A diagram technique for calculation of correlation functions of ferromagnet being described by the Ising model is proposed here and its zero approximation corresponds to the selfconsistent field. The essential dependence of a numerical factor of a diagram upon its structure is the important peculiarity of this technique in comparison with the usual diagram technique for fermion or boson operators. It leads to the special character of analytical series corresponding to the certain diagram series and to the special ways of their summarizing. The exact forma1 expression for the average spontaneous moment in terms of irreducible many-particle correlation functions had been obtained. A certain series of graphs for spontaneous moment which corresponds to the Gaussian fluctuations of the selfconsistent field being taken into account had been extracted. A relation between the proposed diagram technique and the expansion of functional integrals for the correlation functions of the Ising model had been obtained as well. In conclusion the referencies to authors papers where a diagram technique for Heisenberg and s-d models are developed are given there.
A diagram technique for calculation of correlation functions of ferromagnet being described by the Ising model is proposed here and its zero approximation corresponds to the selfconsistent field. The essential dependence of a numerical factor of a diagram upon its structure is the important peculiarity of this technique in comparison with the usual diagram technique for fermion or boson operators. It leads to the special character of analytical series corresponding to the certain diagram series and to the special ways of their summarizing. The exact forma1 expression for the average spontaneous moment in terms of irreducible many-particle correlation functions had been obtained. A certain series of graphs for spontaneous moment which corresponds to the Gaussian fluctuations of the selfconsistent field being taken into account had been extracted. A relation between the proposed diagram technique and the expansion of functional integrals for the correlation functions of the Ising model had been obtained as well. In conclusion the referencies to authors papers where a diagram technique for Heisenberg and s-d models are developed are given there.
An alternative formulation of the cumulant expansion method of Frank and Mitran (see ibid., vol.10, p.2641 (1977) and Solid State Commun., vol.24, p.343 (1977)) is developed for the self-consistent calculation of the two-spin correlation function of the three-dimensional Ising model. The present technique is much simpler and more accurate since the cumulant series is summed exactly and certain other approximations made by Frank and Mitran are avoided. In addition to obtaining critical temperatures for the zero-field ferromagnet which are within 1% of the exact values, the entire phase boundary of the antiferromagnet in an external field is accurately given by the present method. Approximate critical exponents are also obtained. The present method requires considerably less numerical effort than real space renormalisation group methods and gives better results.
A new diagram technique based on the generalized Wick theorem has been elaborated for the systems with strong electronic correlations.Coulomb repulsion of the electrons of the Hubbard model is considered as the main part of Hamiltonian and is taken into account in a zero order approximation.The hopping matrix elements are considered as a perturbation.One-particle Matsubara-Green function of the model has been investigated and Dyson equation has been obtained.New elements of the theory which are characteristic of this new approach are the local many-particle irreducible Green functions, or Kubo cumulants.They become zero when Coulomb interaction is zero.The main task of this paper is the summing of ladder diagrams which take into account the most essential charge and spin fluctuations of the system.The integral equations, which sum such diagrams, have been established for two different channels.The coherent potential approximation has been used to simplify and solve these equations.On this basis a metal-dielectric phase transition has been investigated.
The authors construct a diagram technique for the (t-J) model Hamiltonian expressed in terms of Hubbard operators. This technique combines features of the diagram technique for normal Fermi systems and those for the Heisenberg model with spin operators. The general graphic structure is identified for one-particle Green functions composed of Hubbard operators. The (t-J) model has logical consistency, is reasonably simple and can serve as a working tool in exploring various properties of this model. To an approximation that resides in summing ladder-type diagrams with antiparallel electron lines, the authors calculate the system's magnetic susceptibility. A formula is derived which reflects the dual nature of the magnetic states in the Hubbard model; this duality manifests itself in the presence of a localised and an itinerant contribution in the bare susceptibility. The authors trace how the relative value of the localised contribution varies as the electron concentration is increased from 0 to 1. With n>2/3, the paramagnetic phase of the system becomes unstable with respect to the occurrence of ferromagnetic or antiferromagnetic ordering. A magnetic phase diagram for zero temperature is constructed on the (t/U, n) plane.
