Frequency Dependent Functional Renormalization Group for Interacting Fermionic Systems

TL;DR

This paper develops a frequency-dependent functional renormalization group method for interacting fermionic systems, enabling more accurate and stable calculations of correlation functions and susceptibilities across various models.

Contribution

It introduces a systematic frequency and momentum expansion of fRG equations, extending channel decomposition to the frequency domain with linear integral equations.

Findings

Stable solutions for various Hamiltonians

Converged vertex and self-energy in models

Efficient computational scheme

Abstract

We derive an expansion of the functional renormalization (fRG) equations that treats the frequency and momentum dependencies of the vertices in a systematic manner. The scheme extends the channel-decomposed fRG equations to the frequency domain and reformulates them as a series of linear integral equations in the particle-particle, particle-hole and particle-hole exchange channels. We show that the linearity of the equations offers numerous computational advantages and leads to converged, stable solutions for a variety of Hamiltonians. As the expansion is in the coupling between channels, the truncations that are necessary to making the scheme computationally viable still lead to equations that treat contributions from all channels equally. As a first benchmark we apply the two-loop fRG equations to the single impurity Anderson model. We consider the sources of error within the fRG, the computational cost associated with each, and how the choice of regulator affects the flow of the fRG. We then use the optimal truncation scheme to study the Extended Hubbard Hamiltonian in one and two dimensions. We find that in many cases of interest the fRG flow converges to a stable vertex and self-energy from which we can extract the various correlation functions and susceptibilities of interest.