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

Abstract

The dual-fermion approach provides a formally exact prescription for calculating properties of a correlated electron system in terms of a diagrammatic expansion around dynamical mean-field theory (DMFT). Most practical implementations, however, neglect higher-order interaction vertices beyond two-particle scattering in the dual effective action and further truncate the diagrammatic expansion in the two-particle scattering vertex to a leading-order or ladder-type approximation. In this work we compute the dual-fermion expansion for the two-dimensional Hubbard model including all diagram topologies with two-particle interactions to high orders by means of a stochastic diagrammatic Monte Carlo algorithm. We benchmark the obtained self-energy against numerically exact Diagrammatic Determinant Monte Carlo simulations to systematically assess convergence of the dual-fermion series and the validity of these approximations. We observe that, from high temperatures down to the vicinity of the DMFT Néel transition, the dual-fermion series converges very quickly to the exact solution in the whole range of Hubbard interactions considered (4 ≤ U/t ≤ 12), implying that contributions from higher-order vertices are small. As the temperature is lowered further, we observe slower series convergence, convergence to incorrect solutions, and ultimately divergence. This happens in a regime where magnetic correlations become significant. We find however that the self-consistent particle-hole ladder approximation yields reasonable and often even highly accurate results in this regime.

Truncated unity parquet solver

“…2 -a regime known as pseudogap 61 . This behavior of the self-energy in the two dimensional Hubbard model, incoherent state → bad metal → pseudogap is consistently obtained by a wide range of computational methods to study correlated electron systems 27,30,59,62,63 and therefore believed to be intrinsic to the Hubbard model. A further lowering of T (β = 30) indicates a completely gapped Fermi surface.…”

Parquetlike equations for the Hedin three-leg vertex

“…where we used definition (17) to identify the singlet Hedin vertexλ s . Now we insert the SBE decomposition (13c) on the right-hand-side and are left with,…”

Determinant Diagrammatic Monte Carlo Algorithm in the Thermodynamic Limit

“…Major recent achievements of DiagMC are the study of the normal phase of the unitary fermi gas [13], the determination of the ground state phase diagram of the Hubbard model up to filling factor 0.7 and interactions U/t ≤ 4 [14], and the settlement of the Bose-metal issue [15]. Given the power and the versatility of the technique, systematic diagrammatic extensions of Dynamical Mean Field Theory are now being studied [16,17].In DiagMC [10,11,18,19] one expresses the perturbative expansion in terms of connected Feynman diagrams, which can be written directly in the TL. One then performs a random walk in the space of topologies and of integration variables of Feynman diagrams.…”

“…Major recent achievements of DiagMC are the study of the normal phase of the unitary fermi gas [13], the determination of the ground state phase diagram of the Hubbard model up to filling factor 0.7 and interactions U/t ≤ 4 [14], and the settlement of the Bose-metal issue [15]. Given the power and the versatility of the technique, systematic diagrammatic extensions of Dynamical Mean Field Theory are now being studied [16,17].…”