Fortschritte der Physik / Progress of Physics: Band 26, Heft 6 1978 [Reprint 2021 ed.] 9783112519127, 9783112519110

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Zeitschrift „Fortschritte der Physik 4 * Herausgeber: Prof. Dr. Frank Kaschluhn, Prof. Dr. Artur Lösche, Prof. Dr. Rudolf Ritschi, Prof. D r . Robert Rompe, im Auftrag der Physikalischen Gesellschaft der Deutschen Demokratischen Republik. Verlag: Akademie-Verlag, DDR • 108 Berlin, Leipziger Straße 3 - 4 ; F e r o r u f : 22 36221 und 2236229; Telex-Nr. 114420; 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 Deutschen Demokratischen Republik. Cesamtherstellung: V E B Druckhaus „Maxim Gorki", DDR - 74 Altenburg, Carl-von-Oesietzky-Stra0e 30/31. Erscheinungsweise: Die Zeitschrift „Fortschritte der Physik" erscheint monatlich. Die 12 Hefte eines Jahres bilden einen Band. Bezugspreis je B a n d : 180,— M zuzüglich Versandspesen (Preis für die D D R : 120,— M). Preis je Heft 15,— M (Preis für die D D R : 1 0 , - M). Bestellnummer dieses Heftes: 1027/26/6. (C) 1978 by Akademie-Verlag Berlin. Printed in the German Democratic Republic. AN (EDV) 57618

Fortschritte der Physik 26, 3 5 7 - 3 9 6 (1978)

Quantum Field Theory of Crystals and Dislocations* M. WADATI

Institute of Applied. Physics, Tsukuba University, Ibaraki, 300—31,

Japan

and H . MATSUMOTO, Y . TAKAHASHI a n d H . UMEZAWA

Theoretical Physics Institute, University of Alberta, Edmonton, Alberta, Canada T6G 2J1

Abstract We present a quantum field theoretical treatment of crystals from the viewpoint of the spontaneous breakdown of the spatial translation invariance. I t is proved t h a t at least three massless Goldstone particles exist and they are identified as the acoustic phonons. We carry out the boson transformation to show that the classical phenomenological theory of crystals can be derived. The classical momentum and stress tensor are expressed in terms of the classical displacement field. The method is extended to a microscopic derivation of the theory of crystal dislocations. I t is shown t h a t the Burgers vector is quantized and is a special case of the so-called topological quantum number. An explicit expression of a general solution of the dislocation is obtained.

Contents 1. 2. 3. 4. 5. 6. 7. 8. 9.

Introduction Periodic Functions Crystal Phonons and Dislocations Theory of Crystal Dislocations Spectral Representation of the Density Correlation Function Functional Formalism and the Ward-Takahashi Relations Point Group Reduction Formulae and Dynamical Map The Dynamical Rearrangement of Translational Symmetry

Appendix A. Remarks on the Violation of Time-Reversal Invariance Appendix B . The Explicit Solution of Infinitely Long Straight Dislocation Appendix-C. Proof, of (9.12)

357 360 363 371 378 380 382 385 388 391 392 393

* This work was partly supported b y the National Research Council of Canada and Faculty of Science of the University of Alberta. 27

Zeitschrift „Fortschritte der Physik", Heft 6

358

M . WADATI, H . MATSUMOTO, Y . TAKAHASHI a n d H . TJMEZAWA

1. Introduction The quantum field theory has by now been developed sufficiently to supply us with the most complete technique for handling systems composed of a large number of interacting particles. On the other hand, hardly anything has been done up to the present on the theory of crystals from the field theoretical view point: so far only the displacement of ions located in given lattice points has been quantized to form an ensemble of phonons and the ions themselves have been treated merely as unquantized objects [J]. Accordingly, crystal dislocations have been treated purely phenomenologically from the classical view point [2]. I t is the purpose of the present paper to provide a fully quantized field theoretical treatment of crystals and dislocations. In order to make the paper reasonably selfcontained, we have presented technical details associated with the basic concepts such as the idea of dynamical map, dynamical rearrangement of symmetry and the boson transformation, and the derivation of the Ward-Takahashi relations in terms of the pathintegral formalism [3—6], However, we have separated these technical details, which appear at the later half of the paper, from essential physical arguments and necessary mathematical expressions associated with them. The physics of crystals is a very old subject. The traditional approach [2] is to introduce a self-consistent periodic potential due to ions arranged in a lattice form. Small displacement of ions from their equilibrium position is then treated as a field. The quanta associated with the field are the phonons. That is to say, the phonons are quantum cooperative modes among a large number of ions. In this approach the distribution of ions is considered to be given. Hence, only phenomenological treatment is possible for classical objects such as dislocations of crystals [2], I n this paper, a microscopic theory of crystals and dislocations is formulated from the view point of fully quantized field theory. Before embarking on, it may be helpful to point out that the macroscopic aspects of crystal dislocations shares many features with that of the type I I superconductors. The macroscopic objects such as vortices can be created in superconductors which are controlled basically by certain quantum cooperative modes (i.e. phase quantum). Such macroscopic objects can be treated in terms of quantum field theoretical method which we have developed in the past several years [4, 6, 7]. In our theory, macroscopic objects are created in microscopic systems by means of a boson transformation, the meaning of which will become clear in due course. This method proved its validity in the analysis of vortices in the type I I superconductors [4]. According to our theory, classical objects are macroscopic manifestation of condensation of bosons associated with a quantum cooperative mode. Intuitively, when a large number of bosons are condensed, the quantum fluctuation becomes negligible (i.e. Anjn 1) where the system behaves as a classical object. Such a condensation process can be mathematically formulated in terms of a field transformation, which we call the boson transformation. An effective use of the boson transformation will be demonstrated in this paper to derive a theory of crystal dislocations. Our model and ideas of crystals and dislocations are as follows. We start with a system consisting of one kind of molecules represented by a quantized field ip(x). Hereafter we shall refer to this field as a molecule field. This field is self-interacting. The type of its interaction needs not be specified. Such a system can be characterized, for example, by a Lagrangian f[y>(x)]

