Series studies of critical exponents in continuous dimensions. [Ising-like model]

Using results we have previously derived for the high‐temperature susceptibility expansion of classical models as a closed‐form function of lattice dimension, we study the dimensional dependence of the critical exponent γ and critical temperature, as well as the correction‐to‐scaling exponent Δ1, for Ising‐like (n=1) models. The numerical results are obtained by extrapolation of 10‐th order series on loose‐packed hypercubical lattices, and 8‐th order series on close‐packed hypertriangular lattices. The critical exponent γ increases monotonically with decreasing dimension, d, for d<4, and apparently tends to infinity at d=1; while the critical temperature decreases monotonically and smoothly to zero at d=1. Detailed contact is made with the ε‐expansion estimates for critical exponents obtained in the context of renormalization group theory.Using results we have previously derived for the high‐temperature susceptibility expansion of classical models as a closed‐form function of lattice dimension, we study the dimensional dependence of the critical exponent γ and critical temperature, as well as the correction‐to‐scaling exponent Δ1, for Ising‐like (n=1) models. The numerical results are obtained by extrapolation of 10‐th order series on loose‐packed hypercubical lattices, and 8‐th order series on close‐packed hypertriangular lattices. The critical exponent γ increases monotonically with decreasing dimension, d, for d<4, and apparently tends to infinity at d=1; while the critical temperature decreases monotonically and smoothly to zero at d=1. Detailed contact is made with the ε‐expansion estimates for critical exponents obtained in the context of renormalization group theory.

AIP conference proceedingsJan 1, 19764 citations

Critical phenomena in a binary lattice gas. II. Series analysis

J. Anthony Gualtieri, John C. Wheeler

The critical behavior of a two-component lattice-gas model (the 1-2 model) in two and three dimensions is investigated by means of standard and nonstandard methods of series analysis on exact series expansions for the thermodynamic properties of the model. The model exhibits a critical line that passes from the Ising critical point at finite temperature to a critical point at zero temperature. In two dimensions the Ising values of the critical exponents found along the critical line and at zero temperature support the expectation that the model lies in the Ising universality class. In three dimensions Ising values of the critical exponents are found away from zero temperature, but near and at zero temperature the series analysis suggests a crossover to non-Ising values of the critical exponents. This would imply that the model belongs to the Ising universality class at nonzero temperatures and to a different universality class at zero temperature---a quite unexpected result. Various possible explanations for this unusual behavior of the series results are discussed.The critical behavior of a two-component lattice-gas model (the 1-2 model) in two and three dimensions is investigated by means of standard and nonstandard methods of series analysis on exact series expansions for the thermodynamic properties of the model. The model exhibits a critical line that passes from the Ising critical point at finite temperature to a critical point at zero temperature. In two dimensions the Ising values of the critical exponents found along the critical line and at zero temperature support the expectation that the model lies in the Ising universality class. In three dimensions Ising values of the critical exponents are found away from zero temperature, but near and at zero temperature the series analysis suggests a crossover to non-Ising values of the critical exponents. This would imply that the model belongs to the Ising universality class at nonzero temperatures and to a different universality class at zero temperature---a quite unexpected result. Various possible explanations for this unusual behavior of the series results are discussed.

Physical review. B, Condensed matterSep 1, 19792 citations

Critical exponents of random Ising-like systems in general dimensions

Yurij V. Holovatch, Mykola A. Shpot

Journal of Statistical PhysicsFeb 1, 199238 citations

Approximate renormalization-group calculations of critical exponents in continuous dimensions

Sheldon L. Katz, Michel Droz, J. D. Gunton

The Kadanoff lower-bound renormalization transformation is used to obtain approximate values of the critical exponents for the spin-1/2 Ising model on a simple hypercubic $d$ -dimensional lattice for all integral values of $z={2}^{d}$ between 3 and 16. The results for $\ensuremath{\gamma}, \ensuremath{\delta}$, and the thermal and magnetic correction to scaling exponents are compared with corresponding results obtained from the $\ensuremath{\epsilon}$ expansion and series calculations.The Kadanoff lower-bound renormalization transformation is used to obtain approximate values of the critical exponents for the spin-1/2 Ising model on a simple hypercubic $d$-dimensional lattice for all integral values of $z={2}^{d}$ between 3 and 16. The results for $\ensuremath{\gamma}, \ensuremath{\delta}$, and the thermal and magnetic correction to scaling exponents are compared with corresponding results obtained from the $\ensuremath{\epsilon}$ expansion and series calculations.

Physical review. B, Solid stateFeb 1, 197719 citations

Accurate critical exponents for Ising like systems in non-integer dimensions

J. C. Le Guillou, Jean Zinn‐Justin

Current physics sources and commentsJan 1, 199028 citations

The Kadanoff lower-bound renormalization transformation is used to obtain approximate values of the critical exponents for the spin-1/2 Ising model on a simple hypercubic

d

-dimensional lattice for all integral values of

z={2}^{d}

between 3 and 16. The results for $\ensuremath{\gamma}, \ensuremath{\delta}$, and the thermal and magnetic correction to scaling exponents are compared with corresponding results obtained from the $\ensuremath{\epsilon}$ expansion and series calculations.