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

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.

Extended Hubbard model with the renormalized Wannier wave functions in the correlated state II: quantum critical scaling of the wave function near the Mott-Hubbard transition

“…The role of these correlations is indeed negligible if the fundamental correlation function d ≡ n i↑ n i↓ vanishes or is very small, as also discussed earlier. 13 It vanishes in the Mott insulating state (for R > R c ). The microscopic parameters of this model are expressed via the Wannier functions in the standard manner 9,12 , {w i (r)} ≡ {w(r − R i )}, as follows: the bare atomic energy is ǫ a ≡ w i |H 1 |w i , the hopping integral t ij ≡ w i |H 1 |w j , intraatomic-interaction magnitude U ≡ w i w i |V 12 |w i w i , and interatomic-interaction magnitude K ij ≡ w i w j |V 12 | w i w j , where H 1 is the Hamiltonian for a single particle in the system, and V 12 represents interparticle interaction.…”

“…However, one has to realize, that calculation of r involves a three-dimensional integration over space in the situation when the Wannier functions are strongly anisotropic. 13 Because of this specific reason, we regard the scaling of w 0 (r) maximum shown in Fig. 5 as more direct, as since involves the wave function characteristic which is not averaged out and, obviously, independent of the spatial direction.…”

“…In Section 2 we introduce briefly the method devised before. 9,12,13 In Sec. 3 we introduce an original critical scaling of the single-particle wave function characteristics and of the ground-state energy.…”

Optimized Wannier Functions for Hubbard Chain

“…This energy is important as we study the system evolution as a function of the interatomic distance R [1] and reach the proper atomic limit when R → ∞. To solve this problem we use the optimized single-particle wave functions method [2].…”

Correlated Electron Systems of Different Dimensionalities

“…In order to determine the ground state energy we use the Gutzwiller ansatz [6] (see also [7] where we used the exact Lieb-Wu solution as well as the Gutzwiller wave function approximation (GA) for the Hubbard chain)…”