From Infinite to Two Dimensions through the Functional Renormalization Group

We present a novel scheme for an unbiased, nonperturbative treatment of strongly correlated fermions. The proposed approach combines two of the most successful many-body methods, the dynamical mean field theory and the functional renormalization group. Physically, this allows for a systematic inclusion of nonlocal correlations via the functional renormalization group flow equations, after the local correlations are taken into account nonperturbatively by the dynamical mean field theory. To demonstrate the feasibility of the approach, we present numerical results for the two-dimensional Hubbard model at half filling.

Figure 1

Schematic illustration of the ${\mathrm{DMF}}^2\mathrm{RG}$ approach, showing the evolution of the Gaussian part $G_{\mathrm{\Lambda }}^0$ of the action from DMFT to its exact expression for a two-dimensional system. The (truncated) flow equations for the self-energy ${\mathrm{\Sigma }}^{\mathrm{\Lambda }}$ and the two-particle vertex ${\mathrm{\Gamma }}^{\mathrm{\Lambda }}$ are explicitly given in terms of Feynman diagrams.

Figure 2

Flow of the largest component ($g_{\max }$) of the two-particle vertex function, i.e., in our case, $\mathrm{\Gamma }$ in the particle-hole crossed channel, for zero transfer frequency (${\nu }_2-{\nu }_1^{\prime }=0$), antiferromagnetic momentum transfer [${\mathbf{k}}_2-{\mathbf{k}}_1^{\prime }=(\pi ,\pi )$] and ${\mathbf{k}}_1=(0,\pi )$, ${\mathbf{k}}_2=(\pi ,0)$ computed by fRG, with interaction cutoff ${\mathrm{\Lambda }}_{\text{int}}$[26] (inset) and ${\mathrm{DMF}}^2\mathrm{RG}$ (main panel) for the two-dimensional half-filled Hubbard model at $U=1$, at different (inverse) temperatures.

Figure 3

Comparison of the results for the imaginary part of the fermionic self-energy of the two-dimensional Hubbard model for $U=1$, and $\beta =10$, calculated within DMFT ($\mathbf{k}$ independent, in black) and ${\mathrm{DMF}}^2\mathrm{RG}$, for different $\mathbf{k}$ vectors [the color coding of the different $\mathbf{k}$ is defined in the inset, note that the values of $\mathrm{Im}\mathrm{\Sigma }(\mathbf{k},i{\omega }_n)$ for $\mathbf{k}=(0,0)$ and ($\pi$, $\pi$) coincide because of the particle-hole symmetry]. Upper inset: Scheme of the 8 patches discretization used for the calculations. Lower inset: $T$ dependence of the momentum-resolved static spin susceptibility $S(\mathbf{q},i\mathrm{\Omega }=0)$.

Figure 4

Comparison of the imaginary part of the self-energy for $U=0.5$, 0.75, $n=1$, and $\beta =10$, calculated by fRG and ${\mathrm{DMF}}^2\mathrm{RG}$, for different $\mathbf{k}$ vectors (color coding as in Fig. 3).