Causality is a fundamental requirement for physical consistency in many-body approximations. In the context of Green's functions, causality requires a positive spectral function and analyticity in the upper half of the complex-frequency plane. These properties are not guaranteed by approximate methods, and formal proofs of causality for the underlying approximation schemes are often difficult to establish. The recently rediscovered Nevanlinna--Pick criterion provides a practical way to test causality directly from Matsubara-frequency data, without performing analytic continuation to the real-frequency axis. In this work, we use the Nevanlinna--Pick criterion to investigate the causality of the ladder dual-fermion approach, a diagrammatic extension of dynamical mean-field theory designed to incorporate nonlocal correlations. The analysis is performed for the spinless Falicov--Kimball model, which provides a useful testing ground because several quantities can be computed exactly or semi-analytically, avoiding Monte Carlo noise. This is particularly important since the Nevanlinna--Pick criterion is highly sensitive to noise in the input data. In addition, we derive several analytical properties of the Pick matrix, providing insight into the structure of the criterion when applied to Green's functions. For small systems, apparent causality violations are observed, but these are found to be associated with finite-size effects. For larger system sizes, no robust signs of causality violations are found, neither within the full ladder dual-fermion approach nor within the leading dual-fermion correction.
Strong electronic correlations generally require non-perturbative treatment. Local correlations are captured by dynamical mean-field theory while nonlocal correlations can be treated with diagrammatic extensions such as the Dual Fermion approach. Dual Fermion is built on physically motivated, but in principle uncontrolled approximations, so careful benchmarking is needed to understand the strengths and limitations of the method. In this work, we benchmark ladder Dual Fermion and dynamical mean-field theory for the Falicov-Kimball model with the exact classical Monte Carlo solution. We focus on the thermodynamics, electronic structure and susceptibility, especially at the combined frequency and momentum structure, and find that Dual Fermion clearly outperforms dynamical mean-field theory. Somewhat surprisingly, Dual Fermion is not as accurate for the relation between orbital density versus chemical potential in the doped system. These results demonstrate the need for rigorous benchmarking of diagrammatic extensions of dynamical mean-field theory for models with inequivalent orbitals, which is essential for modelling materials.
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.
GW+DMFT-type schemes replace the local part of a
GW self-energy by a separately computed local self-energy. A causal double counting removes a non-causal term from the hybridization (Lee and Haule, 2017), but the resulting total self-energy has been flagged as not guaranteed to be causal (Zhu and Chan, 2021). From the Herglotz representation, the spliced self-energy is causal if and only if its spectral measure is positive semi-definite. For translation-invariant single-band systems this becomes a momentum-resolved bound: the local weight may fall short of the local
GW weight by at most the momentum minimum of the
GW weight. On finite clusters, a mismatch between
GW and exact poles generically produces indefinite residues. On the Matsubara axis, a negative damping eigenvalue at one frequency certifies non-causality; passing does not establish causality. We examine these diagnostics in an idealized setting, not self-consistent
GW+DMFT: one-shot
GW on the exact Green's function of small Hubbard clusters with the exact local self-energy inserted. At all 528 parameter sets, our computed
GW and exact self-energies pass both tests, as expected. The splice fails the pointwise test at 54 points, none at half filling, all at the lowest Matsubara frequency, and already at
U/t=0.5; there the local
GW damping is 1.3-9 times the exact one. A block Pick test on eight frequencies, run with a second imaginary-time code, flags 481 of the 528 sets; for the other 47 it is inconclusive. Moment checks miss all of these, because the spliced high-frequency moment is exact. The Dyson Green's function is then non-causal too, yet passes the pointwise test everywhere tested. This version replaces a withdrawn one whose reported causality violations of
GW itself came from a sign error in the polarization; with the sign corrected, the computed
GW shows the expected positivity.
We introduce a Julia implementation of the recently proposed Nevanlinna analytic continuation method. The method is based on Nevanlinna interpolants and inherently preserves the causality of a response function due to its construction. For theoretical calculations without statistical noise, this continuation method is a powerful tool to extract real-frequency information from numerical input data on the Matsubara axis. This method has been applied to first-principles calculations of correlated materials. This paper presents its efficient and full-featured open-source implementation of the method including the Hamburger moment problem and smoothing.
We introduce a Julia implementation of the recently proposed Nevanlinna analytic continuation method. The method is based on Nevanlinna interpolants and inherently preserves the causality of a response function due to its construction. For theoretical calculations without statistical noise, this continuation method is a powerful tool to extract real-frequency information from numerical input data on the Matsubara axis. This method has been applied to first-principles calculations of correlated materials. This paper presents its efficient and full-featured open-source implementation of the method including the Hamburger moment problem and smoothing.
Analytical continuation (AC) connects theoretical calculations and experimentally measurable quantities. The recently proposed Nevanlinna AC method is capable of accurately reproducing the sharp features of spectral functions at high frequencies while maintaining the causality of the response function. However, their use is currently limited to fermions. Here, we present an extension of this method to bosons using the hyperbolic tangent trick, allowing us to transform bosons into auxiliary fermions to which the Nevanlinna analytic continuation can be applied.