A functional-integral formulation of the Ising model is used as a link between the usual approach in terms of summation over spin variables and field-theory-like formulas. The latter take the form of Feynman-diagram expansions, Dyson integral equations, or Schwinger functional-derivative field equations. Basic to the theory is a spinless-nucleon-like Green's function, related to the Ising spin-spin correlation function. By means of an infinite renormalization technique, a mesonlike propagator, related to local energy-energy correlations, is bootstrapped. Finally, a vertex function, associated with energy-spin-spin correlations, is introduced. To zeroth order this theory bears similarity to the spherical model, but vital differences are also noted. A brief discussion is presented of the relation of methods employed in field theory for the treatment of the infrared divergence, and approximations which might be of value for critical correlations.
On the basis of a regular perturbation theory in the form of diagrammatic technique with X-operators, a ferromagnetic state in the Hubbard model with infinite on-site Coulomb interaction U is studied for a wide range of electron concentration n. Treating the kinetic energy of electrons as perturbation, we get the exact graphic representations for three Green’s functions [Formula: see text] and [Formula: see text] for electrons with spin ↑ and ↓ (spontaneous magnetic moment is in ↑ direction) and [Formula: see text] for spin waves. All of them are expressed through the unique system of four-point and three-point vertex parts for effective electron-magnon and electron-electron interactions. These vertex parts are calculated in two approximations: a low-density approximation for n≪1 or nh=1–n≫1, and the generalized random-phase approximation (GRPA), which was suggested earlier by us for the description of paramagnetic phase in this model. An important result in both cases is the understanding of essential difference of electron states with spin ↑ and ↓. The state for ↑ has coherent character in all region of electron concentration (n>nc) where a ferromagnetic state exists, while the state for spin ↓ has mixed characters including both coherent and incoherent contribution. For the saturated ferromagnetism (n>ns) when nh≪1, [Formula: see text] is an entirely incoherent (nonquasiparticle) state. Appearance of incoherent state is probably a general property of strongly correlated systems distinguishing its behavior from the Fermi liquid. We show also that the Hubbard-I approximation has no region of applicability for the electron with spin ↓ in a ferromagnet and we found a new factor describing the correlation-narrowing of ↓ spin electron band. Green’s function [Formula: see text] calculated in two different limits n≪1 and nh≪1, are jointed together to allow us to calculate ns which lies in the intermediate concentration regime. For simple cubic lattice we found ns=0.87. In the limit nh≪1 our results reduce to the earlier known ones, including Nagaoka’s result for the spin wave energy. We point out the way to construct a rigorous theory for ferromagnetic state in the intermediate range of electron concentration. In the conclusion several important problems are discussed in connection to the continuation of the present work.
Averaging over interaction fields and spin correlations in the Ising modelIn the theory of phase transitions in magnetic systems, there are a small number of exact results.One of them is the well-known Onsager solution for the Ising model on a square lattice.The problem of phase transitions in magnets is a special case of the problem of many bodies.There are many approaches and methods in this area, including the method of averaging over exchange fields.The application of the distribution function over interaction fields to the study of the properties of a system of many interacting particles has been used for a long time.In previous works of the authors, the method of averaging over exchange fields was applied to the analysis of the magnetic properties of pure and diluted magnets.In this paper, we formulate and prove the relations on which the method of averaging over exchange fields can be based on clusters of several spins.It turns out that based only on the possibility of representing thermodynamic means as means of the distribution function of the fields, some nontrivial exact relations can be obtained.The paper considers the Ising model on simple lattices with coordination numbers of three and four.Exact expressions are found that connect the average products of three spins with spontaneous magnetization in the system.Inequalities are obtained that limit the limiting value of the Curie temperature and the maximum value of spontaneous magnetization at a given temperature.