By M. Springford

Lifshitz, I. M. and Kaganov, M. I. Geometric recommendations within the electron conception of metals. -- Wilkins, J. W. figuring out quasi-particles. -- Chambers, R. G. The generalized path-integral method of delivery difficulties. -- Pippard, A. B. The Shoenberg influence

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36) and express the free energy in terms of these variables. 3 Antiferromagnetism 23 becomes t JrO = L. L. [( Yk + a hs) aL aku + (Yk k u ta hs ) bL bku + tk (aL bku + bL aku)] . 37) This is diagonalized by the following canonical transformation: { akU = A ku cos (h - B ku a sin Ok, bku = A ku a sin Ok + B ku COS Ok, tan20k = 2tk/ hs. ()= L. L. E ku (AkuAku u k + Btu Bk- u), E ku -- Yk + a (2 tk + "41 h2) s 1/2 . 1 shows the energy dispersion for a one-dimensional example with the Brillouin zone Qoundary at ± KI2.

L. [( Yk + a hs) aL aku + (Yk k u ta hs ) bL bku + tk (aL bku + bL aku)] . 37) This is diagonalized by the following canonical transformation: { akU = A ku cos (h - B ku a sin Ok, bku = A ku a sin Ok + B ku COS Ok, tan20k = 2tk/ hs. ()= L. L. E ku (AkuAku u k + Btu Bk- u), E ku -- Yk + a (2 tk + "41 h2) s 1/2 . 1 shows the energy dispersion for a one-dimensional example with the Brillouin zone Qoundary at ± KI2. , JrO = L. L. 40) Q being the antiferromagnetic wave vector (Q = KI2 in Fig. 1). For h s =1= 0 the energy gap is formed at the zone boundary.

3 SCR Theory of Ferromagnetic Metals As indicated above, the necessity of self-consistent renormalization is a natural consequence of logical considerations toward a consistent theory beyond . HF-RPA. 13], where the local moment picture is clearly inadequate and the HF-RPA theory cannot explain the CW law consistently. 15). 14]. 3 SCR Theory ofFerromagnetic Metals 51 removing an interaction vertex. Although calculating i(q, w) = Xo(q, w)1 [1 + A(q, w)] is possible within certain approximations, as discussed in Chap.