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arXiv:nucl-th/0106016 · Physical Review Letters

Ambiguities in the Implementation of the Impulse Approximation for the Response of Many-Fermion Systems

Omar Benhar A, Adelchi Fabrocini, Stefano Fantoni

Published Jul 11, 2001• 28 citations
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Abstract

Within the impulse approximation, the response of a many-body system at large momentum transfer can be directly related to ground state properties. Although the physics assumptions underlying impulse approximation are well defined, their implementation involves ambiguities that may cause significant differences in the calculated responses. We show that, while minimal use of the impulse approximation assumptions naturally leads to write the response in terms of the spectral function, the alternative definition in terms of the momentum distribution involves a more extended use of the same assumptions.

DOI
10.1103/physrevlett.87.052501

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Within the impulse approximation, the response of a many-body system at large momentum transfer can be directly related to ground state properties. Although the physics assumptions underlying impulse approximation are well defined, their implementation involves ambiguities that may cause significant differences in the calculated responses. We show that, while minimal use of the impulse approximation assumptions naturally leads to write the response in terms of the spectral function, the alternative definition in terms of the momentum distribution involves a more extended use of the same assumptions.
First the structure function of the many fermion systems is written in terms of the fluctuation of the target density and too the value of this function at very small momentum transfer is calculated. It is shown that the values of the structure function of these systems at very small momentum transfer calculated from two different ways which one is the mathematical straight way and other way is the sum rules of the quantum fluids, are not equal. This topic transparently is a paradox for the many-fermion systems and must be considered. Finally for removing this paradox, we offer a new defination for the response function or the structure function of the many fermion systems.
The last decade has seen a large increase in the number of electronic-structure calculations that involve adding a Hubbard term to the local-density approximation band-structure Hamiltonian. The Hubbard term is then determined either at the mean-field level or with sophisticated many-body techniques such as using dynamical mean-field theory. We review the physics underlying these approaches and discuss their strengths and weaknesses in terms of the larger issues of electronic structure that they involve. In particular, we argue that the common assumptions made to justify such calculations are inconsistent with what the calculations actually do. Although many of these calculations are often treated as essentially first-principles calculations, in fact, we argue that they should be viewed from an entirely different point of view, namely, as based on phenomenological many-body corrections to band-structure theory. Alternatively, it may also be considered that they are just based on a Hubbard model that is more complex than the simple one- or few-band models traditionally used in many-body theories of solids.
A variational formulation for the calculation of interacting fermion systems based on the density-matrix functional theory is presented. Our formalism provides for a natural integration of explicit many-particle effects into standard density-functional-theory-based calculations and it avoids the ambiguities of double-counting terms inherent in other approaches. Like the dynamical mean-field theory, we employ a local approximation for explicit correlations. Aiming at the ground state only, we trade some of the complexity of Green's-function-based many-particle methods against efficiency. Using short Hubbard chains as test systems, we demonstrate that the method captures ground-state properties, such as left-right correlation, beyond those accessible by mean-field theories.
The Momentum Average (MA) approximation is a variational approximation that has been applied to analytically calculate the spectral function of single polarons and bipolarons, and was found to quantitatively match numerical results apart from the strongly adiabatic limit. The MA approximation was recently extended to a low carrier concentration of spinless fermions in 1D by making the additional assumption that the N-fermion ground-state is approximated well by the mean-field ground-state. This allowed the spectral function to be calculated analytically outside the Migdal limit. Good agreement was found with Density Matrix Renormalization Group simulations for fermion concentrations up to 0.15. In this thesis, the same approximation is extended to calculate the spectral function of a Holstein polaron for low carrier concentrations of spinless fermions on square and cubic lattices. Polaron effective mass and quasiparticle weight were found to depend strongly on the concentration x, especially in 3D. The 2D mass renormalizations were found to give a close match (< 5% error) to Bold Diagrammatic Monte Carlo (BDMC) results from the literature at all x < 0.2 when coupling is weak. However, there were significant quantitative differences between MA and BDMC results at large coupling. A possible explanation for this discrepancy is the presence of a soft excitation close to the polaron ground state. In addition, an efficient numerical scheme to calculate the real-space free electron propagator for a noninteracting Fermi sea is presented, which is used to calculate the self-energy for the 3D Green’s function at finite concentrations.
The Momentum Average (MA) approximation is a variational approximation that has been applied to analytically calculate the spectral function of single polarons and bipolarons, and was found to quantitatively match numerical results apart from the strongly adiabatic limit. The MA approximation was recently extended to a low carrier concentration of spinless fermions in 1D by making the additional assumption that the N-fermion ground-state is approximated well by the mean-field ground-state. This allowed the spectral function to be calculated analytically outside the Migdal limit. Good agreement was found with Density Matrix Renormalization Group simulations for fermion concentrations up to 0.15. In this thesis, the same approximation is extended to calculate the spectral function of a Holstein polaron for low carrier concentrations of spinless fermions on square and cubic lattices. Polaron effective mass and quasiparticle weight were found to depend strongly on the concentration x, especially in 3D. The 2D mass renormalizations were found to give a close match (< 5% error) to Bold Diagrammatic Monte Carlo (BDMC) results from the literature at all x < 0.2 when coupling is weak. However, there were significant quantitative differences between MA and BDMC results at large coupling. A possible explanation for this discrepancy is the presence of a soft excitation close to the polaron ground state. In addition, an efficient numerical scheme to calculate the real-space free electron propagator for a noninteracting Fermi sea is presented, which is used to calculate the self-energy for the 3D Green’s function at finite concentrations.