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Revista brasileira de fisica

Correlation effects in the ising model in an external field

Henrique E. Borges, P.R Silva

Published Jan 1, 1983• 0 citations
OpenAlex

Abstract

The thermodynamic properties of the spin - 1/2 Ising Model in an external field are evaluated through the use of the exponential differ- ential operator method and Callen's exact relations. The correlations effects are t reated in a phenomenological approach and the results are compared with other treatments.

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•Jan 1, 1983•0 citations
A new method incorporating the effects of many-body static spin correlations into an effective-field theory is discussed. The method is based on the introduction of a differential operator and the concept of the correlated effective field into two exact Callen identities of the Ising model. The resulting statistical theory is shown to have an accuracy equivalent to that of the Bethe-Peierls method. It is shown that the correlated-effective-field parameter at the transition temperature has a universal function of $\frac{1}{(z\ensuremath{-}1)}$, where zzz is the number of nearest neighbors.
Within a new type of correlated-effective-field theory we calculate the effective-field parameters (including their response to external field), critical temperatures, susceptibility, short-range order parameters, and specific heat associated with an anisotropic, square-lattice, spin-\textonehalf{} Ising ferromagnet. The present formalism, which is shown to recover the Bethe-Peierls thermodynamics for the pure ZZZ-coordinated-lattice, spin-\textonehalf{} Ising model, yields satisfactory results; in particular, the crossover from a pure Ising ferromagnet on a square lattice to that on a linear chain is reasonably exhibited in the behavior of the susceptibility and the specific heat.
A new type of effective-field theory of the Ising model is presented. The differential-operator method is introduced into the exact spin correlation function identity obtained by Callen. The Curie temperatures are evaluated by using two different types of effective Hamiltonians. It is also shown how the Zernike and the Bethe-Peierls equations can be reproduced within one framework depending on the choice of effective fields in an effective Hamiltonian. The spin correlation function and the specific heat are presented.
The method of the two-time temperature-dependent Green's function has been used to analyze the Ising model of a ferromagnet in an external magnetic field. The selection of a particular Green's function enables us to write an exact expression for the equation of motion. We are then led to a differential difference equation for the correlation function corresponding to the Green's function. No decoupling assumptions have been made, so the equation is exact for both arbitrary spin and range of interaction. It is shown how various approximate theories may be extracted from our formalism. The exact differential difference equation may be reduced to a partial differential equation. The latter form allows us to generate relations among the magnetization and spin-spin correlation functions. These relations are given in detail for the case of spin \textonehalf{} and zzz nearest neighbors.
Abstran We study the dilution of the spin-1/2 transverse Ising Model by means oic an effective fleld type treatment based on an extension of Callen's relation (Phys. Lett. 4, 161 (1963)) to the present model. The thermodynanics of the diluted model is obtained and the resultsareshown to be an improvement over the standard mean field treatment. Wealsocompare the results w ith the Monte Carlo calculation for the spin-oo transverse Ising Model. The transverse Ising model has been used tp describe phase transitionc in ferroelectrics, ferromagnets and cooperative Jahn-Teller systems with an applied external magnetic field. This model hamiltonian was f i rst proposed by de ~ennes' to represent the basic features of hydrogen-boncled ferroelectrics of the KH2P04 family. In these systems the Ising term corresponds to the interaction between the protons at different lattice sites and the transverse field accounts for the possibility of the protons occupy one of the two minima of a doublepotential well in a given site. The mdel can also be applied to study rare - earth compounds with s inglet c rystal field ground states, as has been done by Wang and cooper2. In these systems there is a competition between the exchange interactlon represented by the Ising term and the crystal field represented by the transverse field. There will be magnetic ordering if the ratio between these two terms exceeds a certain value. Cooperative Jahn-Teller systems with an applied external magnetic field is another example where this model hami I tonian works we11 3. Here the phase transition is driven by the interaction between localized orbital electronic states and the crystal lattice. Even at low temperatures sufficiently high applied external magnetic field can retrieve the high temperature phase. We refer to a paper by Stinchcombe 4 which gives a