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Ferroelectrics

The order parameter and induced susceptibility ofc a magnetic of a magnetic field dependent ising model

Ljubisav Novaković

Published Jan 1, 1980• 0 citations
Publisher

Abstract

. A two-dimensional Ising model with nearest neighbour exchange interactions among electrons, in the field of a finite intensity, is investigated. By including interactions along a given direction into the structure of the transfer matrix, we pass over from the direct lattice space to the pseudo-spin space. A field dependent unitary transformation in the pseudo-spin space is applied in order to evaluate the partition function exactly. There are two important consequences of the general field dependence of the partition function: (a) The order parameter undergoes a finite discontinuity at the transition temperature; and (b) The induced susceptibility discovers a logarithmic singularity just below the transition temperature. Neither consequence seems to conform with an existing theory of a two-dimensional ferromagnet.

DOI
10.1080/00150198008207025

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. A two-dimensional Ising model with nearest neighbour exchange interactions among electrons, in the field of a finite intensity, is investigated. By including interactions along a given direction into the structure of the transfer matrix, we pass over from the direct lattice space to the pseudo-spin space. A field dependent unitary transformation in the pseudo-spin space is applied in order to evaluate the partition function exactly. There are two important consequences of the general field dependence of the partition function: (a) The order parameter undergoes a finite discontinuity at the transition temperature; and (b) The induced susceptibility discovers a logarithmic singularity just below the transition temperature. Neither consequence seems to conform with an existing theory of a two-dimensional ferromagnet.
The partition functions for two-dimensional nearest neighbour Ising model in a non-zero magnetic field have been derived for finite square lattices with the help of graph theoretical procedures, show-bit algorithm, enumeration of configurations and transfer matrix methods. A new recurrence relation valid for all lattice sizes is proposed. A novel method of computing the critical temperature has been demonstrated. The magnetization and susceptibility for infinite lattices in the presence of magnetic field is estimated. The critical exponent δ pertaining to the magnetization has also been computed.
A study of the two-dimensional Ising model on a rectangular lattice decorated with v-dimensional classical vector spins or Ising spins of magnitude s in an external field is presented. Exact expressions are obtained for the magnetisation and correlation functions, and the phase diagram is also determined exactly. It is shown that when there is no direct interaction between the spins of the host lattice there is a critical field h c , above which the system does not present long range order, and that this h c does not depend on v or s. The isothermal susceptibility related to the decorating spins is also obtained exactly, and it is shown that it diverges logarithmically at the transition temperature only for non-vanishing external field. A discussion is presented on the solution of equivalent models, and as a by-product it is shown how to map the model onto the two-dimensional eight-vertex model with temperature dependent parameters in an arbitrary staggered field.
The form factor representation of the correlation function of the 2D Ising model on a cylinder is generalized to the case of arbitrary arrangement of correlating spins. The magnetic susceptibility on a lattice, one of whose dimensions ( N ) is finite, is calculated in both the ferromagnetic and the paramagnetic region of the parameters of the model. The structure of singularities of susceptibility in the complex temperature plane at finite values of N and the transition to the thermodynamic limit N →∞ are discussed.
Making use of the Stratonovich linearization procedure for bilinear exponents and certain eigenvalue properties of the generalised Jacobi matrices, an exact expression has been obtained, in a closed form, for the partition function of the d (=3) dimensional, nearest neighbour coupled, Ising model for finite continuous spins(=D) in the presence of an external field. A Gaussian measure has been chosen for integration over the spin space. The method admits complete generalisation with respect to the dimensionality, the range of interaction and the sign of the interaction constant J (ferro- or antiferromagnetic). All thermodynamical quantities of interest can be obtained by taking suitable partial derivatives of the partition function. The critical indices can be ascertained for the cases of d=3 for which the theory predicts a second-order phase transition. For the ferromagnetic case(J>0), the spontaneous magnetisation vanishes identically the temperature T>T (the transition temperarture) and the magnetic susceptibility follows essentially the Curie Weiss Law.
The statistical mechanics of finite $L\ifmmode\times\else\texttimes\fi{}L$ Ising square lattices in a field $\ifmmode\pm\else\textpm\fi{}h$ of random sign is investigated numerically by a modified recursive transfer matrix method for 6< L< 166<~L<~166< L< 16. Our results are consistent with the absence of a spontaneous magnetization for $h\ensuremath{\ne}0$ even in the ground state. The singularities occurring at T=0T=0T=0 in the range 0< h< 4J0<~h<~4J0< h< 4J, JJJ being the exchange constant, are discussed in terms of a cluster expansion. For nonzero TTT, less than the critical temperature of the pure two-dimensional Ising model, and hhh sufficiently small, the system exhibits a nonzero spin-glass order parameter of the Mattis type although there is no magnetization. The ferromagnetic correlation function becomes long ranged for $h\ensuremath{\rightarrow}0$ and is calculated from the domain-wall density.
The magnetic critical exponent of the two-dimensional axial next-nearest-neighbor Ising model is estimated by means of the finite-size scaling of the order parameter. For a finite system without ordering field, this magnitude is computed from the two leading eigenvectors of the transfer matrix. The ferromagnetic-paramagnetic transition temperatures estimated with this method exhibit a small lattice-size dependence. Moreover, they are in good agreement with those obtained using phenomenological renormalization, and also with analytical approximations. The correlation-length critical exponent is calculated by means of phenomenological renormalization. The results found with these numerical techniques indicate that the ferromagnetic-paramagnetic transition of the model seems to be continuous, in contradiction with a previous result obtained using the persistence-length criterion, which indicated that the transition is always first order. We found that the phase transition is first order only when the ratio of the next-nearest-neighbor and nearest-neighbor interactions equals one-half.