Reduction formula for fermion loops and density correlations of the 1D Fermi gas

Fermion N-loops with an arbitrary number of density vertices N > d+1 in d spatial dimensions can be expressed as a linear combination of (d+1)-loops with coefficients that are rational functions of external momentum and energy variables. A theorem on symmetrized products then implies that divergencies of single loops for low energy and small momenta cancel each other when loops with permuted external variables are summed. We apply these results to the one-dimensional Fermi gas, where an explicit formula for arbitrary N-loops can be derived. The symmetrized N-loop, which describes the dynamical N-point density correlations of the 1D Fermi gas, does not diverge for low energies and small momenta. We derive the precise scaling behavior of the symmetrized N-loop in various important infrared limits.Comment: 14 pages, to be published in Journal of Statistical Physic

Longitudinal fluctuations in the Berezinskii-Kosterlitz-Thouless phase

We analyze the interplay of longitudinal and transverse fluctuations in a $U(1)$ symmetric two-dimensional $\phi^4$-theory. To this end, we derive coupled renormalization group equations for both types of fluctuations obtained from a linear (cartesian) decomposition of the order parameter field. Discarding the longitudinal fluctuations, the expected Berezinskii-Kosterlitz-Thouless (BKT) phase characterized by a finite stiffness and an algebraic decay of order parameter correlations is recovered. Renormalized by transverse fluctuations, the longitudinal mass scales to zero, so that longitudinal fluctuations become increasingly important for small momenta. Within our expansion of the effective action, they generate a logarithmic decrease of the stiffness, in agreement with previous functional renormalization group calculations. The logarithmic terms imply a deviation from the vanishing beta-function for the stiffness in the non-linear sigma model describing the phase fluctuations at three-loop order. To gain further insight, we also compute the flow of the parameters characterizing longitudinal and transverse fluctuations from a density-phase representation of the order parameter field, with a cutoff on phase fluctuations. The power-law flow of the longitudinal mass and other quantities is thereby confirmed, but the stiffness remains finite in this approach. We conclude that the marginal flow of the stiffness obtained in the cartesian representation is an artifact of the truncated expansion of momentum dependences.Comment: Updated version. Substantial changes in Title, Abstract, Conclusion. New Section

Parametrization of Nambu vertex in a singlet superconductor

We analyze general properties of the effective Nambu two-particle vertex and its renormalization group flow in a spin-singlet superconductor. In a fully spin-rotation invariant form the Nambu vertex can be expressed by only three distinct components. Solving exactly the flow of a mean-field model with reduced BCS and forward scattering interactions, we gain insight into the singularities in the momentum and energy dependences of the vertex at and below the critical energy scale for superconductivity. Using a decomposition of the vertex in various interaction channels, we manage to isolate singular momentum and energy dependences in only one momentum and energy variable for each term, such that the singularities can be efficiently parametrized.Comment: 21 pages, 5 figure

Effective interactions and fluctuation effects in spin-singlet superfluids

We derive and evaluate one-loop functional flow equations for the effective interactions, self-energy and gap function in spin-singlet superfluids. The flow is generated by a fermionic frequency cutoff, which is supplemented by an external pairing field to treat divergencies associated with the Goldstone boson. To parametrize the singular momentum and frequency dependences of the effective interactions, the Nambu interaction vertex is decomposed in charge, magnetic, and normal and anomalous pairing channels. The one-loop flow solves reduced (mean-field) models for superfluidity exactly, and captures also important fluctuation effects. The Ward identity from charge conservation is generally violated, but can be enforced by projecting the flow. Applying the general formalism to the two-dimensional attractive Hubbard model, we obtain detailed results on the momentum and frequency dependences of the effective interactions for weak and moderate bare interactions. The gap is reduced by fluctuations, with a stronger reduction at weaker interactions, as expected.Comment: 51 pages, 20 figure

Fermi Surface of the 2D Hubbard Model at Weak Coupling

We calculate the interaction-induced deformation of the Fermi surface in the two-dimensional Hubbard model within second order perturbation theory. Close to half-filling, interactions enhance anisotropies of the Fermi surface, but they never modify the topology of the Fermi surface in the weak coupling regime.Comment: 4 pages, LaTeX2e, 5 embedded EPS figures, accepted to be published in Z. Phys.