We consider formulations of the functional renormalization-group (fRG) flow for correlated electronic systems with the dynamical mean-field theory as a starting point. We classify the corresponding renormalization-group schemes into those neglecting one-particle irreducible six-point vertices (with respect to the local Green’s functions) and neglecting one-particle reducible six-point vertices. The former class is represented by the recently introduced DMF 2 RG approach [31], but also by the scale-dependent generalization of the one-particle irreducible representation (with respect to local Green’s functions, 1PI-LGF) of the generating functional [20]. The second class is represented by the fRG flow within the dual fermion approach [16, 32]. We compare formulations of the fRG approach in each of these cases and suggest their further application to study 2D systems within the Hubbard model.
We consider the classical
O(N)-symmetric models confined in a
d-dimensional slablike geometry and subject to periodic boundary conditions. Applying the one-particle-irreducible variant of functional renormalization group (RG), we compute the critical Casimir forces acting between the slab boundaries. The applied truncation of the exact functional RG flow equation retains interaction vertices of arbitrary order. We evaluate the critical Casimir amplitudes ${\ensuremath{\Delta}}_{f}(d,N)$ for continuously varying dimensionality between two and three and
N=1,2. Our findings are in very good agreement with exact results for
d=2 and
N=1. For
d=3, our results are closer to Monte Carlo predictions than earlier field-theoretic RG calculations. Inclusion of the wave function renormalization and the corresponding anomalous dimension in the calculation has negligible impact on the computed Casimir forces.