Efficient Perturbation Theory for Quantum Lattice Models

We present a novel approach to long-range correlations beyond dynamical mean-field theory, through a ladder approximation to dual fermions. The new technique is applied to the two-dimensional Hubbard model. We demonstrate that the transformed perturbation series for the nonlocal dual fermions has superior convergence properties over standard diagrammatic techniques. The critical Néel temperature of the mean-field solution is suppressed in the ladder approximation, in accordance with quantum Monte Carlo results. An illustration of how the approach captures and allows us to distinguish short- and long-range correlations is given.

Figure 1

Diagrams for the dual self-energy ${\Sigma }^d$. The exact two- and three-particle impurity vertices are depicted by squares and hexagons, respectively. The lines are dual Green functions. Diagrams (a) and (c) are purely local, while the remaining diagrams give nonlocal contributions. (g) Diagrammatic representation of the Bethe-Salpeter equation in the electron-hole channel.

Figure 2

Comparison of the leading eigenvalue of the Bethe-Salpeter equation in the electron-hole channel for a wave vector $\mathbf{q}=(\pi ,\pi )$ and $\Omega =0$ obtained within different approximations with quantum Monte Carlo data taken from Ref. 29 for an $8\times 8$ lattice. An eigenvalue of ${\lambda }_{\max }$ indicates the antiferromagnetic instability.

Figure 3

Leading eigenvalue of the Bethe-Salpeter equation for dual fermions as a function of $U$ in comparison to DMFT. The diagrams considered in each of the calculations are shown in the legend (cf. Fig. 1). The arrows point to the data for the same parameters as in Fig. 2.

Figure 4

Dynamical susceptibility $\chi (\mathbf{q},\omega )$ for $U/t=4$, and $T/t=0.19$.

Figure 5

Left: Metallic and insulating DOS within the coexistence region of the Mott transition $U/t=6.25$, $T/t=0.08$. Right: DMFT (dashed line), DF [diagrams (a) and (b)] (thin line), and LDFA (thick line) DOS at $U/t=4$, $T/t=0.19$. The LDFA DOS exhibits the antiferromagnetic pseudogap.