Exact renormalization group flow equations for nonrelativistic fermions: Scaling toward the Fermi surface

Abstract

We construct exact functional renormalization group ͑RG͒ flow equations for nonrelativistic fermions in arbitrary dimensions, taking into account not only mode elimination, but also the rescaling of the momenta, frequencies, and fermionic fields. The complete RG flow of all relevant, marginal, and irrelevant couplings can be described by a system of coupled flow equations for the irreducible n-point vertices. Introducing suitable dimensionless variables, we obtain flow equations for generalized scaling functions which are continuous functions of the flow parameter, even if we consider quantities which are dominated by momenta close to the Fermi surface, such as the density-density correlation function at long wavelengths. We also show how the problem of constructing the renormalized Fermi surface can be reduced to the problem of finding the RG fixed point of the irreducible two-point vertex at vanishing momentum and frequency. We argue that only if the degrees of freedom are properly rescaled is it possible to reach scale-invariant non-Fermi-liquid fixed points within a truncation of the exact RG flow equations.

Self-consistent Fermi surface renormalization of two coupled Luttinger liquids

“…However, the calculation of the FS in these works is either not self-consistent 5,9 in the sense discussed in the classic book by Nozières 11 , or the self-consistency has not been properly implemented within the framework of the RG 10 . The self-consistent definition of the FS is follows naturally from the fact that the shape of the FS is a characteristic property of a zero temperature RG fixed point 12,13,14,15 . We emphasize that in our approach the concept of a flowing FS depending on the RG flow parameter 5,9 never appears: we construct the true FS as a RG fixed point, which by definition does not flow.…”

Ferromagnetism in one-dimensional metals: Breakdown of the Hartree-Fock approximation and possible first-order phase transition

“…This calculation is more complicated than in the case of two Fermi points, because a self-consistent treatment within renormalized perturbation theory requires the introduction of four counterterms, one for each Fermi point 14,15 . Vollhardt et al 21 pointed out that a large asymmetry in the DOS with a peak at the lower band edge tends to favor ferromagnetism by minimizing the increase of kinetic energy due to the spin polarization.…”

Functional renormalization group in the broken symmetry phase: momentum dependence and two-parameter scaling of the self-energy

“…The other strategy, which was pioneered by Morris [3] and has been preferentially used in the condensed matter community to study non-relativistic fermions [9,10], is based on the expansion of Γ in powers of the fields, leading to an infinite hierarchy of coupled integro-differential equations for the one-particle irreducible vertices. This approach has the advantage of providing information on the momentum-and frequency dependence of the vertices.…”