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Renormalized Ring Diagram Energy in a Fermion System of Hard Spheres : The Off Energy Shell Effect

修一郎 山崎, 輝也 山西, S. Yamasaki, T. Yamanishi

Published Mar 20, 2004• 0 citations
Publisher

Abstract

The off energy shell effect on the contribution to the energy from the renormalized three-particle ring diagram is studied for the ground state of a system of hard sphere fermions. The linked cluster formalism supplemented with infinite partial summation method, such as the K-matrix method, fails in giving a finite value to the energy of interest. In place of the K-matrix, we make use of the effective potential which improves the pseudopotential introduced by Fermi and exploited by Lee, Huang and Yang. Our potential prevents two spheres from penetrating each other. It gives a finite energy correction whose dominant part varies as a^4 loga with the small value of the core radius a. Correction term of order a^4 is also presented.

DOI
10.57375/00000157

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The off energy shell effect on the contribution to the energy from the renormalized three-particle ring diagram is studied for the ground state of a system of hard sphere fermions. The linked cluster formalism supplemented with infinite partial summation method, such as the K-matrix method, fails in giving a finite value to the energy of interest. In place of the K-matrix, we make use of the effective potential which improves the pseudopotential introduced by Fermi and exploited by Lee, Huang and Yang. Our potential prevents two spheres from penetrating each other. It gives a finite energy correction whose dominant part varies as a^4 loga with the small value of the core radius a. Correction term of order a^4 is also presented.
The off energy shell effect on the contribution to the energy from the renormalized three-particle ring diagram is studied for the ground state of a system of hard sphere fermions. The linked cluster formalism supplemented with infinite partial summation method, such as the K-matrix method, fails in giving a finite value to the energy of interest. In place of the K-matrix, we make use of the effective potential which improves the pseudopotential introduced by Fermi and exploited by Lee, Huang and Yang. Our potential prevents two spheres from penetrating each other. It gives a finite energy correction whose dominant part varies as a^4 loga with the small value of the core radius a. Correction term of order a^4 is also presented.
The energy shift in the ground state system of hard shoere fermions is studied with particular emphasis on the off energy-shell sefl energy effect of excited particled induced by three- aprticle correlation. The energy shift is known to diverge when one attenppts to analyze it in terms of K- or G-matrix which describes the two-particle coreelation successfully. Use is made of a new method which deals with the hard sphere interaction in terms of effective potential. The energy shift of tnterest is expressed as a convergent integral which varies as a^4log a for small values of the sphere radius a. The contribution to the integral of order a^4 is also identified.
The three-particle correlation energy induced by the polarization effects in the ground state system of hard core fermions is studied with the aid of an effective potential which prevents the cores from penetrating each other. The energy varies as a^4 log a with the small value of the core radius a and is not analytic at a = O. The nonanalyticity is responsible for the failure of the linked cluster expansion formalism, complemented with the partial summation method, in giving a finite value to the three-particle correlation energy in quantum fluid. It shows that t, he expansion of the energy beyond third order terms in t, he coupling const, ant, is not possible. The energy is examined more carefully and components behaving as a^4 are identified. Nonanalyitic component besides the a^4 log a term and dominating the a^4 term, such as a^4 log |log a|, is not found.
A new pseudopotential is proposed for the description of the hard core repulsive interaction. The pseudopotential prevents long wavelength spheres from penetrating each other in contrast to Fermi's pseudopotential. The proposed pseudopotential is applied to the ground state of a dilute system of hard sphere fermions in the calculation of the three-body correlation energy, which has been reported to have divergent terms requiring cutoff in an earlier work. The corresponding terms obtained here are found to converge and depend as a4log a on the sphere radius a.
A new method is proposed for treating the hard core potential with a small core radius a. This is an alternative to the method based on Fermi's pseudopotential. Various shortcomings of the latter method are removed with the present method. These shortcomings include the penetrability of hard spheres and incompatibility with the completeness of the unperturbed states, and they result in the invalidity of the pseudopotential method even in applications to effects that are of low order in powers of a. The present method is developed taking account of the three-particle correlation, which is a higher-effect plagued by problems of divergence and the nonanalyticity of the relevant energy as a function of the potential strength. Treatment of the three-particle correlation requires a careful preliminary study of the two-particle interaction in many-body systems, particularly when a hard core potential is involved. In this report, we present basic ideas and obtain improvement over the methods appearing in the literature.