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:1706.01388 · Physical Review Letters

Fierz Convergence Criterion: A Controlled Approach to Strongly Interacting Systems with Small Embedded Clusters

Thomas Ayral, J. Vučičević, Olivier Parcollet

Published Oct 16, 2017• 51 citations
arXivHTMLPDF

Abstract

We present an embedded-cluster method, based on the triply irreducible local expansion formalism. It turns the Fierz ambiguity, inherent to approaches based on a bosonic decoupling of local fermionic interactions, into a convergence criterion. It is based on the approximation of the three-leg vertex by a coarse-grained vertex computed from a self-consistently determined cluster impurity model. The computed self-energies are, by construction, continuous functions of momentum. We show that, in three interaction and doping regimes of the two-dimensional Hubbard model, self-energies obtained with clusters of size four only are very close to numerically exact benchmark results. We show that the Fierz parameter, which parametrizes the freedom in the Hubbard-Stratonovich decoupling, can be used as a quality control parameter. By contrast, the GW+extended dynamical mean field theory approximation with four cluster sites is shown to yield good results only in the weak-coupling regime and for a particular decoupling. Finally, we show that the vertex has spatially nonlocal components only at low Matsubara frequencies.

DOI
10.1103/physrevlett.119.166401

Related papers

Embedded multi-boson exchange: A step beyond quantum cluster theories

Dominik Kiese, Nils Wentzell, Igor S. Krivenko +3 more

We introduce a diagrammatic multi-scale approach to the Hubbard model based on the interaction-irreducible (multi-boson) vertex of a small cluster embedded in a self-consistent medium. The vertex captures short-range correlations up to the length scale of the cluster, while long-range correlations are recovered from a set of diagrammatic equations for the Hedin three-leg vertex. By virtue of the crossing symmetry, the Fierz decoupling ambiguity of the Hubbard interaction is resolved exactly. Our benchmarks for the half-filled Hubbard model on the square lattice are in very good agreement with numerically exact diagrammatic Monte Carlo simulations. Published by the American Physical Society 2024

arXiv:2406.15629•Physical Review Research•Nov 18, 2024•6 citations

Nonlocal collective electronic fluctuations: From model systems to realistic materials

Viktor Harkov

The main focus of the present thesis lies on the development and application of numerical approaches based on quantum lattice models to describe collective excitations in strongly correlated materials. To this end, an efficient and balanced dual triply irreducible local expansion (D-TRILEX) approach that captures the major effects of spatial correlations in strongly correlated fermionic systems is presented, which is based on partial bosonization of the collective electronic fluctuations. Thereby, a specific choice of the bare interaction in the corresponding channel of instability allows a simultaneous description of collective excitations in different channels without facing the Fierz ambiguity problem which usually arises in methods based on partial bosonization. Compared to more elaborate methods, D-TRILEX does not require the computation of the numerically expensive four-point vertex. Furthermore, taking into account only the leading longitudinal contribution of the bosonic modes to the self-energy reduces the complexity of the diagrammatic structure drastically making the method applicable to more complex realistic multi-orbital systems. As the main result of this thesis, the D-TRILEX approach is introduced and its limit of applicability is studied by applying it to the two-dimensional single-band Hubbard model. Performing a comprehensive diagrammatic analysis by comparing D-TRILEX with methods, which consider different sets of diagrams and different approximations for the four-point vertex, we find that contributions that are not considered by the partially bosonized approximation of the four-point vertex have only a minor effect on electronic degrees of freedom and the longitudinal bosonic modes taking into account in D-TRILEX self-energy are the leading ones in a broad range of control parameters. After investigating the applicability of the D-TRILEX approach we apply the method to study collective electronic effects in the hole-doped InSe monolayers using a realistic electronic-structure model. Thereby, we find that due to the weakly screened long-range Coulomb interaction the system shows a charge density waves phase for the broad range of considered parameters, however, inside this regime we have found a coexisting ferromagnetic phase. D-TRILEX can be considered as an approximation of the more elaborate dual boson (DB) theory, which accounts for local correlations within the dynamical mean-field theory (DMFT) and nonlocal correlations perturbatively. As a further result of this thesis, we investigate the weak coupling regime of the two-dimensional single-band Hubbard model within the ladder DB approximation and compare our results to the numerically exact diagrammatic quantum Monte Carlo (DiagMC) method. We find that the DB method qualitatively captures all the different regimes. Even quantitatively DB shows a good agreement with the exact results for single- and two-particle quantities at high and moderate temperatures. However, the ladder approximation slightly underestimates the strong antiferromagnetic fluctuations that appear in the system at low temperatures. As the final result of this thesis, we perform a parametrization of the local four-point vertex in terms of the (three-point) Hedin vertex and screened interactions in the framework of single-boson exchange (SBE) decomposition. Similar to the D-TRILEX approach the computational costs are reduced by neglecting the interaction-irreducible part of the four-point vertex. The approximated vertex is used to calculate the DMFT susceptibility for different interaction strengths at different temperatures. We find that the approximation shows a good agreement in the limits of weak and strong couplings. Thereby, our study highlights the importance of the Hedin vertex for local vertex corrections.

