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Progress of Theoretical Physics

Reducibility of an Ising Ferromagnet from Griffiths "Equalities"

N. Seto

Published Mar 1, 1976• 2 citations
Publisher

Abstract

For an Ising ferromagnet, it is shown that the correlation function between a pair of spins vanishes if and only if, irrespective of an external magnetic field, the pair decouples from each other. For the correlation functions between two- and two-body spins as well as one- and two-body spins, it is proved, in the presence of a suitable magnetic field, the vanishing of these implies the reducibility of the system.

DOI
10.1143/ptp.55.683

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For an Ising ferromagnet, it is shown that the correlation function between a pair of spins vanishes if and only if, irrespective of an external magnetic field, the pair decouples from each other. For the correlation functions between two- and two-body spins as well as one- and two-body spins, it is proved, in the presence of a suitable magnetic field, the vanishing of these implies the reducibility of the system.
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.
Expressions for general correlation functions in Ising ferromagnets are derived in terms of the Lee-Yang theorem. By the use of these expressions, the asymptotic forms of correlation functions in the presence of magnetic field are analyzed, assuming those in the absence of the external field. In particular, the critical indices of spin-spin, energy-density-energy-density, energy-density-spin correlation functions are related with the critical indices of thermal susceptiblity χT0, specific heat, susceptibility χH0, and magnetization. These results agree in form with those predicted by the scaling law approach.
For spin 1 / 2 Ising models with ferromagnetic pair interactions J ij which are taken to be statistically independent, non-negative, random variables, the authors show that the averaged two-spin correlation function ( sigma p sigma q ) is less than or equal to the value ( sigma p sigma q ) J for the corresponding non-random system in which each J ij is replaced by its average value J ij . The inequality implies for sufficiently high temperatures that the averaged isothermal susceptibility chi <or= chi j . With additional assumptions about the existence and nature of the phase transitions for the systems, one concludes that the critical temperature tau c for the averaged system is not greater than the corresponding critical temperature tau (J) for the nonrandom system. In the process of obtaining the results it is noticed that Griffiths' inequalities ( sigma p sigma q )>or=0 and delta ( sigma p sigma q )/ delta J rs >or=0 imply that delta 2 ( sigma p sigma q )/ delta J rs 2 <or=0.
According to the traditional solutions, there is no magnetisation in the one-dimensional Ising model in the absence of external magnetic field. It is shown that there is a mistake within them. In them, local order of spins was not taken into account and the partition function and the correlation function were calculated wrongly. In this paper, that mistake is corrected and a new solution is presented. According to it, there is ferromagnetism in that model. Analogously, solutions of the two-dimensional and three-dimensional Ising models are obtained.