Phase transitions of classical Heisenberg ferromagnets with arbitrary parameter of anisotropy

The existence of a phase transition of the first kind is proved for anisotropic classical Heisenberg ferromagnet in two or more dimensions and with arbitrary parameter of anisotropy α, |α|<1; a similar fact is proved for much more general lattice spin systems.     

Critical temperatures of the anisotropic Heisenberg ferromagnets

High temperature series expansions for the fluctuation in the long range order M_x is obtained through order T^?7 for a spin-$\text{1}\text{2}$ anisotropic Heisenberg Hamiltonian. The critical temperatures at which the spontaneous magnetization occurs on the xy-plane are determined.     

Green's-function theory for disordered anisotropic Heisenberg ferromagnets

The Green's function theory for disordered Heisenberg Ferromagnets developed by Montgomery et al. has been extended for anisotropic systems. The densities of spin-wave states for such systems exhibit split-bands. This behavior differs qualitatively from that of the isotropic case and offers a partial explanation for the neutron scattering experiment of Buyers et al. in which they observe two well defined branches of magnetic excitations in disordered substitutional anti-ferromagnetic alloys.     

Phase transitions in ferromagnets and singularities

This paper is devoted to the investigation of critical phenomena from the mathematical point of view. An interesting question in this field is universality (i.e. independence of the type of material and, to some extent, of the type of process) of the so-called critical exponents. We try to show that one can compute the critical exponents using structural stability arguments. The universality is then implied automatically as a consequence of similarity of models and of the structure of stable singularities. The considerations lead to some new results: information about the generic shape of hysteresis loops and universal form of the equation of state in the critical region.     

Nonlinear spin excitations in a classical Heisenberg anisotropic helimagnet

The nonlinear spin excitations in an anisotropic helimagnet in the presence of a constant magnetic field are investigated in the classical continuum limit. The helical character is introduced into the model, in analogy with the twist in a cholesteric liquid crystal. After deriving a class of spin wave solutions for the stereographic representation of the Landau-Lifshitz equation, in-plane stationary spin configurations are obtained and their stability is analysed. When the external magnetic field is along the anisotropic axis, modulational instability is observed in the spin lattice, and when the external magnetic field is normal to the anisotropic axis, the spin configurations are unstable. The perturbed spin components show fluctuations in the tail region, while the velocity and amplitude of the soliton remain unaltered.     

Phase transitions in anisotropic magnetic superlattices

The paper reports a theoretical study of phase transitions in antiferromagnetic superlattices. The boundaries of stability of the ferro-and antiferromagnetic phases have been calculated by a model taking into account the layered structure of these materials. The expressions obtained are precise in the limit of a large number of layers. In the case of a small number of layers, the problem reduces to a simple numerical procedure of solving a transcendental equation. The critical field is shown to depend substantially on the biquadratic non-Heisenberg exchange constant. Fiz. Tverd. Tela (St. Petersburg) 39, 178–180 (January 1997)     

Phase transitions in classical vector models

The equation of state obtained earlier /6/ for 3-dimensional Ising ferromagnets in the first order in$\text{1}\text{z}$ (z is the number of the nearest neighbours) is generalized with regard to the classical D-component vector model. The results obtained are valid in a wide temperature range |τ|?[z²(D+2)]^?1 (τ=(T_c-T)/T_c) and become exact in the limit D → ∞.     

A constant-magnetization ensemble for the classical anisotropic Heisenberg model

A constant-magnetization ensemble is introduced in order to study classical, anisotropic Heisenberg systems. Existence, uniform convergence, and convexity properties are proved for an appropriate thermodynamic potential. The thermodynamic equivalence of this ensemble with the more common canonical ensemble is also established. In a subsequent paper, this formulation is used to obtain an exact statistical mechanical solution of classical Heisenberg systems with long-range Kac interactions.This research was carried out at The Department of Physics, Case Western Reserve University, Cleveland, Ohio, and was supported in part by the U.S. Atomic Energy Commission. This work is based on portions of a dissertation by K. M. submitted in partial fulfillment of the Ph.D. degree, Case Western Reserve University.     

