Parquet approximation and one-loop renormalization group: Equivalence on the leading-logarithmic level

Abstract

We demonstrate how to devise a Matsubara-formalism based one-loop approximation to the flow of the functional renormalization group (FRG) that reproduces identically the leading-logarithmic parquet approximation. This construction of a controlled fermionic FRG approximation in a regime not accessible by perturbation theory generalizes a previous study from the real-time zero-temperature formalism to the Matsubara formalism and thus to the de facto standard framework used for condensed-matter FRG studies. Our investigation is based on a simple model for the absorption of x-rays in metals. It is a core part of our construction to exploit that in a suitable leading-logarithmic approximation the values of the particle-hole susceptibility on the real and on the imaginary frequency axis are identical.

Fulfillment of sum rules and Ward identities in the multiloop functional renormalization group solution of the Anderson impurity model

“…already provides the correct asymptotic behavior (24). This is similar to the sum rule of χ σσ , where Eqs.…”

“…Therefore, the Σ asymptote [Eq. (24)] is violated when using a 1 or multiloop vertex flow while keeping the standard self-energy flow. This problem is circumvented by including the multiloop corrections to the self-energy flow [16], which guarantee a perfect equivalence to the SDE and, thereby, that the correct asymptote will be restored.…”

“…Whether or not mfRG yields quantitative improvements over the 1 truncation depends on the context. For a zerodimensional model with a logarithmically divergent perturba-tion theory, it was recently shown [24] that the leading logarithms can be obtained in the 1 truncation, in which case the higher-loop contributions incorporated via mfRG thus are subleading. Similarly, 1 fRG treatments of the interacting resonant level model [25][26][27][28] as well as inhomogeneous Tomonaga-Luttinger liquids [29][30][31][32][33][34][35] should yield a proper summation of the leading logs governing the infrared behavior of these systems.…”

Multiloop flow equations for single-boson exchange fRG

“…For completeness, we illustrate here how these relations arise within the present framework. The starting point is the general relation between the four-point correlator G (4) and the four-point vertex Γ ,…”

“…. a e-mail: kugler@physics.rutgers.edu (corresponding author) e.g., from a perturbative [2] or leading-log [4] perspective. Another truncation scheme is given by the multiloop fRG approach, mfRG, which includes all contributions of the six-point vertex to the flow of the fourpoint vertex and self-energy that can be computed with numerical costs proportional to the 1 flow [5][6][7].…”