Physical review. B, Condensed matter

Exact diagonalization of many-fermion Hamiltonian with wave-function renormalization

Adam Rycerz, Józef Spałek

Published Jan 18, 2001• 14 citations

Abstract

We propose a method of determining states for correlated fermions in Fock space, which we supplement with the Lagrange-Euler equation for the single-particle wave functions ${{w}_{i}(\mathbf{r})}$ contained in the microscopic parameters. The equation is derived by treating the ground state energy of the interacting system as a functional of ${{w}_{i}(\mathbf{r})}.$ The method is applied to a linear chain up to $N=10$ atoms with all pair interactions and all hopping integrals included. Renormalized Wannier functions, the parameters, the distribution function, and the dimerization energy, are all discussed as a function of the lattice parameter.

DOI: 10.1103/physrevb.63.073101

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Physical review. B, Condensed matter

Exact diagonalization of many-fermion Hamiltonian with wave-function renormalization

Adam Rycerz, Józef Spałek

Published Jan 18, 2001• 14 citations

Publisher

Abstract

We propose a method of determining states for correlated fermions in Fock space, which we supplement with the Lagrange-Euler equation for the single-particle wave functions

{{w}_{i}(\mathbf{r})}

contained in the microscopic parameters. The equation is derived by treating the ground state energy of the interacting system as a functional of

{{w}_{i}(\mathbf{r})}.

The method is applied to a linear chain up to

N=10

atoms with all pair interactions and all hopping integrals included. Renormalized Wannier functions, the parameters, the distribution function, and the dimerization energy, are all discussed as a function of the lattice parameter.

DOI: 10.1103/physrevb.63.073101

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Optimization of single-particle basis for exactly soluble models of correlated electrons

Józef Spałek, Robert Podsiadły, W. Wójcik +1 more

We determine the explicit form of the single-particle wave functions ${{w}_{i}(\mathbf{r})}$ 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 ${{w}_{i}(\mathbf{r})}$ 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 ${\mathrm{H}}_{2}$ 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 $N<~8$ 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.We determine the explicit form of the single-particle wave functions ${{w}_{i}(\mathbf{r})}$ 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 ${{w}_{i}(\mathbf{r})}$ 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 ${\mathrm{H}}_{2}$ 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 $N<~8$ 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.

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Physical ReviewOct 15, 196338 citations

We propose a method of determining states for correlated fermions in Fock space, which we supplement with the Lagrange-Euler equation for the single-particle wave functions

{{w}_{i}(\mathbf{r})}

contained in the microscopic parameters. The equation is derived by treating the ground state energy of the interacting system as a functional of

{{w}_{i}(\mathbf{r})}.

The method is applied to a linear chain up to

N=10

We determine the explicit form of the single-particle wave functions

{{w}_{i}(\mathbf{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

{{w}_{i}(\mathbf{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

{\mathrm{H}}_{2}

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<~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.

N>~8.

During this process the Fermi-Dirac distribution is smeared out, the effective band mass increases by \ensuremath{\sim}50%, and the spin-spin correlation functions reduce to those for a Heisenberg antiferromagnet. Values of microscopic parameters such as the hopping and the kinetic-exchange integrals, as well as the magnitude of both intraatomic and interatomic Coulomb and exchange interactions are calculated. We also determine the values of various local electron-lattice couplings, and show that they are comparable to the kinetic exchange contribution in the strong-correlation limit. The magnitudes of the dimerization and the zero-point motion are also discussed. Our results provide a canonical example of a tractable strongly correlated system with a precise, first-principles description as a function of interatomic distance of a model system involving all hopping integrals, all pair-site interactions, and exact one-band Wannier functions.

We describe briefly a new method of approach to the interacting fermion and boson systems. Namely, we determine the explicit form of the single-particle wave functions {w

_{i}

(r)} appearing in the microscopic parameters of models in the second-quantization representation. The method is illustrated on the examples of H

_{2}

molecule and He atom, the Hubbard chain, and a ring of N

\leq

10 atoms.

A self-consistent procedure is presented for the determination of the properties of a many-fermion system, taking into account all two-particle correlations. We consider a system of

N

fermions, interacting through two-body forces and write the wave function in the form, $\ensuremath{\Psi}={\ensuremath{\Psi}}_{0}+\ensuremath{\Sigma}{\mathrm{ij}}^{}{f}^{(2)}(\mathrm{ij})$, where ${\ensuremath{\Psi}}_{0}$ is a determinant of one-particle wave functions and

{f}^{(2)}(\mathrm{ij})

is an antisymmetrized product of ($N\ensuremath{-}2$) one-particle functions and of one two-particle function. By introducing two-particle functions for each electron pair, all two-particle correlations are taken into account. It is shown that for the best one- and two-particle functions a system of coupled integrodifferential equations can be derived. These equations are derived by varying the expectation value of the Hamiltonian with respect to the one- and two-particle functions, taking into account the normalization and orthogonality as subsidiary conditions. After eliminating the Lagrangian multipliers, we have obtained the following result. We obtained

N

one-particle equations for the

N

one-particle wave functions and one-particle orbital parameters. These equations are characterized by a potential (and exchange operator) in which, besides the Hartree-Fock type potential terms, there are also the potentials arising from the two-particle functions, where the latter occur in diagonal, as well as in nondiagonal form. For the two-particle functions and the orbital parameters associated with them, we have obtained two-particle equations in which the equation for the function ${\ensuremath{\varphi}}_{\mathrm{ij}}$ contains all one-particle functions and all the other two-particle functions. It is shown that the system of coupled one- and two-particle equations can be solved with a self-consistent procedure. The method can be applied to systems with any number of particles.