Frequency-Dependent Functional Renormalization Group for Interacting Fermionic Systems

We derive an expansion of the functional renormalization group (fRG) equations that treats both the frequency and momentum dependencies of the vertex in a systematic manner by recasting the fRG equations as a series of Bethe-Salperter equations in the particle-particle, particle-hole, and particle-hole exchange channels. The linearity of the equations offers numerous computational advantages and leads to stable solutions at both the one- and two-loop levels. As the expansion is in the coupling between channels, the truncations that are necessary to make the scheme computationally tractable still lead to equations that treat contributions from all channels equally. We consider the sources of error within the truncations, the computational costs associated with them, and how the choice of regulator affects the flow of the fRG. As benchmarks, we apply different trunctions of the fRG equations to the single-impurity Anderson model. We then use the optimal truncation to study the one-dimensional bond-charge Hamiltonian and the two-dimensional extended Hubbard model. We find that in many cases, the fRG converges to a stable vertex and self-energy from which one can extract the various correlation functions and susceptibilities of interest.