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Physical Review

Self-Consistent Pair Correlations in Mott-Type Insulators

Gábor Kemény, Laurent G. Caron

Published Jul 15, 1967• 20 citations
Publisher

Abstract

A simple cubic hydrogenic lattice with the Hubbard Hamiltonian is used as a model for Mott-type insulators. A new self-consistent approximation is developed by the decoupling of four-particle Green's functions, leading to a higher-order Hartree-like eigenvalue equation for electron-hole Green's functions. This equation is an improvement over the previously used $\ensuremath{\Omega}$ approximation. The solution of the eigenvalue equation leads to the conjecture that an exact solution (i.e., one without decoupling) would result in each atom having exactly one electron.

DOI
10.1103/physrev.159.768

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A simple cubic hydrogenic lattice with the Hubbard Hamiltonian is used as a model for Mott-type insulators. A new self-consistent approximation is developed by the decoupling of four-particle Green's functions, leading to a higher-order Hartree-like eigenvalue equation for electron-hole Green's functions. This equation is an improvement over the previously used \ensuremathΩ\ensuremath{\Omega}\ensuremathΩ approximation. The solution of the eigenvalue equation leads to the conjecture that an exact solution (i.e., one without decoupling) would result in each atom having exactly one electron.
We consider the Hubbard model on the infinite-dimensional Bethe lattice and construct a systematic series of self-consistent approximations to the one-particle Green's function, G(n){\mathit{G}}^{(\mathit{n})}G(n)(\ensuremath{\omega}), n=2,3,.... The first n-1 equations of motion are exactly fullfilled by G(n){\mathit{G}}^{(\mathit{n})}G(n)(\ensuremath{\omega}) and the nth equation of motion is decoupled following a simple set of decoupling rules. G(2){\mathit{G}}^{(2)}G(2)(\ensuremath{\omega}) corresponds to the Hubbard-III approximation. We present analytic and numerical results for the Mott-Hubbard transition at half filling for n=2,3,4.
We calculate the one-particle density of states for the Mott-Hubbard insulating phase of the Hubbard model on a Bethe lattice in the limit of infinite coordination number. We employ the Kato-Takahashi perturbation theory around the strong-coupling limit to derive the Green function. We show that the Green function for the lower Hubbard band can be expressed in terms of polynomials in the bare hole-hopping operator. We check our technique against the exact solution of the Falicov-Kimball model and give explicit results up to and including second order in the inverse Hubbard interaction. Our results provide a stringent test for analytical and numerical investigations of the Mott-Hubbard insulator and the Mott-Hubbard transition within the dynamical mean-field theory. We find that the Hubbard-III approximation is not satisfactory beyond lowest order, but the local-moment approach provides a very good description of the Mott-Hubbard insulator at strong coupling.
Starting from the exact solution of the Hubbard model in the atomic limit, and treating the hopping term as a perturbation by means of a diagram technique, we discuss the electronic structure of the model in some simple approximations. We show that a metal-insulator transition is obtained above a critical value of the on-site Coulomb interaction in a self-consistent evaluation of the one-particle Green's function to one-loop order. A relationship with results in infinite dimensions is briefly discussed.
A self-consistent equation is derived for the order parameter of quantum phases in an optical lattice within a mean-field approximation to the Bose–Hubbard model. Analyzing the solutions of the self-consistent equation in terms of the number fluctuations of quasiparticles, the one-body density matrix of atoms, and the normalized correlation function of quasiparticles, we have identified two types of Mott-insulator phases that are characterized by different values of the order parameter, quasiparticle fluctuations, and correlations. We have also identified the quantum critical points separating these two types of Mott-insulator phases and the corresponding quantum critical lines on the phase diagram.
An exact expression for the Green's function G(k,\ensuremath{\omega}) of one hole in the U\ensuremath{\rightarrow}\ensuremath{\infty} one-dimensional Hubbard model is derived from the known Bethe-ansatz solution of the model. Due to the spin-charge decoupling of the one-hole excitation spectrum and the Fermi-like statistics of the spin excitations, G shows branch-cut singularities at \ensuremath{\omega}=\ifmmode\pm\else\textpm\fi{}2 sink and a nontrivial dependence on the momentum of the hole. As a result, the hole can propagate although the quasiparticle weight vanishes.
A new many-body theory perturbationally consistent to all orders of the perturbation is presented and applied to the Hubbard problem as an example. Use is made of the recent Kim and Wilson commutator projection operator to derive the Dyson equation for the one-particle Green's function and an exact set of coupled differential equations for the damping or self-energy term which are solved self-consistently. The Hubbard III result follows in our first approximation and the Esterling-Lange (EL)-type result in the second. Since we keep the hopping finite throughout the calculation, no degeneracy problem is encountered as in EL.