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Exact Diagonalization of Many-Fermion Hamiltonian Combined with Wave-Function Readjustment: Application to One-Dimensional Systems

Józef Spałek, Robert Podsiadły, Adam Rycerz, W. Wójcik

Published Dec 1, 2000• 0 citations
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Abstract

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.

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Exact Diagonalization of Many-Fermion Hamiltonian Combined with Wave-Function Readjustment II. Metallicity and Electron Localization in Nanoscopic Systems

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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.

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

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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.

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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.

Physical review. B, Condensed matter•Jun 15, 2000•40 citations

An ab-initio many-body method for electronic structure calculations of solids. I. Description of the method

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In this work we reexamine the many-fermion problem in arbitrary dimensions. It is shown that in two dimensions or higher, the Hamiltonian of interacting fermions can be separated into individual nonintersecting sectors labeled by the wave-vector $\vec q$q⃗. Within each sector the Hamiltonian maps onto a generalized version of the one-dimensional Luttinger model that resembles a boson string. These are chain-like quadratic forms in boson operators that are readily diagonalized in the absence of “exchange” corrections. Moreover, in a simple example involving SU(2) fermions, that of the Hubbard model, we show that it can be possible also to incorporate exchange terms and express them entirely within an enlarged set of string variables.

Journal of Mathematical Physics•Aug 16, 2012•2 citations
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 {wi_{i}i​(r)} appearing in the microscopic parameters of models in the second-quantization representation. The method is illustrated on the examples of H2_{2}2​ molecule and He atom, the Hubbard chain, and a ring of N ≤\leq≤ 10 atoms.
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 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 wi(r){{w}_{i}(\mathbf{r})}wi​(r) contained in the microscopic parameters. The equation is derived by treating the ground state energy of the interacting system as a functional of wi(r).{{w}_{i}(\mathbf{r})}.wi​(r). The method is applied to a linear chain up to N=10N=10N=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.
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.
We propose a new, alternative method for ab-initio calculations of the electronic structure of solids, which has been specifically adapted to treat many-body effects in a more rigorous way than many existing ab-initio methods. We start from a standard band-structure calculation for an effective one-particle Hamiltonian approximately describing the material of interest. This yields a suitable set of one-particle basis functions, from which well localized Wannier functions can be constructed using a method proposed by Marzari and Vanderbilt. Within this (minimal) basis of localized Wannier functions the matrix elements of the non-interacting (one-particle) Hamiltonian as well as the Coulomb matrix elements can be calculated. The result is a many-body Hamiltonian in second quantization with parameters determined from first principles calculations for the material of interest. The Hamiltonian is in the form of a multi-band Hamiltonian in second quantization (a kind of extended, multi-band Hubbard model) such that all the standard many-body methods can be applied. We explicitly show how this approach can be solved in the simplest many-body approximation, the mean-field Hartree-Fock approximation (HFA), which takes into account exact exchange and corrects for self-interaction effects.
Using eigen-functional bosonization method, we study quantum many-particle systems, and show that the quantum many-particle problems end in to solve the differential equation of the phase fields which represent the particle correlation strength. Thus, the physical properties of these systems are completely determined by the differential equation of the phase fields. We mainly focus on the study of D-dimensional electron gas with/without transverse gauge fields, two-dimensional electron gas under an external magnetic field, D-dimensional boson systems, a D-dimensional Heisenberg model and a one-band Hubbard model on a square lattice, and give their exact (accurate for Heisenberg model) functional expressions of the ground state energy and action, and the eigen-functional wave functions of the fermions/bosons. With them, we can calculate a variety of correlation functions of the systems, such as single particle Green's functions and their ground state wave functions. In present theoretical framework, we can unifiably represent the Landau Fermi liquid, non-Fermi liquid (D≥2D\geq 2D≥2) and Tomonaga-Luttinger liquid.
In this work we reexamine the many-fermion problem in arbitrary dimensions. It is shown that in two dimensions or higher, the Hamiltonian of interacting fermions can be separated into individual nonintersecting sectors labeled by the wave-vector q⃗\vec qq​q⃗. Within each sector the Hamiltonian maps onto a generalized version of the one-dimensional Luttinger model that resembles a boson string. These are chain-like quadratic forms in boson operators that are readily diagonalized in the absence of “exchange” corrections. Moreover, in a simple example involving SU(2) fermions, that of the Hubbard model, we show that it can be possible also to incorporate exchange terms and express them entirely within an enlarged set of string variables.