Abstract
In this chapter we describe the Gaussian approximation of the fluctuating field in the functional integral method. First, we present the simplest Gaussian approximation based on the saddle-point method. This approximation leads to the Stoner mean-field equations and RPA dynamic susceptibility. The optimal Gaussian approximation in the DSFT utilizes a quadratic approximation of the free energy based on a variational principle, which we describe in a rather general form here. The optimal Gaussian approximation allows to take both quantum nature (dynamics) and spatial correlation (nonlocality) of thermal fluctuations of the electron spin density.
The only form of functional integrals we can evaluate is the Gaussian quadrature... (N.N. Bogoliubov and D.V. Shirkov, Introduction to the Theory of Quantized Fields, 3rd edn., Wiley, New York, 1980)
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Notes
- 1.
In the DSFT we omit the band index ν in the Green functions, because we consider N d degenerate d bands.
- 2.
By formula (8.49) the free energy \(\mathcal{F} = -T\ln Z\) should be written as
$$\displaystyle{ \mathcal{F} = T\ln \int \mathrm{e}^{-F_{0}(V )/T}\,\mathrm{D}V - T\ln \int \mathrm{e}^{-F(V )/T}\,\mathrm{D}V, }$$and the same first term should appear in \(\mathcal{F}_{\mathrm{mod}}\). Since the extra terms cancel in (9.25), we omit them for brevity.
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Melnikov, N.B., Reser, B.I. (2018). Gaussian Approximation. In: Dynamic Spin-Fluctuation Theory of Metallic Magnetism. Springer, Cham. https://doi.org/10.1007/978-3-319-92974-3_9
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