3 = i f i ( z ) jtwix)

~

1 giW7 ^Vfc)

1

• ^ ( z ) - ~2 j ^ V M

fHv)

v

i

x

-

V) W^j)

(1.1)

Quantum Field Theory of Crystals and Dislocations

359

which leads to the Heisenberg equation d »' Jt

1 = -

2M

+ J ¿ W f r ) V(x

-

y) xp{y) v(x).

(1.2)

Here, V{x — y) is the potential between two molecules, M is the mass of a molecule and the time coordinates of x and y are taken to be equal. Athough we have written down an explicit Lagrangian, the argument to be presented in this paper is valid regardless of its details, provided that (i) it satisfies the spatial translational invariance f

= / d*xX[ip(x + cx)]1)

(1.3)

and (ii) the self-interaction is such that the crystal can be formed. The latter assumption can be expressed mathematically as [5] where and v{x) is a periodic function

0 the information so supplied will remain as spontaneously broken translational invariance, i.e. the lattice structure. The Ward-Takahashi relation emerging from (1.7) shows that there are at least three phonons whose energy vanishes at the momentum zero limit, which implies that these are the Goldstone bosons. Here, we introduce the quasi-particle picture [4, 7]. In the quasi-particle picture, the Hilbert space which specifies the matrix elements of Heisenberg operators is the Fock space of free quanta whose energy spectra are actually observable. Such free quanta are called quasi-particle. Heisenberg operators are represented by means of the matrix elements in this Hilbert space. This is called the quasiparticle 1

) In what follows, we mean by (x) = y,[x-,xi°(x),yfi{x)].

(3.19)

As will be proved in section 9, the spatial translation of the Heisenberg operator y>(x)

ip{x +

Xi°(a; x) = Xi°(x + «) + E i

%

Vll

y>°(x) -> y°(a; x) = yfl(x + )*(-&)

xSfr)

+ •••.

(3.44)

Considering the fact that a>2jk(k) is of second order in k for small k, it is reasonable to define the elastic constants C\f(k) by (^(fc) Û,»(fc) vWh

= E Ctfik) l,m

ktkm =

°{x)).

(3.65)

Since the observables are single-valued and the functions (3-67) c « which implies that the Burgers vectors are quantized. The integration in (3.67) is carried out along the closed loop c. The relation (3.67) will be called the quantization condition of the Burgers vector. We point out that the Burgers vector is a special case of so-called topological quantum numbers. It may be worth mentioning here that the flux quantization of the vortices in superconductor is another example of the topological quantum numbers.

Quantum Field Theory of Crystals and Dislocations

371

4. Theory of Crystal Dislocations In the last section we have shown how the boson transformation creates a macroscopic object in the system of phonons and quasi-molecules (crystal). The macroscopic objects are described by the displacement u(z), which is a c-number function satisfying the equation E MB) ui(x) = 0 (4-!) )

with AiW

5 + E vt l,m

=

(4.2)

where en(-iV)

= [rfi-iV)

(4-3)

v(-iF)h-

Since the ela-stic constants satisfy the symmetry conditions (3.48), there are only 21 independent elements among them, when fc = 0. The Burgers vector is quantized as (3.67). The energy and the momentum of macroscopic objects are, respectively, {PiUdx)) C%(-iV) Uf(x))\, «ì(») + E (FiWiCx)) c\?(-iV) {7m[v mUj(x))y (4.4)

W[u] = i-2' Zf&x E [