Monte Carlo studies of critical phenomena in spin lattice systems

Lothar Schülke, Bo Zheng

Chin‐Kun Hu

Analytic calculations, Monte Carlo (MC) simulations [N. Metropolis and S. Ulam, J. Am. Stat. Asso 44, 335 (1949)] and renormalization group (RG) theory [K. G. Wilson, Phys. Rev. B 4, 3174 and 3184 (1971)] are important methods of statistical physics to study phase transitions and critical phenomena. This paper gives a brief historical review on analytic, MC, and RG approaches to some lattice critical systems in memory of late Professors Shang-keng Ma (1940-1983) and Yu-Ming Shih (1942-2005), who played some key roles in early developments of statistical physics of critical phenomena in Taiwan. The paper first introduces some developments in the study of critical phenomena from late 19th century to early 70s in the 20th century when renormalization group theory was formulated. The paper then reviews some topics on analytic, Monte Carlo, and RG approach to some lattice models, including variational RG approach to the Ising model, slow relaxation of a spin glass model at low temperatures, iterative method for quantum spin models, percolation theory of critical phenomena of Ising-like models and hard-core particle model, histogram Monte Carlo method and histogram Monte Carlo RG method, universal finite-size scaling functions of lattice models, critical slowing down of the Ising model, finite-size corrections for critical lattice models, and self-organized critical systems. Such topics are closely related to the author's own research experience. Finally, the paper outlines some very recent developments, summarizes main results of this paper, and points out some interesting problems for further studies, e.g. why a biological system can maintain in a non-equilibrium state for a very long time.

N. De La Espriella, J. Madera Yancez, M. S. Páez Mesa

"We used a Monte Carlo simulation to analize the magnetic behavior of Ising model of mixed spins S A i = ß3=2 ; ß1=2 and æ B j = ß 5 = 2 ; ß 3 = 2 ; ß 1 = 2 , on a square lattice. Were studied the possible critical phenomena that may emerge in the region around the multiphase point (D= j J 1 j = ° 3 ;J 2 = j J 1 j = 1) and the dependence of the phase diagrams with the intensities of the anisotropy field of single ion (D= j J 1 j ) and the ferromagnetic coupling of exchange spin S A i (J 2 = j J 1 j). The system displays first order phase transitions in a certain range of the parameters of the Hamiltonian, which depend on D= j J 1 j and j J 2 = j J 1 j . In the plane (D= j J 1 j ;k B T= j J 1 j), the de- crease of j D= j J 1 jj , implies that the critical temperature, T c , increases and the first order transition temperature, T t , decreases. In the plane (J2 = j J 1 j ;k B T= j J 1 j ), T c increases with the increasing of J 2 = j J 1 j , while that T t decreases."

Wolfhard Janke E

Ibragimkhan K. Kamilov, Akai Kurbanovich Murtazaev, Kh. K. Aliev

The current state of Monte Carlo work on phase transitions and critical phenomena is reviewed. Both classical and quantum Monte Carlo results are discussed with emphasis on statistical lattice-model studies involving the highly accurate calculation of critical exponents. It is shown that finite-size scaling is an effective method whereby not only simple lattice models but also those of complex practical magnetic materials can be treated. For models allowing crossover phenomena, it is shown that both finite-size scaling theory and conventional power-law functions must be used in carrying out a Monte Carlo analysis.

Gregory A. Kohring