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Acta Physica Polonica A

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

J. Kurzyk, Józef Spałek, W. Wójcik

Published Apr 1, 2007• 9 citations
Publisher

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, 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.

DOI
10.12693/aphyspola.111.603

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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, 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.
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.
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.
We summarize briefly the main results obtained within the proposed EDABI method combining Exact Diagonalization of (parametrized) many-particle Hamiltonian with Ab Initio self-adjustment of the single-particle wave function in the correlated state of interacting electrons. The properties of nanoscopic chains and rings are discussed as a function of their interatomic distance R and compared with those obtained by Bethe ansatz for infinite Hubbard chain. The concepts of renormalized orbitals, distribution function in momentum space, and of Hubbard splitting as applied to nanoscopic systems are emphasized.
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 determine the explicit form of the single-particle wave functions wi(r){{w}_{i}(\mathbf{r})}wi​(r) appearing in the microscopic parameters of models in the second-quantization representation. Namely, the general form of the renormalized wave equation is derived from the Lagrange-Euler principle by treating the system ground-state energy of an exact correlated state as a functional of wi(r){{w}_{i}(\mathbf{r})}wi​(r) and their derivatives. The method is applied to three model situations with one orbital per atom. For the first example---the Hubbard chain---the optimized basis is obtained only after the electronic correlation has been included in the rigorous Lieb-Wu solution for the ground-state energy. The renormalized Wannier wave functions are obtained variationally starting from the atomic basis for the s-type wave functions. The principal characteristics such as the ground-state energy and the model parameters are calculated as a function of interatomic distance. Second, the atomic systems such as the H2{\mathrm{H}}_{2}H2​ molecule or He atom can be treated in the same manner and the optimized orbitals are obtained to illustrate the method further. Finally, we illustrate the method by solving exactly correlated quantum dots of N< 8N<~8N< 8 atoms with the subsequent optimization of the orbitals. Our method may be regarded as the next step in analyzing exactly soluble many-body models that provides properties as a function of the lattice parameter and defines at the same time the renormalized wave function for a single particle.