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

Reviews of Modern Physics

Degenerate Mass Operator Perturbation Theory in the Hubbard Model

D. M. Esterling, Robert Lange

Published Oct 1, 1968• 35 citations
Publisher

Abstract

The "atomic limit" (kinetic energy zero) of the Hubbard Hamiltonian has extreme degeneracy under very general circumstances. The kinetic energy or hopping term removes this degeneracy. It has been shown by Harris and Lange that there are important first-order manifestations of this fact. For instance, nonlocal correlations exist which invalidate usual Green's function truncation schemes. We present a mass operator perturbation scheme devised on the principle that, first of all, the role of kinetic energy as a breaker of degeneracy must be accounted for. A sort of Taylor's expansion of the mass operator in terms of the kinetic energy is found, the coefficients of which are functionals of the exact Green's functions at zero kinetic energy. The ambiguity due to the degeneracy is removed from these functions self consistently and thereby the degeneracy properly treated. We present a Green's function which accounts for all first-order effects of the kinetic energy.

DOI
10.1103/revmodphys.40.796

Related papers

Hubbard Model. I. Degeneracy in the Atomic Light

D. M. Esterling, Robert Lange

The atomic limit of the Hubbard Hamiltonian is extremely degenerate. The ground-state degeneracy, in particular, makes the zero-temperature Green's function for this limit ambiguous. The manifestations of the degeneracy in the one- and two-particle Green's functions are found. The degeneracy gives rise to nonlocal spatial correlation of physical significance which must be taken into account in narrow-band-regime perturbation schemes for the model.

Physical review. B, Solid state•Mar 1, 1970•12 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

Green's function for the strong coupling limit of the degenerate Hubbard model

S. Cojocaru, M. I. Vladimir

The asymptotic expression for the one-particle Green's function in the strong coupling limit of the double degenerate Hubbard model containing multiplet terms is obtained for arbitrary band filling within the irreducible Green's function approach. An explicit analytic expression is obtained for the mean-field Greens's function, conserving the sum rules for the first few spectral moments. Stability of homogeneous phases is analised in terms of a single flip spin and orbital excitations. It is shown that degeneracy has an additional enhancing effect on correlation driven interband transfer of spectral weight. Discovery of superconductivity in fullerides and of materials with colossal magnetoresistance during recent years have increased the interest for electron systems with physical properties being essentially dependent on orbital degeneracy and strong correlation effects. In many cases the orbitally degenerate Hubbard model (DHM) incorporates the key features of such materials [1], although there are even fewer exact results established for DHM than for the nondegenerate Hubbard model (NHM). In view of complexity of correlated systems, corrections to the atomic limit of one-particle excitation spectrum and spectral weights of correlated bands found by Harris and Lange for NHM [2] by means of perturbation theory based on canonical transformations provide an important exact result which serves as a test for different approximations. A similar derivation for DHM has not been carried out yet. In the present paper the Green's function for the strong coupling limit of the DHM is obtained by means of irreducible Green's function approach [3] - [6] which is closely related to canonical perturbation theory because of its spectral moment conserving construction. As was explicitly proved in [7] for the NHM, the first couples of spectral moments of correlated bands obtained within mean-field decoupling are identical to those of canonical perturbation theory to first order in / tU ( t is the n.n. hopping amplitude and U the Coulomb repulsion).

The Moldavian Journal of the Physical Sciences•Jun 1, 2003•0 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

Projection Operator Formalism: Application to the Hubbard Model

A. J. Fedro, Robert S. Wilson

We present a method of doing many body theory based on the commutator projection operator discussed by Kim and Wilson. Such a method immediately yields the Dyson equation leaving one the task of evaluating the self energy term. Such a term was previously handled by ordinary perturbation theory. We find, on the other hand, an exact set of differential equations involving the self energy and its time derivatives in Zubarev‐like fashion which can be solved self‐consistently in every order of perturbation in contrast to chain breaking procedures which are not in general perturbationally correct. We apply our method to the Hubbard model and find the Esterling and Lange (EL) type result. That is, the form of EL is correct but their T − L(k⇒) expression is not complete. It is worthwhile to point out here that our calculation is done with finite hopping so we do not encounter the degeneracy problem of EL.

