Renormalization group flow for fermionic superfluids at zero temperature

TL;DR

This paper uses the functional renormalization group to analyze quantum fluctuations in fermionic superfluids, revealing significant suppression of the order parameter and universal infrared behavior in two and three dimensions.

Contribution

It introduces a truncated flow equation approach for fermion-boson systems, capturing fluctuation effects on both universal and non-universal properties of superfluids.

Findings

Fermionic gap is significantly reduced by fluctuations.

Longitudinal fluctuations vanish linearly in 2D and logarithmically in 3D.

Transverse Goldstone mode causes strong renormalization of longitudinal fluctuations.

Abstract

We present a comprehensive analysis of quantum fluctuation effects in the superfluid ground state of an attractively interacting Fermi system, employing the attractive Hubbard model as a prototype. The superfluid order parameter, and fluctuations thereof, are implemented by a bosonic Hubbard-Stratonovich field, which splits into two components corresponding to longitudinal and transverse (Goldstone) fluctuations. Physical properties of the system are computed from a set of approximate flow equations obtained by truncating the exact functional renormalization group flow of the coupled fermion-boson action. The equations capture the influence of fluctuations on non-universal quantities such as the fermionic gap, as well as the universal infrared asymptotics present in every fermionic superfluid. We solve the flow equations numerically in two dimensions and compute the asymptotic behavior analytically in two and three dimensions. The fermionic gap \Delta is reduced significantly compared to the mean-field gap, and the bosonic order parameter \alpha, which is equivalent to \Delta in mean-field theory, is suppressed to values below \Delta by fluctuations. The fermion-boson vertex is only slightly renormalized. In the infrared regime, transverse order parameter fluctuations associated with the Goldstone mode lead to a strong renormalization of longitudinal fluctuations: the longitudinal mass and the bosonic self-interaction vanish linearly as a function of the scale in two dimensions, and logarithmically in three dimensions, in agreement with the exact behavior of an interacting Bose gas.