•May 4, 2023•0 citations

Mott physics and collective modes: An atomic approximation of the four-particle irreducible functional

Thomas Ayral, Olivier Parcollet

We discuss a generalization of the dynamical mean field theory (DMFT) for strongly correlated systems close to a Mott transition based on a systematic approximation of the fully irreducible four-point vertex. It is an atomic-limit approximation of a functional of the one- and two-particle Green functions, built with the second Legendre transform of the free energy with respect to the two-particle Green function. This functional is represented diagrammatically by four-particle irreducible (4PI) diagrams. Like the dynamical vertex approximation $(\mathrm{D}\mathrm{\ensuremath{\Gamma}}\mathrm{A})$, the fully irreducible vertex is computed from a quantum impurity model whose bath is self-consistently determined by solving the parquet equations. However, in contrast with $\mathrm{D}\mathrm{\ensuremath{\Gamma}}\mathrm{A}$ and DMFT, the interaction term of the impurity model is also self-consistently determined. The method interpolates between the parquet approximation at weak coupling and the atomic limit, where it is exact. It is applicable to systems with short-range and long-range interactions.

arXiv:1605.09048•Physical review. B./Physical review. B•Aug 31, 2016•62 citations

Momentum-space cluster dual-fermion method

Sergei Iskakov, Hanna Terletska, Emanuel Gull

Recent years have seen the development of two types of nonlocal extensions to the single-site dynamical mean field theory. On one hand, cluster approximations, such as the dynamical cluster approximation, recover short-range momentum-dependent correlations nonperturbatively. On the other hand, diagrammatic extensions, such as the dual-fermion theory, recover long-ranged corrections perturbatively. The correct treatment of both strong short-ranged and weak long-ranged correlations within the same framework is therefore expected to lead to a quick convergence of results, and offers the potential of obtaining smooth self-energies in nonperturbative regimes of phase space. In this paper, we present an exact cluster dual-fermion method based on an expansion around the dynamical cluster approximation. Unlike previous formulations, our method does not employ a coarse-graining approximation to the interaction, which we show to be the leading source of error at high temperature, and converges to the exact result independently of the size of the underlying cluster. We illustrate the power of the method with results for the second-order cluster dual-fermion approximation to the single-particle self-energies and double occupancies.

arXiv:1712.03295•Physical review. B./Physical review. B•Mar 12, 2018•14 citations

Two-particle correlations in a dynamic cluster approximation with continuous momentum dependence: Superconductivity in the two-dimensional Hubbard model

Peter Staar, Thomas Maier, T. C. Schulthess

The DCA${}^{+}\phantom{\rule{0.222222em}{0ex}}$algorithm was recently introduced by Stear, Maier, and Schulthess [Phys. Rev. B 88, 115101 (2013)] to extend the dynamic cluster approximation (DCA) with a continuous lattice self-energy in order to achieve better convergence with cluster size. Here we extend the DCA${}^{+}\phantom{\rule{0.222222em}{0ex}}$algorithm to the calculation of two-particle correlation functions by introducing irreducible vertex functions with continuous momentum dependence consistent with the DCA${}^{+}\phantom{\rule{0.222222em}{0ex}}$self-energy. This enables a significantly more controlled and reliable study of phase transitions than with the DCA. We test the new method by calculating the superconducting transition temperature ${T}_{c}$ in the attractive Hubbard model and show that it reproduces previous determinantal quantum Monte Carlo results. We then calculate ${T}_{c}$ in the doped repulsive Hubbard model, for which previous DCA calculations could only access the weak-coupling ($U=4t$) regime for large clusters. We show that the new algorithm provides access to much larger clusters and delivers asymptotically converged results for ${T}_{c}$ for both the weak ($U=4t$) and intermediate ($U=7t$) coupling regimes, and thereby enables the accurate determination of the exact infinite cluster size result.

