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Physical review. B, Condensed matter

Moment-conserving decoupling approach for many-body systems

Hong‐Gang Luo, Shun-Jin Wang

Published Dec 15, 1999• 1 citation
Publisher

Abstract

A nonperturbative, controllable, and moment-conserving decoupling approach is proposed. Our assumption is that a higher-order Green function (GF) essentially contains two-part information. One part relates to the single-particle GF, which is involved in the corresponding moment; the other part is a higher-order quantity, which is independent of the single-particle GF. Thus the higher-order GF may be separated into two corresponding terms. We derive the chain of equations of motion (EOM) of the latter term from the usual EOM for GF's. To solve the equations a truncation approximation is made at the level desired. The single-band Hubbard model is studied using this approach. Moment-conserving solutions up to the order $m=2$ are obtained. The properties of a solution corresponding to the Hubbard-I approximation level are discussed in detail. The quasiparticle bands and the momentum distribution are in excellent agreement with the results from the quantum Monte Carlo simulation.

DOI
10.1103/physrevb.60.15480

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A nonperturbative, controllable, and moment-conserving decoupling approach is proposed. Our assumption is that a higher-order Green function (GF) essentially contains two-part information. One part relates to the single-particle GF, which is involved in the corresponding moment; the other part is a higher-order quantity, which is independent of the single-particle GF. Thus the higher-order GF may be separated into two corresponding terms. We derive the chain of equations of motion (EOM) of the latter term from the usual EOM for GF's. To solve the equations a truncation approximation is made at the level desired. The single-band Hubbard model is studied using this approach. Moment-conserving solutions up to the order m=2m=2m=2 are obtained. The properties of a solution corresponding to the Hubbard-I approximation level are discussed in detail. The quasiparticle bands and the momentum distribution are in excellent agreement with the results from the quantum Monte Carlo simulation.
Using a Green's-function equation of motion approach the two-dimensional Hubbard model is studied. The decoupling of the hierarchy is performed according to the conserving of the first four spectral moments. Beyond the Roth's two-pole approximation, the damping effect occurs naturally. The energy dispersions at various fillings, the spectral weight, and the density of states are calculated. Comparisons with the quantum Monte Carlo results and the cluster perturbation theory show that our approach can capture some essential features of the Hubbard model.
Using a Green's-function equation of motion approach the two-dimensional Hubbard model is studied. The decoupling of the hierarchy is performed according to the conserving of the first four spectral moments. Beyond the Roth's two-pole approximation, the damping effect occurs naturally. The energy dispersions at various fillings, the spectral weight, and the density of states are calculated. Comparisons with the quantum Monte Carlo results and the cluster perturbation theory show that our approach can capture some essential features of the Hubbard model.
Moment-conserving decoupling procedures for the one-electron Green's function (electron propagator) are investigated. By analysis of one particular system (Hubbard model) it becomes clear that the moment-conserving techniques so far proposed for interacting systems are all special cases of [1, 0], [2, 1], or [3, 2] Padé approximants differing only in which terms they omit, how they define self-consistency, and which operator algebra they utilize. The number of formally conserved moments appears not to be a particularly useful criterion of the accuracy of a decoupling procedure, since a two-moment scheme (Hartree-Fock) may be numerically closer to the exact answer than some three-moment procedures. Use of the complete Padé approximant, as suggested by Lukman and Goscinski for the particle-hole propagator, appears to be the most consistent method.
Several widely used methods for the calculation of band structures and photo emission spectra, such as the GW approximation, rely on many-body perturbation theory. They can be obtained by iterating a set of functional differential equations (DEs) relating the one-particle Green's function (GF) to its functional derivative with respect to an external perturbing potential. In this work, we apply a linear response expansion in order to obtain insights into various approximations for GF calculations. The expansion leads to an effective screening while keeping the effects of the interaction to all orders. In order to study various aspects of the resulting equations, we discretize them and retain only one point in space, spin and time for all variables. Within this one-point model we obtain an explicit solution for the GF, which allows us to explore the structure of the general family of solutions and to determine the specific solution that corresponds to the physical one. Moreover, we analyze the performances of established approaches like GW over the whole range of interaction strength, and we explore alternative approximations. Finally, we link certain approximations for the exact solution to the corresponding manipulations of the DE which produces them. This link is crucial in view of a generalization of our findings to the real (multidimensional functional) case where only the DE is known.
A well-established method to deal with highly correlated systems is based on the expansion of the Green's function in terms of spectral moments. In the context of the Composite Operator Method one approximation is proposed: a set of n composite fields is assumed as fundamental basis and the dynamics is considered up to the order s. The resulting Green's function has a sn-pole structure. The truncation of the hierarchy of the equations of motion is made at the s-th order and the first s-1 equations are treated exactly. A theorem, which rules the conservation of the spectral moments, is presented. The procedure is applied to the Hubbard model and a recurrence relation for the calculation of its electronic spectral moments is derived.
We introduce a new approach to strongly correlated systems, based on equations of motion for composite ("quasiparticle") operators. Conserving approximations to the appropriate Green's functions are generated through functional differentiation. For illustration, we apply this scheme to the infinite-UUU, single-band Hubbard model. The analogue of the Hartree-Fock approximation becomes exact in the limit of infinite spin degeneracy (NNN), while the random-phase approximation leads to the corresponding 1N\frac{1}{N}N1​ fluctuations.