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Springer proceedings in physics

Coherent-Anomaly Method in Two-Dimensional Antiferromagnetic Ising Models on Square and Triangular Lattices

Adam Lipowski, Masuo Suzuki

Published Jan 1, 1992• 0 citations
Publisher
DOI
10.1007/978-3-642-84821-6_37

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arXiv:cond-mat/9812341•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Mar 1, 1999•16 citations
The coherent-anomaly method (CAM) is applied to the kinetic Ising model. Dynamical cluster-mean-field approximations are formulated to obtain a series of the dynamical mean-field critical coefficients. The coherent-anomaly scaling relations are derived on the basis of the scaling form of the generating function in nonequilibrium systems to estimate the exponent Δ of the critical slowing down. The dynamical critical exponent is estimated as Δ⋍2.15(±0.02){\varDelta}{\backsimeq}2.15({\pm}0.02)Δ⋍2.15(±0.02) for the kinetic Ising model on the two-dimensional triangular lattice.
A systematic procedure of constructing a series of cluster-mean-field approximations is established to investigate the coherent anomalies of the mean-field critical coefficients \barχ( L , T c ( L )) in the two- and three-dimensional Ising models. The values of critical exponents as well as the critical temperatures are estimated by means of the new method (Coherent Anomaly Method) as T c * =2.230(0.038) J / k B , γ=1.785(0.116) for the two-dimensional square lattice and T c * =4.529(0.058) J / k B , γ=1.258(0.068) for the three-dimensional simple cubic lattice. The results suggest that the present approach to study critical phenomena is very promising.
Transfer-matrix scaling methods have been used to study critical properties of field-induced phase transitions of two distinct two-dimensional antiferromagnets with discrete-symmetry order parameters: triangular-lattice Ising systems (TIAF) and the square-lattice three-state Potts model (SPAF-3). Our main findings are summarized as follows. For TIAF, we have shown that the critical line leaves the zero-temperature, zero-field fixed point at a finite angle. Our best estimate of the slope at the origin is ${(dT}_{c}{/dH)}_{T=H=0}=4.74\ifmmode\pm\else\textpm\fi{}0.15.$ For SPAF-3 we provided evidence that the zero-field correlation length diverges as $\ensuremath{\xi}(\stackrel{\ensuremath{\rightarrow}}{T}0, H=0)\ensuremath{\simeq}\mathrm{exp}{(a/T}^{x}),$ with $x=1.08\ifmmode\pm\else\textpm\fi{}0.13,$ through analysis of the critical curve at $H\ensuremath{\ne}0$ plus crossover arguments. For SPAF-3 we have also ascertained that the conformal anomaly and decay-of-correlations exponent behave as (a) $H=0:c=1, \ensuremath{\eta}=1/3.$ (b) $H\ensuremath{\ne}0:c=1/2, \ensuremath{\eta}=1/4.$