arXiv:1402.4329•Physical Review B•May 27, 2014•26 citations

Continuous momentum dependence in the dynamical cluster approximation

Urs R. Hähner, Thomas Maier, T. C. Schulthess

The dynamical cluster approximation (DCA) is a quantum cluster extension to the single-site dynamical mean-field theory that incorporates spatially nonlocal dynamic correlations systematically and nonperturbatively. The ${\mathrm{DCA}}^{+}$ algorithm addresses the cluster shape dependence of the DCA and improves the convergence with cluster size by introducing a lattice self-energy with continuous momentum dependence. However, we show that the ${\mathrm{DCA}}^{+}$ algorithm is plagued by a fundamental problem when its self-consistency equations are formulated using the bare Green's function of the cluster. This problem is most severe in the strongly correlated regime at low doping, where the ${\mathrm{DCA}}^{+}$ self-energy becomes overly metallic and local, and persists to cluster sizes where the standard DCA has long converged. In view of the failure of the ${\mathrm{DCA}}^{+}$ algorithm, we propose to complement DCA simulations with a post-interpolation procedure for single-particle and two-particle correlation functions to preserve continuous momentum dependence and the associated benefits in the DCA. We demonstrate the effectiveness of this practical approach with results for the half-filled and hole-doped two-dimensional Hubbard model.

