The energy of the spin-glass state of a binary mixture at T = 0 and its variational properties

In the random-bond model of Ising spins, the concept of a multiple-bond distribution of effective field was introduced in the pair approximation. The integral equation for a single-bond distribution was derived intuitively. The variational energy at T = 0 is expressed in terms of two parameters μ and η where μ is the probability of zero effective field in the single-bond distribution and η is the magnetization per spin. For η = 0, the energy of the spin-glass state corresponds to a local minimum as a function of μ, for an even z (number of the nearest neighbours) and to an inflection point for an odd z. It was shown that the spin-glass state corresponds to a local minimum with respect to μ and η for z = 4, to an inflection point with respect to μ and a local minimum with respect to η for z = 3. It is conjectured that the maximum of the energy of the spin-glass state of Sherrington and Kirkpatrick is attributed not to the replica method, but to the mean field approximation. Stationary properties of the energy as a function of both μ and η were examined in detail.