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physica status solidi (b)

On a Self‐Consistent Approximation Procedure with Application to the Hubbard Model

K. Elk

Published Aug 1, 1974• 9 citations
Publisher

Abstract

Abstract Considering a very general derivation of Dyson's equation an approximation procedure is discussed, which exhibits a summary of different known methods. Concerning the Hubbard model a generalization of Roth's procedure follows. In particular it is possible to obtain a self‐consistent method.
DOI
10.1002/pssb.2220640209

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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.
We develop a self-consistent method to solve the basic equation for the self-energy correction of the Hubbard model obtained in the preceding paper. The term $\ensuremath{\pi}[\ensuremath{\Delta}]$ involving second functional derivatives is neglected and the quantities $〈{N}_{R\ensuremath{\sigma}}(t)〉$ and $〈{C}_{R\ensuremath{\sigma}}^{\ifmmode\dagger\else\textdagger\fi{}}(t){C}_{{R}^{\ensuremath{'}\ensuremath{'}}\ensuremath{\sigma}}(t)〉$ are initially assumed to be independent of the external fields $\ensuremath{\epsilon}(\ensuremath{\sigma})$ and $\ensuremath{\epsilon}(\overline{\ensuremath{\sigma}})$. Under these restrictions, the complete self-energy correction is shown to be expanded in powers of $\ensuremath{\epsilon}(\ensuremath{\sigma})$ and $\ensuremath{\epsilon}(\overline{\ensuremath{\sigma}})$ in the form $\ensuremath{\Sigma}{R{R}^{\ensuremath{'}}\ensuremath{\sigma}}^{}(t{t}^{\ensuremath{'}})={\ensuremath{\xi}}_{0}(R{R}^{\ensuremath{'}}\overline{\ensuremath{\sigma}}t){\ensuremath{\delta}}_{t{t}^{\ensuremath{'}}}+\ensuremath{\Sigma}{n=0}^{}\ensuremath{\Sigma}{{R}^{\ensuremath{'}\ensuremath{'}}}^{}\ensuremath{\Sigma}{{R}^{\ensuremath{'}\ensuremath{'}\ensuremath{'}}}^{}{\ensuremath{\xi}}^{(n)}(R{R}^{\ensuremath{'}};{R}^{\ensuremath{'}\ensuremath{'}}{R}^{\ensuremath{'}\ensuremath{'}\ensuremath{'}})\ensuremath{\epsilon}({R}^{\ensuremath{'}\ensuremath{'}}{R}^{\ensuremath{'}\ensuremath{'}\ensuremath{'}}\ensuremath{\sigma}t){\ensuremath{\delta}}_{t{t}^{\ensuremath{'}}}$, where ${\ensuremath{\xi}}_{0}$ consists of all possible terms linear in $\ensuremath{\epsilon}(\overline{\ensuremath{\sigma}})$, while ${\ensuremath{\xi}}^{(n)}$ is made up of all possible terms of the nnnth degree in $\ensuremath{\epsilon}(\overline{\ensuremath{\sigma}})$. Equations for ${\ensuremath{\xi}}_{0}$ and ${\ensuremath{\xi}}^{(n)}$ are solved exactly and the resulting series is summed analytically, yielding a compact and complete analytic solution for the restricted equation. The part which is linear in $\ensuremath{\epsilon}$ is shown to be equal to the perturbation result obtained in the preceding paper, confirming the claim that the perturbation result is exact through terms linear in $\ensuremath{\epsilon}$. The method is extended and the effect of $\frac{\ensuremath{\delta}〈N〉}{\ensuremath{\delta}\ensuremath{\epsilon}}$ and $\frac{\ensuremath{\delta}〈{C}^{\ifmmode\dagger\else\textdagger\fi{}}C〉}{\ensuremath{\delta}\ensuremath{\epsilon}}$ is included. The effect is found to eliminate the difficulty that the value of one of the terms in the self-energy correction is abnormally overestimated in the previous result in the split-band, half-filled limit.
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.