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arXiv:1501.04960 · Physical Review B

On the dangers of partial diagrammatic summations: Benchmarks for the two-dimensional Hubbard model in the weak-coupling regime

Jan Gukelberger, Li Huang, Philipp Werner

Published Jun 9, 2015• 74 citations
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Abstract

We study the two-dimensional Hubbard model in the weak-coupling regime and compare the self-energy obtained from various approximate diagrammatic schemes to the result of diagrammatic Monte Carlo simulations, which sum up all weak-coupling diagrams up to a given order. While dynamical mean-field theory provides a good approximation for the local part of the self-energy, including its frequency dependence, the partial summation of bubble and/or ladder diagrams typically yields worse results than second-order perturbation theory. Even widely used self-consistent schemes such as $GW$ or the fluctuation-exchange approximation (FLEX) are found to be unreliable. Combining the dynamical mean-field self-energy with the nonlocal component of $GW$ in $GW + \mathrm{DMFT}$ yields improved results for the local self-energy and nonlocal self-energies of the correct order of magnitude, but here, too, a more reliable scheme is obtained by restricting the nonlocal contribution to the second-order diagram. FLEX + DMFT is found to give accurate results in the low-density regime, but even worse results than FLEX near half-filling.

DOI
10.1103/physrevb.91.235114

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We study the two-dimensional Hubbard model in the weak-coupling regime and compare the self-energy obtained from various approximate diagrammatic schemes to the result of diagrammatic Monte Carlo simulations, which sum up all weak-coupling diagrams up to a given order. While dynamical mean-field theory provides a good approximation for the local part of the self-energy, including its frequency dependence, the partial summation of bubble and/or ladder diagrams typically yields worse results than second-order perturbation theory. Even widely used self-consistent schemes such as GWGWGW or the fluctuation-exchange approximation (FLEX) are found to be unreliable. Combining the dynamical mean-field self-energy with the nonlocal component of GWGWGW in GW+DMFTGW + \mathrm{DMFT}GW+DMFT yields improved results for the local self-energy and nonlocal self-energies of the correct order of magnitude, but here, too, a more reliable scheme is obtained by restricting the nonlocal contribution to the second-order diagram. FLEX + DMFT is found to give accurate results in the low-density regime, but even worse results than FLEX near half-filling.
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 N\'eel transition, the dual fermion series converges very quickly to the exact solution in the whole range of Hubbard interactions considered ($4\ensuremath{\le}U/t\ensuremath{\le}12$), 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.
We introduce a numerical algorithm to stochastically sample the dual fermion perturbation series around the dynamical mean field theory, generating all topologies of two-particle interaction vertices. We show results in the weak and strong coupling regime of the half-filled Hubbard model in two dimensions, illustrating that the method converges quickly where dynamical mean field theory is a good approximation, and show that corrections are large in the strong correlation regime at intermediate interaction. The fast convergence of dual corrections to dynamical mean field results illustrates the power of the approach and opens a practical avenue towards the systematic inclusion of nonlocal correlations in correlated materials simulations. An analysis of the frequency scale shows that only low-frequency propagators contribute substantially to the diagrams, putting the inclusion of higher order vertices within reach.
The dynamical mean-field theory (DMFT) combined with the fluctuation exchange (FLEX) method, namely, FLEX+DMFT, is an approach for correlated electron systems to incorporate both local and nonlocal long-range correlations in a self-consistent manner. We formulate FLEX+DMFT in a systematic way starting from a Luttinger-Ward functional, and apply it to study the ddd-wave superconductivity in the two-dimensional repulsive Hubbard model. The critical temperature (Tc)({T}_{c})(Tc​) curve obtained in the FLEX+DMFT exhibits a dome structure as a function of the filling, which has not been clearly observed in the FLEX approach alone. We trace back the origin of the dome to the local vertex correction from DMFT that renders a filling dependence in the FLEX self-energy. We compare the results with those of GW+DMFT, where the Tc{T}_{c}Tc​-dome structure is qualitatively reproduced due to the same vertex correction effect, but a crucial difference from FLEX+DMFT is that Tc{T}_{c}Tc​ is always estimated below the N\'eel temperature in GW+DMFT. The single-particle spectral function obtained with FLEX+DMFT exhibits a double-peak structure as a precursor of the Hubbard bands at temperatures above Tc{T}_{c}Tc​.
We show how to use diagrammatic techniques to compute the weak-coupling perturbation series of the self-consistent solution to a dynamical mean field theory (DMFT) problem. This approach constitutes an alternative to using diagrammatic techniques directly as an impurity solver. It allows one to bypass the need of multiple perturbative series resummations within the DMFT self-consistency loop. It can be applied at or out of equilibrium, with any diagrammatic formalism, such as real times, imaginary times, or Matsubara frequencies formalisms. As a proof of principle, we illustrate our method with the half-filled Hubbard model on the Bethe lattice in the DMFT approximation, using quantum quasi Monte Carlo (QQMC) to obtain the impurity perturbation series on the real time axis.
The accurate determination of magnetic phase transitions in electronic systems is an important task of solid state theory. While numerically exact results are readily available for model systems such as the half-filled 3D Hubbard model, the complexity of real materials requires additional approximations, such as the restriction to certain classes of diagrams in perturbation theory, that reduce the precision with which magnetic properties are described. In this work, we examine the description of magnetic properties in second order perturbation theory, GW, FLEX, and two TMatrix approximations to numerically exact CT-QMC reference data. We assess finite-size effects and compare periodic lattice simulations to cluster embedding. We find that embedding substantially improves finite size convergence. However, by analyzing different partial summation methods we find no systematic improvement in the description of magnetic properties, with most methods considered in this work predicting first-order instead of continuous transitions, leading us to the conclusion that non-perturbative methods are necessary for the accurate determination of magnetic properties and phase transitions.
We describe a recent implementation of the combined GWGWGW and dynamical mean field method (GWGWGW+DMFT) for the two-dimensional Hubbard model with onsite and nearest-neighbor repulsion. We clarify the relation of the GWGWGW+DMFT scheme to alternative approaches in the literature, and discuss the corresponding approximations to the free-energy functional of the model. Furthermore, we describe a numerically exact technique for the solution of the GWGWGW+DMFT equations, namely, the hybridization expansion continuous-time algorithm for impurity models with retarded interactions. We compute the low-temperature phase diagram of the half-filled extended Hubbard model, addressing the metal-insulator transition at small intersite interactions and the transition to a charge-ordered state for stronger intersite repulsions. GWGWGW+DMFT introduces a nontrivial momentum dependence into the many-body self-energy and polarization. We find that the charge fluctuations included in the present approach have a larger impact on the latter than on the former. Finally, within the GWGWGW+DMFT framework, as in extended DMFT, the intersite repulsion translates into a frequency dependence of the local effective interaction. We analyze this dependence and show how it affects the local spectral function.