Logarithmic singularity

That analyticity was the source of the problem should have been obvious from the work of Onsager (1944) [16] who obtained an exact solution for the two-dimensional Ising model in zero field and found that the heat capacity goes to infinity at the transition, a logarithmic singularity tiiat yields a = 0, but not the a = 0 of the analytic theory, which corresponds to a finite discontinuity. (Wliile diverging at the critical point, the heat capacity is synnnetrical without an actual discontinuity, so perhaps should be called third-order.)... 

Added in Proof.] We do not here discuss the logarithmic singularities which occur in the virial expansion and have recently been reported by I. Oppenheim and K. Kawasaki [Phys. Rev. 139A, 1763 (1965)]. 

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).)...

Fig. 6.6 The wave-vector dependence of the energy-wavenumber characteristic, ( ) which has a node at q0 and a weak logarithmic singularity in its slope at q = 2kF. Also shown are a set of degenerate cubic reciprocal lattice vectors that are centred on q0. A tetragonal distortion would lift their degeneracy away from the node at q0 as shown, thereby lowering the band-structure energy. (After Heine and Weaire (1970).)...

The energy-wave-number characteristic, ( )> depends only on the density of the free-electron gas and the nature of the pseudopotential core but not on the structural arrangement of the atoms. Its behaviour as a function of the wave vector, q, is illustrated in Fig. 6.6, where we see that it vanishes at q0 as expected. It also has a weak logarithmic singularity in its slope at q = 2kF. 

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... 

In practice, however, the electron-hole symmetry relation (16) is never perfectly satisfied. There are deviations due to small corrections neglected in the spectrum (2). Following a lattice compression, the electron spectrum is altered and these deviations magnify under pressure and tend to suppress the logarithmic singularity (17) T, thus rapidly decreases and even vanishes above some critical pressure (Fig. 5). 

The model is based at the outset from the observation that a ID non-interacting Fermi gas with a two-point Fermi surface at k° gives rise to two infrared logarithmic singularities in the response of the system to correlate pair of particles either in the electron-electron (Cooper) or in the 2k electron-hole (Peierls) scattering channel. These are of the form... 

In a similar way the leading cut off dependence of Fp(Fig. 11.1c) can be absorbed into a renormalization factor Z2/Z multiplying the density insertion itself. Finally consider structures of type (L, M) = (0,1) (Fig. 11.1a), which for d — 4 show 5 — 2. Mass subtraction reduces 5 to 5 = 0, which still corresponds to logarithmic singularities. These singularities can be absorbed into a redefinition of — c (Eq. (7.25))... 

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