arXiv:2002.06866•Physical review. B./Physical review. B•May 11, 2020•4 citations
We present an embedded-cluster method, based on the triply irreducible local expansion formalism. It turns the Fierz ambiguity, inherent to approaches based on a bosonic decoupling of local fermionic interactions, into a convergence criterion. It is based on the approximation of the three-leg vertex by a coarse-grained vertex computed from a self-consistently determined cluster impurity model. The computed self-energies are, by construction, continuous functions of momentum. We show that, in three interaction and doping regimes of the two-dimensional Hubbard model, self-energies obtained with clusters of size four only are very close to numerically exact benchmark results. We show that the Fierz parameter, which parametrizes the freedom in the Hubbard-Stratonovich decoupling, can be used as a quality control parameter. By contrast, the GW+extended dynamical mean field theory approximation with four cluster sites is shown to yield good results only in the weak-coupling regime and for a particular decoupling. Finally, we show that the vertex has spatially nonlocal components only at low Matsubara frequencies.
We introduce a diagrammatic multi-scale approach to the Hubbard model based on the interaction-irreducible (multi-boson) vertex of a small cluster embedded in a self-consistent medium. The vertex captures short-range correlations up to the length scale of the cluster, while long-range correlations are recovered from a set of diagrammatic equations for the Hedin three-leg vertex. By virtue of the crossing symmetry, the Fierz decoupling ambiguity of the Hubbard interaction is resolved exactly. Our benchmarks for the half-filled Hubbard model on the square lattice are in very good agreement with numerically exact diagrammatic Monte Carlo simulations. Published by the American Physical Society 2024
The main focus of the present thesis lies on the development and application of numerical approaches based on quantum lattice models to describe collective excitations in strongly correlated materials. To this end, an efficient and balanced dual triply irreducible local expansion (D-TRILEX) approach that captures the major effects of spatial correlations in strongly correlated fermionic systems is presented, which is based on partial bosonization of the collective electronic fluctuations. Thereby, a specific choice of the bare interaction in the corresponding channel of instability allows a simultaneous description of collective excitations in different channels without facing the Fierz ambiguity problem which usually arises in methods based on partial bosonization. Compared to more elaborate methods, D-TRILEX does not require the computation of the numerically expensive four-point vertex. Furthermore, taking into account only the leading longitudinal contribution of the bosonic modes to the self-energy reduces the complexity of the diagrammatic structure drastically making the method applicable to more complex realistic multi-orbital systems. As the main result of this thesis, the D-TRILEX approach is introduced and its limit of applicability is studied by applying it to the two-dimensional single-band Hubbard model. Performing a comprehensive diagrammatic analysis by comparing D-TRILEX with methods, which consider different sets of diagrams and different approximations for the four-point vertex, we find that contributions that are not considered by the partially bosonized approximation of the four-point vertex have only a minor effect on electronic degrees of freedom and the longitudinal bosonic modes taking into account in D-TRILEX self-energy are the leading ones in a broad range of control parameters. After investigating the applicability of the D-TRILEX approach we apply the method to study collective electronic effects in the hole-doped InSe monolayers using a realistic electronic-structure model. Thereby, we find that due to the weakly screened long-range Coulomb interaction the system shows a charge density waves phase for the broad range of considered parameters, however, inside this regime we have found a coexisting ferromagnetic phase. D-TRILEX can be considered as an approximation of the more elaborate dual boson (DB) theory, which accounts for local correlations within the dynamical mean-field theory (DMFT) and nonlocal correlations perturbatively. As a further result of this thesis, we investigate the weak coupling regime of the two-dimensional single-band Hubbard model within the ladder DB approximation and compare our results to the numerically exact diagrammatic quantum Monte Carlo (DiagMC) method. We find that the DB method qualitatively captures all the different regimes. Even quantitatively DB shows a good agreement with the exact results for single- and two-particle quantities at high and moderate temperatures. However, the ladder approximation slightly underestimates the strong antiferromagnetic fluctuations that appear in the system at low temperatures. As the final result of this thesis, we perform a parametrization of the local four-point vertex in terms of the (three-point) Hedin vertex and screened interactions in the framework of single-boson exchange (SBE) decomposition. Similar to the D-TRILEX approach the computational costs are reduced by neglecting the interaction-irreducible part of the four-point vertex. The approximated vertex is used to calculate the DMFT susceptibility for different interaction strengths at different temperatures. We find that the approximation shows a good agreement in the limits of weak and strong couplings. Thereby, our study highlights the importance of the Hedin vertex for local vertex corrections.
We discuss a generalization of the dynamical mean field theory (DMFT) for strongly correlated systems close to a Mott transition based on a systematic approximation of the fully irreducible four-point vertex. It is an atomic-limit approximation of a functional of the one- and two-particle Green functions, built with the second Legendre transform of the free energy with respect to the two-particle Green function. This functional is represented diagrammatically by four-particle irreducible (4PI) diagrams. Like the dynamical vertex approximation $(\mathrm{D}\mathrm{\ensuremath{\Gamma}}\mathrm{A})$, the fully irreducible vertex is computed from a quantum impurity model whose bath is self-consistently determined by solving the parquet equations. However, in contrast with $\mathrm{D}\mathrm{\ensuremath{\Gamma}}\mathrm{A}$ and DMFT, the interaction term of the impurity model is also self-consistently determined. The method interpolates between the parquet approximation at weak coupling and the atomic limit, where it is exact. It is applicable to systems with short-range and long-range interactions.
Recent years have seen the development of two types of nonlocal extensions to the single-site dynamical mean field theory. On one hand, cluster approximations, such as the dynamical cluster approximation, recover short-range momentum-dependent correlations nonperturbatively. On the other hand, diagrammatic extensions, such as the dual-fermion theory, recover long-ranged corrections perturbatively. The correct treatment of both strong short-ranged and weak long-ranged correlations within the same framework is therefore expected to lead to a quick convergence of results, and offers the potential of obtaining smooth self-energies in nonperturbative regimes of phase space. In this paper, we present an exact cluster dual-fermion method based on an expansion around the dynamical cluster approximation. Unlike previous formulations, our method does not employ a coarse-graining approximation to the interaction, which we show to be the leading source of error at high temperature, and converges to the exact result independently of the size of the underlying cluster. We illustrate the power of the method with results for the second-order cluster dual-fermion approximation to the single-particle self-energies and double occupancies.
The DCA+{}^{+}\phantom{\rule{0.222222em}{0ex}}+algorithm was recently introduced by Stear, Maier, and Schulthess [Phys. Rev. B 88, 115101 (2013)] to extend the dynamic cluster approximation (DCA) with a continuous lattice self-energy in order to achieve better convergence with cluster size. Here we extend the DCA+{}^{+}\phantom{\rule{0.222222em}{0ex}}+algorithm to the calculation of two-particle correlation functions by introducing irreducible vertex functions with continuous momentum dependence consistent with the DCA+{}^{+}\phantom{\rule{0.222222em}{0ex}}+self-energy. This enables a significantly more controlled and reliable study of phase transitions than with the DCA. We test the new method by calculating the superconducting transition temperature Tc{T}_{c}Tc​ in the attractive Hubbard model and show that it reproduces previous determinantal quantum Monte Carlo results. We then calculate Tc{T}_{c}Tc​ in the doped repulsive Hubbard model, for which previous DCA calculations could only access the weak-coupling (U=4tU=4tU=4t) regime for large clusters. We show that the new algorithm provides access to much larger clusters and delivers asymptotically converged results for Tc{T}_{c}Tc​ for both the weak (U=4tU=4tU=4t) and intermediate (U=7tU=7tU=7t) coupling regimes, and thereby enables the accurate determination of the exact infinite cluster size result.
The dynamical cluster approximation (DCA) is a quantum cluster extension to the single-site dynamical mean-field theory that incorporates spatially nonlocal dynamic correlations systematically and nonperturbatively. The DCA+{\mathrm{DCA}}^{+}DCA+ algorithm addresses the cluster shape dependence of the DCA and improves the convergence with cluster size by introducing a lattice self-energy with continuous momentum dependence. However, we show that the DCA+{\mathrm{DCA}}^{+}DCA+ algorithm is plagued by a fundamental problem when its self-consistency equations are formulated using the bare Green's function of the cluster. This problem is most severe in the strongly correlated regime at low doping, where the DCA+{\mathrm{DCA}}^{+}DCA+ self-energy becomes overly metallic and local, and persists to cluster sizes where the standard DCA has long converged. In view of the failure of the DCA+{\mathrm{DCA}}^{+}DCA+ algorithm, we propose to complement DCA simulations with a post-interpolation procedure for single-particle and two-particle correlation functions to preserve continuous momentum dependence and the associated benefits in the DCA. We demonstrate the effectiveness of this practical approach with results for the half-filled and hole-doped two-dimensional Hubbard model.