The U n i v e r s i t y of B r i t i s h Columbia FACULTY OF GRADUATE STUDIES PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY of WING GAY TAM B.Sc, ( S p e c i a l ) U n i v e r s i t y of Hong Kong, 1960 M . S c , U n i v e r s i t y of B r i t i s h Columbia, 1964 IN ROOM 301, PHYSICS (HENNINGS) BUILDING TUESDAY-,. SEPTEMBER 19TH, 1967 AT 3:30 P„M„ COMMITTEE IN CHARGE Chairman: Bo N. Moyls R. Guccione - Gush L . de S o b r i n o E l Macskasy W. P. Rastau D. L I . W i l l i a m s Opechowski E x t e r n a l Examiner: G.F.. K o s t e r M a s s a c h u s e t t s I n s t i t u t e o f Technology Cambridge, Mass. Research S u p e r v i s o r : W. Opechowski MAGNETIC OPERATOR GROUPS OF AN ELECTRON IN A CRYSTAL ABSTRACT The problem of an e l e c t r o n i n a c r y s t a l presence of a u n i f o r m magnetic field i n the is investigated u s i n g group t h e o r y method. A group of o p e r a t o r s commuting w i t h the H a m i l t o n i a n of an e l e c t r o n magnetic field constructed. t i o n s that electric and a c r y s t a l e l e c t r i c p o t e n t i a l i s T h i s group (a magnetic i s homomorphic to the group space group) of space time t r a n s f o r m a - leave the magnetic f i e l d potential subgroup o£~7$\ invariant. t h a t under mapped onto the l a t t i c e T It i n the presence of a u n i f o r m The p r o p e r t y of the the above homomorphism i s of h turns out the s t r u c t u r e of and the c r y s t a l i s studied i n detail.. depends on the magni- tude and the o r i e n t a t i o n of the magnetic f i e l d , so t h a t , i n f a c t one has to d e a l w i t h an i n f i n i t e c l a s s of groups. In particular, i t i s u s e f u l to d i v i d e t h i s c l a s s o f groups, i n t o two then r e f e r r e d fields, subclasses: infinite one s u b c l a s s i s to as c o r r e s p o n d i n g to " r a t i o n a l " magnetic the o t h e r as c o r r e s p o n d i n g to " i r r a t i o n a l " magnetic field, —* The group J j " i s a g e n e r a l i s a t i o n of the "magnetic t r a n s l a t i o n group" r e c e n t l y s p e c i a l case of a symmetric "physical" i n t r o d u c e d by Zak f o r the gauge. He a l s o constructed i r r e d u c i b l e r e p r e s e n t a t i o n s of the "magnetic t r a n s l a t i o n group" f o r the s p e c i a l case of a I n t h i s case a group 0" ' " r a t i o n a l " magnetic f i e l d . always has a maximal A b e l i a n subgroup w i t h index. a finite (The term " p h y s i c a l " r e p r e s e n t a t i o n simply means a r e p r e s e n t a t i o n which can be generated by f u n c t i o n s of s p a t i a l c o o r d i n a t e s . ) In t h i s t h e s i s no such r e s t r i c t i o n the " p h y s i c a l " i s introduced: **** irreducible representations of are also corir s t r u c t e d f o r the case of i r r a t i o n a l . m a g n e t i c f i e l d , i n which case the index of a maximal A b e l i a n group i s always i n f i n i t e ; representations Using the " p h y s i c a l " i r r e d u c i b l e are then always i n f i n i t e dimensional. a complete s e t o f Landau f u n c t i o n s the basis functions generating representations of "physical" irreducible are found f o r the s p e c i a l case when the c r y s t a l netic f i e l d sub- i s simple is parallel turns out when the f i e l d c u b i c and the mag- to a l a t t i c e vector. It i s " i r r a t i o n a l " the b a s i s f u n c t i o n s , and the energy spectrum depends o n l y on one o f the parameters l a b e l l i n g irreducible representations The of the " p h y s i c a l " . problem o f p e r t u r b a t i o n produced by a weak p e r i o d i c p o t e n t i a l on the Landau l e v e l s f o r a f r e e e l e c t r o n i n a magnetic f i e l d In t h i s c o n n e c t i o n i s also considered. we make p l a u s i b l e - the v a l i d i t y of c e r t a i n q u i t e g e n e r a l s e l e c t i o n r u l e s f o r an arbitrary periodic potential. GRADUATE STUDIES F i e l d of Study: S o l i d State Physics Elementary Quantum Mechanics E l e c t r o m a g n e t i c Theory Quantum Theory of S o l i d s Advanced Magnetism S p e c i a l R e l a t i v i t y Theory Group Theory Methods i n Quantum Mechanics • D i f f e r e n t i a l Equations I n t e g r a l Equations F.A. Kaempffer G.M. V o l k o f f W. Opechowski W. Opechowski H. Schmidt W. Opechowski M. A n v a r i E. Macskas'w PUBLICATION W. G. Tarn and W. Opechowski, "Magnetic Space Group., of an E l e c t r o n i n a C r y s t a l " , P h y s i c s L e t t e r s , 23, 212 (1966). AWARDS 1965-60 1963-64 1965-67 Hong Kong Government B u r s a r y U n i v e r s i t y F e l l o w s h i p (UBC) NRC S t u d e n t s h i p MAGNETIC OPERATOR GROUPS OF AN ELECTRON IN A CRYSTAL by WING GAY TAM B. Sc. ( S p e c i a l ) , Hong Kong U n i v e r s i t y , , 1960 M. Sc., U n i v e r s i t y of B r i t i s h Columbia, 1964 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF . DOCTOR OF PHILOSOPHY . i n the Department - of. PHYSICS We accept t h i s t h e s i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1967 In p r e s e n t i n g for thesis an a d v a n c e d d e g r e e a t that the Study. thesis Library for the U n i v e r s i t y make it that freely of British available permission for of for representatives. by h.i;s of of this thesis for permission. Pff^SiCf Columbia It financial is the requirements Columbia, I reference and extensive or The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, Canada Date fulfilment p u r p o s e s may be g r a n t e d b y t h e my w r i t t e n Department agree in p a r t i a l scholarly publication without shall I further Department or this copying of this Head o f my understood gain agree shall that not be copying allowed i ABSTRACT In t h i s t h e s i s the problem of an e l e c t r o n i n a c r y s t a l i n the presence of a uniform magnetic f i e l d i s i n v e s t i g a t e d u s i n g group theory method. A group of operators 771 commuting w i t h the Hamiltonian of an e l e c t r o n i n the presence of a uniform magnetic f i e l d and a c r y s t a l e l e c t r i c p o t e n t i a l i s c o n s t r u c t e d . T h i s group i s homomorphic to the group M (a magnetic space group) of space time t r a n s f o r m a t i o n s t h a t leave the magnetic f i e l d and the c r y s t a l e l e c t r i c p o t e n t i a l i n v a r i a n t . property of the subgroup J of Yf\ "that under the above homomorphism i s mapped onto the l a t t i c e T in detail. The of M i s studied I t t u r n s out the s t r u c t u r e of J" depends on the magnitude and the o r i e n t a t i o n of the magnetic f i e l d , so t h a t , i n f a c t one has to deal w i t h an i n f i n i t e c l a s s of groups. In p a r t i c u l a r , i t i s u s e f u l t o d i v i d e t h i s i n f i n i t e c l a s s of groups i n t o two subclasses: one subclass i s then r e f e r r e d to as corresponding to " r a t i o n a l " magnetic f i e l d s , the other as corresponding t o " i r r a t i o n a l " magnetic f i e l d . The group J " i s a g e n e r a l i s a t i o n of the "magnetic t r a n s l a t i o n group" r e c e n t l y introduced by Zak f o r the s p e c i a l case of a symmetric gauge. He a l s o constructed " p h y s i c a l " i r r e d u c i b l e r e p r e s e n t a t i o n s of the "magnetic t r a n s l a t i o n group" f o r the s p e c i a l case of a " r a t i o n a l " magnetic f i e l d . In t h i s case a group w i t h a f i n i t e index. J* always has a maximal A b e l i a n subgroup (The term " p h y s i c a l " r e p r e s e n t a t i o n simply means a r e p r e s e n t a t i o n which can be generated by ii functions of restriction spatial is introduced: representations irrational coordinates.) of irreducible "physical" are also T magnetic f i e l d , maximal A b e l i a n the In this t h e s i s no such irreducible constructed for t h e case i n w h i c h case t h e i n d e x o f subgroup i s representations always i n f i n i t e ; are then always the of a "physical" infinite dimensional. Using a complete functions of generating J " are found f o r simple set o f Landau f u n c t i o n s "physical" the irreducible special It turns basis functions functions, o u t when t h e f i e l d are and t h e the parameters countably representations representations is is infinite parallel to a "irrational" sets of of the "physical" is lattice the Landau energy s p e c t r u m depends o n l y labelling basis c a s e when t h e c r y s t a l c u b i c and t h e m a g n e t i c f i e l d vector. the on one of irreducible J" , The p r o b l e m o f p e r t u r b a t i o n p r o d u c e d b y a weak periodic potential i n a magnetic on t h e L a n d a u l e v e l s field we make p l a u s i b l e selection rules is also considered. the v a l i d i t y for for of an a r b i t r a r y a free In this certain quite periodic electron connection general potential. iii TABLE OF CONTENTS Page ABSTRACT i LIST OF TABLES v LIST OF SYMBOLS vi ACKNOWLEDGEMENTS viii I. INTRODUCTION AND SUMMARY 1 II. 1 SYMMETRY GROUP OF A UNIFORM MAGNETIC FIELD 8 II. 2 SYMMETRY GROUP OF THE HAMILTONIAN II. 3 THE GROUP OF OPERATORS II. 4 SYMMETRY GROUP OF THE HAMILTONIAN $ & 35 II. 5 STRUCTURE OF THE GROUP ' 41 III. l CLIFFORD'S METHOD OF CONSTRUCTING IRREDUCIBLE j-f 0 (j ' J 12 19 . REPRESENTATIONS OF A GROUP 55 III. 2 IRREDUCIBLE REPRESENTATIONS OF J" III.3 PERIODIC BOUNDARY CONDITION 84 III. 4 THE GROUP IRCl AND ITS CO-REPRESENTATIONS 90 IV. l a THE LANDAU FUNCTIONS 93 IV.lb BASIS FUNCTIONS FOR THE CASE OF RATIONAL MAGNETIC FIELD IV.lc 59 98 BASIS FUNCTIONS FOR THE CASE OF IRRATIONAL MAGNETIC FIELD 104 IV. Id COMPARISON OF BASIS FUNCTIONS 107 IV.2 ANALOGUES OF BLOCH THEOREM 117 IV. 3 PERIODICITY OF THE ENERGY SPECTRUM 126 V. l PERTURBATION CALCULATIONS FOR THE CASE OF IRRATIONAL MAGNETIC FIELD 136 iv V.2 SELECTION RULES FOR THE CASE OF IRRATIONAL MAGNETIC FIELD V.3 V.4 / 150 PERTURBATION CALCULATIONS FOR THE CASE OF RATIONAL MAGNETIC FIELD 154 COMPARISON OF PERTURBATION MATRICES 171 BIBLIOGRAPHY 175 APPENDIX I 177 APPENDIX I I 183 V LIST OF TABLES TABLE 1 MAXIMAL ABELIAN SUBGROUPS OF THE MAGNETIC TRANSLATION GROUP • Page 54 vi LIST OF SOME OF THE SYMBOLS INTRODUCED IN CHAPTER I I 10(3) : Three dimensional in-homogeneous orthogonal group /\ * Time i n v e r s i o n group (j Symmetry group of the uniform magnetic f i e l d V '. Group of t r a n s l a t i o n s R . Subgroup of primed and unprimed r o t a t i o n s of Q l F : Space group T \ Group of l i t t i c e t r a n s l a t i o n s ( i . e . subgroup of a l l t r a n s l a t i o n s of a space group) M ; That magnetic space group which i s the i n t e r s e c t i o n of Mo- F ®A and G Hamiltonian of a f r e e e l e c t r o n i n a uniform magnetic f i e l d ; Hamiltonian of a f r e e e l e c t r o n i n a uniform magnetic f i e l d and p e r i o d i c e l e c t r i c Q - Group of operators commuting with Mo field homomorphic v to > ( j , \ j ; Group of operators commuting with 'Wc ^ homomorphic to V (fcl * Group of operators commuting with $4,' isomorphic to TfL R Group of operator commuting with j-ig, homomorphic 5 to M - Group of operators commuting with J-lg v t homomorphic <—' to J \/x * Group of sets of loops, each element c h a r a c t e r i s e d by a vector \r and a number X ; (0^ X < Ij Subgroup lattice of each element c h a r a c t e r i s e d V vector Group of s e t s of and a number polygons 1 X by , ( 0 $ X <• \) ACKNOWLEDGEMENTS I wish to express my g r a t i t u d e to P r o f e s s o r W. Opechowski f o r suggesting t h i s problem and f o r h i s continuous guidance and valuable advice throughout the performance of t h i s research. I wish to thank my w i f e , A l i c e , f o r typing t h i s thesis and helping to prepare the manuscript f o r p r i n t i n g . The f i n a n c i a l support given by the National Research Council of Canada i n the form of a studentship i s also gratefully acknowledged. 