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Physical review. B, Condensed matter

Functional-derivative study of the Hubbard model. III. Fully renormalized Green's function

Tadashi Arai, Morrel H. Cohen

Published Apr 15, 1980• 7 citations
Publisher

Abstract

The functional-derivative method of calculating the Green's function developed earlier for the Hubbard model is generalized and used to obtain a fully renormalized solution. Higher-order functional derivatives operating on the basic Green's functions, $G$ and $\ensuremath{\Gamma}$, are all evaluated explicitly, thus making the solution applicable to the narrow-band region as well as the wide-band region. Correction terms $\ensuremath{\Phi}$ generated from functional derivatives of equal-time Green's functions of the type $\frac{{\ensuremath{\delta}}^{n}〈N〉}{\ensuremath{\delta}{\ensuremath{\epsilon}}^{n}}$, etc., with $n\ensuremath{\ge}2$. It is found that the $\ensuremath{\Phi}'\mathrm{s}$ are, in fact, renormalization factors involved in the self-energy $\ensuremath{\Sigma}$ and that the structure of the $\ensuremath{\Phi}'\mathrm{s}$ resembles that of $\ensuremath{\Sigma}$ and contains the same renormalization factors $\ensuremath{\Phi}$. The renormalization factors $\ensuremath{\Phi}$ are shown to satisfy a set of equations and can be evaluated self-consistently. In the presence of the $\ensuremath{\Phi}'\mathrm{s}$, all difficulties found in the previous results (papers I and II) are removed, and the energy spectrum $\ensuremath{\omega}$ can now be evaluated for all occupations $n$. The Schwinger relation is the only basic relation used in generating this fully self-consistent Green's function, and the Baym-Kadanoff continuity condition is automatically satisfied.

DOI
10.1103/physrevb.21.3300

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The functional-derivative method of calculating the Green's function developed earlier for the Hubbard model is generalized and used to obtain a fully renormalized solution. Higher-order functional derivatives operating on the basic Green's functions, GGG and \ensuremathΓ\ensuremath{\Gamma}\ensuremathΓ, are all evaluated explicitly, thus making the solution applicable to the narrow-band region as well as the wide-band region. Correction terms \ensuremathΦ\ensuremath{\Phi}\ensuremathΦ generated from functional derivatives of equal-time Green's functions of the type \ensuremathδn〈N〉\ensuremathδ\ensuremathϵn\frac{{\ensuremath{\delta}}^{n}〈N〉}{\ensuremath{\delta}{\ensuremath{\epsilon}}^{n}}\ensuremathδ\ensuremathϵn\ensuremathδn〈N〉​, etc., with n\ensuremath≥2n\ensuremath{\ge}2n\ensuremath≥2. It is found that the \ensuremathΦ′s\ensuremath{\Phi}'\mathrm{s}\ensuremathΦ′s are, in fact, renormalization factors involved in the self-energy \ensuremathΣ\ensuremath{\Sigma}\ensuremathΣ and that the structure of the \ensuremathΦ′s\ensuremath{\Phi}'\mathrm{s}\ensuremathΦ′s resembles that of \ensuremathΣ\ensuremath{\Sigma}\ensuremathΣ and contains the same renormalization factors \ensuremathΦ\ensuremath{\Phi}\ensuremathΦ. The renormalization factors \ensuremathΦ\ensuremath{\Phi}\ensuremathΦ are shown to satisfy a set of equations and can be evaluated self-consistently. In the presence of the \ensuremathΦ′s\ensuremath{\Phi}'\mathrm{s}\ensuremathΦ′s, all difficulties found in the previous results (papers I and II) are removed, and the energy spectrum \ensuremathω\ensuremath{\omega}\ensuremathω can now be evaluated for all occupations nnn. The Schwinger relation is the only basic relation used in generating this fully self-consistent Green's function, and the Baym-Kadanoff continuity condition is automatically satisfied.
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 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.
Nonperturbative solutions to the Hubbard model are found by using the functional derivative method. A system of closed equations is obtained for the Green's functions of the Hubbard model. Exact expressions for the self-energy are derived which involve only first order functional derivatives. A scheme is proposed for systematically calculating self-energy corrections. We apply the solution to finite rings of two and four lattice sites and compare to the results of numerical calculations on these systems.
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.
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.
I review the way the many-body Green functions are used to renormalize the perturbation theory of correlated fermions. The Green functions are introduced to implement systematically dynamical corrections to the static mean-field theory. The renormalizations enter the perturbation theory via self-consistent evaluations of one-particle and two-particle Green functions. The one-particle self-consistency is discussed within the Baym-Kadanoff construction. We point to an inherent dichotomy of the Baym-Kadanoff approach in the relation between one and two-particle Green functions. They are the Schwinger-Dyson equation, directly derived from the generating Luttinger-Ward functional, and the Ward identity. The latter imposes a severe restriction on the two-particle irreducible vertex and is not guaranteed in the Baym-Kadanoff approach. No approximate theory is capable to obey both relations. I further discuss the two-particle approach with two-particle renormalizations that offers a way to reconcile the Schwinger-Dyson equation and the Ward identity. The two-particle irreducible vertex from the critical Bethe-Salpeter equation becomes a generating functional in the two-particle renormalization of the perturbation theory. The irreducible vertex is used in the Ward identity to determine the order parameter beyond the critical point and in the Bethe-Salpeter equation representing the vertex of the Schwinger-Dyson equation to determine the spectral self-energy in such a way that the thermodynamic critical behavior from both equations is qualitatively the same. Finally, I discuss a static mean-field-like approximation with a renormalized effective interaction determined self-consistently that leads to a qualitatively correct description of strong electron correlations in impurity and lattice models.