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Physics Letters A

Critical-exponent renormalization and first-order transitions

Hans W. Capel, Jacques H. H. Perk, L.W.J. den Ouden

Published Jun 1, 1978• 9 citations
Publisher
DOI
10.1016/0375-9601(78)90388-2

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An improved Monte Carlo renormalization group method is established. Based on a new model, as well as the generalized Glauber dynamics, the first-order phase transition is studied in the Erdös–Rényi network, which is used to represent the complex ferromagnet. Both static and dynamic critical exponents can be evaluated simultaneously, and the static critical exponents are evaluated to be independent of the dimension of the system. Thus by using the hyperscaling relation νd* = 2 − α, the effective dimension d* of complex networks can be found.
Phase transitions of the J_{1}-J_{2} Ising model on a square lattice are studied using the higher-order tensor renormalization group (HOTRG) method. This system involves a competition between the ferromagnetic interaction J_{1} and antiferromagnetic interaction J_{2}, and in previous studies, weak first-order and second-order transitions were observed near the ratio g=J_{2}/|J_{1}|=1/2. It has also been suggested that the universality class of the second-order phase transition connected to the first-order transition line for g>1/2 belongs to the Ashkin-Teller class, which is characterized by a continuously varying critical exponent with g, as predicted by field-theoretical and other studies. Our results, based on the HOTRG calculations for significantly larger sizes, indicate that the region of the first-order transition is marginally narrower than that in previous studies. Furthermore, it is suggested that the region where the critical exponent changes does not necessarily coincide with the Ashkin-Teller region.