1 I. INTRODUCTION AND SUMMARY Theoretical investigation conduction electrons uniform magnetic decades, and i s of in a crystal field still the behaviour i n the presence has been c a r r i e d on f o r applying, istics the degeneracy of of the energy eigenstates of the and other the Hamiltonian. quantum m e c h a n i c a l group-theoretical presence aspect of systems. =- <=u.rt Air) invariant. (including to the electron, and A h e r e and i n t h e r e s t the symmetry course t r u e of The p e c u l i a r i t y arises from the f a c t + many of the that the the V(r) SL H>+ V(r) symmetry g r o u p o f and t h e p e r i o d i c sign), of Hamiltonian " t o , e, (As u s u a l , the character- without space-time t r a n s f o r m a t i o n s which leave the uniform magnetic f-{ for o f t h e p r o b l e m o f an e l e c t r o n i n [ f - TM*>] isomorphic c a n be o b t a i n e d This i s of magnetic f i e l d symmetry group o f t h e i s not some by s t u d y i n g t h e of problem. spectrum, the c l a s s i f i c a t i o n the Hamiltonian, four justifying the problem energy l e v e l s , i n t r o d u c i n g any a p p r o x i m a t i o n , group of of a A lot v a r i o u s a p p r o x i m a t e methods o f d e a l i n g w i t h t h e H o w e v e r , many g e n e r a l a s p e c t s of the almost r e c e i v i n g much a t t e n t i o n . w o r k h a s b e e n done i n d e v e l o p i n g , example, of Y, |> electric field potential a r e t h e mass, \/(Y) charge c a n n o n i c a l p o s i t i o n and momentum o f the is the vector potential. We d i s r e g a r d of the thesis the effects due t o the 2 electronic spin). tion, the t r a n s l a t i o n that i s an a r b i t r a r y Hamiltonian This follows operators translation, J-f have a l o n g h i s t o r y *>t £~Tr ^ " 1* ) treatment (which begins w i t h a paper been d e f i n e d unambiguously f o r gauge" the various systematic gaps l e f t of A(r) = £ fj x r group-theoretical the aspects of t i o n of authors. Harper (1955), F i s c h b e c k U963 I , Jannussis (1964). We s h a l l n o t field; d i s c u s s e d b y A s h b y and M i l l e r results. Wannier i s not We s h a l l n o t r e f e r investiga- t h e c a s e when there case has sufficiently explicit II gauge, a group we f i r s t <j of our is of the i m p o r t a n t work for in also been (1965) o n t h e b a s i s to such as (1964), (1962) b e c a u s e h i s u s e o f g r o u p - t h e o r e t i c a l I n Chapter linear this the 196.5). to the Brown consider a generalisation a d d i t i o n t o an e x t e r n a l u n i f o r m m a g n e t i c f i e l d an e x t e r n a l u n i f o r m e l e c t r i c discuss to the problem, II), Zak's of methods purpose. construct, operators for a In I I , III) oecassionally o t h e r a u t h o r s who c o n t r i b u t e d and we h a d t o f i l l c a n be r e g a r d e d a s a H o w e v e r , we s h a l l a l s o r e f e r case o f the problem i n I n doing that t h e r e s u l t s o b t a i n e d b y Zak (1964 I , group I). to present i n t h e t r e a t m e n t s by e a r l i e r our r e s u l t s special of by symmetry "by Zak (1964 t h e s i s we h a v e t r i e d and u n i f o r m w a y . particular, <? a group-theoretical of In this where the (1933))» o n l y v e r y r e c e n t l y h a s t h e "symmetric observa- do n o t commute w i t h Peierls He from the . e Although attempts at )HB immediately an arbitrary commuting w i t h the 3 Hamiltonian }j 0 and h o m o m o r p h i c t o time transformations remains i n v a r i a n t . of space under which a u n i f o r m magnetic field N e x t we f i n d the group a subgroup Q 7H of ^ i s mapped b y t h e a b o v e h o m o m o r p h i c m a p p i n g o n t o t h e subgroup of M under which the magnetic G field which largest and the *~*> periodic potential commute w i t h magnetic Tig % remain i n v a r i a n t . The s u b g r o u p M space g r o u p b e l o n g i n g t o group under which the p e r i o d i c the d e f i n i t i o n s "families" of The s t r u c t u r e of the terms such g r o u p s , of the The o p e r a t o r s of is of course the family of the G potential "magnetic J* in some d e t a i l . invariant H . of T all The s t r u c t u r e isomorphic. H whose members a r e , its is discussed. of J" of isomorphic. Por t h e group i d e n t i c a l w i t h the J ~ is i n t r o d u c e d b y Zak Although t h i s A(^) turns magnetic "magnetic the call J" translation t e r m has o f t e n been used f o r space g r o u p for various groups all the translation We s h a l l a l s o gauge t h e a not = ^ H x r "magnetic above then c h o i c e s o f gauge a r e s y m m e t r i c gauge (1964 I ) . case o f an a r b i t r a r y J Por a f i x e d f i e l d different (1965)). maximal i n general, The r e l a t i o n b e t w e e n t h e g r o u p s corresponding to a magnetic is M I n o t h e r w o r d s , we s h a l l h a v e t o d e a l w i t h class of groups fields and and G u c c i o n e o u t t o d e p e n d on t h e m a g n i t u d e and o r i e n t a t i o n field (for o f Tyi w h i c h u n d e r t h e In particular, A b e l i a n subgroups are l i s t e d . a space space g r o u p " homomorphism i s mapped o n t o t h e l a t t i c e discussed is see O p e c h o w s k i - subgroup o f Yft group" in group". the l a t t i c e ( s e e O p e c h o w s k i and G u c c i o n e the of (1965)) 4 no confusion should a r i s e . In Chapter I I I we construct the " p h y s i c a l " i r r e d u c i b l e r e p r e s e n t a t i o n s of the groups J~ for different orientations and magnitudes of the magnetic f i e l d . The term " p h y s i c a l " (1964 I I ) r e f e r s to those r e p r e s e n t a t i o n , introduced by Zak r . r e p r e s e n t a t i o n s which can be generated by f u n c t i o n s of We have been able to o b t a i n a l l these r e p r e s e n t a t i o n s f o r one subclass of groups "J except The method we have used f o r # t h i s purpose i s based on a c e r t a i n number of theorems due Clifford (1937) to and on a s l i g h t g e n e r a l i s a t i o n of some of these theorems. In a p p l y i n g C l i f f o r d ' s method to our problem the f i r s t step c o n s i s t s i n f i n d i n g i r r e d u c i b l e r e p r e s e n t a t i o n s of a s u i t a b l y chosen maximal A b e l i a n subgroup of a group J From the survey of maximal A b e l i a n subgroups given i n Chapter I I I i t turns out t h a t maximal A b e l i a n subgroups of finite index e x i s t only f o r c e r t a i n magnitudes and o r i e n t a t i o n s of the magnetic f i e l d . We r e f e r to t h a t c l a s s of magnetic f i e l d s as r a t i o n a l ; otherwise we c a l l the magnetic f i e l d irrational. The reason f o r t h i s terminology been used by Brown connection (1964) and Zak (1964 (which has also I) and i n d i f f e r e n t by others) i s best explained by t a k i n g the s p e c i a l but important case of f i e l d p a r a l l e l to a l a t t i c e v e c t o r . By t a k i n g that l a t t i c e v e c t o r a to be p r i m i t i v e (which e n t a i l s no l o s s of g e n e r a l i t y ) we can w r i t e the f o r the f i e l d as f o l l o w s : H = Ac expression 5 H e r e i2 i s the volume o f the p r i m i t i v e u n i t c e l l , a number w h i c h i s i r r a t i o n a l field is irrational It turns dimensional is All t h a t the groups (the i d e n t i t y i n this " I t w o u l d be J" f o r the case. representation In'his not been a b l e one ^ s the We finite i f the v l are infinitely the (1964 irrational to prove t h a t the II) representations restrictions h a v e done t h a t , e x c e p t c l a s s of field i s n o t p h y s i c a l ! ). s e c o n d p a p e r Zak i n t e r e s t i n g to f i n d at a l l . " c a s e of as h a v e no t h e m a g n e t i c t r a n s l a t i o n g r o u p w i t h o u t any magnetic f i e l d ^ rational. "physical"irreducible representations dimensional says: out or r a t i o n a l according physical i r r e d u c i b l e representations irrational the or and on possibly f i e l d s where we representations of obtained have are irreducible. A question r e l a t e d to the b e t w e e n r a t i o n a l and possibility of irrational introducing necessity fields irrational At problem of sense).of the end of the field are and ;we requires and of the briefly co-representations We with the ( i n Wigner's c o n c l u d e t h a t no The difficulties c a s e of an implications of further investigation. V we discuss r e s u l t s p r e s e n t e d i n C h a p t e r s I I and classification imcompatible discuss case of a r a t i o n a l f i e l d . I n C h a p t e r s TV the c h a p t e r we the g r o u p s ^71 a r i s e here' i n the irrational the field. constructing the i s t h a t of p e r i o d i c boundary c o n d i t i o n s . show t h a t p e r i o d i c b o u n d a r y c o n d i t i o n s an of d i s t i n g u i s h i n g eigenfunctions of the III for the the Hamiltonian "J-ig 6 and f o r some g e n e r a l p r o p e r t i e s o f i t s e n e r g y considering the special H c a s e when ± s spectrum parallel by to a l a t t i c e vector. In C h a p t e r representation belonging We call I V we show t h a t e v e r y of a group "physical" c a n be g e n e r a t e d Landau,functions, i n t h e f o r m g i v e n by J o h n s o n and L i p p m a n n out t h a t i n t h e case of i r r a t i o n a l field combinations fields (which certain the ( f i n i t e ) t h e r e s u l t s h a v e been f o u n d periodic i r r e d u c i b l e r e p r e s e n t a t i o n s of of two one s u c h t h e energy parameter. analogues b e e n f o r m u l a t e d by Zak ( 1 9 6 5 ) Schrodinger labelling some are the 'physical In t h e case of independent c o n n e c t i o n we a l s o d i s c u s s has a l r e a d y earlier. V we c o