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Physics Letters A

Upper bounds to the susceptibility and the spin pair correlation function of the Ising ferromagnet

T. Morita

Published Sep 1, 1975• 1 citation
Publisher
DOI
10.1016/0375-9601(75)90170-x

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Journal of Physics C Solid State Physics•Jul 7, 1975•4 citations
Studies a correlation function which is given by the canonical average of a product of one or more spin variables, for the random-bond Ising model, in which the exchange integrals are +J(J>O) and -J with probabilities p and 1-p, respectively. The authors show that an upper bound to the configurational average of the correlation functions, calculated in the thermodynamic limit in the zero external field limit, is the product of the same correlation function for the corresponding ferromagnetic Ising model at the temperature under consideration and the same quantity at the temperature T 1 which is determined by the condition T 1 =2J/ mod k B 1n(p/(1-p)) mod , where k B is the Boltzmann constant. The results are given for the random-bond Ising model of an arbitrary spin S, and also for the diluted random-bond Ising models of an arbitrary spin S, with the pair interaction and with general interaction, and for the diluted and undiluted random-bond n-vector model.
Fisher's rigorous bound chi < chi saw , where chi saw is the susceptibility calculated from the self-avoiding walk (SAW) generating function, is applied to (i) the ferromagnetic Ising alloy, (ii) the dilute Ising ferromagnet, (iii) the layer susceptibility of a free surface, (iv) the Ising ferromagnet in a transverse field, and (v) the dilute Ising ferromagnet in a transverse field. The author obtains an upper bound T UB for the temperature T 0 below which a spontaneous magnetization occurs. The use of a microscopically varying molecule field is analysed. A heuristic argument is given to show that, for the dilute Ising ferromagnet in a transverse field, the critical transverse field at zero temperature jumps from zero to a value at least equal to that of the one-dimensional ground state as x passes through the critical percolation concentration.
It is shown that for any dynamical quantity phi associated with an impurity system the variance, var ( phi ) p calculated after averaging the partition function, is greater than or equal to the corresponding variance var ( phi ) q calculated according to the usual quenched prescription in which the free energy is averaged. In particular, if chi p , chi q denote the respective isothermal susceptibilities subject to the same conditions, then chi p >or= chi q . This is a general result valid for classical and quantum systems in which the susceptibility is proportional to a variance, and it implies that if a quenched spin system displays a phase transition at a temperature of T c q at which a certain variance, say the susceptibility chi q diverges, then chi p must also be infinite at the same temperature. It is also shown that for a large class of spin S ferromagnetic Ising systems which satisfy the conditions of Griffiths' correlation inequalities and other specified conditions, the covariance between any two spins s i and s j is similarly ordered, i.e. cov (s i ,s j ) p >or=cov (s i ,s j ) q .