PHYSICAL REVIE%' B VOLUME 25, NUMBER 8 15 APRIL 1982 Phonon-induced instabilities in mixed-valence systems Juan Giner Institut de Physique, Universite de Liege, Liege 4000, Belgium Franqois Brouers Institut fiir Theoretisehe Physik, Freie Uniuersitiit Berlin, Arnimallee 3, 1000 Berlin 33, West Germany (Received 16 July 1981) We have reconsidered an f dmod-el used by Entel et al. to investigate the effect of phonons on transitions in mixed-valence systems. By reformulating the problem we have found that an abrupt transition occurs between an antiferromagnetic solution with nega- tive renormalized hybridization V to a paramagnetic solution with positive V when the f level moves in the d band. This transition is related to a change of sign of V but by no means to its cancellation. We have shown that, on the contrary, this model favors solu- tions with high absolute values of the renormalized hybridization. I. INTRODUCTION The inAuence of electron-phonon interaction in valence-fluctuation phenomena has been discussed by a number of authors. ' The necessary condi- tion for valence fluctuation is the quasidegeneracy between two ionic states 4f"5d and 4f" '5d +' However, the renormalization of electron energy levels and interactions by electron-phonon as well as electron-electron interactions induces feedback effects which can influence the nature of the tran- sition. Recently, the role of electron-phonon in- teraction has been questioned and there is a debate concerning the relative importance of electron- phonon and Falicov electron-hole interactions on the transitions to mixed-valence states. The purpose of this paper is not to give argu- ments for one or the other point of view. Since the consideration of the coupling of the electrons to a boson field with f dmixing and of -the f- electron d-hole interaction yield model Hamiltoni- ans which have formally similar forms, the con- clusion reached for one model can be used for the other provided the meaning of the parameters is changed. We have noticed that in many cases the difference in the results published in the literature comes more from the difference in the nature of the approximation made than from the nature of the interactions. Here we want to discuss the results of Entel et al. who have studied the effect of the coupling of 4f electrons and longitudinal optical phonons and its influence on phase transition in intermediate-valence compounds within the frame- work of the periodic Anderson model. The phonon-induced 4f Sd interband t-ransitions lead to a renormalization of the hybridization energy V be- tween 4f and conduction electrons. Thus, renor- malized hybridization V can become zero and this is according to these authors one of the require- ments for a discontinuous change of the average number of f electrons per site with the position of the f level in the d band. Their conclusion is that if any abrupt change is to be expected in the Hartree-Fock approximation, it will occur when the renormalized hybridization V tends to zero. Recently, Khomskii et al. have discussed a model including the interaction of the localized f electrons among themselves and with the conduc- tion d electrons as well as the hybridization of con- duction and f states. They have introduced a phenomenological dependence of the parameters, the position of the f level, and the hybridization on the variation of the volume to simulate the electron-lattice coupling. Their conclusion is that first-order transitions are only possible when electron-lattice interaction is taken into account and that, in contrast to the conclusions of Entel et al. , the possibility of jumplike transition exists at nonzero effective hybridization. These remarks and discrepancies have led us to reconsider carefully the model used by Entel et al. We have noted that by reformulating the problem expressing all quantities in terms of three indepen- 25 5214