Renormalized vertex

In a sense, these vertex renormalization factors play about the same role (in a different connection) as the dielectric constants in classical physics. 

It should be noted that these equations are to be solved for each position of the centroid q. The frequency in Eq. (2.27) is the same as the effective frequency obtained for the optimized LHO reference system using the path-integral centroid density version of the Gibbs-Bogoliubov variational method [1, pp. 303-307 2, pp. 86-96], Correspondingly, Eqs. (2.27) and (2.28) are exactly the same as those in the quadratic effective potential theory [1,21-23], The derivation above does not make use of the variational principle but, instead, is the result of the vertex renormalization procedure. The diagrammatic analysis thus provides a method of systematic identification and evaluation of the corrections to the variational theory [3],... 

For low order calculations of the scaling functions a variety of implementations of the RG have been used. The present formulation has grown out of the work [Sch84]. The basic philosophy is the same, but in this earlier work the renormalization scheme was based on field theoretic renormalization conditions1. This amounts to using a non-minimally subtracted theory, where the Z-factors are determined by imposing specific values to certain renormalized field theoretic vertex functions. The renormalized coupling, for instance, is defined as the value of (qi, qa, qg. qi) at some special momenta of order... 

Figure 3.16. A few interactions contributing to the Raman-Stokes process (with creation of one phonon). The propagators are labeled with the notation of the text. The process a is of the first order b could be included by renormalization of the created exciton c would require the renormalization of the interaction (or the vertex).

Figure 1. Illustration of singlet Cooper-pairing due to spin excitations. They are described by a spin susceptibility x(q, <-o) and interact with electrons via a (renormalized) vertex U (U ).

The shaded square represents the renormalized vertex involved in the usual Bethe--Salpeter equation... 

Equ. (14) has the same algebraic structure as has the exact equation for/7, except that it. RPA the matrices are replaced by numbers. From this it is dear that Equ. (16) for TI could have been obtained alternatively by saying that both the bare vertex in the denominator and the renormalized vertex in the nominator of Equ. (14) are small due to the smallness of D0 for kBT > h >0 (< ) (The Coulomb contribution is already omitted here). But to remain consistent, we have to regard the external lines D0 in Equs. (8) or (9) as also being small, i. e. to arrive to D D0, far from the untrivial singularity in D. 

We can associate renormalized vertex functions with the renormalized Green s functions, by proceeding as was done in Chapter 11, Section 5.2.2. It is easy to verify that... 

In order to define a theory that remains finite when a - ac, we have of course to express the renormalized vertex functions in terms of quantities having a physical meaning, and we shall study this point below more precisely. 

In field theory, one renormalizes vertex functions or, in an equivalent way, Green s functions. The vertex functions of field theory have no simple counterparts in polymer theory because the p-reducible diagrams which are factorizable... 

Fig. 12.9. Vertices or insertions requiring the introduction of renormalization factors. A wave vector is injected by the (dashed) interaction line at each vertex. If this wave vector vanishes, the interaction line may be omitted.

To renormalize the partition functions associated with such configurations, it is neoessary to introduce a renormalization factor 3E(zlt..., zM) for the vertex. Such renormalization factors must be used... 

In field theory, one uses a parameter related to the behaviour of the renormalized four-leg vertex and incorrectly called the renormalized interaction . In a very similar way, in polymer theory, the second virial coefficient can be used to define the interaction between two polymers. We proceed as follows. 

In order to justify the conjecture made by Cardy and Hamber, Nienhuis used a cascade of models (including an unsolved six-vertex model)4 and equivalences that are more or less exact finally, he came down to a two-dimensional Coulomb gas. This gas is made of positively and negatively charged particles in interaction, the interaction potential being proportional to Inr where r is the distance between two charges. Then, Nienhuis could apply to this system approximate renormalization techniques which enabled him to predict the critical properties of the system. 

Let us now assume that the chain interacts with itself. 2(6,0 S) is the sum of the contributions of the ring diagrams (see Fig. 13.6). As these diagrams have no free ends, their renormalization is trivial the only renormalization factor is the swelling 3t0(z). In fact, these ring diagrams are characterized by the insertion of one two-leg vertex (represented by a point in Fig. 13.6) but, as was shown in Chapter 12, Section 3.3.3.4, the renormalization factor of two-leg vertices can be chosen equal to unity. 

In the limit a ae, the vertex functions diverge, but the divergences can be absorbed by renormalization. Thus, in agreement with (12.3.3), we set... 

Let us now consider these renormalized vertex functions. In the critical domain, they depend only on the correlation length . The nature of this dependence can be found by examining the dimension equations of various quantities. The normal dimensions of cp(r) and of its Fourier transform [Pg.620]

Renormalized coupling constant in field theory number proportional to the four-leg vertex function (for vanishing external wave vectors)... 

The renormalized vertex (138) is connected with the regularized expression... 

The next step is the actual renormalization procedure. The crucial observation for both the physical interpretation as well as the technical success of this step is the fact that the divergent contributions to the three relevant functions have the same structure as the corresponding free propagators and the free vertex The divergent part of is just proportional to and m, but not e.g. to p, the divergent part of repeats the tensor structure of D y, Eq.(203), and the divergent part of F is proportional to the free vertex 7 (but does not... 

The collection of diagrams above represents all possible contributions to the centroid-constrained correlation function. In fact, all the decorations attached to the intermediate vertices can be removed if the vertex is renormalized. This operation is carried out by replacing all the AK s by AV s, giving... 

In contrast to the Raman scattering results a temperature-dependent softening of the mode is observed between 300 and 5 K in phononic neutron scattering (Reichart and Niicker 1983). This has been attributed by Thalmeier (1984) to the temperature dependence of the vertex corrections, which have to be taken into account in higher-order perturbation theory. The vertex corrections yield a temperature-dependent renormalization of the phonon frequency proportional to the difference in occupation of the F and Fg CEF levels. 

---