F. Ravndal, Scaling and Renormalization Groups, Nordita, Kobenhaven, 1976, v+284ページ, 21×14.5cm.

Silvio R. A. Salinas

J. F. Rentrop

Technological advances in the fabrication of mesoscopic structures allow for an experimental investigation of low-dimensional interacting many-body systems in nonequilibrium situations. From a theoretical perspective, the accurate description of such systems constitutes a major challenge which has in general not been solved. In light of possible applications, such as quantum computation, but also for a better understanding of correlation effects themselves, it is thus important to put a research focus on methodological advances.The so called functional renormalization group (fRG) has turned out to be a successful and versatile tool in the application to the aforementioned systems. Being an RG method, it has the associated advantages, e.g. a successive integrating out of energy scales and avoiding divergences that perturbation theory would have. In this thesis, we present two paradigmatic extensions of the fRG approach.Typically, the fRG is used to investigate the flow of the generating functional of the one-particle-irreducible (1PI) vertex functions. Typical approximations in this framework bring about certain shortcomings, such as the violation of conservation laws. In order to overcome some of these shortcomings, we formulate the fRG for the generating functional of the two-particle-irreducible (2PI) vertex functions. This step constitutes our first extension of the fRG approach. While typically only nonconserving schemes are generated in 1PI fRG, we can define nonconserving and conserving approximation schemes within 2PI fRG. Partially, these are found to be equivalent to known diagrammatic approximations. We apply the various 2PI fRG schemes to the quantum anharmonic oscillator, as a “toy model”, and to the Anderson impurity model, as a full fermionic quantum many-body problem. Overall, we find that the devised 2PI fRG schemes (even the conserving ones) offer no great advantage over established 1PI methods. As an exception, the nonconserving lowest-order 2PI fRG scheme in which the flow parameter is introduced in the interaction produces better results than 1PI fRG schemes which are comparable in computational effort.Our second extension of the fRG approach is to apply it to models which cannot (directly) be formulated in second quantization. Having a localized spin, the Kondo model is a simple example of such models which nevertheless shows nontrivial (but well-understood, e.g. by Anderson’s Poor Man’s Scaling) correlation effects. It has hardly been treated with fRG in the literature. In this thesis, we present how to overcome obstacles arising in the formulation of the Kondo model in functional integral formalism. Instead of the 1PI fRG, we use the Polchinski fRG which is based on connected amputated Green functions. This makes it possible to make a close connection to Poor Man’s Scaling which employs the concept of the T-matrix. We reproduce the results of Poor Man’s Scaling and also devise a systematic extension to it by keeping the frequency dependence of the flowing amputated Green functions in the RG equations. With this scheme, we do not find a significant qualitative improvement over Poor Man’s Scaling. However, we have layed the groundwork for more successful future extensions of the fRG applied to the Kondo model.

Dirk Kreimer

Stefan Alexander Maier, Andreas Eberlein, Carsten Honerkamp A

We present a functional renormalization group (fRG) formalism for interacting fermions on lattices that captures the flow into states with commensurate spin-density-wave order. During the flow, the growth of the order parameter is fed back into the flow of the interactions and all modes can be integrated out. This extends previous fRG flows in the symmetric phase that run into a divergence at a nonzero RG scale, i.e., that have to be stopped at the ordering scale. We use the corresponding Ward identity to check the accuracy of the results. We apply our method to a model with two Fermi pockets that have perfect particle-hole nesting. The results obtained from the fRG are compared with those in random-phase approximation.

Francesco Caravelli, Diego A. R. Dalvit

In the previous chapters, we introduced the scaling hypothesis and saw how Kadanoff's block-spin idea provides a conceptual bridge between microscopic models and macroscopic critical behavior. We also saw how Widom's scaling form and the resulting relations among critical exponents emerge from the assumption of scale invariance near the critical point. We are now ready to formalize these ideas into a mathematical framework: the renormalization group (RG).

Eduardo Fradkin