Correlated starting points for the functional renormalization group

We present a general frame to extend functional renormalization group (fRG) based computational schemes by using an exactly solvable interacting reference problem as starting point for the RG flow. The systematic expansion around this solution accounts for a nonperturbative inclusion of correlations. Introducing auxiliary fermionic fields by means of a Hubbard-Stratonovich transformation, we derive the flow equations for the auxiliary fields and determine the relation to the conventional weak-coupling truncation of the hierarchy of flow equations. As a specific example we consider the dynamical mean field theory (DMFT) solution as reference system, and discuss the relation to the recently introduced ${\mathrm{DMF}}^2\mathrm{RG}$ and the dual-fermion formalism.

Figure 1

Schematic representation of the fRG flow in a general “parameter space.” In conventional schemes the fRG flow starts from the generating functional of an uncorrelated problem ${\mathcal{G}}_{\mathrm{ini}}$. The approximations due to the truncation lead to deviations of the result at the end of the flow from the exact final generating functional ${\mathcal{G}}_{\mathrm{fin}}$. This truncation error can be reduced by using a correlated reference system (provided its generating functional ${\mathcal{G}}_R$ is exactly known) as a starting point for the flow.

Figure 2

Overview of the relation between the flow in physical and auxiliary space; $\mathcal{T}$ denotes the transformation to the auxiliary fields.

Figure 3

Diagrammatic representation of the physical flow equations for ${\Sigma }^{\Lambda }$ and ${\gamma }_2^{\Lambda }$ that correspond to solving the conventional flow equations for the auxiliary fields using the second-order truncation.

Figure 4

Relation between the physical and auxiliary scale-dependent three-particle vertex functions.

Figure 5

Effects of the truncation in the physical and auxiliary flow.

Figure 6

Diagrammatic structure of Eq. (A39). The factors ${\zeta }^{\Lambda }$ appended at each leg of ${\gamma }_{3,a}^{\Lambda }$ are implicitly included in the diagram.