Spin-wave theory analytic solution of a Heisenberg model with RKKY interactions on a Bethe lattice q Jose  Rogan a , Miguel Kiwi a,b, * a Departamento de Fõ Âsica, Facultad de Ciencias, Universidad de Chile, Casilla 653, Santiago 1, Chile b Facultad de Fõ Âsica, Universidad Cato Âlica de Chile, Casilla 306, Santiago 6904411, Chile Received 29 December 2000; accepted 25 January 2001 by C.E.T. Gonc Ëalves da Silva; received in ®nal form by the Publisher 22 March 2001 Abstract An analytic solution for the Heisenberg Hamiltonian with long-range RKKY interactions on a Bethe lattice is obtained in the semi-classical approximation (S ! 1). The main dif®culty that has to be overcome is the exponential growth of the number of neighbors in a Bethe lattice. We suggest a way of handling this problem and derive physically meaningful results. q 2001 Elsevier Science Ltd. All rights reserved. PACS: 75.10.b; 75.30.Kz; 75.10.m Keywords: A. Magnetically ordered materials; D. Exchange and superexchange; D. Spin dynamics; D. Phase transitions 1. Introduction The Heisenberg Hamiltonian [1,2] has been a powerful tool in the description and understanding of a large variety of magnetic systems [3,4], ever since it was independently put forward, three quarters of a century ago, by Heisenberg [1,2] and Dirac [5,6]. However, and in spite of numerous and signi®cant efforts, few analytic results have been achieved [7]. This is in sharp contrast with the wealth of approximate and numerical results that have been obtained during this time span. A signi®cant feature of most of the treatments implemen- ted so far is that they are limited to nearest-neighbor inter- actions. At least three reasons explain this state of affairs: (i) generally the magnetic exchange interaction originates in the direct overlap of orbitals on neighboring ions and thus decays rapidly with distance; (ii) the considerable additional complications the inclusion of longer-range interactions implies; and (iii) the fact that the magnitude of the longer- range interactions was not well established until very recently. However, the powerful development of the ab initio computational machinery has allowed extraction of precise values of exchange integrals, even beyond the ten ®rst neighbors. An example of the latter is the computation by Zhou et al. [8] of the J k 's, for 1 # k # 11, of antiferro- magnetic fcc Fe and Mn. These results are quite surprising: not only does the sign of the J k 's display an oscillating behavior, but also a much slower decay with k than expected. In fact, the values of uJ 11 /J 1 u, 0.06 for Fe and 0.014 for Mn were obtained. Even more remarkable is the fact that, for this fcc structure, the absolute value of the computed next-nearest-neighbor exchange parameter uJ 2 u for Fe is larger in magnitude than J 1 < 20.134 mRy, yield- ing uJ 2 /J 1 u< 1.42. The same happens with the sixth-neighbor for which uJ 6 /J 1 u< 1.10. The above situation is not at all limited to particular elementary metals. In fact, in FeF 2 and MnF 2 , both materials of interest for the manufacture of spin-valve devices [9], also the ®rst-neighbor exchange is slightly ferromagnetic, while the second-neighbor one is antiferro and signi®cantly larger in magnitude [10]. In addition, there are several systems, such as the rare earths and their alloys, where the exchange interaction is well known to be long-ranged. The magnetic order of these materials is adequately described by the RKKY inter- action, initially introduced by Ruderman and Kittel [11] to describe the indirect interaction of two nuclei via their Solid State Communications 118 (2001) 485±490 0038-1098/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S0038-1098(01)00139-9 PERGAMON www.elsevier.com/locate/ssc q Supported by the Fondo Nacional de Desarrollo Cientõ ®co y Tecnolo Âgico (FONDECYT, Chile) under grant #8990005. * Corresponding author. Address: Facultad de Fõ Âsica, Universidad Cato  lica de Chile, Casilla 306, Santiago 6904411, Chile. Tel.: 156- 2-686-4476; fax: 156-2-553-6468. E-mail address: mkiwi@puc.cl (M. Kiwi).