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arXiv:cond-mat/0001326

Improved Theory of Single-Particle Properties Using Sea-bosons

Girish S. Setlur

Published Jan 21, 2000• 0 citations
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Abstract

In this article, we combine the ideas introduced by us earlier in various proportions to arrive at a simple and yet powerful means of studying single-particle properties of homogeneous Fermi systems in detail without making assumptions regarding the validity or otherwise of Fermi-liquid theory. Novelties include the exact forms of the momentum distribution and spectral functions in the intermediate density regime and a dielectric function that is sensitive to significant qualitative changes in single-particle properties. This time, care is taken to include the particle-hole mode by interpreting the sum over modes as a weighted sum with the dynamical structure factor appearing as the weight.

DOI
10.48550/arxiv.cond-mat/0001326

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In this article, we combine the ideas introduced by us earlier in various proportions to arrive at a simple and yet powerful means of studying single-particle properties of homogeneous Fermi systems in detail without making assumptions regarding the validity or otherwise of Fermi-liquid theory. Novelties include the exact forms of the momentum distribution and spectral functions in the intermediate density regime and a dielectric function that is sensitive to significant qualitative changes in single-particle properties. This time, care is taken to include the particle-hole mode by interpreting the sum over modes as a weighted sum with the dynamical structure factor appearing as the weight.
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We develop a systematic theory of multiparticle excitations in strongly interacting Fermi systems. Our work is the generalization of the time-honored work by Jackson, Feenberg, and Campbell for bosons, that provides, in its most advanced implementation, quantitative predictions for the dynamic structure function in the whole experimentally accessible energy/momentum regime. Our view is that the same physical effects, namely, fluctuations of the wave function at an atomic length scale are responsible for the correct energetics of the excitations in both Bose and Fermi fluids. Besides a comprehensive derivation of the fermion version of the theory and discussion of the approximations made, we present results for homogeneous 3He^{3}\text{H}\text{e}3He and electrons in three dimensions. We find indeed a significant lowering of the zero-sound mode in 3He^{3}\text{H}\text{e}3He and a broadening of the collective mode due to the coupling to two-particle--two-hole excitations in good agreement with experiments. The most visible effect in electronic systems is the appearance of a ``double-plasmon'' excitation.
Here we try and delienate the properties of the function that corresponds to fluctuations in the momentum distribution. The quantity denoted by $ N(k,k^{'}) $ is quite an interesting object. It satisfies various elegant sum rules and is also quite singular in some respects. All these properties are brought out and a formal connection is found between this object and the momentum distribution of the interacting and non-interacting many-fermion systems. This exercise is quite general in that it does not refer to any particular hamiltonian. It is also quite useful since in an earlier preprint(cond-mat/9810043) we showed how to compute the spectral function and single-particle lifetime of homogeneous Fermi systems where the only undetermined quantity was this function $ N(k,k^{'}) $.
Based on ideas introduced in a previous preprint cond-mat/9701206 we propose an exactly solvable model of bosons interacting amongst themselves via a Van-der Waal-like repulsive interaction, and compute both the filling fraction and the dynamical structure factor exactly. The novelty of this approach involves introducing, analogous to Fermi sea (or surface) displacements, Bose fields that in this case, correspond to fluctuations of the Bose condensate. The exact dynamical structure factor has a coherent part that corresponds to the Bogoliubov spectrum and an incoherent part that is a result of correlations.
In this article, we revisit the question of the validity of Hartree-Fock and random-phase approximations. We show that there is a connection between the two and while the RPA as it is known in much of the physics literature is of limited validity, there is a generalised sense in which the random phase approximation is of much wider applicability including to systems that do not possess Fermi surfaces. The main conclusion is that the Hartree-Fock approximation is a mean-field idea applied to the density operator, and the random-phase approximation is a mean-field idea applied to the number operator. The generalised RPA is used to compute single-particle properties such as momentum distribution and spectral functions. It is found that we have to go beyond the generalised RPA and include fluctuations in the momentum distribution in order to recover a nonzero imaginary part of the one-particle self energy, which is also explicitly computed, all this in any number of spatial dimensions and no bosonization is needed.