Logarithmic divergence

As 11(a) approaches a constant as a oo the function II(p2) as defined by Eq. (10-97) is logarithmically divergent. The once subtracted dispersion relation is well defined, and reads... 

This in general gives a logarithmic divergence, as shown by Nozieres and De Dominicis, so the transition probability is infinite. The exciton state, if one exists, is of course filled, and this model provides one description of the way the hole is screened (see Friedel 1952a, b). 

It is found that the D(t) does show a logarithmic divergence as expected due to the presence of the t 1 tail in the VACF. The value of the slope (Aj) is 0.0091, which is in good agreement with those obtained from the simulations (0.0079 from VACF and 0.0086 from MSD). 

In the case of an ohmic bath, s = 1, the integrand in k(T) scales as 1 /ujp, p = 1, 2 and has thus a logarithmic divergence at the lower integration limit. Thus, the MF contribution would vanish. In other words, no gap would exist on this approximation level. 

Maybe the major achievement provided by the discovery of the fe rfim is the possibility to study the complete set of critical exponents on a ferroic system for the first time after their prediction [9,10]. Table 15.1 shows the results as compared with predictions from theory and simulations. Most remarkably, the order parameter exponent ft (Figure 15.10) clearly deviates from the prediction ft 0 and achieves a value which comes close to that observed recently on the standard rfim system, the dilute uniaxial antiferromagnet Fci ,Zn.,F2, x = 0.15, in an external magnetic field [50], Further, the most disputed value, namely the specific heat exponent a [48] (Figure 15.12) clearly describes the same logarithmic divergence as that found on Fci. Zn. I 2, a 0 [10], which still lacks theoretical confirmation. 

One of the two integrals (4.67) may be evaluated analytically the second one (elliptic) may be calculated numerically by the trapezoidal method with the introduction of a small imaginary part in (4.67) to avoid spurious oscillations due to the finite number of integration points. The resulting density-of-states function is shown in Fig. 4.9. The band is asymmetric because of the equivalent term Vx j it exhibits three van Hove singularity points two discontinuities at the boundaries, and one logarithmic divergence corres-... 

Figure 4.9. Density of states of the naphthalene Nhg triplet excitonic band. We note the asymmetric form due to the coexistence of equivalent and nonequivalent interactions, and the three van Hove points connected with the long-range order two discontinuities at the boundaries (at — 3.S and 5.8 cm1) ane one logarithmic divergence (at —1cm"1), characteristic of 2D dispersion. The optical absorption occurs at the two ends of the band (the a and b Davydov components).

The logarithmic divergencies at co = 0 are truncated by analyticity in the low-energy range. 

Since experiments on polymer mixtures in thin film geometry are often carried out in the intention to examine wetting behavior [69,71,81-83], we discuss here also the behavior of < )s as function of (]). In the semi-infinite case, we have a logarithmic divergence when phase coexistence is approached, (cf. Fig. 6b)... 

Following the above equation it is obvious that if a plot of 1 /T versus 1 /if is linear, the system is categorized as ID. At low frequency, the ID diffusion breaks down because of inter-chain hopping and 2D or 3D behaviour is expected. In two dimensions, /(co) displays a logarithmic divergence, while in three dimensions, it is nearly constant. The crossover between ID and 2D or 3D regimes occurs at cu sD , which is the inter-chain diffusion rate.106... 

The Fermi surface departs from perfect nesting when the second harmonic contribution in Eq. (27) becomes a relevant contribution, namely if t[ T0. Thus the nesting of the Fermi surface is frustrated and the susceptibility Xo( 0 but only a relative (nondivergent as T —> 0) maximum at a... 

Notice that this capillary wave term alone cannot represent the whole correlation because it logarithmically diverges as Ko(es) — log(es) for s -> 0. The singularity 1/p can also be derived from the condition of existence of an inhomogeneous solution to the LMBW equation (2). Indeed, the matrix equation (40) has non-tiivial solutions p 0 at ext -> 0 if and only if... 

A simple feature of logarithmic divergences like (19) is the lack of a particular scale in the energy interval between Ep and T. Another way to put it is to say that... 

Response functions. The elementary Cooper and Peierls logarithmic divergences (19) of the interacting electron gas are also present order by order in the perturbation theory of response functions in the 2kf density-wave and superconducting channels. A scaling procedure can thus be applied in order to obtain the asymptotic properties of the real part of the retarded response functions which we will note Xm.( )- is convenient to introduce auxiliary response functions noted x ( ) [107], which are defined... 

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