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Critical exponents from power spectra

K. B. Lauritsen, Hans C. Fogedby

Journal of Statistical PhysicsJul 1, 199325 citations

A novel method of calculating amplitude ratios by series expansions

Yigal Meir

The author introduces a new method of calculating critical amplitude ratios using series, which is both simple and powerful. This method, which gives estimates for the amplitude ratios that are neither biased by the values of the critical points nor by the critical exponents, is applied to several models. It is shown that this method produces results where no reliable estimates from series expansion exist. In particular one finds 0.025+or-0.001 for AT'/B 2 for the 3D Ising model and 220+or-10 for C + /C - for the two-dimensional percolation model in agreement with, and with more accuracy than, values obtained by other methods.The author introduces a new method of calculating critical amplitude ratios using series, which is both simple and powerful. This method, which gives estimates for the amplitude ratios that are neither biased by the values of the critical points nor by the critical exponents, is applied to several models. It is shown that this method produces results where no reliable estimates from series expansion exist. In particular one finds 0.025+or-0.001 for AT'/B 2 for the 3D Ising model and 220+or-10 for C + /C - for the two-dimensional percolation model in agreement with, and with more accuracy than, values obtained by other methods.

Journal of Physics A Mathematical and GeneralApr 21, 198730 citations

Renormalization and Series Expansion: Techniques to Study Criticality

IMPERIAL COLLEGE PRESS eBooksNov 1, 20100 citations

METHODS OF COMPUTER SIMULATIONS OF PHASE TRANSITIONS IN THE ASHKIN-TELLER MODEL

Grzegorz Musiał, L. Dębski, G. Kamieniarz +1 more

The complementary Monte Carlo and series expansions methods of computer simulations have been described to investigate the critical behaviour of the Ashkin-Teller model in three dimensions.In the first method the invariance of the ratio of the square of the second moment of the order parameter to its fourth moment in the critical region has been exploited and some critical points on the phase boundaries have been calculated in the regions where the continuous transitions are expected.The continuity of the order parameter on the critical lines is verified by a finite size scaling analysis.Large-scale simulations have been performed on SGI Power Challenge XL and L supercomputers using the 64-bit random number generator.The numerically generated series expansions method is described for which the effective algorithm for generation of graphs starting from polygons and based on collapsing the unlinked vertices, is introduced.The new feature of our algorithm is that for each graph we introduce new links between unlinked vertices and we decorate bonds with new vertices, so that more complex graphs in an early stage of the graph generation procedure are obtained.The resulting set of graphs enables the application of the series expansions method and achievement of the precision of allocation of points on the phase diagram comparable to the precision of the Monte Carlo method, i.e. at least 3 decimal digits.The complementary Monte Carlo and series expansions methods of computer simulations have been described to investigate the critical behaviour of the Ashkin-Teller model in three dimensions.In the first method the invariance of the ratio of the square of the second moment of the order parameter to its fourth moment in the critical region has been exploited and some critical points on the phase boundaries have been calculated in the regions where the continuous transitions are expected.The continuity of the order parameter on the critical lines is verified by a finite size scaling analysis.Large-scale simulations have been performed on SGI Power Challenge XL and L supercomputers using the 64-bit random number generator.The numerically generated series expansions method is described for which the effective algorithm for generation of graphs starting from polygons and based on collapsing the unlinked vertices, is introduced.The new feature of our algorithm is that for each graph we introduce new links between unlinked vertices and we decorate bonds with new vertices, so that more complex graphs in an early stage of the graph generation procedure are obtained.The resulting set of graphs enables the application of the series expansions method and achievement of the precision of allocation of points on the phase diagram comparable to the precision of the Monte Carlo method, i.e. at least 3 decimal digits.

Computational Methods in Science and TechnologyJan 1, 20010 citations

Accurate critical exponents from the ε-expansion

J. C. Le Guillou, Jean Zinn‐Justin

Current physics sources and commentsJan 1, 199030 citations

Method of analysis of critical-point singularities from power-series expansions

Gustavo A. Arteca, Francisco Marcelo Fernández, Eduardo Alberto Castro

A simple method is proposed for determining characteristic parameters of critical-point singularities from power-series expansions. It applies to a broad class of functions of physical interest and provides acceptable, simultaneous approximations to the critical point, exponent, and amplitude. Several examples of low- and high-temperature series expansions for thermodynamic properties of spin-(1/2) Ising models are studied, and quite accurate results are obtained for both close- and loose-packed arrays.A simple method is proposed for determining characteristic parameters of critical-point singularities from power-series expansions. It applies to a broad class of functions of physical interest and provides acceptable, simultaneous approximations to the critical point, exponent, and amplitude. Several examples of low- and high-temperature series expansions for thermodynamic properties of spin-(1/2) Ising models are studied, and quite accurate results are obtained for both close- and loose-packed arrays.

Physical Review AFeb 1, 19867 citations

Shifted ratio method for calculation of critical parameters from power series expansions

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