Renormalization group flow of the Luttinger-Ward functional: Conserving approximations and application to the Anderson impurity model

We study the renormalization group flow of the Luttinger-Ward functional and of its two-particle-irreducible vertex functions, given a cutoff in the two-particle interaction. We derive a conserving approximation to the flow and relate it to the fluctuation exchange approximation as well as to nonconserving approximations introduced in an earlier publication [J. F. Rentrop, S. G. Jakobs, and V. Meden, J. Phys. A: Math. Theor. 48, 145002 (2015)]. We apply the different approximate flow equations to the single-impurity Anderson model in thermal equilibrium at vanishing temperature. Numerical results for the effective mass, the spin susceptibility, the charge susceptibility, and the linear conductance reflect the similarity of the methods to the fluctuation exchange approximation. We find the majority of the approximations to deviate stronger from the exact results than one-particle-irreducible functional renormalization group schemes. However, we identify a simple static two-particle-irreducible flow scheme which performs remarkably well and produces an exponential Kondo-like scale in the renormalized level position.

Figure 1

(a) Diagrammatic representation of ${\mathrm{\Phi }}^{\mathrm{FLEX}}$. Our antisymmetrized charge-index notation leads to diagrams with undirected lines and dotlike Hugenholtz vertices. When ${\mathrm{\Phi }}^{\mathrm{FLEX}}$ is expressed in terms of diagrams composed of directed propagator lines and Feynman interaction lines, one can see that it comprises three different channels: one with longitudinal spin fluctuations and density fluctuations, one with transverse spin fluctuations, and one with particle-particle fluctuations (cf. Refs. [21, 35]). (b) Diagrammatic representation of ${\mathrm{\Phi }}^{\mathrm{cfRG}}$.

Figure 2

Numerical data for the cfRG, PUF, and FLEX approximations. (a) Results for the effective mass are compared to 1PI fRG and NRG results from Fig. 6(a) in Ref. [30]. (b) Results for the charge susceptibility are compared to Bethe ansatz results. We found well-converged solutions of the cfRG self-consistency equation only for $U<8.5\mathrm{\Gamma }$. (c) Results for the spin susceptibility are compared to 1PI fRG data from Fig. 6(b) in Ref. [30] and to Bethe ansatz results. (d) Results for the conductance as function of the gate voltage are compared to 1PI fRG data from Fig. 3 in Ref. [41] and Bethe ansatz data from Ref. [43]. We found well-converged solutions of the cfRG self-consistency equation only for $V_{\mathrm{g}}\ge 2\mathrm{\Gamma }$.

Figure 3

(a) The same numerical data for the effective mass as in Fig. 2, but on a logarithmic scale and including higher interaction strengths. The plot shows additionally data obtained with Hamann's approximation. (b) Numerical data for the occupancy of the dot, calculated for certain $\mathrm{\Phi }$-derivable and non-$\mathrm{\Phi }$-derivable schemes in various ways. The Bethe ansatz result (from Fig. 5 in Ref. [30]) is plotted for comparison.

Figure 4

Numerical data for the PUF, StUF, and MUF approximations in comparison to the same NRG, Bethe, and 1PI fRG curves as in Fig. 2. (a) For the effective mass, the PUF and MUF curves are indistinguishable. (b) For the charge susceptibility, the flow of unrestricted MUF does not come to an end for the intermediate regime $U/\mathrm{\Gamma }\approx \pi ...8$. (c) For the spin susceptibility, neither the flow of restricted MUF nor that of unrestricted MUF comes to an end beyond the critical interaction $U>\pi \mathrm{\Gamma }$. (d) For the conductance, the flow does not come to an end (except for StUF) for gate voltages around the plateau edge at such a high interaction strength.

Figure 5

Numerical data for the CUF and the CF approximations in comparison to the same PUF, NRG, Bethe, and 1PI fRG curves as in Fig. 2. For (a) the effective mass and (c) the spin susceptibility, we show CUF data for $\mathrm{\Lambda }/\mathrm{\Gamma }=1,2,4$. For (b) the charge susceptibility and (d) the conductance, we only show $\mathrm{\Lambda }/\mathrm{\Gamma }=2$ data.