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arXiv:1212.6139 · Condensed Matter Physics

Critical behaviour of a 3D Ising-like system in the ρ^6 model approximation: Role of the correction for the potential averaging

Ihor V. Pylyuk, Ulyak

Published Dec 1, 2012• 1 citation
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Abstract

The critical behaviour of systems belonging to the three-dimensional Ising universality class is studied theoretically using the collective variables (CV) method.The partition function of a one-component spin system is calculated by the integration over the layers of the CV phase space in the approximation of the non-Gaussian sextic distribution of order-parameter fluctuations (the ρ 6 model).A specific feature of the proposed calculation consists in making allowance for the dependence of the Fourier transform of the interaction potential on the wave vector.The inclusion of the correction for the potential averaging leads to a nonzero critical exponent of the correlation function η and the renormalization of the values of other critical exponents.The contributions from this correction to the recurrence relations for the ρ 6 model, fixed-point coordinates and elements of the renormalization-group linear transformation matrix are singled out.The expression for a small critical exponent η is obtained in a higher non-Gaussian approximation.

DOI
10.5488/cmp.15.43006

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The critical behaviour of systems belonging to the three-dimensional Ising universality class is studied theoretically using the collective variables (CV) method.The partition function of a one-component spin system is calculated by the integration over the layers of the CV phase space in the approximation of the non-Gaussian sextic distribution of order-parameter fluctuations (the ρ 6 model).A specific feature of the proposed calculation consists in making allowance for the dependence of the Fourier transform of the interaction potential on the wave vector.The inclusion of the correction for the potential averaging leads to a nonzero critical exponent of the correlation function η and the renormalization of the values of other critical exponents.The contributions from this correction to the recurrence relations for the ρ 6 model, fixed-point coordinates and elements of the renormalization-group linear transformation matrix are singled out.The expression for a small critical exponent η is obtained in a higher non-Gaussian approximation.
A microscopic description of the critical behavior of systems belonging to the universality class of the three-dimensional (3D) Ising model is developed within the collective variables (CV) approach. The higher non-Gaussian approximation (the sextic distribution for the modes of spin-moment density oscillations or the ρ6model) is used. A specific feature of the partition function calculation for an Ising-like system is the inclusion of the correction for the potential averaging. This correction leads to the modified recurrence relations (RR) for the ρ6model and a nonzero critical exponent of the correlation function η. The RR between the coefficients of the effective sextic distributions are written and analyzed. A technique for determining the small critical exponent η is elaborated in the higher non-Gaussian approximation. It is shown that the renormalized critical exponent of the correlation length has a tendency to a reduction in the case when the exponent η is taken into account.
The behaviour of a three-dimensional Ising-like system at temperatures above the critical value of T c is studied in the approximation of the sextic distribution for modes of spin density oscillations (ρ 6 model). The original collective variables method is developed in this higher non-Gaussian approximation for calculating the thermodynamic characteristics of the system near T c taking into account the first confluent correction. The calculations are illustrated by an example of a simple cubic lattice and an exponentially decreasing interaction potential. The main distinctive feature of the method is the separate inclusion of the contributions to the thermodynamic functions from the short-wave and long-wave spin density oscillation modes. The dependences of the phase transition temperature, leading critical amplitudes, and correction-to-scaling amplitudes for the specific heat and susceptibility on the microscopic parameters of the system are investigated.
The behavior of a 3D Ising system at temperatures above Tc is studied in the approximation of sextic distribution for modes of spin moment density oscillations (ρ6 model). An original method is developed in this higher non-Gaussian approximation for calculating the thermodynamic characteristics of the 3D Ising model near Tc taking into account the first confluent correction. The contributions to the thermodynamic functions of an Ising ferromagnet (free energy, entropy, internal energy, and specific heat) from the short-wave and long-wave modes of spin density oscillations are considered separately. A nonuniversal factor determined by microscopic characteristics of the system is singled out in the expressions for leading critical amplitudes and the amplitudes of the confluent correction. Numerical estimates of the critical region size, phase transition temperature, leading critical amplitudes, and the amplitudes of the correction to scaling of specific heat and susceptibility of the system are given for different values of effective radius of the exponentially decreasing interaction potential.
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.
The application of the collective variables method to the study of the behaviour of nonuniversal characteristics of the system in the critical region is illustrated by an example of the order parameter. Explicit expressions for the order parameter (the average spin moment) of a three-dimensional uniaxial magnet are obtained in approximations of quartic and sextic non-Gaussian fluctuation distributions (the ρ^4 and ρ^6 models, respectively), taking into account confluent corrections. Some distinctive features appearing in the process of calculating the order parameter on the basis of two successive non-Gaussian approximations are indicated. The dependence of the average spin moment of an Ising-like system on the temperature and microscopic parameters is studied.
An analytic technique for calculating the correlation function and correlation radius of systems belonging to the universality class of the three-dimensional Ising model is elaborated within the collective variables (CV) method. The calculations of these nonuniversal characteristics are performed for the temperatures [Formula: see text], [Formula: see text] and [Formula: see text] ([Formula: see text] is the phase transition temperature) on the microscopic level in the approximation of the quartic distribution for modes of spin moment density oscillations (the [Formula: see text] model). Proceeding from the expression for the Fourier transform of the correlation function, we have also obtained the susceptibility of the system. The correction to scaling and the potential averaging correction are taken into account in our calculations.