The first- and second-order terms of a perturbation expansion of free-energy density of the s-d exchange Hamiltonian with random impurity positions, in the presence and absence of an external magnetic field B, are evaluated. The reduced s-d Hamiltonian Hr(B) is chosen for the Hamiltonian of the...

The Feynman–Kac theorem and Bogolyubov inequality are applied to obtain a lower bound and an upper bound to the free energy of the s–d Hamiltonian with locally smeared interactions between electrons and impurities. The two bounds, which express in terms of the free energy of impurities in a...