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Physical review. B, Solid state

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

A. J. Fedro, R. S. Wilson

Published Mar 15, 1975• 13 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.11.2148

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•Dec 1, 1980•0 citations
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.
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.
An approach for particle-hole correlation functions, based on the so-called self-consistent random-phase approximation is developed. This leads to a fully self-consistent RPA-like theory that satisfies the fff-sum rule and several other theorems. As a first step, a simpler self-consistent approach, the renormalized RPA, is solved numerically in the one-dimensional Hubbard model. The charge and the longitudinal spin susceptibility, the momentum distribution, and several ground-state properties are calculated and compared with the exact results. Especially at half-filling, our approach provides quite promising results and matches the exact behavior apart from a general prefactor. The strong-coupling limit of our approach can be described analytically.
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.
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.
In the standard framework of self-consistent many-body perturbation theory, the skeleton series for the self-energy is truncated at a finite order N and plugged into the Dyson equation, which is then solved for the propagator G_N G N . We consider two examples of fermionic models, the Hubbard atom at half filling and its zero space-time dimensional simplified version. First, we show that G_N G N converges when N\to∞ N → ∞ to a limit G_∞\, G ∞ , which coincides with the exact physical propagator G_{exact} G e x a c t at small enough coupling, while G_∞ ≠ G_{exact} G ∞ ≠ G e x a c t at strong coupling. This follows from the findings of [Phys. Rev. Lett. 114, 156402 (2015)] and an additional subtle mathematical mechanism elucidated here. Second, we demonstrate that it is possible to discriminate between the G_∞=G_{exact} G ∞ = G e x a c t and G_∞≠G_{exact} G ∞ ≠ G e x a c t regimes thanks to a criterion which does not require the knowledge of G_{exact} G e x a c t , as proposed in [Phys. Rev. B 93, 161102 (2016)].
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.