Renormalization Group and Interacting Electron Systems

Alex C. Hewson, D. Meyer

Walter Metzner

V. Meden, Ulrich Schollwöck

The persistent current in a lattice model of a one-dimensional interacting electron system is systematically studied using a complex version of the density-matrix renormalization group algorithm and the functional renormalization group method. We mainly focus on the situation where a single impurity is included in the ring penetrated by a magnetic flux. Due to the interplay of the electron-electron interaction and the impurity the persistent current in a system of N lattice sites vanishes faster than $1/N.$ Only for very large systems and large impurities our results are consistent with the Bosonization prediction obtained for an effective field theory. The results from the density-matrix renormalization group and the functional renormalization group agree well for interactions as large as half the bandwidth, even though as an approximation in the latter method the flow of the two-particle vertex is neglected. This confirms that the functional renormalization group method is a very powerful tool to investigate correlated electron systems. The method will become very useful for the theoretical description of the electronic properties of small conducting ring molecules.

Claudio Chamon, Andreas W. W. Ludwig, Chetan Nayak

We develop a dynamical approach based on the Schwinger-Keldysh formalism to derive a field-theoretic description of disordered and interacting electron systems. Within this formalism we calculate the perturbative renormalization-group (RG) equations for interacting electrons expanded around a diffusive Fermi liquid fixed point, as obtained originally by Finkelstein (Zh. \'Eksp. Teor. Fiz. 84, 168 (1983) [Sov. Phys. JETP 57, 97 (1983)]) using replicas. The major simplifying feature of this approach, as compared to Finkelstein's, is that instead of $\stackrel{\ensuremath{\rightarrow}}{N}0$ replicas, we only need to consider $N=2$ species. We compare the dynamical Schwinger-Keldysh approach and the replica methods, and we present a simple and pedagogical RG procedure to obtain Finkelstein's RG equations.

Seiler, Christian

We apply the functional renormalization group method (FRG) to the problem of interactions in disordered electron systems. To achieve this we develop an approximation that allows us to simplify the FRG equations in inhomogeneous systems in order to make computations feasible. For a model system we demonstrate the efficacy of our method by comparison with exact results for small system sizes. We utilize our method to investigate the phase diagram of this model system.