Two- and three-spin cluster theory of spin-glasses

Abstract

The cluster effect of spin-glasses is studied on the basis of the two- and three-spin cluster theory. In contrast to ordinary ferromagnets, the effective field containing the cluster effect is shown to be different from that containing no cluster effect, even at absolute zero. Numerical calculations are performed for the infinite-range model with 50 and 16 spins. It is shown that the solutions of the two-spin cluster equations at zero temperature are located at the bottom of deep valleys in the energy space. At finite temperatures the relationship between the local energy structure around the ground state and the existence of well behaved solutions is investigated. The Hessian matrix is examined also. The character of the lowest eigenvector of the Hessian matrix is found to be "localized" at low temperatures, while it is "extended" near the transition temperature. The effect of three-spin clusters is shown to play an important role in the temperature region where the three-spin cluster begins to be excited.