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

Bulletin of the American Physical Society

Self-consistent implementation of the multi-band Gutzwiller variational method: Formalism and combination with DFT

Nicola Lanatà, Hugo U. R. Strand, Xi Dai, Bo Hellsing

Published Mar 21, 2011• 0 citations
OpenAlex

Related papers

Efficient implementation of the Gutzwiller variational method

Nicola Lanatà, Hugo U. R. Strand, Xi Dai +1 more

We present a self-consistent numerical approach to solve the Gutzwiller variational problem for general multiband models with arbitrary on-site interaction. The proposed method generalizes and improves the procedure derived by Deng et al. [Phys. Rev. B 79, 075114 (2009)], overcoming the restriction to density-density interaction without increasing the complexity of the computational algorithm. Our approach drastically reduces the problem of the high-dimensional Gutzwiller minimization by mapping it to a minimization only in the variational density matrix, in the spirit of the Levy and Lieb formulation of density functional theory (DFT). For fixed density the Gutzwiller renormalization matrix is determined as a fixpoint of a proper functional, whose evaluation requires only ground-state calculations of matrices defined in the Gutzwiller variational space. Furthermore, the proposed method is able to account for the symmetries of the variational function in a controlled way, reducing the number of variational parameters. After a detailed description of the method we present calculations for multiband Hubbard models with full (rotationally invariant) Hund's rule on-site interaction. Our analysis shows that the numerical algorithm is very efficient, stable, and easy to implement. For these reasons this method is particularly suitable for first-principles studies (e.g., in combination with DFT) of many complex real materials, where the full intra-atomic interaction is important to obtain correct results.

arXiv:1108.0180•Physical Review B•Jan 31, 2012•87 citations

Density matrix approach to Gutzwiller method for degenerate bands

J. P. Julien

Physica B Condensed Matter•Jan 1, 1999•2 citations

The Gutzwiller variational approach to correlated systems: study of three significant cases

Nicola Lanatà

•Oct 30, 2009•0 citations

RTGW2020: A powerful implementation of DFT + Gutzwiller method

Shiyu Peng, Hongming Weng, Xi Dai

In the present paper, we propose an efficient numerical scheme for Gutzwiller method for multi-band Hubbard models with general onsite Coulomb interaction. Following the basic idea of Deng et al. [Phys. Rev. B 79, 075114 (2009)] and extensions by Lanata et al. [Phys. Rev. B 85, 035133 (2012)], the ground state is variationally determined through optimizing the total energy with respect to the variational single particle density matrix (n0), which is called "outer loop". In the corresponding "inner loop" where n0 is fixed, the non-interacting wave function and the parameters contained in the Gutzwiller projector are determined by a two-step iterative approach. All derivatives of the implementation process have been analytically derived, which allows us to apply some advanced minimization or root-searching algorithms for both the inner and outer loops leading to the highly efficient convergence. In addition, an atomic diagonalization method taking the point group symmetry into account has been developed for the customized design of the Gutzwiller projector, making it convenient to explore many interesting orders at a lower cost of computation. As benchmarks, several different types of correlated models have been studied utilizing the proposed method, which are in perfect agreement with the previous results by DMFT and multi-orbital slave-boson mean field method. Compared with the linear mixing method, the newton method with analytical derivatives shows much faster convergence for the inner loop. As for the outer loop, the minimization using analytical derivatives also shows much better stability and efficiency compared with that using numerical derivatives.

arXiv:2111.09166•Nov 17, 2021•0 citations

The Gutzwiller variational approach to correlated systems: study of three significant cases

Nicola Lanatà

•Oct 30, 2009•0 citations

Development in Variational Gutzwiller Theory: From Formalism to Applications in Twisted Bilayer Graphene

Shu Liang

This thesis develops variational Gutzwiller theory for realistic multi-orbital strongly correlated systems and applies it to twisted bilayer graphene (TBG), where correlated insulating, metallic, and superconducting phases appear in close proximity. We first establish an efficient and robust implementation of the multi-orbital Gutzwiller method, including the original-basis formulation, stable variational equations, analytical Jacobians and gradients, practical density-matrix parametrizations, and the use of gauge freedom to reduce computational cost. Benchmark calculations demonstrate that this implementation captures correlation effects beyond Hartree-band model of TBG with correlated 𝑓orbitals and itinerant 𝑐orbitals. The resulting phase diagram reveals a Fermi-liquid dome that separates a weakly correlated BCS-like superconducting regime from a strongly correlated superconducting regime. In the latter, pairing is tied to local spin-valley-orbital structure within a moiré unit cell and is better understood in terms of preformed local pairs that gain itinerancy through charge fluctuations and hybridization. Finally, we develop a linear-response framework within the variational Gutzwiller method and derive an analytical expression for the static compressibility. Applied to symmetric TBG, this formalism naturally explains the cascade-like compressibility minima as a consequence of strong correlations in the localized orbitals. Together, these results establish variational Gutzwiller theory as a powerful framework for studying correlated normal and superconducting phases in moirématerials.

