Random Mixture of the Ising Magnets in a Magnetic Field: Quenched Site and Bond Problems

Abstract

(1974), 120] to obtain the specific heat and the susceptibility of the random mixture of magnets, is applied for the low·field expansion of the free energy and the magnetization. The quartic terms of the free energies of the linear chain and of the infinite Bethe lattice for the site and the bond problems are obtained. The exact solution of the infinite Bethe lattice is equivalent to the Bethe approximation of the ordinary lattices. A divergence of the quartic term of the free energy of the bond problem is discussed in connection with a phase transition relating to the glass-like phase. Transparent formal similarity (which serves as a check and an outlook) between the site and the bond problems is found, and a relation (which serves as an approximation) between the quenched and the annealed systems is discussed. § 1. Introduction and conclusion In a previous paperv a method using projection operators to obtain thermo­ dynamic and magnetic properties of the random mixture of the magnets (the site and the bond problems in the quenched Ising spin systems) was presented. The method was applied to the linear chain and to the infinite Bethe lattice (of which the exact solution is equivalent to the Bethe approximation) giving the free energy and the susceptibility at zero field. A remarkable distinction in the phase diagrams between the site and the bond problems was clarified. The method was also applied to the quenched classical Heisenberg model.) For the free energy and the magnetization at a finite magnetic field, a concentration expansion was carried out and anomalous behavior in the magnetization process of the dilute linear chain at low temperatures was explained.') In this paper the method of Ref. 1) is applied to the low-field expansion of the free energy and the magnetization of the site and the bond problems for the linear chain and for the infinite Bethe lattice, and the quartic terms with respect to the magnetic field is obtained. The divergence of the second derivative of the susceptibility (a 2x/(JH2 ) of the bond problem characterizes the appearance of the glass-like phase introduced by Matsubara and Sakata.4l Transparent formal simi­ larity between the site and the bond problems is found and it serves as a good check. An approximation for the bond model and a relation to the annealed system