Bishop, RF

Piechocki, W

Stevens, GA

English

The University of Manchester - Institutional Repository

The University of Manchester ResearchCorrelated pairs near the Fermi surfaceDOI:10.1007/BFb0037545Document VersionAccepted author manuscriptLink to publication record in Manchester Research ExplorerCitation for published version (APA):Piechocki, W., Bishop, RF., & Stevens, GA. (1984). Correlated pairs near the Fermi surface. In H. Kümmel, & M. L.Ristig (Eds.), Recent Progress in Many-Body Theories (pp. 110-112). (Series on Lecture Notes in Physics; Vol.198). Springer Nature. https://doi.org/10.1007/BFb0037545Published in:Recent Progress in Many-Body TheoriesCiting this paperPlease note that where the full-text provided on Manchester Research Explorer is the Author Accepted Manuscriptor Proof version this may differ from the final Published version. If citing, it is advised that you check and use thepublisher's definitive version.General rightsCopyright and moral rights for the publications made accessible in the Research Explorer are retained by theauthors and/or other copyright owners and it is a condition of accessing publications that users recognise andabide by the legal requirements associated with these rights.Takedown policyIf you believe that this document breaches copyright please refer to the University of Manchester’s TakedownProcedures [http://man.ac.uk/04Y6Bo] or contact uml.scholarlycommunications@manchester.ac.uk providingrelevant details, so we can investigate your claim.Download date:09. Jun. 2022CORRELATED PAIRS NEAR THE FERMI SURFACE W. Piechocki Nuclear Theory Department Institute of Nuclear Research Hoza 69, PL-OO-681 Warsaw, Poland and R.F. Bishop and G.A. Stevens Department of Mathematics UMIST, P.O.Box 88 Manchester M60 IQD, U.K. One of the problems of the fifties, which occupied a lot of many body physicists dealing with strongly interacting Fermi systems of infinite extent like nucleon matter or liquid 3He, was the occurence of a singu- larity at the Fermi surface in the Bethe-Goldstone (BG) equation. It was generally believed, at ~at time, that this singularity was associa- ted with the existence of a superfluid state I . In the next two decades we can observe a few developments 2. The ap- proach I would like to discuss is the generalization of the BG equation. It includes, apart from particle-particle (pp) ladders, hole-hole (hh) ladders and all possible mixed pp and hh ladders. The generalized lad- der equation of Mehta 3 and the Galitskii-Feynman (GF) equation investi- 4 gated by the Manchester group comprise the same type of ladder diagrams with the property that particles and holes are treated completely sym- metrically. Methals equation does not lose the BG equation singularity. The Manchester group, on the other hand, has not noticed any problem near the Fermi surface but has discovered that the GF T-matrix posses- ses a pole in the bound-state region which corresponds to the formation of bound-state pairs in the medium. Results obtained by using symmetrical ladder equation are interesdng because we know that bound composite clusters of fermions comprised of an even number of fermions (particles) and/or an even number of their superfluid-like phase 5. As an example we have the Cooper palrs which are responsible for superconductivity in weakly interacting Fermi sys- tems. We have decided to examine a generalized ladder equation again, but in the context of the coupled-cluster (CC) method; the CC method is flexible and general enough to enable investigation of the pairlng from differenn points of view and with better and better approximation. We have solved 6 the complete ladder (CLAD) equation of the CC method, cor- responding to generalized ladder equations of the perturbation theory, for the separable two-body interaction. We have restricted ourselves to one-term S-wave interaction but the extension to both multi-term separable interactions and to arbitrary higher partial waves would 111 be straightforward. The solution for the states close to the Fermi surface contains both the bound-pairs and the singularity. To under- stand that result better we have examined excited states 7. Using the Emrich's ansatz for the excited state, we have solved the ladder type equation corresponding to the CLAD equation of the ground state. The solution is characterized entirely by the quantities which describe the pairing of the ground state. For the attractive interaction, the excited state has lower energy than the ground state which means that our reference state choosen to be a filled Fermi-sphere plane-wave determinant is not the best possible point of departure. For the re- pulsive interaction, on the other hand, we observe a real excitation for densities above a certain critical density and no excitation below that density; in the latter range of density we also do not observe any pairing in the ground state. Both results, for the ground state and for the excited state, sug- gest the existence of the pairing in a many-fermion strongly interac- ting system. The question arises whether the observed pairing exists in the exact solution of the Schr~dinger equation. To examine that, we should find a suitable truncation scheme of the CC method which would give a possibility to see the pairing in its successive levels 8. We cannot apply the Bochum truncation scheme 9 because it was invented to describe the normal ground state of strongly interacting systems; by introducing a gap at k F in the single-particle energy spectrum, this method relegates the pairing correlations near the Fermi surface to the terms of a very high order. Presently we know how to treat the short-range repulsion of a two- body interaction, but we do not know how to describe properly the Fermi surface. This research was supported in part by the Polish - U.S. Maria Sklodowska-Curie Fund and by the SERC of Great Britain. References I J.S. Bell and E.J. Squires, Adv.Phys.l__O0 (1961) 211 and references therein 2B.H. Brandow, Phys.Rev. 152 (1966) 863; G.A. Baker Jr. and J.L.Gammel, Phys.Rev. C6 (1972) 403; J.P. Jeukenne, A. Lejeune and C. Mahaux, Nucl. Phys. A 245 (1975) 411. 3M.L. Mehta, Nucl. Phys. 20 (1960) 533. 112 4R.F. Bishop and M.R. Strayer, J.Phys. G, LI3 (1977) and reference therin. 5W. Kohn and D. Sherrington, Rev.Mod.Phys. 42 (1970) I. 6R.F. Bishop, W. Piechocki and G.A. Stevens, to be published. 7R.F. Bishop, W. Piechocki and G.A. Stevens, to be published. 8W. Piechocki, Invited talk presented at 'The 7th (Pan-American) Workshop on Condensed Matter Theories', Altenberg, West-Germany, August 22 -27, 1983. 9B.D. Day and J.G. Zabolitzky, Nucl. Phys. A 366 (1981) 221. 

Correlated pairs near the Fermi surface

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Correlated pairs near the Fermi surface

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