Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Electron correlations in narrow bands

Rishabh Kishore

Published Jul 1, 1980• 4 citations
OpenAlex

Abstract

The standard basis operators (SBO) formalism is used to study the problem of electron corre lations in narrow banda described by the Hubbard Hamiltonian, Exact Dyson equation in the matrix farm for the SBO Green's fUnction is obtained by exploiting the self consistent many body theory developed by Fedro and Wilson, and Kishore. It is feund that the present forma liam provides a natural extension of the Roth's two potes approximation scheme and reproduces the earlier spin wave results. Some inconsistencies in the Roth's scheme have also been discussed.

Related papers

Self-consistent many body theory for the standard basis operator Green's functions

R. Micnas

We propose a self-consistent many body theory for the standard basis operator Green 's functicms and obtain an exact Lyson type matriz equation for the interacting mcmy levei systems. A zeroth order crpproximation, which neglects ali the damping effects, is investigated in detail for the anisotropic Heisenberg model, the isotropic quadvopolar system and the Hubbard model. In case of the anisotropic Heisenberg ferromagnet with both exchange and single -ion anisotropy the low-temperature renomnalization of the apin-waves for the uniaxial ordering agrees with the Bloch-Dyson theory. For the spin -1 easy plane ferromagnet, the criticai parameters for the phase transition at zero temperature are determined and compared with other theories. The elementary excitation spectrum of the spin -1 isotropic quadrupolar system is calculated and compared with the random phase approximation and Callen's like deccupling schemes. Finally, the theory is applied to the studW of the single -particle excitation spectrum of the Hubbard model.

•Dec 1, 1980•0 citations

New self-consistent many-body perturbation theory: Application to the Hubbard model

A. J. Fedro, R. S. Wilson

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 $I$ 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.

Physical review. B, Solid state•Mar 15, 1975•13 citations

Electron Correlation in Narrow Energy Bands. I. The Two-Pole Approximation in a Narrow S Band

Laura M. Roth

An improved version of Hubbard's treatment of correlation in a nondegenerate narrow band is obtained by the use of a new Green's-function decoupling scheme. The resulting one-electron Green's function has two poles on the real axis corresponding to a splitting of the electron bands due to the strong correlations between electrons on the same site. This is the same result that Hubbard obtained, but in our case the poles are shifted and their positions agree with the results of Harris and Lange, who used a moment technique. The theory is applied to a simple cubic lattice with nearest-neighbor interaction, and the lattice is found to be ferromagnetic in the strongly correlated limit for a sufficiently large number of electrons per atom. An examination of the low-density limit shows that the two-pole approximation does not reduce to Kanamori's $T$-matrix result, a failing which it shares with Hubbard's theory. A method is outlined for improving the theory so as to give the correct low-density result. In addition, the possibility of obtaining a minimum principle for the theory is explored.

Physical Review•Aug 10, 1969•254 citations

Functional-derivative study of the Hubbard model. I. Perturbation method and first-order approximation

Tadashi Arai, Morrel H. Cohen, Monica Tosi

In order to study the properties of the Hubbard model for narrow bands, a systematic treatment of the equations of motion of the Green's functions appropriate to that model has been developed. Higher-order Green's functions are reduced to functional derivatives of the basic Green's function $G$ and calculated iteratively in a perturbation scheme which takes the Hubbard I solution ${G}_{0}$ as the zeroth-order Green's function. A zeroth-order approximation to the self-energy correction obtained by inserting ${G}_{0}$ into the functional derivatives is compared with various existing solutions. The perturbation scheme is further extended to an infinite order and the self-energy is calculated exactly up to terms linear in the hopping motion $\ensuremath{\epsilon}$, a result which has not been obtained previously. The self-energy correction in this final result is drastically different from the zeroth-order solution, demonstrating the importance of the infinite-order iterative procedure. Finally, the electron correlations included in the final result are discussed in terms of diagrams.

Physical review. B, Solid state•Feb 15, 1977•8 citations

I. Exact relations among correlation functions of inherently nondynamic interacting systems

Stephen B. Haley

The standard basis operator (SBO) equation of motion is solved exactly for the class of interacting many-body Hamiltonians which can be constructed from a complete set of mutually orthogonal projection operators. This class includes the Ising model Hamiltonians. Two integration procedures are presented: (1) term by term integration over the poles of the SBO double-time temperature-dependent Green’s functions in the spectral representation; (2) direct integration of the time-dependent equation of motion of an SBO. An ensemble average expression relating correlation functions among systems in different energy states is derived. For the Ising model this expression relates correlation functions of spin clusters of different orders, and it is analogous to a relation derived from the master equation under equilibrium conditions.

