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Critical behaviour of a three-dimensional one-component magnet in strong and weak external fields at T>Tc

Mykhailo P. Kozlovskii, Ihor V. Pylyuk, O. O. Prytula

Published Jan 1, 2006• 0 citations
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Abstract

The influence of a homogeneous external field on a three-dimensional uniaxial magnet behaviour near the critical point is investigated on the microscopic level by the nonperturbative collective variable (CV) method using the ρ4 model. The research is carried out in the high-temperature region. The critical exponents related with external field are calculated. The analytic explicit expressions for the free energy, average spin moment and susceptibility are obtained without using any phenomenological assumptions and adjusting parameters. The field dependences of the order parameter are plotted for the field region including the neighbourhood of the pseudocritical line. It is established that long-wave fluctuations of the order parameter are responsible for main features of the crossover between the temperature-dependence and field-dependence critical behaviour of the system. The maximum locations on the temperature scale for the long-wave part of the susceptibility are determined for different values of the field.

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arXiv:1307.2011•Condensed Matter Physics•Jan 1, 2013•1 citation
The influence of a homogeneous external field on a three-dimensional uniaxial magnet behaviour near the critical point is investigated on the microscopic level by the nonperturbative collective variable (CV) method using the ρ4 model. The research is carried out in the high-temperature region. The critical exponents related with external field are calculated. The analytic explicit expressions for the free energy, average spin moment and susceptibility are obtained without using any phenomenological assumptions and adjusting parameters. The field dependences of the order parameter are plotted for the field region including the neighbourhood of the pseudocritical line. It is established that long-wave fluctuations of the order parameter are responsible for main features of the crossover between the temperature-dependence and field-dependence critical behaviour of the system. The maximum locations on the temperature scale for the long-wave part of the susceptibility are determined for different values of the field.
The analytical calculation of the free energy of a three‐dimensional Ising‐like system in a homogeneous external field is performed in the higher non‐Gaussian approximation (the ρ6 model) at temperatures above the critical value of Tc (⁠Tc is the phase‐transition temperature in the absence of an external field). The free energy of the system is found by separating the contributions from the short‐ and long‐wave spin‐density oscillation modes taking into account the generalized point of exit of the system from the critical regime as a function of both the temperature and field variables. 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. In this region, the temperature and field effects on the system are equivalent, the scaling variable is of the order of unity and power series are not efficient. The free energy contains the leading terms and terms determining the temperature and field confluent corrections.
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.
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.
We study the behavior of a three-dimensional Ising-like system in an external field near the critical point by using the non-Gaussian spin-density fluctuations, namely, the quartic measure density with the even and odd powers of the variable (the asymmetric ${\ensuremath{\rho}}^{4}$ model). The basic idea of the analytic method for deriving complete expressions of the thermodynamic characteristics (including the scaling functions) is described. The proposed method allows us to perform the calculations on the microscopic level without any adjustable parameters. Explicit expressions for the total free energy, order parameter, susceptibility, entropy, and specific heat of the system are obtained as functions of the temperature and field. The regions of the so-called weak and strong fields are considered for temperatures above and below Tc{T}_{c}Tc​ (Tc{T}_{c}Tc​ is the phase-transition temperature in the absence of an external field). The average spin moment and susceptibility, depending on the field variation and the proximity to Tc{T}_{c}Tc​, are investigated. It is confirmed that the temperature and field fluctuations for the order parameter play the leading roles in the weak and strong fields, respectively.
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 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.