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Journal of Physics A Mathematical and General

A generalized parametric equation of state describing the global behaviour of an Ising-like system

FERDINANDO DE PASQUALE, P. Tombesi

Published Mar 1, 1977• 6 citations
Publisher

Abstract

The authors report a microscopic derivation of the parametric representation of the equation of state for an Ising-like system by means of the skeleton expansion technique. Such a representation explicitly describes the crossover behaviour from the small correlation length region to the critical region. The Gaussian-Ising crossover is obtained at criticality. The derivation is based on differential relations that connect natural thermodynamical variables with the parameters defining the representation. Differentiability properties are checked in the small epsilon limit.

DOI
10.1088/0305-4470/10/3/012

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The authors report a microscopic derivation of the parametric representation of the equation of state for an Ising-like system by means of the skeleton expansion technique. Such a representation explicitly describes the crossover behaviour from the small correlation length region to the critical region. The Gaussian-Ising crossover is obtained at criticality. The derivation is based on differential relations that connect natural thermodynamical variables with the parameters defining the representation. Differentiability properties are checked in the small epsilon limit.
It is well known that the properties of scaling and universality exhibited by the polymer system have a very natural description in terms of the critical behaviour of an analogous magnetic system. This analogy is used to derive renormalisation-group equations directly for the polymer system. Solving these renormalisation-group equations a compact parametric description is obtained of the crossover from the Wilson-Fisher to gaussian fixed point displayed by the mean square size of a polymer (R 2 ) and the second A 2 and third A 3 virial coefficients correct to O( epsilon 2 ) in d=4- epsilon spatial dimensions.
We compute the 2 n -point coupling constants in the high-temperature phase of the two-dimensional Ising model by using transfer-matrix techniques. This provides the first few terms of the expansion of the effective potential (Helmholtz free energy) and of the equation of state in terms of the renormalized magnetization. By means of a suitable parametric representation, we determine an analytic extension of these expansions, providing the equation of state in the whole critical region in the t , h plane.
Equation of State and Thermodynamic Functions of the Ising-like Magnet at T>TcT>T_cT>Tc​
The 3D Ising-like system in the external field is described using the non-perturbative collective variables method. The universal as well as nonuniversal system characteristics are obtained within the framework of this approach. The calculations are carried out on the microscopic level starting from the Hamiltonian. They are valid in the whole h−Th-Th−T plane of the critical region. It is established, that the contributions related with wave vector values $\vk\to0$ exhibit the properties of the total system near the critical point. The behaviour of the susceptibility as function of the temperature in the presence of the field is investigated. The locations of the maximums susceptibility on the temperature scale for different values of the field are established.
The microscopic approach to calculating and investigating the three-dimensional (3D) Ising-like system thermodynamics above and below the phase transition temperature Tc is developed using the collective variables method.Mathematical description is performed within the framework of the higher non-Gaussian approximation ( 6 model) taking into account the corrections to scaling.A nonuniversal factor depending on microscopic parameters of the system is singled out in the expressions for leading critical amplitudes and correction-to-scaling amplitudes.Numerical estimates of the critical region size, phase transition temperature as well as the graphs of temperature dependences of entropy and other thermodynamic characteristics near Tc are given for various values of eective radius of the exponentially decreasing interaction potential.The variation of the free energy of the system at the phase transition point, average spin moment, and speci c heat with increasing ratio of the potential eective radius to the simple cubic lattice constant is traced.The results of calculations and their comparison with the other authors' data show that the 6 model provides a more adequate quantitative description of the critical behaviour of a 3D Ising-like system than the 4 model.
A microscopic approach to the investigation of the three-dimensional Ising-like system thermodynamics below the phase transition temperature T c is schematically presented within the framework of the higher non-Gaussian approximation (ρ 6 model) taking into account the corrections to scaling. A microscopic analogue of the Landau free energy is calculated. Explicit expressions for the thermodynamic characteristics are obtained as functions of the temperature and microscopic parameters of the system. The thermodynamic characteristics near T c and their amplitudes are given for various values of the effective radius of the exponentially decreasing interaction potential (including the values corresponding to the nearest-neighbour interaction, the interaction between the nearest and next-nearest neighbours, and that between the nearest, next-nearest, and third neighbours). The evolutions of the free energy of the system at the phase transition point, average spin moment, and specific heat with increasing ratio of the potential effective radius to the simple cubic lattice constant are plotted. The results of calculations and their comparison with other authors' data show that the ρ 6 model provides a better quantitative description of the critical behaviour of a three-dimensional Ising magnetic than the ρ 4 model.
High-temperature series are computed for a generalized three-dimensional Ising model with arbitrary potential. Three specific "improved" potentials (suppressing leading scaling corrections) are selected by Monte Carlo computation. Critical exponents are extracted from high-temperature series specialized to improved potentials, achieving high accuracy; our best estimates are gamma=1.2371(4), nu=0.630 02(23), alpha=0.1099(7), eta=0.0364(4), beta=0.326 48(18). By the same technique, the coefficients of the small-field expansion for the effective potential (Helmholtz free energy) are computed. These results are applied to the construction of parametric representations of the critical equation of state. A systematic approximation scheme, based on a global stationarity condition, is introduced (the lowest-order approximation reproduces the linear parametric model). This scheme is used for an accurate determination of universal ratios of amplitudes. A comparison with other theoretical and experimental determinations of universal quantities is presented.