Thermodynamic Self-Consistent Theory of Indirect Exchange in Metals

A theory of indirect exchange forces (so-called s-d or s-f exchange) is given, based on Peierl's statistical mechanical variational principle. We consider a set of determinantal states, | M1…MN〉 = | M〉, where | Mi| ≤S, the localized spin, and N is the number of localized spins, the ith spin being quantized along some direction ẑi; in each of these determinants, the conduction electrons occupy general band orbitals φk which are constant over the set. The energies, EM=〈M | H | M〉, are of course functionals of the φk; varying the partition function with respect to the φk then leads to Hartree-Fock-like (HF) equations for the φk which involve the thermal averages M̄i of the localized spins, which in turn are given as functions of the φk. A particular type of solution, a spiral, is shown to exist insofar as symmetry is concerned; in connection with this proof a generalization of the Bloch translational symmetry theorem is given. Neglecting entirely the band-band exchange terms (which would give rise to Overhauser's spin-density waves), the spiral wave vector is the same as in the well known indirect ``s-f'' exchange theory in the limit M̄i→0 (i.e., as the Néel temperature is approached). For finite M̄i, the present approach provides a reasonable generalization, giving the expected energy gaps due to the magnetic long range order. It is shown that the corresponding band structure exhibits spin-density waves even though their source in Overhauser's theory has been eliminated. The temperature dependence of the gaps plus the implied changes in the Fermi surface gives rise to temperature-dependent effective exchange integrals which cause a thermal variation in spiral wavelength.