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

Pylyuk Pylyuk,Ulyak Ulyak

doi:10.5488/cmp.15.43006

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 \rho^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 \eta and the renormalization of the values of other critical exponents. The contributions from this correction to the recurrence relations for the \rho^6 model, fixed-point coordinates and elements of the renormalization-group linear transformation matrix are singled out. The expression for a small critical exponent \eta is obtained in a higher non-Gaussian approximation.

Highlights

We shall use the approach of collective variables (CV) [1, 2], which allows us to calculate the expression for the partition function of a system and to obtain complete expressions for thermodynamic functions near the phase-transition temperature Tc in addition to universal quantities
The term collective variables is a common name for a special class of variables that are specific for each individual physical system
The CV ρk are the variables associated with the modes of spin-moment density oscillations, while the order parameter is related to the variable ρ0, in which the subscript “0” corresponds to the peak of the Fourier transform of the interaction potential

Summary

Method

We shall use the approach of collective variables (CV) [1, 2], which allows us to calculate the expression for the partition function of a system and to obtain complete expressions for thermodynamic functions near the phase-transition temperature Tc in addition to universal quantities (i.e., critical exponents). The integration of partition function begins with the variables ρk having a large value of the wave vector k (of the order of the Brillouin half-zone boundary) and terminates at ρk with k → 0. For this purpose, we divide the phase space of the CV ρk into layers with the division parameter s. In each nth layer (corresponding to the region of wave vectors Bn+1 < k Bn , Bn+1 = Bn /s, s > 1), the Fourier transform of the interaction potential is replaced by its average value

The setup

Basic relations

Discussion and conclusions