. H. Swendsen, Jian-Sheng Wang, Phys. Rev. Lett 58, 86 (1987).
[18] H. W. J. Blote, Nonlocal Monte Carlo Methods (April 1998).
[19] R. Gupta, J. DeLapp, George G. Batrouni, Phys. Rev. Lett 61, 1996 (1988).
[20] M. S. S. Challa, D. P. Landau, Phys. Rev. B 33, 437 (1986).
[21] M. Hasenbusch, Phys. Rev. B 33, 437 (1986).
[22] Wei-Mou Zheng, Commun. Theor. Phys. 20(4) pp. 439-444 (1993).
[23] M. S. S. Challa, D. P. Landau, Phys. Rev. B 34, 1841(1986)
[24] E. Rastelli, S. Regina, A. Tassi, Phys. Rev. B 69, 174407 (2004)
[25] E. Rastelli, S. Regina, A. Tassi, Phys. Rev. B 70, 174447 (2004)
[26] C. Domb, J. L. Lebowitz, Phase Transitions and Critical Phenomena 9, 5 (1984)
[27] G. Kamieniarz, H. W. J. Blote, J. Phys. A: Math. Gen. 26, 201 (1993)
[28] R. B. Griffiths, Phys. Rev. B 7, 545 (1984)
[29] D. P. Landau, R. H. Swendsen, Phys. Rev. Letts 46, 1437 (1980)
[30] H. W. J. Blote, E. Luijten, J. R. Heringa, J. Phys. A: Math. Gen. 28, 6289 (1995)
[2] B. Nienhuis, Phys. Rev. Lett. 49, 1062 (1982).
[3] B. Duplantier, H. Saleur, Phys. Rev. Lett. 59, 539 (1987).
[4] H. W. J. Blote, B. Nienhuis, J. Phys. A: Math. Gen. 22, 1415 (1989).
[5] W.-A. Guo, H. W. J. Blote, Y.-Y. Liu, Commun. Theor. Phys. 41, 911 (2004).
[6] B. Nienhuis, S. Warnaar, H. W. J. Blote, J. Phys. A: Math. Gen. 26, 477 (1993).
[7] B. Nienhuis, A. N. Berker, Eberhard K. Riedel, M. Schick, Phys. Rev. Lett. 43, 737 (1979).
[8] B. Nienhuis, Eberhard K. Riedel, M. Schick, J. Phys. A: Math. Gen. 13, L189 (1980).
[9] B. Nienhuis, J. Phys. A: Math. Gen. 15, L199 (1982).
[10] W.-A. Guo, B. Nienhuis, H. W. J. Blote, Phys. Rev. Lett. 96, 045704 (2006).
[11] J. M. Kosterlitz, D. J. Thouless, J. Phys. C: Solid State Phys. 5, L124 (1972).
[12] J. M. Kosterlitz, D. J. Thouless, J. Phys. C: Solid State Phys. 6, 1181 (1973).
[13] J. M. Kosterlitz, J. Phys. C: Solid State Phys. 7, 1046 (1974).
[14] K. Binder, D. W. Heermann, Monte Carlo Simulation in Statistical Physics (Springer, Berlin, Heidelberg, 2002).
[15] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Zeller, E. Zeller, J. Chem. Phys. 21, 1089 (1953).
[16] U. Wolff, Phys. Rev. Lett 62, 361 (1989).
[17] R. H. Swendsen, Jian-Sheng Wang, Phys. Rev. Lett 58, 86 (1987).
[18] H. W. J. Blote, Nonlocal Monte Carlo Methods (April 1998).
[19] R. Gupta, J. DeLapp, George G. Batrouni, Phys. Rev. Lett 61, 1996 (1988).
[20] M. S. S. Challa, D. P. Landau, Phys. Rev. B 33, 437 (1986).
[21] M. Hasenbusch, Phys. Rev. B 33, 437 (1986).
[22] Wei-Mou Zheng, Commun. Theor. Phys. 20(4) pp. 439-444 (1993).
[23] M. S. S. Challa, D. P. Landau, Phys. Rev. B 34, 1841(1986)
[24] E. Rastelli, S. Regina, A. Tassi, Phys. Rev. B 69, 174407 (2004)
[25] E. Rastelli, S. Regina, A. Tassi, Phys. Rev. B 70, 174447 (2004)
[26] C. Domb, J. L. Lebowitz, Phase Transitions and Critical Phenomena 9, 5 (1984)
[27] G. Kamieniarz, H. W. J. Blote, J. Phys. A: Math. Gen. 26, 201 (1993)
[28] R. B. Griffiths, Phys. Rev. B 7, 545 (1984)
[29] D. P. Landau, R. H. Swendsen, Phys. Rev. Letts 46, 1437 (1980)
[30] H. W. J. Blote, E. Luijten, J. R. Heringa, J. Phys. A: Math. Gen. 28, 6289 (1995)