Functional renormalization group approach to correlated fermion systems

Abstract

Numerous correlated electron systems exhibit a strongly scale-dependent behavior. Upon lowering the energy scale, collective phenomena, bound states, and new effective degrees of freedom emerge. Typical examples include (i) competing magnetic, charge, and pairing instabilities in two-dimensional electron systems, (ii) the interplay of electronic excitations and order parameter fluctuations near thermal and quantum phase transitions in metals, (iii) correlation effects such as Luttinger liquid behavior and the Kondo effect showing up in linear and non-equilibrium transport through quantum wires and quantum dots. The functional renormalization group is a flexible and unbiased tool for dealing with such scale-dependent behavior. Its starting point is an exact functional flow equation, which yields the gradual evolution from a microscopic model action to the final effective action as a function of a continuously decreasing energy scale. Expanding in powers of the fields one obtains an exact hierarchy of flow equations for vertex functions. Truncations of this hierarchy have led to powerful new approximation schemes. This review is a comprehensive introduction to the functional renormalization group method for interacting Fermi systems. We present a self-contained derivation of the exact flow equations and describe frequently used truncation schemes. Reviewing selected applications we then show how approximations based on the functional renormalization group can be fruitfully used to improve our understanding of correlated fermion systems.

Chirald-wave superconductivity in doped graphene

“…For this, the reader is referred to recent reviews by e.g. Metzner et al [94] and Platt et al [95]. We instead just sketch the main ingredients.…”

“…However, at least one work on the honeycomb lattice has employed a cutoff in Matsubara frequency space [44]. These different cutoff variants have been used in other contexts too [94], and it is the general understanding that, besides some known caveats, the choice of the cutoff is mainly a technical question, without any practical importance for the physical results.…”

“…Here the flow has to be stopped as the approximations, like the neglect of self-energy corrections (discussed in more detail in e.g. [94]), eventually renders the scheme invalid. Nevertheless, important information can be obtained form analyzing the diverging coupling function.…”

“…Most studies, see e.g. the reviews [94] and [95], use a band energy cutoff, such that modes with |ε b (k)| ≥ Λ in band b are integrated out for cutoff Λ. However, at least one work on the honeycomb lattice has employed a cutoff in Matsubara frequency space [44].…”

Theoretical prediction of a strongly correlated Dirac metal

“…All these estimates are orders of magnitude at best, therefore we turn again to a quantitative treatment in terms of multi-sublattice fRG 27,28 . Since superconductivity is an itinerant effect, the inherent fRG assumption of weak coupling should be acceptable.…”