n s i d e r t h e p r o b l e m equation corresponding as a p e r t u r b a t i o n p r o b l e m , electron . e i g e n v a l u e s o f iHg e i g e n v a l u e becomes In t h i s convergent) "physical" o f t h e B l o c h t h e o r e m one o f w h i c h In C h a p t e r linear F o r the l a t t e r case a l s o show t h a t t h e e n e r g y fields while i n the f by Zak ( 1 9 6 4 I I I ) e a r l i e r . f u n c t i o n s of the parameters irrational countably a r e shown i n A p p e n d i x I I t o be Landau f u n c t i o n s g e n e r a t e We I t turns generate the sets of i n f i n i t e i r r e d u c i b l e r e p r e s e n t a t i o n s of J* . of certain i r r e d u c i b l e r e p r e s e n t a t i o n s of case of r a t i o n a l b u t we u s e them (1949). s e t s of Landau f u n c t i o n s t h e m s e l v e s physical' of by f u n c t i o n s t o t h e s p a c e o f e i g e n f u n c t i o n s o f t h e H a m i l t o n i a n 'he. these eigenfunction infinite irreducible of the t o t h e H a m i l t o n i a n ^Jg the Hamiltonian i n the uniform magnetic f i e l d being o f an the unperturbed 7 Hamiltonian and the p e r i o d i c p o t e n t i a l tion. We regard the p e r t u r b a t i o n V ( r ) the p e r t u r b a - as weak o r , more p r e c i s e l y , we consider i t s e f f e c t on one a r b i t r a r y Landau l e v e l of f i 0 We make p l a u s i b l e the v a l i d i t y of c e r t a i n s e l e c t i o n r u l e s f o r the general case of any r a t i o n a l or i r r a t i o n a l magnetic f i e l d p a r a l l e l to a l a t t i c e vector. We t r e a t i n some d e t a i l the case of s i n u s o i d a l p e r i o d i c p o t e n t i a l , confirming i n t h i s way the v a l i d i t y of the s e l e c t i o n r u l e s i n that p a r t i c u l a r case. F i n a l l y we d i s c u s s the r e l a t i o n between the p e r t u r b a t i o n m a t r i x f o r the case of r a t i o n a l f i e l d and the perturbation m a t r i x f o r the case of i r r a t i o n a l f i e l d . To conclude the summary of the contents of t h i s t h e s i s the author would l i k e t o emphasize that he i s aware of a t l e a s t one major gap i n the t h e s i s : no l i n k has been established between the r i g o r o u s but r a t h e r formal group- theorectical discussion of the problem and those approximate but powerful methods which s t a r t out from the e l e c t r o n band model. energy We are c u r r e n t l y i n v e s t i g a t i n g , however, the r e l a t i o n between our treatment of the p e r t r u b a t i o n problem, given i n Chapter V , and the treatment of the "magnetic breakdown" (Cohen and F a l i c o v (1961)), i n i t i a t e d by Pippard (1962, 1963; see a l s o Chambers (1965, 1966)) and based on the i d e a of " p e r i o d i c network" of e l e c t r o n o r b i t s . , 8 II.1 SYMMETRY GROUP OF A UNIFORM MAGNETIC FIELD Given a uniform time independent magnetic f i e l d we want to f i n d a l l the space time transformations under which the magnetic f i e l d remains i n v a r i a n t . The group of space tirjie transformations we 'are concerned i s the d i r e c t product XO (1) ® A . lO(i) Sere C i s the three dimensional - orthogonal inhomogeneous group whose elements are denoted by (R/'f) The symbol R stands f o r a proper or improper r o t a t i o n and v* and a r b i t r a r y vector. of the group rotation R IQ (3) i s a transformation which consists of a followed by a t r a n s l a t i o n v. The group A two elements: the i d e n t i t y element element Thus an element has only £ and the time r e v e r s a l E, <Cln t h i s chapter some notations are d i f f e r e n t from those used i n Tarn, and Opechowski. (1966) but we f o l l o w very, c l o s e l y the notations and terminology of Opechowski and Guccione( 1965 )). i s denoted by Consequently a general element of 10(3)® J lR\^)A where A = £ or As i s w e l l known, the product of two elements and ( fo-l^k) ^ s 9^ v e n A o f (&I<?)A 2 t by hence the product of two elements t#2}Vz) (R, I v ) = (R.*a| I .CIS) A, and i s given by A.Aa ^ We neglect those transformations which are time t r a n s l a t i o n s . As we s h a l l see l a t e r that we are i n t e r e s t e d only i n the s t a t i o n a r y states of the p h y s i c a l systems to be studied* For these states invariance under time t r a n s l a t i o n s leads to nothing of p h y s i c a l consequence. Every element (RlOA of 20(3)® A induces a transformation on a given uniform time independent magnetic f i e l d H defined by: 6, £ R TT (m. R where if and A - t A - E' if R i s a proper r o t a t i o n if R i s an improper We can see that ' ( j L i . 0 1, i ) rotation i s consistent with the f a c t that H i s a x i a l vector and i t changes sign under time r e v e r s a l . A symmetry transformation of the magnetic f i e l d i s defined as an element (R|^)A I.QU)8A of H such that [(R\ tf)A] H The set of symmetry transformations of ZOL2>) S> A element (1,1.2) - H of n forms a subgroup Q ' I t i s c l e a r that Q contains the i d e n t i t y (P/o)f . I f (R I ^) A belongs to i t s inverse (-•V CR~'J R"'^) A'' also belongs to Cr [(P,.1 tf ) A, J •[ ( K I <S ) A ] r f a so i f At 2 and « [( M (Ral^Az ^ )M Since * I <* M # 11.1• 3) ' are elements of <3 t h e i r product i s also an element of Cj . We note that the t r a n s l a t i o n subgroup with elements of the form V of XDLz)8> A f o r any vector V" i s a Q • subgroup of C(R|v)E]H If R = RH , The c o n d i t i o n f o r be an element of G> i s that R p a r a l l e l to the d i r e c t i o n of through an angle Cj> by [LR\ir)B]H- C (R | H We denote such a r o t a t i o n implies that FT , R &p a to hold - R ff i.e. = H, r o t a t i o n around an axis p e r p e n d i c u l a r d i r e c t i o n of H . axis p e r p e n d i c u l a r R, »' IRJ space i n v e r s i o n . angle [_ ( R» I Similarly i f [_LR\^)E'] <p , = H /?) must be of the form X l?^ H. H = (p , [ (Rt I &) B J H = we must have The symbol j_ (. £1 I ^) l9o° I denotes H ~ ^ we f o r some <f> . As the group of a l l pure t r a n s l a t i o n s i s a normal subgroup of subgroup of J O ( 3 } ® /A ; i t i s a l s o a normal Q . We can summarise the r e s u l t s of our previous d i s c u s s i o n by s t a t i n g that the s o s e t decomposition of Q respect must to the f o r some angle &)EJH f o r some angle conclude' that then i s an improper r o t a t i o n then To s a t i s f y the c o n d i t i o n £, to H = If R I f we denote by R(2) a r o t a t i o n through (.R I v ) E ( ^ < p £(2) I ^ ) £ implies for However f o r be a two-fold around a given to R<^ and hence the r e l a t i o n 0 •$ (f> < 2.TC ) £ ' ] FT = (Rl^)f- must be a r o t a t i o n around axis it sorpe angle [T. /• i) i s a proper r o t a t i o n then from to V firvd. gives the f o l l o w i n g (lfcpR (a) \o)E' V cosets: . with 11 The quotient group group R <3 / V i s e v i d e n t l y isomorphic to the (fylo)£ , f£fylo)£ , ( &j> #<2) I o) E/ which contains and as elements. fa) denoted by This group R i s often , using the .'international' c r y s t a l l o g r a p h i c symbols. We may also note that the group product of the groups and V R and are subgroups of The product ( RI 0 ) A ' (B 1/ . G | V) We have seen that both and V where E with of C^i V » c a n ^ Also the i n t e r s e c t i o n i d e n t i t y element semidirect product of the groups any Q . in R and (E\»)B , ~ # ^ V contains only the Hence the group ^ £ s be w r i t t e n as a product (R|o)A r v in i (£|D)A i s an element i n Conversely any element G R i s a normal subgroup. element i n £ and ( R l o ) A • (B]Z)E i s a semidirect and ]/, is a 12 II.2 SYMMETRY GROUP OF THE HAMILTONIAN ^ c The Hamiltonian of a f r e e e l e c t r o n i n a uniform magnetic f i e l d i s given., by: and e where (e<c>) are the mass and the charge of the electron respectively; conjugate momentum r vectors Since the magnetic f i e l d potential are the p o s i t i o n and and C Is the v e l o c i t y of l i g h t . f-f i s uniform we w r i t e the vector i n a general l i n e a r gauge such that ' + ia * + a.&yr*a z)£ Si where and &s (1.2.2) 3 61 , <£i and 63 are three u n i t vectors along X, in orthogonal axes r e s p e c t i v e l y . A l s o we may w r i t e the f o l l o w i n g matrix form! fit) ' = a. 0,1 ' x N 3 0*21 a - V - 1 2- = 1, 2, O constants subject only to the c o n d i t i o n that are a r b i t r a r y CUXAI A C O I t can be r e a d i l y seen that the transformations by the elements of (T.2.29 t On. The q u a n t i t i e s N = H . induced , the symmetry droup of the uniform 13 magnetic f i e l d N i n the state*, vector leave the Hamiltonian transformation transformed that to /^( ) (JE\&) E %ir , in r is Owing t o t h e 2 linear general , f) £nl ( " f * ~ f " A (. ^ ~ ^ ) ) • is r in i n v a r i a n t U n d e r the' Qot^'f) i n d u c e d by s p a c e do n o t fact we have J _ 2m, H Hence t h e t r a n s f o r m e d differs Hamiltonian from o n l y by a gauge I n o t h e r w o r d s , we can o b t a i n > "f ) * (. " ^ > ^ ) where from y ky making t h e gauge £(t) + VXC?) transformation. transformation = -Y-/£l^) X(r) - Thus although are identical they describe corresponding exactly to d i f f e r e n t t h e same p h y s i c a l choices of gauge. necessary to change t h e gauge so Under t h e t r a n s f o r m a t i o n vector thus given T is transformed jioLr,^) by: is to system To b r i n g t r a n s f o r m e d H a m i l t o n i a n back t o it not is the only that i n d u c e d by y and transformed to ^> to R<p I 0 ) B the K^> and ^-f ( f?^"'''y, 0 ^ ) 14 n 2 L R t ' ( f - | R tf(RjV)) f 2>H Again one from ^-f<? can show t h a t -^-/o (-v , ^ [ R<f r , Rp'-^ differs merely by a gauge t r a n s f o r m a t i o n . t h i s i t i s convenient to choose a c o o r d i n a t e H the z-axis i s along the d i r e c t i o n of r o t a t i o n around the z-axis through angle it ) 1 To do system such t h a t Hcfj Then and is a i n matrix form can be w r i t t e n as R C( >1 (f> A^Cf o CttfCf 0 0 The f a c t that implies H has / X zero components along that h i - ^is = 0 and (X32 - d3 2 - 0 and ^ directions The expression / f ( r ) - Ryfi (R^r) in (1.2.4) can also be w r i t t e n as where ^ f * i s the matrix defined i n ( 1 . 2 . 2 0 and a, ^, Z column matrix with where (* [| 2J) y y ^ ^) i s a n Y <^iX { X ) vanishes i f and only i f the matrix symmetric. Making use of 7\ ~ f?|A here i s a as i t s three elements. c y c l i c permutation therefore T ( I . 2.5) A : Since of t h e i n t e g ers ~ Rf A I R<j> ?)) " Rcf-A Rf is we obtain = (r.E.g) 0 which, because of we may w r i t e then (IE. 2.fe) i s indeed a symmetric matrix. Now 16 Thus we have proved t h a t . 