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arXiv:2211.01400 · Physical review. B./Physical review. B

Improved effective vertices in the multiorbital two-particle self-consistent method from dynamical mean-field theory

Karim Zantout, Steffen Backes, Aleksandar Razpopov, Dominik Leßnich, Roser Valentí

Published Jun 1, 2023• 12 citations
arXivHTMLPDF

Abstract

In this paper, we present a multiorbital form of the two-particle self-consistent approach (TPSC) where the effective local and static irreducible interaction vertices are determined by means of the dynamical mean-field theory (DMFT). This approach replaces the approximate ansatz equations for the double-occupations $\ensuremath{\langle}{n}_{\ensuremath{\alpha},\ensuremath{\sigma}}^{}{n}_{\ensuremath{\beta},{\ensuremath{\sigma}}^{\ensuremath{'}}}^{}\ensuremath{\rangle}$ by sampling them directly for the same model using DMFT. Compared to the usual Hartree-Fock-like ansatz, this leads to more accurate local vertices in the weakly correlated regime and provides access to stronger correlated systems that were previously out of reach. This approach is extended by replacing the local component of the TPSC self-energy by the DMFT impurity self-energy, which results in an improved self-energy that incorporates strong local correlations but retains a nontrivial momentum dependence. We find that this combination of TPSC and DMFT provides a significant improvement over the multiorbital formulation of TPSC as it allows to determine the components of the spin vertex without artificial symmetry assumptions and opens the possibility to include the transversal particle-hole channel. The new approach is also able to remove unphysical divergences in the charge vertices in TPSC. We find a general trend that lower temperatures can be accessed in the calculation. Benchmarking single-particle quantities, such as the local spectral function with other many-body methods, we find significant improvement in the more strongly correlated regime.

