Nonperturbative renormalization-group approach to fermion systems in the two-particle-irreducible effective action formalism

We propose a nonperturbative renormalization-group (NPRG) approach to fermion systems in the two-particle-irreducible (2PI) effective action formalism, based on an exact RG equation for the Luttinger-Ward functional. This approach enables us to describe phases with spontaneously broken symmetries while satisfying the Mermin-Wagner theorem. We show that it is possible to choose the Hartree-Fock random phase approximation theory as initial condition of the RG flow and argue that the 2PI-NPRG is not restricted to the weak-coupling limit. An expansion of the Luttinger-Ward functional about the minimum of the 2PI effective action including only the two-particle 2PI vertex leads to nontrivial RG equations where interactions between fermions and collective excitations naturally emerge.

Figure 1

Diagrammatic representation of propagators and vertices.

Figure 2

Diagrammatic representation of the Bethe-Salpeter equations satisfied by $W_k^{\left(2\right)}$, ${\mathcal{Y}}_k$, and ${\mathcal{J}}_k$ [Eqs. (27), (29), and (80)]. (Signs and symmetry factors are not shown.)

Figure 3

Diagrammatic representation of the RG equations ${\partial }_k{\Phi }_k\left[\mathcal{G}\right]$, ${\partial }_k{\Phi }_k^{\left(1\right)}\left[\mathcal{G}\right]$, and ${\partial }_k{\Phi }_k^{\left(2\right)}\left[\mathcal{G}\right]$. The double wavy line stands for $\Delta {\mathcal{Y}}_k$. (Signs and symmetry factors are not shown.)

Figure 4

Lowest-order Luttinger-Ward functional ${\Phi }_k\left[\mathcal{G}\right]$ [Eq. (85)] and the corresponding RG equation ${\partial }_k{\Phi }_k\left[\mathcal{G}\right]$ [Eq. (86)]. Pairs of solid lines stand for $\Pi$ and the black dot for $U+R_k$.

Figure 5

Luttinger-Ward functional ${\Phi }_k\left[\mathcal{G}\right]$ to $\mathcal{O}\left({\mathcal{G}}^4\right)$.

Figure 6

$1+f_k\left(\gamma \right)$ vs $\gamma$ for a typical value of $k$ ($\Delta \gamma <k^2<{\gamma }_{\max }$) [Eq. (112)].