# Functional renormalization group for interacting electron systems (fRG)

The functional renormalization group (fRG) is a source of powerful computation tools for interacting Fermi systems, especially for low-dimensional systems with competing instabilities and entangled infrared singularities. Our group participates strongly in the development of fRG methods, with a focus on one-dimensional non-Fermi liquid metals and two-dimensional systems with magnetic and superconducting instabilities, such as the 2D Hubbard model.

# Functional renormalization group methods

Renormalization group methods have a long history in quantum field theory and statistical physics. In the context of solid state physics they have been applied since the 1970s, initially mainly in impurity problems and critical phenomena. There RG schemes were used to sum the leading divergencies in perturbation theory and typically only a small number of coupling constants had to be considered. With the advent of high-temperature superconductivity new interest in non-Fermi liquid behavior in interacting Fermi systems arose, and renormalization group methods have proved valuable in the analysis of both the stability and also the instability of the Fermi liquid state. More recently it has been realized that the so-called functional renormalization group (fRG) is a source of new powerful computation tools for interacting Fermi systems, especially for low-dimensional systems with competing instabilities and entangled infrared singularities. Instead of renormalizing only a small number of coupling constants, the fRG captures in principle the complete vertex or correlation functions of the system, which leads to a much more flexible framework.

At the heart of the fRG methods is an exact hierarchy of differential flow equations for the Green or vertex functions of the system, which is obtained by taking derivatives with respect to an energy scale Λ. Usually the scale parameter Λ is taken to be an infrared cutoff in momentum or frequency space, but other choices such as the temperature are possible as well. Approximations are then constructed by truncating the hierarchy and parametrizing the vertex functions with a manageable set of variables or functions. The truncation procedure is guided by small parameters and physical intuition.

**Figure 1:** First equations in the hierarchy of flow equations for the vertex functions in the one-particle irreducible version of the fRG. The propagators with a slash contribute only at the energy scale Λ.

In our group this approach has so far mainly been applied to the two-dimensional Hubbard model, to impurities in one-dimensional metals, and to quantum criticality in metallic systems. For a comprehensive review of the fRG method and many applications, see Rev. Mod. Phys. **84**, 299 (2012).

# The 2D Hubbard model: magnetism and superconductivity

The Hubbard model serves as a minimal model for a number of fascinating phenomena in condensed matter physics. Originally it has been introduced as a model for ferromagnetism in itinerant electron systems, and for the Mott metal-insulator transition. With the discovery of high-temperature superconductivity in the layered copper oxides in 1986 by Bednorz and Müller and the subsequent seminal suggestions by P.W. Anderson, the one-band Hubbard model and its strong coupling variant, the t-J model, have also become a standard model for theories of high-temperature superconductivity.

**Figure 2:** The Hubbard model describes a single band of tight-binding electrons, supplemented by a Coulomb repulsion U which acts only between two electrons (with opposite spin) situated on the same lattice site. The kinetic energy is given by hopping amplitudes between neighboring sites, which leads to a dispersion relation with a band width W. The local nature of the Coulomb interaction can be justified by screening.

The Mott transition and presumably also high-temperature superconductivity fall into the strong coupling domain of the model, where the interaction strength U is comparable or even larger than the bandwidth W. These parameter values are beyond the range of validity of perturbative methods. However, high-temperature superconductors show several phenomena which can be captured by functional RG approaches. Thus one may understand certain aspects of the problem such as the preferred superconducting order parameter and Fermi surface properties by using functional RG techniques at weaker interaction strengths. Furthermore the weakly interacting Hubbard model is quite interesting in its own right, as it exhibits a number of competing Fermi surface instabilities whose detailed understanding may be useful in other contexts.

By now frequently observed in correlated electrons materials is the occurrence of superconductivity in the vicinity of magnetism. A successful theoretical description should give the correct type of magnetic order and of the superconducting state as well. In this context functional RG methods provide a very useful probe into the low temperature phase of model systems. They have illustrated clearly how d-wave superconductivity can arise near antiferromagnetic spin-density wave ordering or p-wave supercondictivity occurs close to ferromagnetism. Although there are a number of theoretical approaches that produce phase diagrams with the similar qualitative features, functional RG methods have the advantage that they treat the emergence and interference of different ordering tendencies in an unbiased way. Therefore they are ideally suited for a quick and transparent exploration of the parameter space of the relevant model in order to assess the relative strength of the different tendencies and find the dominant effects.

# Local experts

- Andreas Eberlein (a.eberlein@fkf.mpg.de)
- Nils Hasselmann (n.hasselmann@fkf.mpg.de)
- Walter Metzner (w.metzner@fkf.mpg.de)
- Benjamin Obert (b.obert@fkf.mpg.de)
- Hiroyuki Yamase (h.yamase@fkf.mpg.de)

Our group enjoys collaborations with: - M. Salmhofer (Heidelberg)
- C. Honerkamp (Aachen)

# Key publications

- W. Metzner, M. Salmhofer, C. Honerkamp, V. Meden, and K. Schönhammer

Functional renormalization group approach to correlated fermion systems

Rev. Mod. Phys.**84**, 299 (2012)

- J. Bauer, P. Jakubczyk, and W. Metzner
Phys. Rev. B

Critical temperature and Ginzburg region near a quantum critical point in two-dimensional metals**84**, 075122 (2011)

- P. Strack, R. Gersch, and W. Metzner
**Renormalization group flow for fermionic superfluids at zero temperature**

Phys. Rev. B**78**, 014522 (2008)

- J. Reiss, D. Rohe, and W. Metzner
**Renormalized mean-field analysis of antiferromagnetism and d-wave superconductivity in the two-dimensional Hubbard model**

Phys. Rev. B**75**, 075110 (2007)

- T. Enss, V. Meden, S. Andergassen, X. Barnabé-Thériault, W. Metzner, and K. Schönhammer
**Impurity and correlation effects on transport in one-dimensional quantum wires**

Phys. Rev. B**71**, 155401 (2005)

- S. Andergassen, T. Enss, V. Meden, W. Metzner, U. Schollwöck, and K. Schönhammer
**Functional renormalization group for Luttinger liquids with impurities**

Phys. Rev. B**70**, 075102 (2004)

- C.J. Halboth and W. Metzner
**Renormalization group analysis of the two-dimensional Hubbard model**

Phys. Rev. B**61**, 7364 (2000)

- C.J. Halboth and W. Metzner
*d*-Wave Superconductivity and Pomeranchuk Instability in the Two-Dimensional Hubbard Model

Phys. Rev. Lett.**85**, 5162 (2000)