Momentum-space cluster dual-fermion method

Recent years have seen the development of two types of nonlocal extensions to the single-site dynamical mean field theory. On one hand, cluster approximations, such as the dynamical cluster approximation, recover short-range momentum-dependent correlations nonperturbatively. On the other hand, diagrammatic extensions, such as the dual-fermion theory, recover long-ranged corrections perturbatively. The correct treatment of both strong short-ranged and weak long-ranged correlations within the same framework is therefore expected to lead to a quick convergence of results, and offers the potential of obtaining smooth self-energies in nonperturbative regimes of phase space. In this paper, we present an exact cluster dual-fermion method based on an expansion around the dynamical cluster approximation. Unlike previous formulations, our method does not employ a coarse-graining approximation to the interaction, which we show to be the leading source of error at high temperature, and converges to the exact result independently of the size of the underlying cluster. We illustrate the power of the method with results for the second-order cluster dual-fermion approximation to the single-particle self-energies and double occupancies.

Figure 1

The diagram types for the dual self-energy with major contributions. The white squares and hexagons correspond to two- and three-particle impurity vertices, respectively. Lines correspond to the bare dual Green's function.

Figure 2

Comparison of the real (top panel) and imaginary (bottom panel) part of the self-energy at the lowest Matsubara frequency obtained from the different approaches. Red line with open diamonds: 4-site DCA cluster ladder DF approximation with the Laue approximation of Sec. 2c1 (DCA DF, DCA Laue, ladder). Blue line with closed circles: single-site ladder DF. Orange line: single-site second-order DF. Dashed green line: $N_c=4$ DCA. Dashed gray line: $N_c=64$ DCA.

Figure 3

Comparison of the different diagram topologies' contributions to the imaginary part of the self-energy at the lowest Matsubara frequency. Solid blue line corresponds to 4-site DCA cluster ladder DF results with the DCA Laue approximation, the short dashed gray line shows additional corrections by including the three-particle vertex function, and the long dashed red line corresponds to the second-order DF with the DCA Laue approximation.

Figure 4

Comparison of the real (top panel) and imaginary (bottom panel) parts of the self-energy at the lowest Matsubara frequency. Solid red line: 4-site DCA cluster second-order dual-fermion approximation. Solid blue line: single-site second-order dual fermions. Solid orange line: 4-site DCA cluster second-order dual-fermion approximation with the DCA Laue approximation. Dashed green line: 4-site DCA results. Dashed gray line: 64-site DCA results.

Figure 5

Comparison of the double-occupancy $\langle n_i{n}_i\rangle$ calculation obtained from DF DCA method (blue cross) in comparison to the single-site DMFT (black square), single-site dual-fermion method (red cross), DF DCA method with DCA Laue approximation (orange diamond), and data from the 2D Hubbard benchmark [1]; the dashed black line corresponds to DCA results extrapolated to thermodynamic limit. The symbol size for the DCA results corresponds to the stochastic error bars.