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Condensed Matter Physics

Recurrence relations for the three-dimensional Ising-like model in the external field

Mykhailo P. Kozlovskii

Published Jan 1, 2005• 8 citations
Publisher

Abstract

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.

DOI
10.5488/cmp.8.3.473

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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.
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.
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.
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.
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.