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arXiv:0706.1266

Lieb-Wu Solution, Gutzwiller-Wave-Function, and Gutzwiller-Ansatz Approximation, with Adjustable Single-Particle Wave Function for the Hubbard Chain

J. Kurzyk, Józef Spałek, W. Wójcik

Published Jun 9, 2007• 0 citations
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Abstract

The optimized single-particle wave functions contained in the parameters of the Hubbard model (t and U) were determined for an infinite atomic chain. In effect, the electronic properties of the chain as a function of interatomic distance R were obtained and compared for the Lieb - Wu exact solution (LW), the Gutzwiller-Wave-Function approximation (GWF), and the Gutzwiller-ansatz case (GA). The ground state energy and other characteristics for the infinite chain were also compared with those obtained earlier for a nanoscopic chain within the Exact Diagonalization combined with an Ab Initio adjustment of the single-particle wave functions in the correlated state (EDABI method). For the sake of completeness, we briefly characterize also each of the solutions. Our approach completes the Lieb-Wu solution, as it provides the system electronic properties evolution as a function of physically controllable parameter - the interatomic distance.

DOI
10.48550/arxiv.0706.1266

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The optimized single-particle wave functions contained in the parameters of the Hubbard model (t and U) were determined for an infinite atomic chain. In effect, the electronic properties of the chain as a function of interatomic distance R were obtained and compared for the Lieb - Wu exact solution (LW), the Gutzwiller-Wave-Function approximation (GWF), and the Gutzwiller-ansatz case (GA). The ground state energy and other characteristics for the infinite chain were also compared with those obtained earlier for a nanoscopic chain within the Exact Diagonalization combined with an Ab Initio adjustment of the single-particle wave functions in the correlated state (EDABI method). For the sake of completeness, we briefly characterize also each of the solutions. Our approach completes the Lieb-Wu solution, as it provides the system electronic properties evolution as a function of physically controllable parameter - the interatomic distance.
The optimized single-particle wave functions contained in the parameters of the Hubbard model (t and U ) were determined for an infinite atomic chain.In effect, the electronic properties of the chain as a function of interatomic distance R were obtained and compared for the Lieb-Wu exact solution, the Gutzwiller-wave-function approximation, and the Gutzwiller--ansatz case.The ground state energy and other characteristics for the infinite chain were also compared with those obtained earlier for a nanoscopic chain within the exact diagonalization combined with an ab initio adjustment of the single-particle wave functions in the correlated state (exact diagonalization combined with an ab initio method).For the sake of completeness, we briefly characterize also each of the solutions.Our approach completes the Lieb-Wu solution, as it provides the system electronic properties evolution as a function of physically controlable parameter -the interatomic distance.
One-dimensional atomic chain with a variable-range hopping is described within the extended Hubbard model.The Gutzwiller-ansatz approximation is used to determine the optimized single-particle (Wannier) wave functions in the correlated state.Hopping integral up to the third neighbors is taken into account and the results are compared with those for the infinite hopping range.Ground state energy of the system is compared with that making use of the rigorous Lieb-Wu solution with the optimized wave functions.The evolution of the properties as a function of interatomic distance is discussed.
The optimized single-particle wave functions contained in the parameters of the Hubbard model (hopping integral t and intraatomic interaction U ) are determined explicitly in the correlated state for electronic systems of various symmetries and dimensions: Hubbard chain, square and triangular lattices, and the three cubic lattices: SC, BCC, and FCC.In effect, the electronic properties of these structures as a function of the interatomic distance R are obtained.In most cases, the model parameters do not scale linearly with the lattice constant.Also, the atomic part of the total ground state energy changes with the U/t ratio and therefore should be (and is) included in the analysis.The solutions of dimensions D > 1 are analyzed by utilizing the approximate Gutzwiller treatment.
We determine the explicit form of the single-particle Wannier functions {wi(r)} appearing in the parameters of quantum models.The method is illustrated on the example of the Hubbard chain, for which we derive the renormalized wave equation starting from a variational principle and by treating the system ground state energy as a functional of {wi(r)} and their derivatives.In this manner, the optimized basis is obtained only after the electronic correlations have been included in the rigorous Lieb-Wu solution.The results for the ground state energy and the size of the renormalized s-type orbitals, both as a function of interatomic distance, are calculated explicitly.
We apply our new method of approach to electronic states in correlated systems [for a brief review, Part I, see: Acta Phys. Pol. B31, 2879 (2000)] to the analysis of the states and electron localization in nanoscopic chains containing up to N = 12 atoms. The Mott and the Hubbard criteria of localization are explicitly evaluated and the importance of the former is stressed. We also define the many-particle wave function and calculate it explicitly for the hydrogen molecule. Further applications of our method are listed at the end.
We have applied our novel numerical scheme combining Lanczos diagonalization in the Fock space with an ab initio renormalization of the single-particle (Wannier) functions, to study the ground state properties of the Extended Hubbard Model. Through the finite-size scaling we determine the discontinuity of the momentum distribution Fermi surface. Our results imply Fermi-liquid behavior for lattice parameter a ≲\lesssim≲ 3a0_{0}0​ (a0_{0}0​ is the Bohr radius) and zero-temperature transition to the localized spin system for larger . Possible further applications of the method are listed at the end.