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.
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.
Based on the variational Gutzwiller theory, we present a method for the computation of response functions for multiband Hubbard models with general local Coulomb interactions. The improvement over the conventional random-phase approximation is exemplified for an infinite-dimensional two-band Hubbard model where the incorporation of the local multiplet structure leads to a much larger sensitivity of ferromagnetism on the Hund coupling. Our method can be implemented into local-density approximation and Gutzwiller schemes and will therefore be an important tool for the computation of response functions for strongly correlated materials.
Ground-state properties, such as energies and double occupancies, of a one-dimensional two-band Hubbard model are calculated using a first-principles Gutzwiller conjugate gradient minimization theory. The favorable agreement with the results from the density matrix renormalization group theory demonstrates the accuracy of our method. A rotationally invariant approach is further incorporated into the method to greatly reduce the computational complexity with a speedup of approximately 50 times. Moreover, we investigate the Mott transition between a metal and a Mott insulator by evaluating the charge gap. With greatly reduced computational effort, our method reproduces the phase diagram in reasonable agreement with the density matrix renormalization group theory.
The recently developed variational discrete action theory (VDAT) provides a systematic variational approach to the ground state of the quantum many-body problem, where the quality of the solution is controlled by an integer N, and increasing N monotonically approaches the exact solution. VDAT can be exactly evaluated in the $d=\ensuremath{\infty}$ multiorbital Hubbard model using the self-consistent canonical discrete action theory (SCDA), which requires a self-consistency condition for the integer time Green's functions. Previous work demonstrates that N=3 accurately captures multiorbital Mott/Hund physics at a cost similar to the Gutzwiller approximation. Here we employ a gauge constraint to automatically satisfy the self-consistency condition of the SCDA at N=3, yielding an even more efficient algorithm with enhanced numerical stability. We derive closed form expressions of the gauge constrained algorithm for the multiorbital Hubbard model with general density-density interactions, allowing VDAT at N=3 to be straightforwardly applied to the seven-orbital Hubbard model. We present results and a performance analysis using N=2 and N=3 for the SU(2Norb) Hubbard model in $d=\ensuremath{\infty}$ with Norb=2−−8, and compare to numerically exact dynamical mean-field theory solutions where available. The developments in this work will greatly facilitate the application of VDAT at N=3 to strongly correlated electron materials.
A gauge constrained algorithm of VDAT at N=3 for the multi-orbital Hubbard model
The recently developed variational discrete action theory (VDAT) provides a systematic variational approach to the ground state of the quantum many-body problem, where the quality of the solution is controlled by an integer N, and increasing N monotonically approaches the exact solution. VDAT can be exactly evaluated in the d=∞ multi-orbital Hubbard model using the self-consistent canonical discrete action theory (SCDA), which requires a self-consistency condition for the integer time Green's functions. Previous work demonstrates that N=3 accurately captures multi-orbital Mott/Hund physics at a cost similar to the Gutzwiller approximation. Here we employ a gauge constraint to automatically satisfy the self-consistency condition of the SCDA at N=3, yielding an even more efficient algorithm with enhanced numerical stability. We derive closed form expressions of the gauge constrained algorithm for the multi-orbital Hubbard model with general density-density interactions, allowing VDAT at N=3 to be straightforwardly applied to the seven orbital Hubbard model. We present results and a performance analysis using N=2 and N=3 for the SU(2Norb) Hubbard model in d=∞ with Norb=2−8, and compare to numerically exact dynamical mean-field theory solutions where available. The developments in this work will greatly facilitate the application of VDAT at N=3 to strongly correlated electron materials.
We introduce Gutzwiller wave functions for multiband models with general on-site Coulomb interactions. As these wave functions employ correlators for the exact atomic eigenstates, they are exact both in the noninteracting and atomic limits. We evaluate them in infinite lattice dimensions for all interaction strengths without any restrictions on the structure of the Hamiltonian or the symmetry of the ground state. The results for the ground-state energy allow us to derive an effective one-electron Hamiltonian for Landau quasiparticles, applicable for finite temperatures and frequencies within the Fermi-liquid regime. As applications for a two-band model we study the Brinkman-Rice metal-to-insulator transition at half-band-filling, and the transition to itinerant ferromagnetism for two specific fillings, at and close to a peak in the density of states of the noninteracting system. Our results significantly differ from those for earlier Gutzwiller wave functions where only density-type interactions were included. When the correct spin symmetries for the two-electron states are taken into account, the importance of the Hund's-rule exchange interaction is even more pronounced, and leads to paramagnetic metallic ground states with large local magnetic moments. Ferromagnetism requires fairly large interaction strengths, and the resulting ferromagnetic state is a strongly correlated metal.