'jO ^ ( R ^ V , £<j>~' ^ ) a n < ^ d i f f e r by j u s t a gauge transformation. The element CR(2)|o)E of CT as we may recall, i s a r o t a t i o n through an angle around some f i x e d axis perpendicular to the d i r e c t i o n of /-/ followed by time r e v e r s a l . So f a r we have only p a r t i a l l y s p e c i f i e d the coordinate axes by choosing the z - a x i s to be along the d i r e c t i o n of now choose the y-axis to be the axis of r o t a t i o n of f i x e s the coordinate system completely. to H . The matrix Let us /R(2) . This corresponding is: -1 'ft, 0 . 0 0 1 0 0 0 -1 Under the transformation induced by (R{z)lo)E' n ~ -* (2) r and ^ to - R(2) ^ transformed to Hi7\ V i s transformed to ^-f ( #(2) r , - £(2) ^ ) 1 0 2^ > is hence Now 3-/O(Y, 17 Becaus e 2a 2a 0 M 0 2a (• At") 2^33 )'^) °* = + R ( 2 ) / ? ' 1 R(l) Y ) f) So 0 51 r R(2)"^^(2) (^*-+ 0 Cu-ri ^ X I*) + ^(2-) A (R(2) ) ~ i s a symmetric matrix, £UA£ H ( R(2) l 3 = i[ w e m a Y write V^( )(?) then 2 f -1 (^ 2 T h u s - - ;2) f ) v ^ ( f } ] ) 2.Y) 2 also d i f f e r s from : by a gauge transformation. Noting that _y and i s b i l i n e a r i n the operators ^o(v,-j>) we can conclude immediately that under the transformation induced by space i n v e r s i o n "^o (*" / ^ ) remains i n v a r i a n t . we know that any element of R can From Section !• * be w r i t t e n intone of the f o l l o w i n g forms . (ft-pR(z) | $ ) <5 (-If^ R ( a ) | fl) E ' a n d can be w r i t t e n , as belongs to the group (R^,|0)£ V ( E | 4? ) E and (xR^|o)E ; and any element of * (£i 0 ) A [R\ 0) A where belongs to £. ( E | ^ ) E As a r e s u l t of our previous discussions i n t h i s s e c t i o n we conclude that f o r any element Gr of the group Gr we can f i n d a s u i t a b l e •gauge transformation Xl*) ~ A I*) ' /^jC ) such that r under the oombined transformation which c o n s i s t s of the transformation induced by transformation Xl*) followed by the gauge * X(* ) ~ ) the 18 Hamiltonian ^-I„(Y remains i n v a r i a n t . We s h a l l c a l l the gauge transformation a ^compensating'gauge transformation. We ca-n also show that i f XD(2) ® /A (£|^)A i s any element of f o r which there e x i s t s a ^compensating' gauge transformation i n the sense we j u s t described then must be an element of (R | ^ ) A II.3. THE GROUP OF OPERATORS g , In S e c t i o n H. 2 . we have considered the c l a s s i c a l \Ho (. Y^> ^ ) Hamiltonian and observed that d^oC^^) does not remain i n v a r i a n t under a l l the transformations of the group Cr • But, whenever <3t was any transformation of O which under does not remain.invariant, we could always f i n d a s u i t a b l e gauge tranformation su ch that 34 f ) 0 remained i n v a r i a n t under the combined transformation c o n s i s t i n g of Or and At*) — * $ V ) - ^ X a ( 3 . • r In quantum mechanics, to every element Or of the group Or there coresponds i n a w e l l known way an operator rq- which e f f e c t s the transformation induced by Or i n the s t a t e vector space. I f the c l a s s i c a l Hamiltonian considered above i s not i n v a r i a n t under Or then the--corresponding Hamiltonian w i l l not commute with , quantum mechanical However, i f does not commute with the Hamiltonian, we s h a l l show that one can always f i n d an operator Pp^. which when operating on the Hamiltonian, has the same e f f e c t as the gauge transformation fiir ) • 7t(r) - Vy^ir) (j-je ( *> commutes with ) , that i s such that '"r\. P<*. We s h a l l c a l l the 'compensating' gauge transformation operator, and . P<j the usual transformation operator corresponding to G. Before we construct the transformation corresponding \Hot^ operators to the elements of G? such that they commute with w e r eniark that when the spin of the e l e c t r o n i s neglected the Hamiltonian of an e l e c t r o n under the i n f l u e n c e of a magnetic f i e l d andonelectric f i e l d can be regarded as a f u n c t i o n 20 of r and e f f e c t of p e r f o r m i n g i.e. X (Y . Hence the a gauge t r a n s f o r m a t i o n on the Hamiltonian, ) • * (" ) ~ ^ ( (0 e f f e c t of an o p e r a t o r or I f we r ) Y i s the same as (1.3,1) (D- determine the o p e r a t o r f u r t h e r require that O v e r i f y t h a t the o p e r a t o r commute w i t h the and V of (j, (£ | ^) ( X 3 , 1) H A(^)' * ] 5 f a c t o r equal to u n i t y , we . ){%X(*y?\ M group Y V • "J> j constant only readily s a t i s f i e s ( X 3. 1) corresponding Although \Ho(Y 1 f ) AXJ> { ~ J ( ' i t i s c l e a r from to the does not 0 ' ' f \ equations that operator i s u n i q u e l y d e f i n e d by transformation jLxj> j - :jr can {"""£§9^^)] S i n c e the compensating gauge t r a n s f o r m a t i o n ,M|> | ^ We 'usual' t r a n s l a t i o n operator which corresponds to ( I . 2, 3) SLx^> up to a be u n i t a r y the l e t us c o n s i d e r the o p e r a t o r s t r a n s l a t i o n subgroup the r a r b i t r a r i n e s s t h a t remains i s a phase f a c t o r . Now ) which s a t i s f i e s the f o l l o w i n g r e l a t i o n s - (0'' (IT, 3 -1 ) Equations factor. ~ T. 7\L the k i n e e t i c momentum i f we the agree to choose i t s phase can, i n s t e a d of the make c o r r e s p o n d pperator the p r o d u c t t 0 of the , By i n c l u d i n g the compensating gauge t r a n s f o r m a t i o n 21 operators we have thus obtained a set of operators which do commute with However, the group property of the ^ol^'^J. o r i g i n a l set of operators JL-/j> [ " j - f t " ^ j i s now s p o i l e d . For t h i s s e t of modified operators i s no longer closed under operator m u l t i p l i c a t i o n . The product of two operators i s given by: which obviously does not belong to the set. Making use of the i d e n t i t y , which i s a s p e c i a l case of the Baker-Hausdorff formula (see f o r example Weiss and Maradudin (1962)) where A , -B are operators such that [ A, [ O l ] = we can w r i t e Equation (1.3.3) becomes [ B , [A,&1~\ = 0,. ^{ ^ 1** (1.3,4) In obtaining the l a s t expression of use of the f o l l o w i n g = = (3T.3. t ) w e have made relation ^ [ C ^ - ^ ) ( ^ ^ ^ - ^ t J ^ ) + ( ^3-^ «)(^^x-^x^) 3 \ (S * £)-H ? 3 ( l 3. D The l a s t expression i n (JT. 3. 7) gives of course the magnetic f l u x l i n k i n g the t r i a n g l e with the sides irT , ^ and . V, + i} { L E \ ^< <^ 1 ) Let us define a s e t of operators 2 by the f o l l o w i n g equation where X 0 ^ i s a r e a l number s a t i s f y i n g the i n e q u a l i t i e s A. < i . (We use square brackets to denote operators i n contradistinction to the round brackets used to denote space time transformations *) Comparing ttf- '5) 2 c l e a r that we have dropped the f a c t o r from the operator i n ( j . 3 , 6 " ) £,xj> |2TTI,\X} . ( X . 3. & ) . operators satisfying and ( X 3. S) ' i t i s JL*j> j" j ^ : 4? • 7\ (V ) - and introduced the f a c t o r i n s t e a d , to obtain the operator defined i n The purpose of doing that i s to make the s e t of { \_B | 1/ , X ] | 0 X ^ I f o r a l l vectors f" and X form a group without s p o i l i n g the commutativity property of each operator with fto (Y, ~j> ) . Now the product of two elements [ B j % , X,] • [ B\ S > ] 2 [ B | f", , X\ ~] and 24 We i n t r o d u c e number Hence % t h e symbol such t h a t which i s X « K^*^-e.r can be r e w r i t t e n (x,3 defined for a n + d any real ^^ ( < ^ I as |> J"^ , A i l [ E | <?, AO This tion. shows t h e s e t [[{r| The i d e n t i t y v^/X"]^ element is closed under j 0 , 0 ~\ and t h e i n v e r s e is Thus t h e s e t o f {[El^'A-]] f o r m s a g r o u p . and an a r b i t r a r y be d e n o t e d by element we s h a l l the vector introduce such t h a t units ^ JQ. C the range operators denote t h i s of will Aj g r o u p by 4/" henceforth to of its a set.of loops. extremity the l o o p , e q u a l , modulo 1, t o a r e a l 0 £ X < I .,. The m u l t i p l i c a t i o n and Vy the set the o r i g i n and t h e v e c t o r flux: l i n k i n g \/ objects A loop of a space c u r v e j o i n i n g the magnetic is between t h e groups a group of g e o m e t r i c a l is consists AT X] £ the r e l a t i o n Each e l e m e n t l^'A) We s h a l l of [ V~, K~] To e l u c i d a t e U multiplica- - V and measured number law of of X in within is . 25 def ined by PV From ( IT. i . t i ) associative. i t follows that m u l t i p l i c a t i o n i n V> i s The i d e n t i t y element of ]/^ i s (o, o)' and the inverse of 1-X) So Vx i s indeed a is group. I t i s evident that the group group r v to y i s isomorphic to the \/y, I f we Consider the many-to-one mapping from l/x given by ^ > X) ( £ , X, ) ( < t > 0 ~ I because i s mapped to ( B\ 4/I ^ + ) = ( f K ) £ ( * £J~c ^" ^ * ^ ) ) f ) (t\V\) ( £ | <7 ) the product of the Z images of (*0~, , X ) and (jtf^Xz) i f follows that Vx. ^ s homomorphic to V" • The kernel of the homomorphic mapping i s the subgroup r v groups The r e l a t i o n between the r v r v VX and V can be denoted s y m b o l i c a l l y by rv 4r where ^ « - — — • means "isomorphic "homomorphic to". From ( E | V" ) £ of \/ operators • C^'X] (xs,t2) v t o " and ^ means (^If 5.1.2) we see that to an element there corresponds an i n f i n i t e number of o n e f ° each X r i n the range o4\<.\. From the e x p l i c i t form of the operator £ "J- , x j shown i n potential as we see that i t depends on the vector and hence on the p a r t i c u l a r choice of the 26 the gauge. This means that i f we have two d i f f e r e n t choices of the gauge both l i n e a r i n y Aj,^) r e s p e c t i v e l y operators ^jj] by X t O anc * and are w r i t t e n as- then we get two d i f f e r e n t groups of corresponding to replacing } ^(v). in Az(yJ, However the m u l t i p l i c a t i o n law between elements of 4/ ( i 3 i ) (JT,3>io) depends on the magnetic f i e l d H but independent of $C*), I f we denote elements of i]^ £ {r , by - £ l r , X"]^ and those of ^ by i t i s readily v e r i f i e d the the one-to-one correspondence "^Ai—*[^'X],A rv ^\J~ and 4l£ i s an isomorphic mapping between the groups Let H a M H 1/" 2 rv { H and j f ' be two magnetic f i e l d s such that where M i s any integer d i f f e r e n t from 1, and V are two groups of operators { C ^ ^ ^ n j corresponding to the two magnetic f i e l d s . that the group U fC^'^^H'^ We want to show i s homomorphic to 4J Consider the correspondence For every element £to, X i ] ^ ' there e x i s t s an element corresponding to i t i n the sense of an. "onto" mapping from ^ and C^>.1HLS>2]H 3.13}, ("^f*)flfi So is to ^T, Since 5, + ^ / ^ ^ ^ H * 27 to prove we have t o show As ( l ^f x is M r e a l number that t , (_M Also f o r any r e a l + ^ ) j- . the v a l i d i t y ( * t of (\J- i s homomorphic that H if > (H condition elements This of of the form The'implication of form w i l l . IT = o the o p e r a t o r group Up t o operators and inequalities implies X £ 0 , X = V change. 