Fermi singularity

This is an application of Fermi s golden rule. The first term is the square of the matrix element of the perturbation, which appears in all versions of perturbation theory. In the second term 8(x) denotes the Dirac delta function. For a full treatment of this function we refer to the literature [2]. Here we note that S(x) is defined such that S(x) = 0 for x 7 0 at the origin S(x) is singular such that / ( r) dx — 1. The term 8 (Ef — Ei) ensures energy conservation since it vanishes unless... 

Fig. 6.3 The wave-vector dependence of the Lindhard response function. X(<7/2Af), which has been normalized by the constant Thomas-Fermi response function, xtf The dashed curve shows an approximation (eqn (6.89)) to the Lindhard response function that does not include the weak logarithmic singularity in the slope at q 2kF = 1 (From Pettifor and Ward (1984).)...

Figure 1 2 1. The different types of 2.5 Lifshitz electronic topological transition (ETT) The upper panel shows the type (I) ETT where the chemical potential EF is tuned to a Van Hove singularity (vHs) at the bottom (or at the top) of a second band with the appearance (or disappearance) of a new detached Fermi surface region. The lower panel shows the type (II) ETT with the disruption (or formation) of a neck in a second Fermi surface where the chemical potential EF is tuned at a vHs associated with the gradual transformation of the second Fermi surface from a two-dimensional (2D) cylinder to a closed surface with three dimensional (3D) topology characteristics of a superlattice of metallic layers...

The interband pairing term enhances Tc [93-97,102] by tuning the chemical potential in an energy window around the Van Hove singularities, z =0, associated with a change of the topology of the Fermi surface from ID to 2D (or 2D to 3D) of one of the subbands of the superlattice in the clean limit. 

Figure 1 2 6. The Fermi surface of the second (red) and third subband (black) of a 2D superlattice of quantum wires near the type (III) ETT where the third suhhand changes from the one-dimensional (left panel) to two-dimensional (right panel) topology. Going from the left panel to the right panel the chemical potential EF crosses a vHs singularity at Ec associated with the change of the Fermi topology going from EF>EC to EF

The main equation for the d-electron GF in PAM coincides with the equation for the Hubbard model if the hopping matrix elements t, ) in the Hubbard model are replaced by the effective ones Athat are V2 and depend on frequency. By iteration of this equation with respect to Aij(u>) one can construct a perturbation theory near the atomic limit. A singular term in the expansions, describing the interaction of d-electrons with spin fluctuations, was found. This term leads to a resonance peak near the Fermi-level with a width of the order of the Kondo temperature. The dynamical spin susceptibility in the paramagnetic phase in the hydrodynamic limit was also calculated. 

We see that the first order correction to the terminal part contains a Kondo-like singularity, which produces a resonance peak at the Fermi-level. The width of the peak E is determined from Eq.(36) by summing... 

We applied the generating functional approach to the periodic Anderson model. Calculation of the electron GFs gdd, 9ds, 9sd and gss reduces to calculation of only the d-electron GF. For this, an exact matrix equation was derived with the variational derivatives. Iterations with respect to the effective matrix element Aij(to) allow to construct a perturbation theory near the atomic limit. Along with the self-energy, the terminal part of the GF Q is very important. The first order correction for it describes the interaction of d-electrons with spin fluctuations. In the paramagnetic phase this term contains a logarithmic singularity near the Fermi-level and thus produces a Kondo-like resonance peak in the d-electron density of states. The spin susceptibility of d-electrons... 

The second symmetry is the consequence of the local symmetry of the density of the states of the resonant band. Finite, small t modifies strongly the dispersion of the conducting band in the vicinity of the van Hove singularities, adding extra states below the Fermi level. The Fermi energy at the half-filling is therefore shifted from the van Hove singularity towards the (n, w) point when t < 0. 

Thus the band-properties have the symmetry with respect to the doping 5C [8] required to bring the Fermi energy back to the van Hove singularity, as observed in the transport data. 

The RPA structure can be recognized in the denominator. It is built on the logarithmic bare response of the two-dimensional Fermi surface. The dominant singularity is singled out by the most negative of the W (Q0, i ). This signals the occurrence of an instability at the mean-field temperature T°c T exp(-2INaWa) (BCS-like equation). 

Spin and charge excitations are thus decoupled by coulombic interactions in the one-dimensional electron gas. However, the one-dimensional Fermion system is not a Fermi liquid, as indicated by the behavior of the momentum distribution function, which does not exhibit a Fermi step at kF and presents a single-particle density of states vanishing according to a power law singularity at EF. This is a Luttinger liquid [29] with... 

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