Ju l 2 00 4 No new ” Renormalized Magnetic Force Theorem ’

Recently, in Ref.[1] the author suggested a new “Renormalized Magnetic Force Theorem” for the calculation of exchange interaction parameters and adiabatic spin-wave spectra of ferromagnets. While this topic is of significant interest for the computational magnetism community, the main results of this paper seem to be not new, and their usage violates some general concepts of the magnetism theory. The following observations can be made: 1. Formally, the “local force” theorem for spin rotations from the Appendix of Ref. [2] is formulated correctly and is generally applicable. Although in the derivation of exchange parameters the well known long-wave approximation (LWA)[3, 4] was used, but not explicitly stated, the formulation of the theorem itself is correct. The right numerical way to avoid LWA in the calculation of exchange coupling within the local spin density approximation (LSDA) was first suggested by Solovyev[5]. It is evident that in Ref.[1] the author confused errors introduced by the LWA with errors in the force theorem. His Eq.(11) is an expression for the exchange coupling beyond LWA which does not constitute a new form of the ”local force” theorem. 2. The central results of Ref.[1] are given by Eqs.(15), (17) and Eqs.(16), (18). As we have noted, these results are not new. Eqs.(15), (17) follow the standard definition[3] of the exchange coupling as the inverse susceptibility, while Eqs.(16), (18) represent a transform of the celebrated formula for the adiabatic spin wave spectrum in the rigid spin approximation [6], clearly proving that the use of ’constraining fields’ concept does not introduce any new physics in the theory of magnetism. 3. Eqs.(16), (18) are not generally applicable, and their limitations are well documented[7, 8]. In particular, the author claims that the correct spin wave spectrum must be renormalized with a factor (1−ω/∆) (which results from the transformation of the exact adiabatic formula to its LWA limit). However, all the calculations in this paper were done in the adiabatic approximation which is also valid only in the limit of ω ≪ |ε − ε|. Since LWA and the adiabatic approximation are based on similar smallness parameters, any improvement of LWA requires going beyond the adiabatic approximation. Moreover, the random phase approximation (which was used indirectly in Ref.[1]) has analogous smallness criteria. Therefore, both ω and ω̃ from Ref.[1] cannot be used at large q if ω/|ε− ε| is not small. This statement is very general and valid for any adiabatic theory. The error is clearly seen in the calculation of Tc in Ni. Tc in Ref.[1] (630K) is in ideal agreement with experiment, but it contradicts the LSDA itself. The LSDA total energy difference between the ferromagnetic and nonmagnetic states of Ni is about 520-550 K. These values represent the upper limit for any reliable theory of Tc based on LSDA. 4. The author claims that Eq.(7), which is the starting point for all practical results in this paper, was derived in Ref.[2]. This is not true. Eq.(7) was not published previously; it contains intra-atomic dispersion and, generally speaking, is incorrect (together with all formulas where it is used). A somewhat similar expression (but in the rigid spin approximation) was derived in Ref.[4], while in Ref.[2] a formula analogous to the result of Ref.[4] was obtained within multiple scattering theory. The misunderstanding comes from the following statement: “. . . all results given here are straightforwardly extended to continuous variables u(r), by omitting integration over atomic cells ΩR everywhere, and replacing discrete summations overR by continuous integrations over r.” This claim is hardly justified. While the exact formulation of the local force theorem contains the functional variation of the total energy, in the rigid spin approximation (assuming no intra-atomic dispersion) one has to use the corresponding derivative: [9]