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Condensed Matter Physics

3D ISING SYSTEM IN AN EXTERNAL FIELD. RECURRENCE RELATIONS FOR THE ASYMMETRIC \rho^6 MODEL

Ihor V. Pylyuk, Mykhailo P. Kozlovskii

Published Jan 1, 2001• 2 citations
Publisher

Abstract

The 3D one-component spin system in an external magnetic field is studied using the collective variables method. The integration of the partition function of the system over the phase space layers is performed in the approximation of the sextic measure density including the even and the odd powers of the variable (the asymmetric ρ⁶ model). The general recurrence relations between the coefficients of the effective measure densities are obtained. The new functions appearing in these recurrence relations are given in the form of a convergent series.

DOI
10.5488/cmp.4.1.15

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The 3D one-component spin system in an external magnetic field is studied using the collective variables method. The integration of the partition function of the system over the phase space layers is performed in the approximation of the sextic measure density including the even and the odd powers of the variable (the asymmetric ρ⁶ model). The general recurrence relations between the coefficients of the effective measure densities are obtained. The new functions appearing in these recurrence relations are given in the form of a convergent series.
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 method for calculation of the partition function of lattice model for the magnet in the external field near critical point (CP) is proposed.The recurrence relations and their explicit solution near the critical point are founded.It is shown that dependence on temperature of thermodynamic functions near CP, when the field value comes down to zero, is in good agreement with the previous results obtained using the collective variable method.The phase transition temperature (when h = 0) is calculated and the dependence on parameters of interaction potential is found.
A three-dimensional Ising-like system in a homogeneous external field is studied on the basis of the higher non-Gaussian measure density (the model). The presented solutions of recurrence relations for the coefficients of the effective measure densities and the generalized point of exit of the system from the critical regime are used for calculating the free energy of the system at temperatures ( is the phase transition temperature in the absence of an external field). A calculation technique is based on the first principles of statistical physics and is naturally realized without any general assumptions and without any adjustable parameters. The obtained expression for the free energy does not involve series expansions in the scaling variable and is valid near the critical point not only in the regions of the so-called weak and strong external fields, but also in the crossover region between these fields, where power series in the scaling variable are not effective.
The distribution of zeros of the partition function for one-dimensional lattice system of generalized Ising spins with magnitude unity is studied, where it is assumed that the component spins interact with the nearest neighbours through a ferromagnetic coupling and each spin state has a certain degeneracy. The spin states are denoted by variable σ which can take either of three values, plus and minus unities and zero with the respective degeneracies, D(1), D(0) and D(-1), where it is assumed D(1) = D(-1). zeros of the partition function Z(ζ) of the system as exposed to an external magnetic field are studied on the ζ-plane, where ζ=exp(H) indicates the exponential function of the external field divided by temperature H and it is regarded as a complex number. The distribution of zeros is classified into three typical patterns, corresponding to the respective three regions on the plane of temperature(T) versus degneracy ratio (ρ=D(0)/D(1)): Zeros never lie on the unit circle in the region of larger ρ, all on the unit circle in the region of smaller ρ and only a part on the unit circle in the remaining region. On the basic of this result, it is expected in the case of two- or three-dimensional system that phase trasitions of first and second orders occur in tha cases of ρ larger and smaller than acritical calue ρc, respectively.