Extended Hubbard model with renormalized Wannier wave functions in the correlated state: beyond the parametrized models

TL;DR

This paper extends a method to analyze infinite correlated electronic systems by explicitly determining renormalized Wannier wave functions and model parameters, revealing their complex dependence on interatomic distance and system symmetry.

Contribution

It introduces a procedure to compute renormalized Wannier functions and model parameters directly in the correlated state for various lattice geometries, beyond parametrized models.

Findings

Model parameters do not scale linearly with lattice spacing.

Atomic energy variations with interatomic distance are significant.

Method applicable to different dimensions and lattice types.

Abstract

The method used earlier for analysis of correlated nanoscopic systems is extended to infinite (periodic) s-band like systems described by the Hubbard model and its extensions. The optimized single-particle wave functions contained in the parameters of the Hubbard model (the hopping \textit{t} and the magnitude of the intraatomic interaction \textit{U}) are determined explicitly in the correlated state for the electronic systems of various symmetries and dimensions: Hubbard chain, square and triangular planar lattices, and the three cubic lattices (SC, BCC, FCC). In effect, the evolution of the electronic properties as a function of interatomic distance $R$ is obtained. The model parameters in most cases do not scale linearly with the lattice spacing and hence, their solution as a function of microscopic parameters reflects only qualitatively the system evolution. Also, the atomic energy changes with $R$ and therefore should be included in the model analysis. The solutions in one dimension (\textit{D} = 1) can be analyzed both rigorously (by making use of the Lieb--Wu solution) and compared with the approximate Gutzwiller treatment. In higher dimensions (\textit{D} = 2, 3) only the latter approach is possible to implement within the scheme. The renormalized single particle wave functions are almost independent of the choice of the scheme selected to diagonalize the Hamiltonian in the Fock space in D=1 case. The method can be extended to other approximation schemes as stressed at the end.