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Condensed Matter Physics

The order parameter and susceptibility of the 3D Ising-like system in an external field near the phase transition point

Mykhailo P. Kozlovskii, Romanik

Published Jan 1, 2010• 7 citations
Publisher

Abstract

The present work is devoted to the investigation of the 3D Ising-like model in the presence of an external field in the vicinity of critical point.The method of collective variables is used.General expressions for the order parameter and susceptibility are calculated as functions of temperature and the external field as well as scaling functions of that are explicitly obtained.The results are compared with the ones obtained within the framework of parametric representation of the equation of state and Monte Carlo simulations.New expression for the exit point from critical regime of the order parameter fluctuations is proposed and used for the calculation.

DOI
10.5488/cmp.13.43004

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The present work is devoted to the investigation of the 3D Ising-like model in the presence of an external field in the vicinity of critical point.The method of collective variables is used.General expressions for the order parameter and susceptibility are calculated as functions of temperature and the external field as well as scaling functions of that are explicitly obtained.The results are compared with the ones obtained within the framework of parametric representation of the equation of state and Monte Carlo simulations.New expression for the exit point from critical regime of the order parameter fluctuations is proposed and used for the calculation.
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.
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 application of the method of collective variables to study the behavior of non-universal characteristics of a three-dimensional Ising-like system in the critical region has been illustrated by an example of the correlation function and the susceptibility. An analytic procedure for the calculation of those characteristics has been developed in the quartic-distribution approximation for order parameter fluctuations. The asymptotics of the correlation function at large distances obtained for the critical temperature (T = Tc) is shown to differ qualitatively from that in the T =/ Tc case because of the presence of the critical regime region for all fluctuation modes.
An analytic method for deriving the free energy of a three-dimensional Ising-like system near the critical point in a homogeneous external field is developed in the the ρ6 model approximation. The mathematical description proposed for temperatures T>Tc (Tc is the phase-transition temperature in the absence of an external field) is valid for fields near , where the scaling variable is of the order of unity and power series in this variable are not effective. At the limiting field , the temperature and field effects on the system in the vicinity of the critical point are equivalent. The total free energy is obtained as a function of temperature, field and microscopic parameters of the system without using series expansions in the scaling variable. In addition to leading terms, the expression for the free energy includes the terms determining the temperature and field confluent corrections.
Method of Calculating the Free Energy of Three-Dimensional Ising-Like System in an External Field with the Use of the ρ6ρ^6ρ6 Model
The microscopic approach to calculating the free energy of a three-dimensional Ising-like system in a homogeneous external field is developed in the higher non-Gaussian approximation (the ρ6ρ^6ρ6 model) at temperatures above the critical value of TcT_cTc​ (TcT_cTc​ 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 both the temperature and field fluctuations of the order parameter. Our analytical calculations do not involve power series in the scaling variable and are valid in the whole field-temperature plane near the critical point including the region in the vicinity of the limiting field h~c\tilde h_ch~c​, which divides external fields into the weak and strong ones (i.e., the crossover region). 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 obtained expression for the free energy contains the leading terms and terms determining the temperature and field confluent corrections.
The critical behavior of the three-dimensional n -vector model in the presence of an external field is investigated. The mathematical description is performed with the collective variables method in the framework of the ρ 4 model approximation at the microscopic level without any adjustable parameters. The recurrence relations of the renormalization group as functions of the external field and temperature were found. The analytical expression for the free energy of the system for the temperatures T > T c and different n was obtained. The equation of state of the n -vector model for the general case of small and large external fields was written. The explicit form of the correspondent scaling function for different values of the order parameter was derived. The obtained results are in qualitative agreement with the data of Monte Carlo simulations.