Critical temperature studies of the anisotropic bilayer and multilayer Heisenberg ferromagnets in Pair Approximation

The Pair Approximation method is applied to studies of the bilayer and multilayer magnetic systems with simple cubic structure. The method allows to take into account quantum effects related to non-Ising couplings. The paper adopts the anisotropic Heisenberg model for spin S=1/2 and considers phase transition temperatures as a function of the strength of exchange integrals in line with the role of intra- and interplanar anisotropic interactions in the onset of low-dimensional magnetism. The compensation effect for the Curie temperature is found for asymmetric interactions within the neighbouring planes of the bilayer system. The paper predicts the saturation of the Curie temperature for strong interplanar interactions. However, such an effect for the multilayer system occurs only when the interplanar interactions are purely of isotropic character.     

Phase transitions in anisotropic lattice spin systems

A general method for proving the existence of phase transitions is presented and applied to six nearest neighbor models, both classical and quantum mechanical, on the two dimensional square lattice. Included are some two dimensional Heisenberg models. All models are anisotropic in the sense that the groundstate is only finitely degenerate. Using our method which combines a Peierls argument with reflection positivity, i.e. chessboard estimates, and the principle of exponential localization we show that five of them have long range order at sufficiently low temperature. A possible exception is the quantum mechanical, anisotropic Heisenberg ferromagnet for which reflection positivity isnot proved, but for which the rest of the proof is valid.Work partially supported by U.S. National Science Foundation grant no. MPS 75-11864Work partially supported by U.S. National Science Foundation grant no. MCS 75-21684 A01     

Elementary excitations in easy-plane Heisenberg ferromagnets

We study the influence of the non-diagnonal terms in an easy-plane Heisenberg ferromagnet on the cross-section for inelastic neutron scattering from single spin-wave states. We show that these terms modify the wavevector dependence of the cross-section for unpolarized neutrons, and the created polarization. An appropriate experiment on CsNiF₃, for example, should be a good test of these predictions, and would eventually clarify some uncertainties about the appropriate Boson Hamiltonian for an easy-plane ferromagnet.     

Classical Heisenberg ferromagnets with defects

A ferromagnet (dimensionality d) containing defects (of dimensionality d′) is treated by the phenomenological Landau theory. It is shown that for d′<d?1 no change of the critical exponent of the local magnetization near the defect occurs, in contrast to the case of the local magnetization near a surface (d′=d?1). The case of point defects (d′=0) in classical Heisenberg ferromagnets with nearest neighbor interactions is treated both by the mean field approximation and the Monte-Carlo method with effective field boundary conditions. We obtain rough agreement between these two approaches—two special cases are considered in detail: (a) interaction with, the defect being zero, (b) anti-ferromagnetic interactions across the defect (which is a crude model of superexchange). It is pointed out that it is a reasonable approximation to express the deviation of the local magnetization from the bulk value near the defect in terms of the spin pair correlation functions of the undisturbed system.     

A variational approach for the classical anisotropic Heisenberg model in a crystal field

The variational approach based on the Bogoliubov inequality for the free energy is used to study the three-dimensional anisotropic Heisenberg XXZ model with a crystal field. The magnetization and the phase diagrams are obtained as a function of the parameters of the Hamiltonian. Limiting cases, such as isotropic Heisenberg, XY, and planar rotator models in two and three dimensions, are analyzed and compared to previous results obtained from analytical approximations as well as to those obtained from more reliable approaches such as series expansion and Monte Carlo simulations. A parametric procedure has been used in order to simplify the solutions of the self-consistent coupled equations.     

A proof of the existence of phase transitions in the anisotropic Heisenberg model

It is rigorously proved that the anisotropic Heisenberg spin model, in two or more dimensions, exhibits a first order phase transition at low temperatures and for large anisotropy (strong coupling of the third components of the spins).     

Intrinsic localized modes of a classical discrete anisotropic Heisenberg ferromagnetic spin chain

We report several exact intrinsic localized mode solutions of the classical spin evolution equation of a one-dimensional anisotropic Heisenberg ferromagnetic spin chain in terms of Jacobian elliptic functions. These include one, two and three spin excitations. All these solutions have smooth anticontinuum limits. Their linear stability and semiclassical quantization are also discussed briefly.