Parquet approximation for the $4\times 4$ Hubbard cluster

We present a numerical solution of the parquet approximation, a conserving diagrammatic approach which is self-consistent at both the single-particle and the two-particle levels. The fully irreducible vertex is approximated by the bare interaction thus producing the simplest approximation that one can perform with the set of equations involved in the formalism. The method is applied to the Hubbard model on a half-filled $4\times 4$ cluster. Results are compared to those obtained from determinant quantum Monte Carlo (DQMC), FLuctuation EXchange (FLEX), and self-consistent second-order approximation methods. This comparison shows a satisfactory agreement with DQMC and a significant improvement over the FLEX or the self-consistent second-order approximation.

Figure 1

(Color online) Different classes of diagrams; the solid line represents the single-particle Green’s function and the wavy line represents the Coulomb interaction: here we use the p-h horizontal channel for illustration. (a) Reducible diagrams: can be separated into two parts by cutting two horizontal Green’s function lines. (b) Irreducible diagrams: can only be separated into two parts by cutting two Green’s function lines in the other two channels. (c) Fully irreducible diagrams: cannot be split in two parts by breaking two Green’s function lines in any channel.

Figure 2

(Color online) Schematic illustration for the different steps in solving the parquet approximation equations self-consistently.

Figure 3

(Color online) Single-particle Green’s function $G\left(\tau \right)$ for two different momenta a), $\mathbf{k}=\left(\pi ,0\right)$, b), $\mathbf{k}=\left(0,0\right)$, extracted from the three diagrammatic approaches and the DQMC. For this temperature $\left(T=0.3t\right)$, the PA result (solid line) looks very close to the DQMC one (symbol solid line) as compared to SC second-order (dashed line) or FLEX (dash-dotted line). In the inset of (b) is an enlarged view of the figure.

Figure 4

(Color online) The inverse temperature dependence of local moment. Among the three diagrammatic approaches, the PA result comes closest to the DQMC one.

Figure 5

(Color online) Uniform susceptibility calculated for different methods as a function of inverse temperature. While at the high temperature region, all the diagrammatic method results come close to the DQMC result, the PA shows its advantage clearly in the low temperature region.