Vertex corrections

We have shown for the case of Li that the step in the occupation number function is surprisingly small z 0.1 and provided semi-empirically obtained values for the local-field correction factor. For the case of Al, we showed the additional cancellation of self-energy and vertex correction. 

The absorption spectrum is proportional to the imaginary part of the macroscopic dielectric function. Adopting the same level of approximation that we have introduced to obtain GW quasiparticle energies, i.e. neglecting the vertex correction by putting T = 55, we get the so called random phase approximation (RPA) for the dielectric matrix. Within this approximation, neglecting local field effects, the response to a longitudinal field, for q 0, is ... 

Here a- corresponds to a dispersive correction to the Stark quenching line shape, a-2 represents a small vertex correction (of order a3) to Stark Hamiltonian. Numerical values for these quantities are —1.85 x 10-9 and 5.13 x 10-8, respectively. The details of calculations demand a separate examination and will be discussed in our forthcoming work. 

Fig. 1. The QED contributions of order a/it) to the bound-electron gj factor depicted as Feynman diagrams. Double lines indicate bound fermions, wavy bnes indicate photons. The interaction with the magnetic field is denoted by a triangle. Diagram (a) is also termed SE, ve (self-energy vertex correction), diagrams (c) and (e) SE, wf (self-energy wave-function correction), diagram (b) VP, pot (vacuum-polarization potential correction), and diagrams (d) and (f) VP, wf (vacuum-polarization wave-function correction)...

For the contribution of the first diagram of Fig. 3 to have any physical meaning, it is necessary to calculate it together with the vertex correction (fourth diagram of Fig. 3). We have obtained the following contribution to the effective hamiltonian for the vertex correction ... 

A corollary of the above results is that one should expect isotope effects in the quasiparticle spectrum measured in the pseudogap state, since once localized, the hot quasiparticles can become strongly coupled to the lattice. (When not localized, the coupling of the hot quasiparticles to phonons is markedly reduced by vertex corrections associated with their magnetic coupling.) Any coupling of cold quasiparticles to the lattice would be very much smaller. These conclusions appear consistent with the ARPES results reported by Lanzara at this workshop, ft leads me to predict that no isotope effect will be found for hot quasiparticles in overdoped materials. 

Figure 5.12 Left residual isotropic resistivity p of disordered CojcPdi- ( ) and Co Pti- (o) alloys. Full lines, calculated including vertex corrections broken lines, calculated omitting vertex corrections. Right calculated anomalous magnetoresistance (AMR) ratio Ap/p of CojPdi- ( ) and Co Pti- (o) alloys. Experimental data denoted by open squares, white diamonds, triangles and crosses arise from various sources (Ebert et al. 1996b).

Using Butler s approach in dealing with Equations (5.40) and (5.41) we account for the so-called vertex corrections within the framework of the CPA. For Co Pdi- it was found that their contribution increases from about 2% for 5 at.% Co to about 25% for 80 at.% Co. 

The difference of our model and that of Chui et al (10) and Grest et al (11) can be best seen if we calculate the second order vertex corrections. Two such diagrams are shown in Fig. 2. In the momentum transfer model the analytic contribution corresponding to these diagrams is... 

Retardation effects. If the phonon contribution dominates the bare vertex (6), the retardation effects associated with heavy ions can play an important role in the many-body theory. In order to develop this point in greater detail let us, for reasons of clarity, ignore the Coulomb contribution to the bare vertex (6). Some simple vertex corrections are shown in Fig. 1. These particular diagrams are chosen because in the one-dimensional case (ij = r/ — 0) they all yield the same > g6log2T contribution to the vertex, provided that the retardation effects are neglected. Such a degenerate situation is usually named parquet. 

We present a detailed calculation of the transition temperature of a model, filamentary excitonic superconductor. The proposed structure consists of a linear chain of transition-metal atoms to which is complexed a ligand system of highly polarizable dye molecules. The model is discussed in the light of recent developments in our understanding of one-dimensional metals. We show that for the structure proposed, the momentum dependence of the exciton interaction results in the superconducting state being favoured over the Peierls state, and in vertex corrections to the electron-exciton interaction which are small. The calculation of the transition temperature is based on what we believe to be reasonable estimates of the strength of the excitonic interaction, Coulomb repulsion and band structure. 

Fig. 23. The self-energy-like Feynman diagrams for the hyperfine-structure splitting in second order in e and first order in the external vector field, (a) and (b) are again wave-function corrections and (c) is the modification due to the perturbation of the propagator, the exact vertex correction.

The vertex correction, replacement Eq. (17), leads to the following expression... 

FIG. 2. Feynman graphs representing the binding energy and vertex correction. In this part the divergences occur only in the zero-potential terms, which are grouped together. F denotes the finite remainder. 

---