Annual Conference on Magnetism and Magnetic Materials (MMM)•Jan 1, 1974•0 citations

Self-consistent Dyson equation and self-energy functionals: An analysis and illustration on the example of the Hubbard atom

Walter Tarantino, Pina Romaniello, J. A. Berger +1 more

Perturbation theory using self-consistent Green's functions is one of the most widely used approaches to study many-body effects in condensed matter. On the basis of general considerations and by performing analytical calculations for the specific example of the Hubbard atom, we discuss some key features of this approach. We show that when the domain of the functionals that are used to realize the map between the noninteracting and the interacting Green's functions is properly defined, there exists a class of self-energy functionals for which the self-consistent Dyson equation has only one solution, which is the physical one. We also show that manipulation of the perturbative expansion of the interacting Green's function may lead to a wrong self-energy as a functional of the interacting Green's function, at least for some regions of the parameter space. These findings confirm and explain numerical results of Kozik et al. for the widely used skeleton series of Luttinger and Ward [Phys. Rev. Lett. 114, 156402 (2015)]. Our study shows that it is important to distinguish between the maps between sets of functions and the functionals that realize those maps. We demonstrate that the self-consistent Green's functions approach itself is not problematic, whereas the functionals that are widely used may have a limited range of validity.

arXiv:1703.05587•Physical review. B./Physical review. B•Jul 18, 2017•55 citations
The "atomic limit" (kinetic energy zero) of the Hubbard Hamiltonian has extreme degeneracy under very general circumstances. The kinetic energy or hopping term removes this degeneracy. It has been shown by Harris and Lange that there are important first-order manifestations of this fact. For instance, nonlocal correlations exist which invalidate usual Green's function truncation schemes. We present a mass operator perturbation scheme devised on the principle that, first of all, the role of kinetic energy as a breaker of degeneracy must be accounted for. A sort of Taylor's expansion of the mass operator in terms of the kinetic energy is found, the coefficients of which are functionals of the exact Green's functions at zero kinetic energy. The ambiguity due to the degeneracy is removed from these functions self consistently and thereby the degeneracy properly treated. We present a Green's function which accounts for all first-order effects of the kinetic energy.
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.
The asymptotic expression for the one-particle Green's function in the strong coupling limit of the double degenerate Hubbard model containing multiplet terms is obtained for arbitrary band filling within the irreducible Green's function approach. An explicit analytic expression is obtained for the mean-field Greens's function, conserving the sum rules for the first few spectral moments. Stability of homogeneous phases is analised in terms of a single flip spin and orbital excitations. It is shown that degeneracy has an additional enhancing effect on correlation driven interband transfer of spectral weight. Discovery of superconductivity in fullerides and of materials with colossal magnetoresistance during recent years have increased the interest for electron systems with physical properties being essentially dependent on orbital degeneracy and strong correlation effects. In many cases the orbitally degenerate Hubbard model (DHM) incorporates the key features of such materials [1], although there are even fewer exact results established for DHM than for the nondegenerate Hubbard model (NHM). In view of complexity of correlated systems, corrections to the atomic limit of one-particle excitation spectrum and spectral weights of correlated bands found by Harris and Lange for NHM [2] by means of perturbation theory based on canonical transformations provide an important exact result which serves as a test for different approximations. A similar derivation for DHM has not been carried out yet. In the present paper the Green's function for the strong coupling limit of the DHM is obtained by means of irreducible Green's function approach [3] - [6] which is closely related to canonical perturbation theory because of its spectral moment conserving construction. As was explicitly proved in [7] for the NHM, the first couples of spectral moments of correlated bands obtained within mean-field decoupling are identical to those of canonical perturbation theory to first order in / tU ( t is the n.n. hopping amplitude and U the Coulomb repulsion).
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.
We present a method of doing many body theory based on the commutator projection operator discussed by Kim and Wilson. Such a method immediately yields the Dyson equation leaving one the task of evaluating the self energy term. Such a term was previously handled by ordinary perturbation theory. We find, on the other hand, an exact set of differential equations involving the self energy and its time derivatives in Zubarev‐like fashion which can be solved self‐consistently in every order of perturbation in contrast to chain breaking procedures which are not in general perturbationally correct. We apply our method to the Hubbard model and find the Esterling and Lange (EL) type result. That is, the form of EL is correct but their T − L(k⇒) expression is not complete. It is worthwhile to point out here that our calculation is done with finite hopping so we do not encounter the degeneracy problem of EL.
Perturbation theory using self-consistent Green's functions is one of the most widely used approaches to study many-body effects in condensed matter. On the basis of general considerations and by performing analytical calculations for the specific example of the Hubbard atom, we discuss some key features of this approach. We show that when the domain of the functionals that are used to realize the map between the noninteracting and the interacting Green's functions is properly defined, there exists a class of self-energy functionals for which the self-consistent Dyson equation has only one solution, which is the physical one. We also show that manipulation of the perturbative expansion of the interacting Green's function may lead to a wrong self-energy as a functional of the interacting Green's function, at least for some regions of the parameter space. These findings confirm and explain numerical results of Kozik et al. for the widely used skeleton series of Luttinger and Ward [Phys. Rev. Lett. 114, 156402 (2015)]. Our study shows that it is important to distinguish between the maps between sets of functions and the functionals that realize those maps. We demonstrate that the self-consistent Green's functions approach itself is not problematic, whereas the functionals that are widely used may have a limited range of validity.