古典蒙地卡羅對三維XY模型非線性與異向性效應之研究

吳柏寬

doi:10.6342/NTU.2014.03010

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古典蒙地卡羅對三維XY模型非線性與異向性效應之研究

Classical Monte Carlo Studies on 3D XY Models with Effects of Nonlinearity and Anisotropy

吳柏寬 , Masters　　Advisor：高英哲　　

英文 DOI： 10.6342/NTU.2014.03010

蒙地卡羅模擬； XY模型；圖型處理器；易辛模型；有限尺度標度變換；異向性； Monte Carlo ； XY model ； GPU ； CUDA ； Ising Model ； Finite Size Scaling ； Anisotropy

Abstract │ Reference (32) │ Altmetrics

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In statistical and condensed matter physics, the phase transition and the critical behavior are very important topics. To study on them, the lattice models play important roles. For the theory of universality, the behaviors of models correspond to realistic physical systems. Therefore, the studies of theoretic models are keys to understand the critical behaviors.
The Monte Carlo simulation is a widely-used way to study the phase transitions of lattice models. With the stochastic process, we can sample in the space of states and get the approximations of the thermal observables. According to the finite-size scaling, from the finite-size results, the critical exponents can be extracted and the properties of the transition can be studied.
However, in finite-size cases, there are correction terms cause by the irrelevant fields, so them cause the systematic errors of exponents. To simulate the system with larger sizes more quickly, in this thesis, we parallelize the procedures of Monte Carlo simulations on GPUs.
With the GPU Monte Carlo simulations, we study the simple XY model and the generalized cases with nonlinear interactions between spins. The simple XY model is in the same universality class with the  lambda-transition of Helium-4, and the Zq anisotropy is dangerously irrelevant for q>=4. In the generalized XY models, the behavior approaches the Potts models as the potential of interactions approaching the delta function. In some region of the parameters, the anisotropy is significantly relevant. The transitions may even become first-order as anisotropy increases.

Reference ( 32 ) 〈TOP〉

[1] K. Huang, Statistical Mechanics (Wiley, 1987).
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[2] J. Cardy, Scaling and Renormalization in Statistical Physics (Cambridge University Press, 1996).
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[3] B. Linder, in Thermodynamics and Introductory Statistical Mechanics (John Wiley & Sons, Inc., 2004) pp. 119–126.
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[4] D. P. Landau and K. Binder, A Guide to Monte Carlo Simulations in Statistical Physics (Cambridge University Press, 2005).
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[6] H. G. Katzgraber, arXiv preprint arXiv:0905.1629 (2009).
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