Efficient implementation of the Gutzwiller variational method

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) (http://dx.doi.org/10.1103/PhysRevB.79.075114)], 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.