4/ this = (_ I, 3. IS) of Recalling 0 & X <• ' 0, is the H , "Pj, " ' , -^pp o r of ^- - 0, i, we have j u s t field of not M - '• shown only the feature is operators the group which a peculiar we a r e d e a l i n g of they the with. p o i n t we have c o n s t r u c t e d homomorphic 3'/3)^ c o n n o t be an i s o m o r p h i s m . the r e s u l t this (_!". element ( M M f ° . fyVl J ^ w h e r e change b u t t h e s t r u c t u r e also Consequently t h e homomorphic mapping t h a t when we change t h e m a g n e t i c / immediately. properties t h e homomorphic mapping c o n s i s t s shows t h e mapping [ ij , X-l" ~i-, ^ i s mapped t o t h e i d e n t i t y = 0 Hence t h e k e r n e l any '{J • and o n l y i f must s a t i s f y X follows the kernel C^'X"]^ The e l e m e n t [ 0, to numbers for From t h e s e two s i m p l e (,H\3.|4) Consider )JL — ( H x ) | an i n t e g e r a group to the t r a n s l a t i o n of subgroup \/ 28 of O, -We now wish to construct operators corresponding to the elements of K , R e c a l l i n g that any element of f{ be w r i t t e n i n one of the f o l l o w i n g forms and (JI^P( ) \ 0)£' (lfy|0)E ($y\o)B- can (£^R(s)|o)£ t we want f i r s t of a l l to 2 construct operators corresponding to and ( £ w I E' . 0) Since the Hamiltonian remains i n v a r i a n t under space i n v e r s i o n , the operator corresponding to Cxio)£ i s j u s t the usual operator which we denote by Because we do not take t h e ' e l e c t r o n s p i n into consideration the time i n v e r s i o n operator i n coordinate i representation i s simply K ^ the complex conjugation operator. Taking the magnetic f i e l d to be along the z-axis of the coordinate system and the axis of r o t a t i o n (R(2)|o)£ to be along the y - a x i s , the usual operator corresponding to (R(a>|0)E' commute with is je*j>{-t*( ^HO^Y,"^) ~ K . This operator does not. and we must modify by a s u i t a b l e compensating gauge transformation. From equation ( l , 2. ^'j we f i n d that the operator i s the required operator which performs the necessary gauge transformation. Consequently i s made to Mariwalla (1965) has considered,operators f o r which (1.3. IS") i s a s p e c i a l case . 29 correspond to [ f y \ 0 , 0 ^ ' given by a) It i s a simple matter to v e r i f y that [i| D J [ ! | o, ol - [£l o, o] (i.s.17) The s e t of tranformations subgroup of (R<f I 0 ) E ft The usual r o t a t i o n operator corresponding to i s . ' A^f-fcp ( i f t*j>j-c^(*fv - )j j and to each operator we have a corresponding compensating gauge transformation operator 9((.r<^) = ( An forms an i n f i n i t e 4 A21 t*j>| Xi^, (j) ^ Cftti Cf + #12 £U^> £-*di^> - where dLi±£^(j ) 2" The above expression i s obtained by means of operators ^J>{^;X(^f)) ^ f f - ^ f b ^ " ' ^ ^ xhe ,(Oi^Oir) are not closed under m u l t i p l i c a t i o n . Because we are unable to f i n d a r e l a t i o n between and f. + f*) W£<p.) %( >^) the . 7 y procedure of modifying the s e t of operators ^ f i ^ ^ t ^ A ' ^ f f - i ^ ' f i t 0 f o r m t h e g r o u D ^ 30 which depends on the property that linear i n is cannot be c a r r i e d over to the present case. However:we have found a set of operators given by [Rfl6,0] where 4[-tf[xtV%)-)f^-i^ 4HC^J )]i ; = 31 Ax 0- x-r l{ /Xu^ + and l A^, = <2. x 21 (1,3.20) + a 2^ 2 + #2j£ which forms a group that commutes with the Hamiltonian To show that [^1 -o; ^3 commutes with fa(.~i ]T) fa we make use of the f o l l o w i n g three commutation r e l a t i o n s : - - i( f x " 0 Thus =1 /C and hence f A*) (13.22) (1.3.25) 31 By means of the Baker-Hausdorff formula (1962)) the operator Ft*./^, "' <f) where 2 functions of X, ^, £fc,p | p^J can be w r i t t e n as i s an i n f i n i t e s e r i e s whose terms are Cj> only. [ £<j> | o> D~) regard 0 j (Weiss and Maradudin This means that we can s t i l l as a product of two operators: the one on the r i g h t i s j u s t the ordinary operator f o r r o t a t i o n through an angle f> around the z-axis and the f a c t o r on the l e f t i s the compensating gauge transformation operator. The' group of r o t a t i o n s isomorphic i s obviously to the group of operators £ftpI o , We now want to show that the operators [•0.(2) | 0 0^7 and | 0, o J v isomorphic £ j | - 0 , 0] generate a group which i s to R . Because [ J | o/o] ^ [ l I and 0, oT' - we obtain the r e l a t i o n s [ I I o, o~\ [!?(« 1.0,0]'[a: I o, o~] ' - [ f ? | o, o"] ; w The element [_ I | Oj [_&j> I OJ CQ and thus commutes with (.6(2) I 0; °1 j u s t as Ifop I 0) £. [1:3.25) which commutes with and (/?(2.>1 0) E y By a s t r a i g h t forward c a l c u l a t i o n we obtain that C*(»l o,orf^| 0, 0 ] -[fyj 0, o"f tRt«l o.o]' (13.2)) 32 We see that (Mo)E'(fyi0)£ = (^lo)t' which i s the same r e l a t i o n as (1,3.27) Thus from the r e l a t i o n s (XS.17) (1,3.27') , ( X. 3.l<?), ( I . 3, 2.5") ; (_J, 3.2k) and (X, l ^ J ) we conclude that the group R is isomorphic to the group of operators generated by [j.) D> Lfy*) | 0 , and 0~]' denoted by 0 ; 0 • This group of operators w i l l be . ^ When the vector p o t e n t i a l i s expressed i n the A ( ) =• 2 H * , "the elements of the symmetric gauge i . e . group ()5 = Y can be w r i t t e n i n simpler forms. -* -A. Y For from _^ 2 H* R we can e a s i l y see that the only non- zero matrix elements of CL\i &n a\ Z are #12 and. £21. Using Q.M fa* ( l . 3. i t ) ( (lJ,2D) we f i n d These are j u s t the usual operators corresponding to rv and (_Rp|o)£ of the group R, In other words i f the symmetric gauge i s chosen,-no "compensating" gauge transformation operators are required to make the usual 33 operators corresponding to the elements of the group £ commute with $fo(.r\^), groups '[/ and To complete our discussions on the : we wish to f i n d r e l a t i o n s between the elements of these two groups. The f o l l o w i n g equations follows from d i r e c t computations From these equations we can see that the set of a l l ordered p a i r s of elements V {R. such that [/~ and (R belong / r v r v to the groups 4/ and. r e s p e c t i v e l y , forms a group. We r v s h a l l denote this, group by , In e x a c t l y the same way as we showed that semidirect product of R and \/ we can show that r v G is a ^ is i^v a semidirect product of <fl and., t7> that O I t i s easy to see i s homomorphic to C? To summarize, we have i n t h i s s e c t i o n constructed a r- group of operators which commute with the Hamiltonian • homomorphic to the symmetry group of the r v magnetic f i e l d •, The r e l a t i o n s between the groups of operators r-v/ ^, <R>, V and the corresponding groups of space time transformations are l i s t e d below ^ rv g — * o rv , 1/ — * rv v 9 / £ ^ — • <fc 4 * R 4 p & j y 35 II.4 SYMMETRY GROUP OF THE HAMILTONIAN OF AN ELECTRON IN A UNIFORM MAGNETIC FIELD AND A PERIODIC ELECTRIC FIELD We now c o n s i d e r an e l e c t r o n i n t h e presence of a u n i f o r m magnetic f i e l d and a p e r i o d i c e l e c t r i c f i e l d . I t sHamiltonian is where i s t h e p e r i o d i c p o t e n t i a l i n v a r i a n t under a space rv _^ F. group Hence !/(.*) w i l l remain i n v a r i a n t under t h e rv F ® A . t r a n s f o r m a t i o n s of t h e d i r e c t p r o d u c t group (j ry Since i s t h e symmetry group of t h e u n i f o r m magnetic f i e l d t h e rv r v Q i n t e r s e c t i o n of rv F <S> A and i.e. G n (F ® n j gives a l l t h e space time t r a n s f o r m a t i o n s ( e x c e p t pure time d i s p l a c e m e n t s ) which l e a v e t h e magnetic f i e l d potential and t h e e l e c t r i c invariant. ef ri ( F ® A ) One can e a s i l y show t h a t i s a magnetic space.group i n t h e sense of t h e d e f i n i t i o n g i v e n i n Opechowski • * r~v and Guccione (1965). rv but f o r b r e v i t y , we s h a l l s i m p l y w r i t e The p r o o f t h a t follows. M . ( ' I 0) / I r v Ufyl,*') ^ ; and by r e p l a c i n g ( I R(f> f?(a> \ 0) , (£ e f ( 2 ) |p)\7 which i s o b t a i n e d from t h e elements \ O Q n ( r <£> A ) • for • r-v of M T"' \ rv/ i s i n f a c t a magnetic space group i s as ^1 Rq> R(2) I o ) V and rv L e t us c o n s i d e r t h e f o l l o w i n g s e t of elements of ;, 10(3)' Mj More s p e c i f i c a l l y i t i s of t h e type '( B' I ^ P(2> I 0) £ V V by t XRy " R by R{ ) 2 ( &j> R(a) | o) V \ 0) V . It is r e a d i l y v e r i f i e d t h a t t h i s , s e t of elements form a group which rV we s h a l l denote by Cj< , r v The i n t e r s e c t i o n Cr, ' F ' is, again a space group. Combining t h i s and the f a c t that [E\D)B J i s not an element of Q- and hence i t i s not an element of the intersection G n {F ® A ) we see that (j r\ \ F ® n ) i s a magnetic space group. In the l a s t s e c t i o n we have shown that the group of operators of Q y i s homomorphic to Q , the set of elements of under the homomorphism Q > G Since M i s a subgroup which are mapped onto M forms a subgroup of Let us denote t h i s group by *Jfl _ A l l the elements of commute with ^-/g , ^ Wfl To see t h i s we only have to show that elements of ^^TT commute with V ( ? ) . Owing to the f a c t that the compensating gauge transformation operators which occur in. the elements of Q are functions of therefore they commute with every element of Cj Vi?). f but not ^ From our c o n s t r u c t i o n and hence that of Iffl can be w r i t t e n as a product of two operators: one contains a compensating gauge transformation operator and a phase f a c t o r , and the other i s the usual operator corresponding to some transformetion rs of <j. In the case of the elements of |_ ra- "tffl these l a t t e r f a c t o r s are usual operators corresponding to transformations of so Vet) in i t . M . But M i s a subgroup of w i l l not change under any transformation contained As a r e s u l t VCO commutes with operators of and t h i s implies the Hamiltonian T , Iffl commutes with ''.