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The dual-fermion approach provides a formally exact prescription forcalculating properties of a correlated electron system in terms of adiagrammatic expansion around dynamical mean-field theory (DMFT). Mostpractical implementations, however, neglect higher-order interaction verticesbeyond two-particle scattering in the dual effective action and furthertruncate the diagrammatic expansion in the two-particle scattering vertex to aleading-order or ladder-type approximation. In this work we compute thedual-fermion expansion for the two-dimensional Hubbard model including alldiagram topologies with two-particle interactions to high orders by means of astochastic diagrammatic Monte Carlo algorithm. We benchmark the obtainedself-energy against numerically exact Diagrammatic Determinant Monte Carlosimulations to systematically assess convergence of the dual-fermion series andthe validity of these approximations. We observe that, from high temperaturesdown to the vicinity of the DMFT N\'eel transition, the dual-fermion seriesconverges very quickly to the exact solution in the whole range of Hubbardinteractions considered ($4 \leq U/t \leq 12$), implying that contributionsfrom higher-order vertices are small. As the temperature is lowered further, weobserve slower series convergence, convergence to incorrect solutions, andultimately divergence. This happens in a regime where magnetic correlationsbecome significant. We find however that the self-consistent particle-holeladder approximation yields reasonable and often even highly accurate resultsin this regime.

Diagrammatic Monte Carlo approach for diagrammatic extensions of dynamical mean-field theory -- convergence analysis of the dual fermion technique