Renormalization group schemes

Microscopic disorder We consider a lattice the sites of which have disordered resonance energies, with a distribution of width Ae, but have the same intersite interactions (same dipole orientation and oscillator strength) as the perfect lattice. This is the so-called substitutional disorder model.122 We assume the disorder width to be smaller than the excitonic bandwidth (4< Be) and examine the bottom of the excitonic band, where the emitting and the absorbing K 0 states lie. In a renormalization-group scheme, we split the lattice into isometric domains of n sites, on which the excitation is assumed to be localized, and write the substitutional-disorder hamiltonian in this basis we thereby obtain a new disorder width An Aen-1/2 and a... 

In the following we elaborate the renormalization group scheme for the theory of SAWs on the percolation cluster as described by the effective Hamiltonian developed in section 2.4. Extending the ideas of Meir and Harris [22] in this respect we refer to this as the MH-model. The motivation for this model is to calculate the average of a logarithm - as usual for a quenched average-... 

In good solvents, the mean force is of the repulsive type when the two polymer segments come to a close distance and the excluded volume is positive this tends to swell the polymer coil which deviates from the ideal chain behavior described previously by Eq. (1). Once the excluded volume effect is introduced into the model of a real polymer chain, an exact calculation becomes impossible and various schemes of simplification have been proposed. The excluded volume effect, first discussed by Kuhn [25], was calculated by Flory [24] and further refined by many different authors over the years [27]. The rigorous treatment, however, was only recently achieved, with the application of renormalization group theory. The renormalization group techniques have been developed to solve many-body problems in physics and chemistry. De Gennes was the first to point out that the same approach could be used to calculate the MW dependence of global properties... 

We discuss here the basic ideas of the renormalization group, using the discrete chain model. This is not the most elegant or powerful approach, and in Part Til of this book we will present a much more efficient scheme. However, the present approach is conceptually the simplest, and it allows us to explain all the relevant features dilatation symmetry and scaling, fixed points and universality, crossover. Furthermore, technical aspects like the e-expansion also come up. We are then prepared to discuss the Qualitative concept of scaling in its general form and to work out some consequences. 

To summarize the renormalization group proves two parameter scaling. The two parameters J q3 z however show a more complicated temperature dependence than assumed in the naive two-parameter scheme. The latter is correct only close to the 0-point. Furthermore the scaling functions take two different forms, representing the weak or the strong coupling branch. 

Among the technical methods proper to the one dimensional geometry, one may cite the Bethe ansatz [19], the bosonization techniques [18], and, more recently, the Density-Matrix Renormalization Group (DMRG) method (20, 21] and a closely related scheme which is directy considered in this note, the Recurrent Variational Approach (RVA) [22, 21], The two first methods are analytical and the third one is numerical the RVA method is in between. 

Recently, Oono [119] and Oono and Kohmoto [120] applied the renormalization group theory to polymer hydrodynamics of the Kirkwood-Riseman scheme. They computed and Pe to first order in e, where e = 4 — d with d being the dimensionality of space, and obtained in three dimensions... 

Here, r is a rescaling parameter, which defines the scale of the external momenta in the minimal subtraction scheme. In the same way as in the Callan-Symanzik equation (77) the coefficients in (86) define the renormalization group functions ... 

Experimental values of g and pg are considerably influenced by the polydispersity of polymer samples used however, both g and pg are universal to a first approximation, the former being mostly found to be in the range (2.0 - 2.7) X lO mor and the latter in the range 1.25-1.35 (6.15-5.70 for Fg). Recently, Oono and Kohmoto [10,11] applied renormalization-group theory to the polymer hydrodynamics of the Krikwood-Riseman scheme and computed the values of g =2.36 x lO moP and Fg = 6.20, which compares rather favorably with experimental values. 

Because of an additive separability of tgci (this follows from McWeeny s theory of generalized group functions [42]), renormalizing this matrix is a permissible operation. Thus, after renormalizing to unify, the roci becomes the main object to be analyzed within the ESSA/GCI scheme we propose here. Hence, in most relations of Sect. 14.3 we must make the replacement... 

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