• Let us denote the subgroup of t r a n s l a t i o n s of by F®A M This i s j u s t the l a t t i c e t r a n s l a t i o n subgroup of P 7 L of the space group r . We denote the subgroup of which corresponds to by of If. ' T J" 7TL. J" i s also a subgroup r—v 'IT Making use of the isomorphism between and the rv be denoted by L J I r^> Tx 4 — v T> ( t / X) the form , d e f i n i t i o n the element f to -J i . e . L "T> have i s any l a t t i c e vector. C^, X) of 7> I L s to Lpr Now the elements of where V> we.can f i n d a subgroup of which i s isomorphic * T L ^v V*, group of sets of loops rv a From i t s set of loops each 'of which c o n s i s t s of a space curve j o i n i n g the o r i g i n of the l a t t i c e vector c to i t s extremity and the vector —t and such t h a t the magnetic f l u x l i n k i n g the loop, measured i n units ^ j£ c i s equal, modulo 1 , to X, Among the t o t a l i t y of loops which are contained i n the group TX l e t us s e l e c t L a subset i n the f o l l o w i n g way. From any given l a t t i c e vector £ we pick out those loops which c o n s i s t of curves made up of l a t t i c e vectors j o i n i n g the o r i g i n of and the vector -t , t to i t s extremity In other words the loops we picked out are polygons whose sides are l a t t i c e vectors. to such loops as polygons. define the group V/. In e x a c t l y the same way as we we can define elements are w r i t t e n as We s h a l l r e f e r ( t , X) 7> and as a group whose (t , A) i s a set of polygons constructed out of l a t t i c e vectors one of which i s Besides, the magnetic f l u x l i n k i n g each polygon of the s e t , measured i n u n i t s • ^c/e- i s equal, modulo s a t i s f i e s the i n e q u a l i t i e s o * X < 'I . \ to f X and As we s h a l l see i n d e t a i l l a t e r on, the r e s t r i c t i o n to sets of polygons w i l l introduce a. r e s t r i c t i o n to the p o s s i b l e set of values that -t 7x can have i n the i n t e r v a l ( 0, I ) . of those elements of V > whose t r a n s l a t i o n s are l a t t i c e vectors and whose A loops. consists are defined by polygons instead of any (t , A) To each element an operator C^^xJ of \7" > of denote t h i s group by J" there corresponds The set of operators of J Ll which correspond to elements of with t h i s group. The group T\ i s i t s e l f a group. We and we s h a l l be mainly concerned In order to explain why t h i s group has a s p e c i a l r o l e to play i n our future d i s c u s s i o n we must r e c a l l that the p h y s i c a l system we want to study i s an electron i n the presence of uniform magnetic f i e l d and a p e r i o d i c e l e c t r i c field. Since the magnetic f i e l d and the p e r i o d i c e l e c t r i c f i e l d are i n v a r i a n t under the group of l a t t i c e t r a n s l a t i o n s we want to construct a group of operators corresponding to l a t t i c e t n s l a t i o n s such that they commute with r a The product of compensating gauge transformation operator ^ ~ /Kl^*)' operator -^J-fg Y \ and the ordinary t r a n s l a t i o n JU/J> --ji. $' ^ \ (, ^ ) ( i n that order) commutes with but, as shown e a r l i e r (see Jtxj> { ^ A ( ^ ) ' of a l l operators of the form does not form a group. (IT. ) the set *\ f ] To modify t h i s set of operators so that they do form a group we repeat the procedure used to obtain operator of . However we are now only concerned with l a t t i c e t r a n s l a t i o n s and we do not need a l l the values of X w i t h i n the range ^y(?[--^ ( £ • f - | to make the operators y ] *"\ minimal set of values of A l 2 7 r K ^ \ form a group. Within the range 0 The A <i 39 i s j u s t the set a r i s i n g from magnetic f l u x e s l i n k i n g a l l the This point w i l l be described i n Section JT,S. p o s s i b l e polygons. I t i s not d i f f i c u l t to see the f o l l o w i n g r e l a t i o n s are true. J M e- > CM T rv J~ And the group The r e s t r i c t i o n on A, and va)e denote by \J , i s i n f a c t a subgroup of fffl does not a f f e c t operators of the subgroup of ffl ?TL such that L & rv consists of those operators i n M s~**r g •'G X which under the homomorphism r-^ i s mapped onto M are determined and furthermore the values of by polygons. V From the f a c t that ~Yfl j 4 J fsj j -j- ^ M / T • —> T H E O R V D E R O F can at most be 24. i t follows T H E 9 r o u P For an axis of symmetry of a c r y s t a l l a t t i c e can at most be s i x f o l d . I f the l a t t i c e has Inversion symmetry, a hexagonal axis p a r a l l e l to the d i r e c t i o n of the uniform magnetic f i e l d and a two-fold axis perpendicular to the magnetic f i e l d then the order of i s equal to 24. or l e s s . Thus the order of Tfl j J M / T must also be 24 This conclusion also f o l l o w s from the f a c t that the l a r g e s t admissiable magnetic point group i n the sense of Opechowski and Guccione(1965) i s 24. In t h i s s e c t i o n we have obtained a group of operators commuting with the Hamiltonian 7^1 ^ B I ? > / f ) rv of Q which commutes with the Hamiltonian making the set of operators ^{'i I as a subgroup /4 (Y < ^> ) « 0 ~ t ^(^j In form the group HJ —> factors JO(j> O ^ X < {" ' " A1 [f't where X 2n '• translations J*\{-\ V ,'we have found i t necessary to introduce i s any r e a l number such that S i m i l a r l y , fox the group of l a t t i c e T we can make the set of operators ) ^ ) ] " I by introducing f a c t o r s J form a group j^j? { 2TT-u X j . need a smaller set of values f o r X T But i n t h i s case we w i t h i n the range Also we have obtained r e l a t i o n s s i m i l a r to ! those i n (I". 3.32) namely! ?ft * M , J * T II.5 STRUCTURE OF THE GROUP a, Let that - a . $3 > z Q) ' of the l a t t i c e * J be any three l a t t i c e vectors such t (&2X<%) be the: volume of a p r i m i t i v e u n i t c e l l H The uniform magnetic f i e l d can be expressed as: where ^, ^ t 7^ are three dimensionless r e a l numbers. We want to f i n d the values of Aof T\ f o r the elements where each element i s a set of polygons. ( t , A.) Consider the magnetic f l u x passing through an a r b i t r a r y t r i a n g l e formed by the l a t t i c e vectors ' £ / y = -ft^a, + ^ fH a 2 t = ^1 < + ^ 2 ^ and + ^1/% a a -(T^+tr , + ), I t s value i s • (fx?*; The magnetic f l u x passing through any polygon i s the a l g e b r a i c sum of the f l u x e s passing through the t r i a n g l e s into which the polygon can be decomposed. This i s therefore a q u a n t i t y of the form 2 l e i " (• where. U | , U 2 and + U 4 a ) are i n t e g e r s . Consequently a l l p o s s i b l e values of .X i s given by the set of 42 - A- (*3 -' I* 'Us*) The set i s e i t h e r f i n i t e or i n f i n i t e by countable. In the general l i n e a r gauge the element is given.by and A ° ( f ^ i+ ^ a If instead of w r i t i n g we w r i t e X AL ) it) - r ~ 2H * and the group ^|Of for a l l U t l U t j 2 ; and /^(O i n the symmetric gauge becomes i d e n t i c a l with the group f i r s t introduced by Zak (1964 I) and c a l l e d by him the "magnetic t r a n s l a t i o n group". The term."magnetic t r a n s l a t i o n group" has been used byOpechowski and Guccione(1965) and others to mean the subgroup of a l l the primed and unprimed t r a n s l a t i o n s of any magnetic space groupi Since i n t h i s t h e s i s the t r a n s l a t i o n subgroup of the magnetic space group M 3 X ( ^ ) i n the general l i n e a r gauge • (the symmetric gauge) then Y t H x t So i f we choose to w r i t e + consists of unprimed t r a n s l a t i o n s (ordinary t r a n s l a t i o n s ) only,we s h a l l be able to avoid using 43 the term "magnetic t r a n s l a t i o n group" f o r groups other than We have stressed before that the s t r u c t u r e of the group depends on the magnetic f i e l d . obvious f o r the group , i t , A,] • I t From the m u l t i p l i c a t i o n r u l e A,] it ('A,+ 0" we note that group This feature i s even more A 2 t£f c H^xt )) 2 f J (1.5.5) i s i n general non-Abelian but i f we choose the magnetic f i e l d equal to ^^p^ 0.3 becomes an Abelian group. f o r example, then Because of t h i s p e c u l i a r feature we are i n t e r e s t e d i n conditions under which two groups 3" 1 and 3^2 corresponding to two d i f f e r e n t given magnetic f i e l d s may have s i m i l a r s t r u c t u r e s , more p r e c i s e l y , when one of the two groups i s homomorphic to another and.when they are isomorphic. In S e c t i o n H. 3> we have proved that when where M Hz ~ M Hi i s an integer 4 \ then the corresponding group i s homomorphic to •if we replace \J-^ { f~ and by .—' and . t h i s case we can make a stronger statement. ^ mapping between [ r e \7i A ] Moreover i n The homomorphic. - 0% and H r w This statement w i l l c e r t a i n l y be true r** U", 'l - j • | i s the same as [t, ( J . 3. <3) ( M A ) ]H f 2 and also the k e r n e l of the homomorphism are elements which can be w r i t t e n as t 0 , A = ( where ft = O I, 2,* • » |M~f j y In contrast to the s i t u a t i o n when we discussed the r e l a t i o n 44 rv '" ' •• rv N and U between the groups a now the values of A are r e s t r i c t e d to the set ( X 5 - 2 ) (.X 5". 0, { \ > i s given by C, i, • • / "'M and A- - ( + Tl* M J"/ j «X are isomorphic. X = 0 | only then the two groups Otherwise they w i l l be homomorphic. , r^v An example where u-^Uu ^ - i n t e g e r s ^7^. + consists of a s i n g l e element and Hi Hence i f the i n t e r s e c t i o n of the two sets of { \ < A = (tf/M)f , numbers when f o r J" 3~i and O2 are isomorphic i s when j ^2 ^ 3 are a l l i r r a t i o n a l numbers. In t h i s way, we f i n d c e r t a i n s u f f i c i e n t , but not necessary, conditions f o r two groups vTi• 3% t In f a c t , consider the case when Hi where M/ , M2 , M.3 H2 are such that are even i n t e g e r s , i t i s easy to see the r^y corresponding groups and to be isomorphic. r-— Cf\ and C^L a r e isomorphic and i s not a s p e c i a l case of that discussed before. To prove ( X 5. 0 i n the form values of X" J i i s isomorphic to J ) of then ( X 5 . 2 ) J~i . 2 we assume gives a l l the p o s s i b l e The p o s s i b l e values of are given by: ( % l^i * Mi) + (% M ) ^ ^ l ^ + M j ) ) ^ . + Hi i s 2 * of J^. The set of values of A i s thus i d e n t i c a l with the set of A Moreover because = ^ H, ' ( t! x T ) + • inte^ 2 CT the groups and ( Oz are isomorphic. Let us summarise the above discussions by considering the two s u f f i c i e n t conditions of isomorphism f o r two magnetic —* Hi t r a n s l a t i o n s groups f o r the s p e c i a l case when both H a r 2 as e p a r a l l e l to the same l a t t i c e vector. Q3 By choosing the shortest l a t t i c e vector along the magnetic f i e l d s .