Singularities in the X-Ray Absorption and Emission of Metals. II. Self-Consistent Treatment of Divergences

Abstract

The many-body approach of the preceding paper is generalized to include self-energy and vertex renormalization is self-consistent fashion. The algebra of parquet diagrams is developed in terms of an unknown divergent parameter (related to the deep-hole Green's function 9), and of an unknown irreducible interaction R. The resulting interaction operator y is used to calculate 9 and R; the corresponding coupled equations are solved self-consistently in the weak coupling limit. The spectral densities of the deep-hole Green's function and of the x-ray response function are calculated explicitly; in the weak coupling limit, the Mahan singularity found in the preceding paper is unaffected by renormalization. The self-consistent formalism presented here describes the reaction of divergent fluctuations on themselves, and should, therefore, be useful in other more complicated problems, such as the Kondo effect.

Anderson orthogonality and the numerical renormalization group

“…The quantity of present interest, | G I |G F |, is simply a particular example of such an overlap. As a consequence, the long-time decay of G I |e −iĤFt |G I is often governed by ∆ AO , too, 3,5 showing power-law decay in time with an exponent depending on ∆ AO . This will be elaborated in a separate publication.…”

“…models describing a Fermi sea coupled to localized quantum degrees of freedom. Examples are the Mahan exciton (ME) and the Fermi-edge singularity (FES) [2][3][4][5] in absorption spectra, and the Kondo effect 6 arising in magnetic alloys 7 or in transport through quantum dots. 8 For all of these, the low-temperature dynamics is governed by the response of the Fermi sea to a sudden switch of a local scattering potential.…”

Scaling Theory vs Exact Numerical Results for Spinless Resonant Level Model

“…We find a distinct change in the resistivity at a given temperature T almost exactly at U f c = U * f c (T ), where U * f c is the characteristic Coulomb interaction given in Eq. (19). The resistivity shows substantial U f c dependence in the range of |U f c | < |U * f c |, i.e.…”

Effect of three-particle correlations in low-dimensional Hubbard models

“…Mahan [18], Nozières and coworkers have shown that the deep level Green's function defined as above decays as a power-law function of time [19], with an exponent proportional to the square of the phase shift at the Fermi energy due to the local potential. This result has been interpreted in terms of Anderson's orthogonality catastrophe [20,21], and rederived in the simpler language of Tomonaga bosons [22].…”