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Physical review. B, Solid state

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

Tadashi Arai, Morrel H. Cohen, Monica Tosi

Published Feb 15, 1977• 8 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.15.1817

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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
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 \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ, 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 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 $\ensuremath{\Gamma}$, 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 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.
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.
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.
The equations for the Green's functions of the theory of superconductivity are developed in an iterative scheme in which the two-particle function is rewritten as the functional derivative of the single-particle function with respect to external-source terms. When a source coupled to electron pairs in addition to an external potential coupled to the charge density is included in the Hamiltonian, the possibility for a gap in the single-particle excitations is found. The first order in the iteration scheme for the solution to the Green's function equations is independent of the way in which the two-particle function is generated, as the final result must be. Equations for the vertex functions are obtained and used to find the linear response of the current to an applied electromagnetic field. An exact solution to the vertex equation of interest is used to show that no current will be induced by a static longitudinal vector potential. In addition, all the functions calculated are found to have translationally invariant solutions. Thus, the original invariances of the Hamiltonian are restored when the source terms are set equal to zero. Corrections to the self-energy are estimated by using another of the solutions to the vertex equations.
Employing the equation of motion method for the Green's function, we have obtained the self-energy for the Hubbard model in the weak correlation limit up to the 2nd order in U, the on-site interaction strength. The quasi-particle spectral density of states is determined numerically, using the local approximation which is known to be exact for the infinite dimensional Hubbard model. Further, assuming a linear dispersion for the bare electronic spectra for the 1D Hubbard model, we have evaluated analytically the imaginary part of the self-energy, Im Sigma, near the Fermi energy (E(F)) and the renormalization coefficient at E(F), Z(F). It is found that the 1D Hubbard model system exhibits a non-Fermi liquid behavior within the 2nd order perturbational treatment.
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.