-f H\ ~ we may w r i t e If -a A —-s, "V °-2, . r i M = M H| where M 2 i s an integer> we can see. from the f i r s t s u f f i c i e n t c o n d i t i o n of isomorphism and C7a are isomorphic f o r a l l M i r r a t i o n a l number. where ox, N such-that M CFi , and If ^ the groups when ^ i s an i s r a t i o n a l , and equal to are r e l a t i v e l y prime i n t e g e r s , then M and N (say) must be are also r e l a t i v e l y prime f o r the groups "to be isomorphic. From the second s u f f i c i e n t c o n d i t i o n of isomorphism i t i s evident, i f Hz = H + J^JQ ^ ^3 where M \T, and i n t e g e r , that the corresponding groups isomorphic whatever the value of ^ i n ^1 . i s any even ^J z a r e 46 We w i l l use t h e s e r e s u l t s when we c o n s i d e r t h e d i s p e r s i o n law i n Chapter IV. F o r t h e purpose of c o n s t r u c t i n g i r r e d u c i b l e representations !J of t h e magnetic t r a n s l a t i o n group we s h a l l have t o know i t s A b e l i a n subgroup, e s p e c i a l l y i t s maximal A b e l i a n subgroups. An A b e l i a n subgroup of (7 is r e f e r r e d t o as maximal i f i t i s n o t a p r o p e r subgroup of a n o t h e r A b e l i a n subgroup of sj • A n e c e s s a r y c o n d i t i o n f o r two elements [t X J / i / y of J" [-fe-,>X] and t o belong, t o t h e same A b e l i a n subgroup i s t h a t they commute. A simple c a l c u l a t i o n show t h a t this condition implies that Writing £ » Oa, ^ and u s i n g e x p r e s s i o n 71^+ (TT.S.l) for H equation (1,6^7) becomes rv. S i n c e t h e s t r u c t u r e of depends on t h e o r i e n t a t i o n as w e l l as t h e magnitude of t h e magnetic f i e l d we s h a l l now c o n s i d e r t h e f o l l o w i n g cases (a) separately: t h e magnetic f i e l d i s p a r a l l e l t o some l a t t i c e vector (b) t h e magnetic f i e l d i s p a r a l l e l t o some l a t t i c e plane but not along any l a t t i c e vector and (c) the magnetic f i e l d i s not p a r a l l e l to any lattice plane (This l i s t exhausts a l l the p o s s i b i l i t i e s ) . Case (a) Since the magnetic f i e l d i s along a l a t t i c e vector, we can choose the shortest l a t t i c e vector along the d i r e c t i o n of H as a basic p r i m i t i v e vector and denote i t by us choose two other vectors are non-coplanar &i, 0-2 such that Q3 . 0-\, di and Let and ^3 the volume of a primitive unit c e l l . Here we s t i l l have to d i s t i n g u i s h two d i f f e r e n t s i t u a t i o n s namely: ti (a 1) ~^Q_ ~N = relatively and \j (aii) — where 4 n . / N -3 are prime integers ~=t~ ^ a where ^ 3 i s an i r r a t i o n a l number . Case (a i ) From ( X 5. 2) we can see that the set of p o s s i b l e values of X U$ =• 0 , ± f i s given i n t h i s case by t ±2, 2 . and on. by the symbol i t follows the set of values of X and £ Ov, <£, •+ s i m p l i f i e s to where L e t u s denote the. highest common f a c t o r , of the integers Condition X = (^ ^^ (1.5.7') f° r , X'] (2,^) then i s given by "two elements [^0,+^+^ to commute can now be \] 48 [ n,%2 ~ " f ^z^i) i>te^er - (1,5,7") From t h i s c o n d i t i o n i t i s r e a d i l y seen that i f N integers such that [ and N , are /\j = H'hl' then the s e t of elements with N'd, + n t4% 2 forms an A b e l i a n subgroup. Abelian subgroup of n ^ a> ± f, ± » 0, 2 , •• • Moreover t h i s i s a maximal \J". Suppose there i s an Abelian £ \jhM 'ct, + OuAi'^ •+ subgroup of (J- .which contains \~\ j / as a proper subgroup then i t must contain an element £ •+' K/' or a m u l t i p l e of assume that [Va, ^if^s , X"] nz&2+- i s not a m u l t i p l e of elements of But OYV and 3 N/', nJ( N" = ^v, - Q , v KI cannot be a m u l t i p l e of 'Ha'cfc + ^ 3 0~\ , From =. Cnreg&r _ jp %," / ( ^ ,X"] /N a commute with -^•N'V = ~ ^2 ^4 V - C^e^'er /) =0 otherwise the element w i l l be an element of the Abelian. • 4* £ {\jK,H%+*^i% 'V If vi'/ V requirement t h a t , the element , X l J . Now from the [_ "h[ Clz. + 'HJ'&j , \ V must we obtain the c o n d i t i o n = Tit = Crvfeje-r By e x a c t l y the same Cn/fceger. argument we have j u s t used to show ^ \_H"&2, Hence the c o n d i t i o n can only be s a t i s f i e d when group ~ i t must, i n 3 N are by assumption r e l a t i v e l y prime and i s not a m u l t i p l e of Let us Since £ [ Ov, Is/'o! + ^ N " ^ + 7) tf , X~j J i t implies that i s not commute with a l l the m u s t p a r t i c u l a r , commute with the element (1.5.7") N , 'V /V ' . 'ft/ i s not a m u l t i p l e of ^j'fij + ^ 3 5j , X'J 4 such that e i t h e r M*\ - 0 i s true only when we can see that ^1 = 0 . However the r e s u l t n£ 71/= £ w/'^f, + ^'^ s ^l'&z J A" 2 + 2 does not belong to the , X ~] j £ [ 'H.N'a, + ^ N"5^ + Abelian group j [' yi h'a, shows that assumption that 0 contradicts the t + H N' ^+ / / XJ] y 2 is , This maximal a J", Abelian subgroup of Case (a i i ) For X H = • i s given by Since ^ unless X : <2s the set of a l l p o s s i b l e values of \ = U (x^)f ~ ( is irrational X^i = ^ 3 . with 3 = = 0, ± 1, ±-2, • • • ((T ~ ^')l)f ^ 0 Thus the set of a l l p o s s i b l e values of i s countably i n f i n i t e . For the present case the condition f o r the elements [71, a, + n^d^ + ^Xj and [ Ov,' + + ,V to J commute i s As ^ i s i r r a t i o n a l t h i s c o n d i t i o n can only be s a t i s f i e d when yi\^2 - subgroups of and [[ = 0 , Cf I t follows at once the maximal Abelian are the f o l l o w i n g : ,X]j where ££ -7^<£, + ^s 3 a Ol, l a ^ 7 i = ; 3 s , | 0,^*2, Case (b) When the magnetic f i e l d i s p a r a l l e l to some l a t t i c e plane but i s not p a r a l l e l to any l a t t i c e vector we may assume without loss, of g e n e r a l i t y that the plane i s spanned by The magnetic f i e l d can now be w r i t t e n as (X 2 J #3 50 H where j ^ - ^2 mi and ^ ^ * are non-zero r e a l numbers. The r a t i o must be an i r r a t i o n a l q u a n t i t y , otherwise the 3 magnetic f i e l d would be d i r e c t e d along a l a t t i c e vector. There are again only two d i f f e r e n t p o s s i b i l i t i e s to be considered: (b i ) one of the q u a n t i t i e s ^ 2 , ^3 i s rational and the other i r r a t i o n a l ; and ( b i i ) both of them i r r a t i o n a l and such that the ratio: Case (b i ) H = j ~ J ^ (^2^2 + -jj #3 ) Let us assume that that ^2 i s an i r r a t i o n a l number and prime i n t e g e r s . i s given by + i f "T?)f (5 where The c o n d i t i o n f o r the elements £ fc/fa! •+ H2<?2. / ^2 = Since and ^ N are r e l a t i v e l y The set of a l l p o s s i b l e values This i s r e a d i l y seen to be a countably and ^, + ^ 3 «3 ( ^3 ^ - ^1 , A'J ^3 ) + U _ U a such X can take =• 0 , ± I, * 2, - > - 2 i n f i n i t e set. £ oa, a, +• oa Zt + on^Oj , A J 2 2 to commute i s TT (u, W - ) C n-tecj e r i s i r r a t i o n a l t h i s condition implies ^ (n n' { 2 - 7v V, 2 ) = Cntejer, I t i s not d i f f i c u l t to see the maximal Abelian subgroup are then the f o l l o w i n g : { OiN'SU 711 , n where , ^ " $ 2 x > n 3 x ] J vi a [ [ 2 +^a 2 x]) 3t = 0 , ± 1 , * 2 , • * - • a.r\d N ' N " = N . Case . ( b i i ) H can be w r i t t e n as W = [ej^Q For t h i s case where both ^ and ^ are i r r a t i o n a l . X = p o s s i b l e values of i s again 3 LL XJ u =• 3 ^^3) Ihe. set of a l l Hi ^ + ^ <^ ^ count ably i n f i n i t e because + 0, * From the 1> ±-2, commutativity c o n d i t i o n ) + ^ ( n, K - 7v >v;) » \x ( n n,' - ->i, 3 i>fe^ e r 4 we can deduce that the maximal Abelian subgroups are : { [yiza2-i- msa3 , XIJ [ [<m£, and , X l j . Case (c) In t h i s l a s t case the magnetic f i e l d i s not p a r a l l e l to any l a t t i c e plane. vectors where 0L\ ) ^ ' \ f three r a t i o s irrational. 1 y Hence f o r any choice of the basic p r i m i t i v e , <%2 > ^ "jfe a r %> I *f* . e n o n - z e r o r e a l numbers such that the °2?/^ 3 a n d / a r e a 1 1 We may now have only one of the f o l l o w i n g two cases (c i ) two of the q u a n t i t i e s / ^ i • > ^3 one r a t i o n a l • ( c i i ) a l l three q u a n t i t i e s are i r r a t i o n a l are i r r a t i o n a l 52 Case (c i ) ' _ i In t h i s case l e t H where ^i,^ ^ j^jx be given by are i r r a t i o n a l q u a n t i t i e s such that the r a t i o N i s also i r r a t i o n a l , and prime i n t e g e r s . can take i . e. U, , (t , U 2 3 are again relatively I t i s easy to see that the set of values £3 ^ = + 0, X 7 ^ "^)f 2 \, i Z , - condition for' the elements £ ^ d, •+ ^^Q-i* ^3 ^ where ' - N' N" = o r i s countably i n f i n i t e . [ 'fri + VhAi + oa a 3 3 ,X ] The and / ^ J to commute i s now given by. Hence the maximal Abelian subgroup of {'0,A/'£ + ^ N V , A, , X]J J are A ] ) and N • Case ( c i i ) Since the magnetic f i e l d i s where , ^2 , ^ 3 %'h'l 1 of X are a l l i r r a t i o n a l and so are the r a t i o s X /^ 1 2, 3nt ^ / i s again countably i n f i n i t e . the elements [ M< to commute i . e . + + y X] the set of a l l p o s s i b l e values From the c o n d i t i o n f o r and [W + + *k<&, J 53 ^ + it - 7 i ox.'J j 2 = integer f o l l o w s t h a t t h e maximal A b e l i a n groups o f {[> £,A-7y, and t We s u m m a r i s e J~ are ( C ^ ^ , X ] J . t h e above r e s u l t s f o r t h e d i f f e r e n t o r i e n t a t i o n and m a g n i t u d e s of the magnetic f i e l d i n Table 1 . We n o t e t h a t a l l t h e A b e l i a n s u b g r o u p s we above a r e n o r m a l . A l s o we s e e t h a t o n l y have t h e maximal A b e l i a n subgroups group, a f i n i t e index. obtained i n t h e c a s e (a i ) of the magnetic translation FT M A X I M A L ABPLIAN SUBGROUPS TABLE 1 MAXIMAL ABELIAN SUBGROUPS OF THE MAGNETIC TRANSLATION GROUP A 55 III.l. CLIFFORD'S METHOD OF CONSTRUCTING IRREDUCIBLE REPRESENTATIONS OF A GROUP In t h i s chapter we construct what we s h a l l c a l l the 'physical irreducible 7 representations of CT f o r a l l d i f f e r e n t cases we discussed" i n S e c t i o n 11,5, To do t h i s we s h a l l make use of a method due to C l i f f o r d (1937). C l i f f o r d ' s method make i t p o s s i b l e , i n p r i n c i p l e , to construct a l l the f i n i t e dimensional I r r e d u c i b l e representations of an a r b i t r a r y f i n i t e ! ~ • • or i n f i n i t e group i f . a l l the f i n i t e i r r e d u c i b l e represen•' ~> t a t i o n s of .some normal subgroup f-j of i t are known. An o u t l i n e of C l i f f o r d ' s method can be found i n Lomont's book (Lomont (1959)) we s h a l l use e s s e n t i a l l y the same terminology as Lomont. following In p a r t i c u l a r we s h a l l need the definitions..* "Conjugate representations": and A 2 = {b (H)j L2; Two representations ]> (H) (where i s the matrix of C0 belonging to the group element H Ac ) of a normal subgroup H of a group Cj are conjugate to each other r e l a t i v e to Q i f there e x i s t s an element Q for a l l H i n /H , ^ Sub "Orbit": such that ~X> (H) -~b[Q H<=() of Q An o r b i t of a normal group — j-j of < 3 ? i s a maximal set of i n e q u i v a l e n t i r r e d u c i b l e representations of H which are mutually conjugate r e l a t i v e to £7 , " L i t t l e group": Let A = | ^ H ) j b t i o n of the normal subgroup [_ and of Q H e a n irreducible | H of a group < 3 , i s c a l l e d the l i t t l e group of A. representaThe subgroup r e l a t i v e to i f the representations conjugate to A r e l a t i v e to L 56 are a l l equivalent. rv "Subduced r e p r e s e n t a t i o n " : Let Q _ <~ G subgroup of matrices i n P r-v be a group and /{ and V . which are images of A P ft i s r e f e r r e d to as the representation of of n, P subduced by the be a group, H an i r r e d u c i b l e representation of l i t t l e group of A of G , "Allowable r e p r e s e n t a t i o n " : Let G> group, A Those i n v i r t u e of homomorphic mapping form a representation representation P G , a representation of be a ' rv r e l a t i v e to G> H. and H a normal L "the An i r r e d u c i b l e representation "ft i s said to be an allowable representation • rv of j_ A • ~' r e l a t i v e to j-j and ^ if subduces a m u l t i p l e of H. on "Induction": Given a representation of a subgroup of <3 . /\ one can always construct a representation of G> by the method c a l l e d " i n d u c t i o n ^ We are not going to give a general d e s c r i p t i o n of t h i s method, which can be found i n Lomont (1959). We s h a l l e x p l a i n the method i n connection with the s p e c i a l cases we consider l a t e r . C l i f f o r d proved the v a l i d i t y of h i s method f o r the case i n which: ( l ) the i r r e d u c i b l e representations of the normal rv subgroup ^ from which one constructs the i r r e d u c i b l e representations of the whole group (3 are f i n i t e dimensional and.