We present a simple method for summing so-called parquet diagrams of fermionic many-body systems with competing instabilities using the functional renormalization group. Our method is based on partial bosonization of the interaction multi-channel Hubbard-Stratonovich transformations. A simple truncation of the resulting flow equations, retaining only the frequency-independent parts of the two-point and three-point vertices amounts to solving coupled Bethe-Salpeter equations for the effective interaction to leading logarithmic order. We apply our method by revisiting the X-ray problem and deriving the singular frequency dependence of the X-ray response function and the particle-particle susceptibility. Our method is quite general and should be useful in many-body problems involving strong fluctuations in several scattering channels.

We calculate the renormalized Fermi surface and the quasiparticle properties in the Fermi liquid phase of three-dimensional dipolar fermions to second order in the dipole-dipole interaction. Using parameters relevant to an ultracold gas of erbium atoms, we find that the second-order corrections typically renormalize the Hartree-Fock results by less than 1%. On the other hand, if we use the second-order correction to the compressibility to estimate the regime of stability of the system, the point of instability is already reached for a significantly smaller interaction strength than in the Hartree-Fock approximation.

In two dimensions the Fourier transform of the interaction between two point dipoles has a term which grows linearly in the modulus |q| of the momentum. As a consequence, in second order perturbation theory the self-energy of two-dimensional dipolar fermions is ultraviolet divergent. We show that for electric dipoles this divergence can be avoided if one takes into account that physical dipoles consist of two opposite charges which are separated by a finite distance. Using this regularization, we calculate the self-energy, the renormalized chemical potential, and the renormalized Fermi surface of dipolar fermions in two dimensions in second order perturbation theory. We find that in the Fermi liquid phase the second order corrections weaken first order effects.

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Using Popov's hydrodynamic approach we derive an effective Euclidean action for the longwavelength phase fluctuations of superfluid Bose gases in D dimensions. We then use this action to calculate the damping of phase fluctuations at zero temperature as a function of D. For D > 1 and wavevectors |k| 2mc (where m is the mass of the bosons and c is the sound velocity) we find that the damping in units of the phonon energy E k = c|k| is to leading order γ k /E k = AD(k D 0 /2πρ)(|k|/k0) 2D−2 , where ρ is the boson density and k0 = 2mc is the inverse healing length. For D → 1 the numerical coefficient AD vanishes and the damping is proportional to an additional power of |k|/k0; a self-consistent calculation yields in this case γ k /E k = 1.32 (k0/2πρ) 1/2 |k|/k0. In one dimension, we also calculate the entire spectral function of phase fluctuations.