Infinite-Volume Limit of Stochastic s-d System with Single-Component Impurity and s-Electron Spins

Abstract

A Hamiltonian [Formula: see text], with locally smeared Ising-type s-d exchange between s-electrons and magnetic impurities, in a dilute magnetic alloy, is investigated. The Feynman-Kac theorem, Laplace expansion and Bogolyubov inequality are applied to obtain a lower and upper bound (lb and ub) on the system’s free energy per conducting electron [Formula: see text]. The two bounds differ, in the infinite-volume limit by a term [Formula: see text], linear in impurity concentration: lb[Formula: see text], ub[Formula: see text], [Formula: see text] denoting the Hamiltonian of the approximating mean-field s-d system. [Formula: see text] represents randomly positioned impurities interacting with a mean field implemented by the gas of conduction s-electrons, the latter interacting with the field of barriers and wells (according to the s-electron’s spin orientation) localized at the impurity sites. The inequality [Formula: see text] demonstrates increasing accuracy of the mean-field [Formula: see text]-theory, with decreasing impurity concentration.