Lattice Monte Carlo studies of quantum critical phenomena in graphene

Akai Kurbanovich Murtazaev

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.

Jong Yeon Lee, Johannes Hofmann, Eslam Khalaf +2 more

G. Baskaran

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.

Seizo Watarai, Akira Nakanishi

Monte Carlo simulations are performed on the simple cubic Ising lattice with both the dipolar and the nearest neighbor interactions. Applying the finite size scaling method, cross over effect of critical phenomena, which has been predicted by the renormalization group theory, can be observed from the Monte Carlo data.