Journal of Mathematical Physics•May 1, 1978•0 citations

Electron Correlation in a Narrow Band Metal a New Method of Calculation

Samuel P. Bowen

A generalization of Feenberg's Perturbation Theory is shown to give exact expressions for the temperature dependent retarded greens functions for a many body problem. The theorem establishing this relationship is sketched and the result briefly applied to the Hubbard model.

Annual Conference on Magnetism and Magnetic Materials (MMM)•Jan 1, 1974•0 citations
The standard basis operators (SBO) formalism is used to study the problem of electron corre lations in narrow banda described by the Hubbard Hamiltonian, Exact Dyson equation in the matrix farm for the SBO Green's fUnction is obtained by exploiting the self consistent many body theory developed by Fedro and Wilson, and Kishore. It is feund that the present forma liam provides a natural extension of the Roth's two potes approximation scheme and reproduces the earlier spin wave results. Some inconsistencies in the Roth's scheme have also been discussed.
We propose a self-consistent many body theory for the standard basis operator Green 's functicms and obtain an exact Lyson type matriz equation for the interacting mcmy levei systems. A zeroth order crpproximation, which neglects ali the damping effects, is investigated in detail for the anisotropic Heisenberg model, the isotropic quadvopolar system and the Hubbard model. In case of the anisotropic Heisenberg ferromagnet with both exchange and single -ion anisotropy the low-temperature renomnalization of the apin-waves for the uniaxial ordering agrees with the Bloch-Dyson theory. For the spin -1 easy plane ferromagnet, the criticai parameters for the phase transition at zero temperature are determined and compared with other theories. The elementary excitation spectrum of the spin -1 isotropic quadrupolar system is calculated and compared with the random phase approximation and Callen's like deccupling schemes. Finally, the theory is applied to the studW of the single -particle excitation spectrum of the Hubbard model.
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.
An improved version of Hubbard's treatment of correlation in a nondegenerate narrow band is obtained by the use of a new Green's-function decoupling scheme. The resulting one-electron Green's function has two poles on the real axis corresponding to a splitting of the electron bands due to the strong correlations between electrons on the same site. This is the same result that Hubbard obtained, but in our case the poles are shifted and their positions agree with the results of Harris and Lange, who used a moment technique. The theory is applied to a simple cubic lattice with nearest-neighbor interaction, and the lattice is found to be ferromagnetic in the strongly correlated limit for a sufficiently large number of electrons per atom. An examination of the low-density limit shows that the two-pole approximation does not reduce to Kanamori's TTT-matrix result, a failing which it shares with Hubbard's theory. A method is outlined for improving the theory so as to give the correct low-density result. In addition, the possibility of obtaining a minimum principle for the theory is explored.
In order to study the properties of the Hubbard model for narrow bands, a systematic treatment of the equations of motion of the Green's functions appropriate to that model has been developed. Higher-order Green's functions are reduced to functional derivatives of the basic Green's function GGG and calculated iteratively in a perturbation scheme which takes the Hubbard I solution G0{G}_{0}G0​ as the zeroth-order Green's function. A zeroth-order approximation to the self-energy correction obtained by inserting G0{G}_{0}G0​ into the functional derivatives is compared with various existing solutions. The perturbation scheme is further extended to an infinite order and the self-energy is calculated exactly up to terms linear in the hopping motion $\ensuremath{\epsilon}$, a result which has not been obtained previously. The self-energy correction in this final result is drastically different from the zeroth-order solution, demonstrating the importance of the infinite-order iterative procedure. Finally, the electron correlations included in the final result are discussed in terms of diagrams.
The standard basis operator (SBO) equation of motion is solved exactly for the class of interacting many-body Hamiltonians which can be constructed from a complete set of mutually orthogonal projection operators. This class includes the Ising model Hamiltonians. Two integration procedures are presented: (1) term by term integration over the poles of the SBO double-time temperature-dependent Green’s functions in the spectral representation; (2) direct integration of the time-dependent equation of motion of an SBO. An ensemble average expression relating correlation functions among systems in different energy states is derived. For the Ising model this expression relates correlation functions of spin clusters of different orders, and it is analogous to a relation derived from the master equation under equilibrium conditions.