We consider fermion-dimer scattering in the presence of a large positive scattering length in the frame of functional renormalization group equations. A flow equation for the momentum dependent fermion-dimer scattering amplitude is derived from first principles in a systematic vertex expansion of the exact flow equation for the effective action. The resummation obtained from the nonperturbative flow is shown to be equivalent to the one performed by the integral equation by Skorniakov and Ter-Martirosian (STM). The flow equation approach allows to integrate out fermions and bosons simultaneously, in line with the fact that the bosons are not fundamental but build up gradually as fluctuation induced bound states of fermions. In particular, the STM result for atom-dimer scattering is obtained by choosing the relative cutoff scales of fermions and bosons such that the fermion fluctuations are integrated out already at the initial stage of the RG evolution.

Magnetic and superconducting instabilities in the two-dimensional t-t'-Hubbard model are discussed within a functional renormalization group approach. The fermionic four-point vertex is efficiently parametrized by means of partial bosonization. The exchange of composite bosons in the magnetic, charge density and superconducting channels accounts for the increase of the effective couplings with increasing length scale. We compute the pseudocritical temperature for the onset of local order in various channels.Comment: 11 pages, 14 figures, equivalent to published versio

The phases with spontaneously broken symmetries corresponding to antiferromagnetic and d-wave superconducting order in the two-dimensional t − t -Hubbard model are investigated by means of the functional renormalization group. The introduction of composite boson fields in the magnetic, charge density, and superconducting channels allows an efficient parametrization of the four-fermion vertex and the study of regimes where either the antiferromagnetic or superconducting order parameter, or both, are nonzero. We compute the phase diagram and the temperature dependence of the order parameter below the critical temperature, where antiferromagnetic and superconducting order show a tendency of mutual exclusion.

The temperature dependence of d-wave superconducting order for two dimensional fermions with d-wave attraction is investigated by means of the functional renormalization group with partial bosonization. Below the critical temperature T c we find superconductivity, a gap in the electron propagator and a temperature dependent anomalous dimension. At T c the renormalized "superfluid density" jumps and the approach to T c from above is characterized by essential scaling. These features are characteristic for a phase transition of the KosterlitzThouless (KT) type.The two-dimensional Hubbard model [1] is often associated with high temperature superconductivity [2]. Its ground state is assumed to be a d-wave superconductor for an appropriate range of doping [3,4]. Under the hypothesis of d-wave superconductivity we study within a simplified effective model the phase transition from the high temperature "symmetric phase" to the low temperature superconducting phase by means of the functional renormalization group. As characteristic features we find (i) essential scaling as the critical temperature T c is approached from above, (ii) a massless Goldstone mode for T < T c leading to superfluidity, (iii) a temperature dependent anomalous dimension η for T < T c , (iv) a jump in the renormalized superfluid density at T c and (v) a gap in the electron propagator for T < T c which vanishes non-analytically as T → T c .The features (i)-(iv) are characteristic for a KT phase transition [5] which is a natural candidate for a universality class with a global U(1) symmetry. Actually, while in the infinite volume limit the "bare" order parameter vanishes in accordance with the Mermin-Wagner theorem, a renormalized order parameter [6] spontaneously breaks the U(1) symmetry for T < T c , leading to an infinite correlation length for the Goldstone mode. Vortices arise as topological defects for T < T c . In the Hubbard model the fermionic excitations (single electrons or holes) are an important ingredient for driving the transition. Finally, coupling our model to electromagnetic fields the photon acquires a mass through the Higgs mechanism while the Goldstone mode disappears from the spectrum, being a gauge degree of freedom. Superfluidity is replaced by superconductivity. Recently, various experimental data have been interpreted as signatures of KT phase fluctuations [7,8,9,10]. The direct relation between real high temperature superconductors and the two-dimensional Hubbard model is difficult, however.The problem of d-wave superconductivity in the Hubbard model can be separated into two qualitative steps in the renormalization flow. In the first step the fluctuations with momenta |p 2 − p 2 F | > Λ 2 (with p F on the Fermi-surface) have to generate a strong effective interaction in the d-wave channel. This phenomenon has indeed been found in renormalization group studies [11,12,13,14,15,16,17,18,19], the d-wave coupling being triggered by spin wave fluctuations. It also has been found in various other approaches, see e.g. [20,21,22]....

Within the two-dimensional repulsive t −t ′ -Hubbard model, an attractive coupling in the d-wave pairing channel is induced by antiferromagnetic fluctuations. We investigate this coupling using functional renormalization group equations. The momentum dependent d-wave coupling can be bosonized by the use of scale dependent field transformations. We propose an effective coarse grained model for the Hubbard model which is based on the exchange of antiferromagnetic and d-wave collective bosons.