DOI
10.1103/physrevb.107.235101

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In this paper, we present a multiorbital form of the two-particle self-consistent approach (TPSC) where the effective local and static irreducible interaction vertices are determined by means of the dynamical mean-field theory (DMFT). This approach replaces the approximate ansatz equations for the double-occupations \ensuremath⟨n\ensuremathα,\ensuremathσn\ensuremathβ,\ensuremathσ\ensuremath′\ensuremath⟩\ensuremath{\langle}{n}_{\ensuremath{\alpha},\ensuremath{\sigma}}^{}{n}_{\ensuremath{\beta},{\ensuremath{\sigma}}^{\ensuremath{'}}}^{}\ensuremath{\rangle}\ensuremath⟨n\ensuremathα,\ensuremathσ​n\ensuremathβ,\ensuremathσ\ensuremath′​\ensuremath⟩ by sampling them directly for the same model using DMFT. Compared to the usual Hartree-Fock-like ansatz, this leads to more accurate local vertices in the weakly correlated regime and provides access to stronger correlated systems that were previously out of reach. This approach is extended by replacing the local component of the TPSC self-energy by the DMFT impurity self-energy, which results in an improved self-energy that incorporates strong local correlations but retains a nontrivial momentum dependence. We find that this combination of TPSC and DMFT provides a significant improvement over the multiorbital formulation of TPSC as it allows to determine the components of the spin vertex without artificial symmetry assumptions and opens the possibility to include the transversal particle-hole channel. The new approach is also able to remove unphysical divergences in the charge vertices in TPSC. We find a general trend that lower temperatures can be accessed in the calculation. Benchmarking single-particle quantities, such as the local spectral function with other many-body methods, we find significant improvement in the more strongly correlated regime.
Theoretical methods that are accurate for both short-distance observables and long-wavelength collective modes are still being developed for the Hubbard model. Here, we benchmark an approach that combines dynamical mean-field theory (DMFT) observables with the two-particle self-consistent theory (TPSC). This offers a way to include non-local correlations in DMFT while also improving TPSC. The benchmarks are published diagrammatic quantum Monte Carlo results for the two-dimensional square lattice Hubbard model with nearest-neighbor hopping. This method (TPSC+DMFT\mathrm{TPSC}+\mathrm{DMFT}TPSC+DMFT) is relevant for weak to intermediate interaction, satisfies the local Pauli principle, and allows us to compute a spin susceptibility that satisfies the Mermin-Wagner theorem. The DMFT double occupancy determines the spin and charge vertices through local spin and charge sum rules. The TPSC self-energy is also improved by replacing its local part with the local DMFT self-energy. With this method, we find improvements for both spin and charge fluctuations and for the self-energy. We also find that the accuracy check developed for TPSC is a good predictor of deviations from benchmarks for this model. TPSC+DMFT\mathrm{TPSC}+\mathrm{DMFT}TPSC+DMFT can be used in regimes where quantum Monte Carlo is inaccessible. In addition, this method paves the way to multiband generalizations of TPSC that could be used in advanced electronic structure codes that include DMFT.
Nonlocal correlations play an essential role in correlated electron systems, especially in the vicinity of phase transitions and crossovers, where two-particle correlation functions display a distinct momentum dependence. In nonequilibrium settings, the effect of nonlocal correlations on dynamical phase transitions, prethermalization phenomena, and trapping in metastable states is not well understood. In this paper, we introduce a dynamical mean field theory (DMFT) extension to the nonequilibrium two-particle self-consistent (TPSC) approach, which allows to perform nonequilibrium simulations capturing short- and long-ranged nonlocal correlations in the weak-, intermediate-, and strong-correlation regimes. The method self-consistently computes local spin and charge vertices, from which a momentum-dependent self-energy is constructed. Replacing the local part of the self-energy by the DMFT result within this self-consistent scheme provides an improved description of local correlation effects. We explain the details of the formalism and the implementation, and demonstrate the versatility of DMFT+TPSC with interaction quenches and dimensional crossovers in the Hubbard model.
The Emery model, or three-band Hubbard model, is a Hamiltonian that is thought to contain much of the physics of cuprate superconductors. This model includes two noninteracting ppp orbitals and one interacting ddd orbital per unit cell. Few methods that can solve multiorbital interacting Hamiltonians reliably and efficiently exist. Here, we introduce an application of the two-particle self-consistent (TPSC) approach to the Emery model. We construct this method within the framework of the TPSC +++ DMFT method, which can be seen as a way to introduce nonlocal corrections to dynamical mean-field theory (DMFT). We show that interacting orbital densities, rather than the noninteracting ones, must be used in the calculations. For the Emery model, we find that at constant bare interaction UUU, the vertex for spin fluctuations, Usp{U}_{sp}Usp​, decreases rapidly with filling. This may be one of the factors that contributes to electron-doped cuprates appearing less correlated than hole-doped ones. More generally, our work opens the road to the application of the TPSC approach to spin fluctuations in multiorbital models.
We suggest a new approach for treatment of local and non-local interactions in correlated electronic systems, which is based on the combination of the (extended) dynamical mean-field theory ((E)DMFT) and the two-particle irreducible functional renormaliztion-group (2PI-fRG) method. The considering approach uses self-energy and the two-particle irreducible vertices, obtained from (E)DMFT, as an input of 2PI-fRG approach. Using 2PI vertices may improve the applicability of various truncations and fulfilling conservation laws in comparison with one-particle irreducible approaches. In case of purely local interaction in a certain truncation of the DMFT+2PI-fRG equations, the obtained equation for the self-energy has a similar, although not identical, structure to that in the ladder dynamic vertex approximation (D$\Gamma $A). For the non-local interactions, in a simplest truncation we reproduce the results for the two-particle vertices/susceptibilities in the ladder approximation of the dual boson approach, but obtain different equation for the self-energy with a correct treatment of the one-particel reducible vertices of higher orders. The proposed scheme is rather general and can be applied to study various phenomena in strongly-correlated electronic systems, e.g. as a tool describing ab initio screening of the Coulomb interaction in strongly correlated systems.
We propose an approach describing correlated electronic systems within the two-particle irreducible functional renormalization-group (2PI-fRG) approach, which uses initial self-energy and the two-particle irreducible vertices, obtained from (extended) dynamical mean-field theory [E(DMFT)]. Using the 2PI-fRG approach allows us to treat both local and nonlocal interactions. In the case of purely local interaction the corresponding equations have similar (although not identical) structure to the earlier developed DMFT+fRG (DMF2RG)({\mathrm{DMF}}^{2}\mathrm{RG})(DMF2RG) approach. In a simplest truncation, neglecting scale dependence of the two-particle irreducible vertices, we reproduce the results for the two-particle vertices/susceptibilities in the ladder approximation of the dual-boson (DB) or the dynamic vertex approximation approach; in more sophisticated truncations the method allows us to consider nonlocal corrections to the self-energy as well as the interplay of charge and spin correlations. The proposed scheme is tested on the two-dimensional standard and extended $U\text{\ensuremath{-}}V$ half-filled Hubbard models. For the standard Hubbard model we obtain nonlocal self-energy, which is in agreement with numerical studies; for the extended Hubbard model we find the boundary of charge instability, which agrees well with the results of the DB approach. We also ascertain that the effect of spin correlations on electron interaction in the charge channel, not considered previously in the DB approach, only slightly reduces critical next-nearest-neighbor interaction of charge instability of the extended Hubbard model at the considered finite low temperature, yielding better agreement with dynamic cluster approximation. The considered method is rather general and can be applied to study various phenomena in strongly correlated electronic systems.
Extending many-body numerical techniques which are powerful in the context of simple model calculations to the realm of realistic material simulations can be a challenging task. Realistic systems often involve multiple active orbitals, which increases the complexity and numerical cost because of the large local Hilbert space and the large number of interaction terms or sign-changing off-diagonal Green’s functions. The two-particle self-consistent approach (TPSC) is one such many-body numerical technique, for which multi-orbital extensions have proven to be involved due to the substantially more complex structure of the local interaction tensor. In this paper we extend earlier multi-orbital generalizations of TPSC by setting up two different variants of a fully self-consistent theory for TPSC in multi-orbital systems. We first investigate the strengths and limitations of the approach analytically and then benchmark both variants against dynamical mean-field theory (DMFT) and D-TRILEX results. We find that the exact behavior of the system can be faithfully reproduced in the weak-coupling regime, while at stronger couplings the performance of the two TPSC variants strongly depends on details of the system.