Fortschritte der Physik / Progress of Physics: Band 24, Heft 7 1976 [Reprint 2021 ed.] 9783112520482, 9783112520475

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FORTSCHRITTE DER PHYSIK H E R A U S G E G E B E N IM AUFTRAGE D E R P H Y S I K A L I S C H E N

GESELLSCHAFT

DER DEUTSCHEN D E M O K R A T I S C H E N VON F. KASCHI.UHN,

LÖSCHE, R. R I T S C H L UND R. ROMPE

HEFT?

A K A D E M I E

REPUBLIK

• 1976 • B A N D 2 1

- V E K L A G

EVP 1 0 , - M 31728

B E R L I N

BEZUGSMÖGLICHKEITEN Bestellungen sind zu ricliteu — in «1er DDR an eine Buchhandlung oder an den Akademie-Verlag. D D R - 1 0 8 Berlin, Leipziger Straße 3—4 — im sozialistischen Ausland an eine Buchhandlung f ü r fremdsprachige Literatur oder an den zuständigen Postzeitungsvertrieb — in der BRD und Wcsllierlin an eine Buchhandlung oder an die Ausliefcrungsstclle KUNST UND WISSEN, Erich Bieber, 7 Stuttgart 1, Wilhelmstraßc 4—6 — in Osterreich an den Globus-Buchvertrieb, 1201 Wien, llöchstädtplatz 3 — im übrigen Ausland an den Internationalen Buch- und Zeitschriftenhandel; den Buchcxpovt, Volkseigener Außenhandelsbetrieb der Deutschen Demokratischen Republik. D D R - 7 0 1 Leipzig, Postfach 160, oder an den Akademie-Verlag, D D R - 1 0 8 Berlin, Leipziger Straße 3—4

Zeitschrift „Fortschritte der P h y s i k " Herausgeber: Prof. D r . F r a u k Kaschluhn, Prof. Dr. Artur Lösclic, Prof. Dr. Rudolf Ritsehl, Prof. Dr. Robert Rompc, im Auftrag der Physikalischen Gesellschaft der Deutschen Demokratischen Republik. Verlag: Akademie-Verlag, D D R - 108 Berlin, Leipziger StraOe 3—4; Fernruf: 2200441; Telex-Nr. 114420; Postscheckkonto: Berlin 35021; B a n k : Staatsbank der D D R , Berlin, Konto-Nr.: 6836-26-20712. Chefredakteur: Dr. Lutz Rothkirch. Anschrift der Redaktion: Sektion Physik der Humboldt-Universität zu Berlin, D D R -104 Berlin, Hessische Straße 2. Veröffentlicht unter der Lizenznummer 1324 des Presseamtes beim Vorsitzenden des Ministerrates der Deutscheu Demokratischen Republik. Gesamthcrstcllung: V E B Druckhaus „Maxim Gorki", D D R - 7 4 Altcnburg, Carl-von-Ossietzky-Straße 30/31. Erscheinungsweise: Die Zeitschrift „Fortschritte der Physik* 4 erscheint monatlich. Die 12 Hefte eines J a h r e s bilden einen Band. Bezugspreis je B a n d ; 180,— M zuzüglich Versandspesen (Preis f ü r die D D R : 120,— M). Preis je H e f t 15,— M (Preis für die D D R : 1 0 , - M). Bestellnummer dieses H e f t e s : 1027/24/7. © 1976 b y Akademie-Verlag Bcrliu. Printcd in the German Democratic Rcpublic.

Fortschritte der Physik 24, 3 5 7 - 4 0 4 (1976)

A Rigorous Formulation of the Boson Method in Superconductivity H . MATSUMOTO a n d H .

Department of Physics,

UMEZAWA

University of Alberta, Edmonton,

Alberta T60

2J1,

Canada

Abstract A rigorous formulation of the boson method in superconductivity is presented. Assuming the local gauge invariance and presence of a phase-dependent order parameter and utilizing the path-integral formalism, we derive a set of macroscopic equations which control all the superconducting states. These equations are model-independent. They contain certain functions and parameters which should be calculated for a given model. Using this rigorous formulation, we study what kind of approximations are involved in the usual formulation of the boson method and how these approximations can be improved. Contents 1. Introduction 2. Notations and Summary of the Results 3. The Functional Formalism 4. The Quasi-Particles 5. The Dynamical Maps and Physical States 6. The Non-Singular Boson Transformation 7. The Dynamical Maps and Dynamical Rearrangement 8. The Singular Boson Transformation and Macroscopic Equations 9. The Boson Formulation 10. The Boson Transformation and Ground State Energy 11. The Vortex Solutions Appendix A. The Quantum Field Theory of Free Plasmon Field Appendix B. The Hartree Approximation in a Scalar Model Appendix C. The Ghost in the Random Phase Approximation

357 362 371 373 379 382 384 385 388 391 397 398 401 402

1. Introduction In referent [Í], one of us (H.U.) presented a set of macroscopic equations which determine the observables such as electromagnetic field, current and energy in super-conductors under given boundary conditions. The formulation based on these equations is called the boson method in superconductivity. A merit of the boson method lies in the fact that solving the macroscopic equation under practical boundary conditions is relatively easy. This is because most of the complicated computations were already made in the course of deriving the macroscopic equations. This is in contrast to, for example, the theory based on the G O R ' K O V equation [2], which 26

Zeitschrift „Fortschritte der Physik", Heft 7

358

H . MATSTJMOTO a n d H . U M E Z A W A

is a non-linear and non-local equation. Although the derivation of the Gor'kov equation in the Green's function formalism was a simple one, solving this equation has presented a considerable amount of difficulty when the order parameter is not small. I t should, however, be pointed out that there has been made a steady progress informulating various computational technics based on the Gor'kov equation or on the Bogoliubov equation [3]. An essential step in the derivation of the boson method is to identify a field operator which is associated with the collective mode. The space-time dependent order state is then created by the boson condensation which can be mathematically expressed by a transformation of the field operator of the collective mode called the boson transformation [1, 6, 9]. The boson method has been used in our analysis of magnetic properties of type I I superconductors and also in our study of the Josephson phenomena [1, 11]. The results have been in reasonable agreement with experiments. I t is therefore important to answer the question asking what kind of approximations are made in the boson method in its present form and how it is possible to improve the approximation. In this paper, we try to answer this question by deriving a rigorous form of the macroscopic equations. In the derivation of the macroscopic equations in ref. [2], several approximations had to be made including the following ones: (a) The electron-electron interaction mediated by the phonons was assumed to be a point interaction. (b) The electromagnetic vector potential was not quantized. (c) The pair (i.e. random phase) approximation was applied to the Bethe-Salpeter equation for the wave function of the collective mode. (d) The effect of impurities was not taken into account. In this paper we will show that all these approximations can be avoided when the functional formalism [4, 5, 6] is employed and we will present a rigorous formulation of the boson method in superconductivity. Our considerations in the following sections begin with the quantum field theory of the electron fields