•Jan 1, 2026•0 citations
We present a self-consistent numerical approach to solve the Gutzwiller variational problem for general multiband models with arbitrary on-site interaction. The proposed method generalizes and improves the procedure derived by Deng et al. [Phys. Rev. B 79, 075114 (2009)], overcoming the restriction to density-density interaction without increasing the complexity of the computational algorithm. Our approach drastically reduces the problem of the high-dimensional Gutzwiller minimization by mapping it to a minimization only in the variational density matrix, in the spirit of the Levy and Lieb formulation of density functional theory (DFT). For fixed density the Gutzwiller renormalization matrix is determined as a fixpoint of a proper functional, whose evaluation requires only ground-state calculations of matrices defined in the Gutzwiller variational space. Furthermore, the proposed method is able to account for the symmetries of the variational function in a controlled way, reducing the number of variational parameters. After a detailed description of the method we present calculations for multiband Hubbard models with full (rotationally invariant) Hund's rule on-site interaction. Our analysis shows that the numerical algorithm is very efficient, stable, and easy to implement. For these reasons this method is particularly suitable for first-principles studies (e.g., in combination with DFT) of many complex real materials, where the full intra-atomic interaction is important to obtain correct results.
In the present paper, we propose an efficient numerical scheme for Gutzwiller method for multi-band Hubbard models with general onsite Coulomb interaction. Following the basic idea of Deng et al. [Phys. Rev. B 79, 075114 (2009)] and extensions by Lanata et al. [Phys. Rev. B 85, 035133 (2012)], the ground state is variationally determined through optimizing the total energy with respect to the variational single particle density matrix (n0), which is called "outer loop". In the corresponding "inner loop" where n0 is fixed, the non-interacting wave function and the parameters contained in the Gutzwiller projector are determined by a two-step iterative approach. All derivatives of the implementation process have been analytically derived, which allows us to apply some advanced minimization or root-searching algorithms for both the inner and outer loops leading to the highly efficient convergence. In addition, an atomic diagonalization method taking the point group symmetry into account has been developed for the customized design of the Gutzwiller projector, making it convenient to explore many interesting orders at a lower cost of computation. As benchmarks, several different types of correlated models have been studied utilizing the proposed method, which are in perfect agreement with the previous results by DMFT and multi-orbital slave-boson mean field method. Compared with the linear mixing method, the newton method with analytical derivatives shows much faster convergence for the inner loop. As for the outer loop, the minimization using analytical derivatives also shows much better stability and efficiency compared with that using numerical derivatives.
This thesis develops variational Gutzwiller theory for realistic multi-orbital strongly correlated systems and applies it to twisted bilayer graphene (TBG), where correlated insulating, metallic, and superconducting phases appear in close proximity. We first establish an efficient and robust implementation of the multi-orbital Gutzwiller method, including the original-basis formulation, stable variational equations, analytical Jacobians and gradients, practical density-matrix parametrizations, and the use of gauge freedom to reduce computational cost. Benchmark calculations demonstrate that this implementation captures correlation effects beyond Hartree-band model of TBG with correlated 𝑓orbitals and itinerant 𝑐orbitals. The resulting phase diagram reveals a Fermi-liquid dome that separates a weakly correlated BCS-like superconducting regime from a strongly correlated superconducting regime. In the latter, pairing is tied to local spin-valley-orbital structure within a moiré unit cell and is better understood in terms of preformed local pairs that gain itinerancy through charge fluctuations and hybridization. Finally, we develop a linear-response framework within the variational Gutzwiller method and derive an analytical expression for the static compressibility. Applied to symmetric TBG, this formalism naturally explains the cascade-like compressibility minima as a consequence of strong correlations in the localized orbitals. Together, these results establish variational Gutzwiller theory as a powerful framework for studying correlated normal and superconducting phases in moirématerials.