Renormalization additive

This equation describes not only the crystal growth, but with an additional noise term it also describes the evolution of the surface width and is called the Edward-Wilkinson model. An even better treatment has been performed by renormalization methods and other techniques [44,51-53]. 

An additional approximation is introduced here elements of the H2hp,2ph block are neglected. Since this block vanishes identically when HF reference states are used, the present approximation may be regarded as an improvement to the so-called 2p-h TDA [7, 23, 24] method with orbital and reference-state renormalizations [25, 26, 27]. 

The lattice gas model of Bell et al. [33] neither gave any detailed mechanism of the orientational ordering nor separated the contributions of the headgroup and the acyl chain. Lavis et al. [34] discussed Ref. 33 critically and concluded that the sharp kink point in the isotherm at transition was an artifact of the mean field approximation used. An improved correspondence to experimental data was claimed by the use of the real-space renormalization group method [35]. The same authors returned to the problem [35] and concluded that in addition to the orientation of the molecules, chain melting had to be included in a model which could interpret the phase transitions. 

Here the L" and R" renormalization tensors are uniquely specified by the additional orthonormality constraints... 

Here p iaa occ, L() (respectively p iaa unocc, L()) represents the probability of the atomic configuration of site i, where the orbital a with spin a is occupied (resp. unoccupied) and where L[ is a configuration of the remaining orbitals of this site. This result is similar to the expression obtained by Biinemann et al. [22], but it is obtained more directly by the density matrix renormalization (5). To obtain the expression of the qiaa factors, an additional approximation to the density matrix of the uncorrelated state was necessary. This approximation can be viewed as the multiband generalization of the Gutzwiller approximation, exact in infinite dimension [23]... 

Mass renormalization requires [15] that an additional term vj/yvv /5m be added where 5m is the difference between the physical and bare masses [77]. 

As A x was supposed stationary the integral is independent of time. The effect of the fluctuations is therefore to renormalize A0 by adding a constant term of order a2 to it. The added term is the integrated autocorrelation function of At. In particular, if one has a non-dissipative system described by A0, this additional term due to the fluctuations is usually dissipative. This relation between dissipation and the autocorrelation function of fluctuations is analogous to the Green-Kubo relation in many-body systems 510 but not identical to it, because there the fluctuations are internal, rather than added as a separate term as in (2.1). 

Finally, note that the relaxation equation [Eq. (76)] is usually written in terms of the hydrodynamic modes. In many problems of chemical interest, nonhydrodynamic modes such as intramolecular vibration, play an important role [50]. Presence of such coupling creates an extra channel for dissipation. Thus, the memory kernel, T, gets renormalized and acquires an additional frequency-dependent term [16, 43]. 

Even for d < 4 the question of existence of the continuous chain limit is not completely trivial. The problem is most easily analyzed by taking a Laplace transform with respect to the chain length, which results in the held theoretic representation of polymer theory. In field theory it is not hard to show that the limit — 0 can be taken only after a so-called additive renormalization we first have to extract some contributions which for — 0 would diverge. The extracted terms can be absorbed into a 1 renormalization he. a redefinition of the parameters of the model. Transfer riling back to polymer theory we find that this renormalization just shifts the chemical potential per segment. We thus can prove the following statement after an appropriate shift of the chemical potential the continuous chain limit for d < 4 can be taken order by order in perturbation theory. In this sense the continuous chain model or two parameter theory are a well defined limit of our model of discrete Gaussian chains. 

We indeed encountered additive renormalization before. In our calculation of f.ip(n) (Sect. 5.4.3) we had to subtract the single chain part from the one loop correction, before we could replace the sum over discrete segments by the Debye function, which is calculated for a continuous chain. (See Eq. (3.22).) In the result (5.6 ) this introduced the term -n/ c (f) to shift the chemical potential. All the... 

This redefinition is another example of additive renormalization. In principle it can be implemented perturbatively to all orders. As a result, not the mathematics of our model but the interpretation of changes. 0K sums up contributions from all higher interactions and thus should be viewed as a parameterization of the interaction among whole strands of the chains rather than as a true two-segrnent interaction. This also makes clear that there is no hope to calculate 0e from a realistic model. 

As mentioned above, the matter is quite involved and we will only sketch the arguments. Discussing some low order terms we will demonstrate the additive renormalizations due to higher order interactions (Sect. 10.1). Then we will discuss the general ideas on the structure of the renormalization group, defining important concepts like relevance or irrelevance of interactions or the critical manifold (Sect. 10.2). Concerning the field theoretic realization of the RG, we will summarize some results (Sect. 10.3). 

This argument is nothing but the additive renormalization discussed in Sect 7.2 in the context of the continuous chain limit, now interpreted on the level of the interaction. From the field theoretic formulation we easily see that it holds to all orders. We thus may split the two-body interaction into a one-body part which effectively takes care of interactions within small fractions of the chains, and a remainder conforming to the estimate (10.1). 

We finally note that this discussion gains additional importance with respect to the continuous chain limit. In Chap. we have shown that we can construct the continuous chain model only after an additive renormalization. which essentially extracts a one-body part from the two-body potential. If we... 

This on the one hand produces additional renormalization factors and on the other hand identifies Zn as... 

We know that the additively renormalized bare theory (the left hand side in Eq. (11.10)) exists in that limit. Also the -factors attain finite limits Z(ti), Zu(u), Zn(u). Indeed they are constructed as power series in u, the coefficients for t > 0,6 > 0 taking the form of polynomials in / r) . Thus no problem results from setting t = 0. Since the renormalized theory is finite for d = 4, whereas the bare continuous chain model diverges for d —> 4, showing poles in 5, also the Z-factors must diverge for d —> 4. In the NCL we can therefore formulate the theorem of renormalizability as follows ... 

Similar problems are abundant as soon as we leave the region of small momenta and isolated chains. As a final example we consider the semidilute limit. Using the unrenormalized loop expansion in Sect, 5.4.3 we have calculated the first order correction to fip(n). We found a correction of order where c is the segment concentration. The form of this term is due to screening and has nothing to do with the critical behavior treated by renormalization and -expansion. It thus should not be expanded in powers of e. We can trace it back to the occurrence of the size of the concentration blobs as an additional length scale. 

---