Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2008.08098 · Annalen der Physik

Two‐Particle Self‐Consistent Method for the Multi‐Orbital Hubbard Model

Karim Zantout, Steffen Backes, Roser Valentí

Published Jan 15, 2021• 20 citations
arXivHTMLPDF

Abstract

Abstract One of the most challenging problems in solid state systems is the microscopic analysis of electronic correlations. A paramount minimal model that encodes correlation effects is the Hubbard Hamiltonian, which—regardless of its simplicity—is exactly solvable only in a few limiting cases and approximate many‐body methods are required for its solution. In this review, an overview on the non‐perturbative two‐particle self‐consistent method (TPSC), which was originally introduced to describe the electronic properties of the single‐band Hubbard model, is presented. A detailed derivation of the multi‐orbital generalization of TPSC is introduced here and particular features of the method on exemplary interacting models in comparison to dynamical mean‐field theory results are discussed.

DOI
10.1002/andp.202000399

Related papers

Multi-orbital two-particle self-consistent approach – Strengths and limitations

Jonas B. Profe, Jiawei Yan, Karim Zantout +2 more

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.

arXiv:2410.00962•SciPost Physics•Jul 21, 2025•4 citations

Two-particle self-consistent approach for multiorbital models: Application to the Emery model

C. Gauvin-Ndiaye, J. Leblanc, S. Marin +3 more

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 $p$ orbitals and one interacting $d$ 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 $U$, the vertex for spin fluctuations, ${U}_{sp}$, 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.

arXiv:2308.14091•Physical review. B./Physical review. B•Apr 2, 2024•5 citations

Improved two-particle self-consistent approach for the single-band Hubbard model in two dimensions

C. Gauvin-Ndiaye, C. Lahaie, Yu. M. Vil’k +1 more

The two-particle self-consistent approach (TPSC) is a method for the one-band Hubbard model that can be both numerically efficient and reliable. However, TPSC fails to yield physical results deep in the renormalized classical regime of the bidimensional Hubbard model where the spin correlation length becomes exponentially large. We address the limitations of TPSC with improved approaches that we call TPSC+ and TPSC+SFM (spin fluctuation mediated). In this work, we show that these improved methods satisfy the Mermin-Wagner theorem and the Pauli principle. We also show that they are valid in the renormalized classical regime of the two-dimensional Hubbard model, where they recover a generalized Stoner criterion at zero temperature in the antiferromagnetic phase. We discuss some limitations of the TPSC+ approach with regards to the violation of the $f$-sum rule and conservation laws, which are solved within the TPSC+SFM framework. Finally, we benchmark the TPSC+ and TPSC+SFM approaches for the one-band Hubbard model in two dimensions and show how they have an overall better agreement with available diagrammatic Monte Carlo results than the original TPSC approach.

arXiv:2305.19219•Physical review. B./Physical review. B•Aug 18, 2023•13 citations

Nonlocal corrections to dynamical mean-field theory from the two-particle self-consistent method

N. Martin, C. Gauvin-Ndiaye, A.–M. S. Tremblay

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 ($\mathrm{TPSC}+\mathrm{DMFT}$) 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. $\mathrm{TPSC}+\mathrm{DMFT}$ 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.

arXiv:2211.01919•Physical review. B./Physical review. B•Feb 28, 2023•14 citations

Dynamical mean field theory extension to the nonequilibrium two-particle self-consistent approach

Olivier Simard, Philipp Werner

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.

arXiv:2302.14134•Physical review. B./Physical review. B•Jun 27, 2023•9 citations

Local-density approach and quasiparticle levels for generalized Hubbard Hamiltonians

Pablo Jauralde Pou, Rúben Pérez, Fernando Flores +5 more

This paper presents a general method to describe and analyze electron correlation effects in local-orbital electronic structure calculations using a generalized Hubbard Hamiltonian. In our approach, we first introduce a local density formalism where the total energy of the system is obtained as a function of the orbital occupancies ${{n}_{i}}$ associated with each local orbital; in particular, exchange and correlation local potentials are presented for a multilevel case. In parallel, using the dynamical mean field approximation, a many-body solution is obtained by means of a local self-energy that appropriately interpolates between the low and high correlation limits. We also show that the local density and the many-body solutions are linked through charge consistency conditions. These two solutions are applied to a multilevel Anderson impurity and to a multiband Hubbard lattice, our results showing the high accuracy of the approach presented in this paper. Further on, we discuss how to apply our previous analysis to the case of crystals and molecules and analyze several examples: bulk Si, and HF and ${\mathrm{H}}_{2}\mathrm{O}$ molecules. The good results obtained for these cases show that our approach for the description of correlation effects offers an interesting alternative to the well-established density functional methods based on the calculation of the electron density $\ensuremath{\rho}(\stackrel{\ensuremath{\rightarrow}}{r}).$

Physical review. B, Condensed matter•Aug 15, 2000•59 citations
Abstract One of the most challenging problems in solid state systems is the microscopic analysis of electronic correlations. A paramount minimal model that encodes correlation effects is the Hubbard Hamiltonian, which—regardless of its simplicity—is exactly solvable only in a few limiting cases and approximate many‐body methods are required for its solution. In this review, an overview on the non‐perturbative two‐particle self‐consistent method (TPSC), which was originally introduced to describe the electronic properties of the single‐band Hubbard model, is presented. A detailed derivation of the multi‐orbital generalization of TPSC is introduced here and particular features of the method on exemplary interacting models in comparison to dynamical mean‐field theory results are discussed.
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.
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.
The two-particle self-consistent approach (TPSC) is a method for the one-band Hubbard model that can be both numerically efficient and reliable. However, TPSC fails to yield physical results deep in the renormalized classical regime of the bidimensional Hubbard model where the spin correlation length becomes exponentially large. We address the limitations of TPSC with improved approaches that we call TPSC+ and TPSC+SFM (spin fluctuation mediated). In this work, we show that these improved methods satisfy the Mermin-Wagner theorem and the Pauli principle. We also show that they are valid in the renormalized classical regime of the two-dimensional Hubbard model, where they recover a generalized Stoner criterion at zero temperature in the antiferromagnetic phase. We discuss some limitations of the TPSC+ approach with regards to the violation of the fff-sum rule and conservation laws, which are solved within the TPSC+SFM framework. Finally, we benchmark the TPSC+ and TPSC+SFM approaches for the one-band Hubbard model in two dimensions and show how they have an overall better agreement with available diagrammatic Monte Carlo results than the original TPSC approach.
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.
This paper presents a general method to describe and analyze electron correlation effects in local-orbital electronic structure calculations using a generalized Hubbard Hamiltonian. In our approach, we first introduce a local density formalism where the total energy of the system is obtained as a function of the orbital occupancies ni{{n}_{i}}ni​ associated with each local orbital; in particular, exchange and correlation local potentials are presented for a multilevel case. In parallel, using the dynamical mean field approximation, a many-body solution is obtained by means of a local self-energy that appropriately interpolates between the low and high correlation limits. We also show that the local density and the many-body solutions are linked through charge consistency conditions. These two solutions are applied to a multilevel Anderson impurity and to a multiband Hubbard lattice, our results showing the high accuracy of the approach presented in this paper. Further on, we discuss how to apply our previous analysis to the case of crystals and molecules and analyze several examples: bulk Si, and HF and H2O{\mathrm{H}}_{2}\mathrm{O}H2​O molecules. The good results obtained for these cases show that our approach for the description of correlation effects offers an interesting alternative to the well-established density functional methods based on the calculation of the electron density $\ensuremath{\rho}(\stackrel{\ensuremath{\rightarrow}}{r}).$