(2) the l i t t l e groups of the i r r e d u c i b l e rv of H rv r e l a t i v e to <3 r-v and H representations r v have f i n i t e index in. Gj t H i s ' r e s u l t s can be then summarised i n the f o l l o w i n g theorem. (Our statement i s almost i d e n t i c a l with that of Lomont (1959 page 231).) Fundamental Theorem Let Gr be any group and H a normal subgroup whose i r r e d u c i b l e representations are a l l f i n i t e dimensional. The . l i t t l e group of any i r r e d u c i b l e representation of H r e l a t i v e to Gr and H has a f i n i t e index i n <3\ Let A i r r e d u c i b l e representation of . H r e l a t i v e to Gr and H be L , be any and the l i t t l e group of A I f y i s an allowable representation of j__ then (1) . the representation of <3 induced by y (2) i s irreducible i f only one l i t t l e group, per o r b i t i s used to induce the i r r e d u c i b l e representations of G\ then each i r r e d u c i b l e representation of Gr i s found once and only once. For the c o n s t r u c t i o n of the p h y s i c a l irreducible representations, a term which we s h a l l define i n Section of G III.2 we always s t a r t from the i r r e d u c i b l e representations of a normal A b e l i a n subgroup. t i o n s of an Abelian Since the i r r e d u c i b l e representa- group are.one dimensional, the f i r s t assumption i n the above i s s a t i s f i e d . However, except i n the case when the magnetic f i e l d i s r a t i o n a l , the second assumption i s not v a l i d . In other words the Abelian subgroups are, apart from t h i s case, a l l of countably ' i n f i n i t e index i n the d i s c r e t e i n f i n i t e group G. For these cases we have adapted some of C l i f f o r d ' s r e s u l t s (Appendix I) so that we can s t i l l use t h i s method to construct''physical' i r r e d u c i b l e representations of G We may mention that i n the case of space groups the l i t t l e groups are commonly c a l l e d by p h y s c i s t s the (space) groups of £ -vector and that the method of constructing the i r r e d u c i b l e representations of space groups described by Koster (1957) i s a s p e c i a l case of C l i f f o r d ' s method. 59 III.2 IRREDUCIBLE REPRESENTATIONS OF .. J <v T The i r r e d u c i b l e representations of depend on the • o r i e n t a t i o n of the magnetic f i e l d as w e l l as i t s magnitude. This i s because the s t r u c t u r e of the group j" i t s e l f depends on the o r i e n t a t i o n and magnitude of the magnetic f i e l d . Consequently we have to f i n d the i r r e d u c i b l e representations -0 of f o r the d i f f e r e n t cases enumerated i n Section X S separately. H = —— rr Case ( a l ) •— where jeia N u ^, N are r e l a t i v e l y prime integers From Table 1 we see that the group where , 'Hj. and ^3 X = ( T ' N " )f with 0 , ± 1, * 2, - • • are equal to a - O, ±-1, £ L^i^i + ^K^c + Vi3,X]j 2, — • • and i s one of the rv maximal Abelian subgroups of rv by J CJ. Let us denote t h i s group ^r ^ and the coset decomposition of C/ ^0 according to J" is : N-1 The i r r e d u c i b l e - representations of J are easy to obtain. For apart from the f a c t that i t i s Abelian i t can be regarded r~ . as a d i r e c t product of four simpler Abelian subgroups of • 'J i.e.. 60 The bounded i r r e d u c i b l e representations of the Abelian [t^ groups a ,ol) t t {lp^zUa it bl) and ( j / ^ ^ o l ) are a l l given by: A where < k i ) ['^?,0l K*, , k*2, 2? 4 Ki, Ka. , < I. group of order » ^ | ( - 2 T A ^ ^ ) (JT. 2.5) are a l l r e a l numbers such that The group - ^ ^[o,Xjj Its is a finite cyclic inequivalent irreducible representations are The l a b e l 5 0, I, 2, •> ' • i s any of the numbers Since any element -[n. ^ + ^Ma + ^ a y j 2 ~ '• ) sy o f j» can be w r i t t e n as a product of four elements each belonging to o] j , '{i*zH% ol\ . [C^K.ol] / and X]] respectively i . e . - I ^ O ] [^N« , 2 0 i r % C the i r r e d u c i b l e representations of 0] J L°<0 + 2 ^ ' ^ ] (*2.5) are j u s t the d i r e c t 61 of i r r e d u c i b l e representations of Each i r r e d u c i b l e representa- t i o n of -J Let 2TTc s( i ^ | { - 2 m ( ^ , ' » i + Kz7i + = a £, , 1^ , i n the usual way ^ and b ± 3 j =• y a Sl s£ = . * : We 2TZ t t = ^ —^ * * Sl t 3 defind a vector [kX be w r i t t e n as = ' a —» ** > S2 a by + 2,'^) of CT can follows: be any s o l u t i o n of the Schrodinger equation 2irCX we have the group the form [ D.'xl { C o , X l J t^-irc X ) . = e ™ ^ , f o r elements of are j u s t numerical phase f a c t o r s of Hence i n the space of state vectors spanned by a complete set of s o l u t i o n s of s ; (JL.2,fc) ) j i a then the i r r e d u c i b l e representation Let 01, ^ and be the p r i m i t i v e r e c i p r o c a l l a t t i c e vectors , defined Ki, Ka, i s l a b e l l e d by four q u a n t i t i e s (iff. 2.8-) the 62 vectors transform according to'the i r r e d u c i b l e representations of { iV,Xl\ with l a b e l .5 - I and no other. 7 r e f e r r e d to i r r e d u c i b l e representations of .. S = I as " p h y s i c a l representations". a p h y s i c a l representation of 7 Zak ( \<fU Z) and 7 with In other words, or any of its.subgroups i s a representation which can be generated by some subspace of the space of s o l u t i o n s of (IT.2.S) , From now on we s h a l l be concerned only with these representations. We s h a l l now show that the l i t t l e group of <SK*,«>'> {7°) 7 r e l a t i v e to A 7° and i s 7° To do t h i s we f i r s t f i n d those representations ..of conjugate to where ^ ( K " * * , K»,0 j J y } (JC. 2.k) J Since ^ - 0, i > 2, - • • , N - I . ' Comparing equations relation 1 itself. and therefore (lE,2,lo) we obtain the 63 = But J ° i^.^Z.^ ^ * f c ^ A [ ,A l + ^ ^ + ^ 3 ^ 5 (nr.2,11) i s an a r b i t r a r y element of thus By assumption satisfies ^ N Y OC are r e l a t i v e l y prime, and since ^ < /\j } /N ^ cannot be an integer. Therefore except ^ =. o , By d e f i n i t i o n the l i t t l e group r e l a t i v e to /jl^^'^'^n with elements. (7 With the a i d of ^ J i s a subgroup of- 7 such that (HT. 2. 12) group r e l a t i v e to is and. ' we can conclude that the l i t t l e [J'] f J a n d itself. The f a c t that \7~° i s i t s e l f the l i t t l e group . J° ' 64 A r e l a t i v e to I ^ J , .J > and T makes the T c o n s t r u c t i o n of i r r e d u c i b l e representations of p a r t i c u l a r l y simple. . For to f i n d the i r r e d u c i b l e representations of J" by using C l i f f o r d ' s method one has to obtain the allowable representations of the l i t t l e group. In the present case a l l the i r r e d u c i b l e representations of the l i t t l e group are allowable as a t r i v i a l consequence of the d e f i n i t i o n . From each'physical." allowable representation of the l i t t l e group we can obtain a p h y s i c a l i r r e d u c i b l e t i o n of 3~ by " i n d u c t i o n " . the s e t of elements i s such that . .Q" , !J If t L$&2> { °1 > &] \ i s any element of -0,1,2,- with ^ = we note that 0.1,2, - * • , NM give a l l the d i s t i n c t cosets of CT l e t us define a f u n c t i o n L f >j ( H . 2. i) From representa- .IN/-4 ) CT and ^7" any element of ^^'''^ ^ ^~ ^ ) 1 such that Therefore ( T ( X "C) ^ 'f J ( r ) ' ^ (K«,Ka, Ks,0 Let ^ is a e re P { if ) J » [ ov,*! 4 res N x M matrix with elements ® ~ation of n+ ;J* induced by i s defined by B + ^ + ^ i ^ i , >Q where 0 £ ^ <N . 65 Because the product [ - ^ , o] Oi«! + K N * - ^ ^ * i s an element of j 1 (J~ ° only 'if ' i f^.tj' ^ M = • yU.4j > N where ,• £ = By means of From f 1 H (;X.2,|4) p + . ( IT , 2. £> ) H ^ "jf ^ ~ and j i^il^i^e) (^^N) or ^ + if i t follows that >^ and 0< ft" < N we obtain / . ' aire [ x + + ^= , 0] , X] [^ ' -i ^ ' U ^ y ) j 66 Hence from each p h y s i c a l i r r e d u c i b l e representation ^ («ikz, i ) Kj, q representation from £ g- w e 2>^' (-HI. 2.< °). obtain a p h y s i c a l i r r e d u c i b l e of J"-. I t i s r e a d i l y seen that a p h y s i c a l ' i r r e d u c i b l e representation 1 cannot be conjugate to a representation which i s not a p h y s i c a l " representation. two. .representations •CT K-/ + ^ ( ^jsT ) | X and )^ tfjji- are .* ^ , (0 4 j <N) 0, ; K 4 ( i K ' ' , I) ?£)f, K 3> 0 • -| -_ /| = i f "V 0 r hence two representations k| / | Wc,'- K,. | = K ^ 0 / (X.2.l*; Relation A ^' ^' and are i n e q u i v a l e n t except for j = o ,• A ' LK K i / ^ and ' A ''' '^ 1K belong, to the same b r b i t i f . and only i f and of the representation • A ^ " ' ^ *' ') } "^T, + Q" A ' f o r some' i n t e g e r shows that -.the representations ^ ( the conditions f o r the can take any of the set of values i s conjugate to "ff, A to be conjugate r e l a t i v e to K, Since ( X 2. I 0) Also from 0 .Ka' = K ,. K$ = K a £> ^ l i 3 other words we r I n .i i/ can c h a r a c t e r i s e those o r b i t s formed by the p h y s i c a l .representations of ^ o• CT'- by the', t r i p l e t . ( K * ,*') w i t h K K, 67 s a t i s f y i n g the conditions 0 4 K, < ^ 0 4 f o r every member of the o r b i t can be denoted by f o r some A of the s e t Case ( a i i ) : *z * \ , < \- , 0 A^ + ^' ' ^' Kl 1y (o, l , 2 ---- [sf-i), / H - 7-77T ^ FL 3 ) where ^ I i s an i r r a t i o n a l t number In s p i t e of the f a c t that the magnetic t r a n s l a t i o n group i s d i f f e r e n t from the one we j u s t considered because, the rV magnetic f i e l d s are d i f f e r e n t we s t i l l use -J" The group of. J , i s a maximal Abelian subgroup We denote t h i s group by 7° i s also an i n v a r i a n t subgroup of case ( a i ) (J C7 , j c a n ] ^- ^ [ t ^ ^ j O l \ d observe, that . In contrast to the (_C^^> 0 1 j , f [ ^ 3 ^ > 3 ^ ] i - e- • In t h i s case the group *>><j>!±. 2-rrC n be regarded as the d i r e c t product of the three simpler Abelian groups: ' { L o/ a i s an i n v a r i a n t subgroup of i n f i n i t e index and The group and to denote i t . ' . { i s a group generated by The i r r e d u c i b l e representations of £(jhi#»,o'lj and { [ 0,X l j are given by 68 A , S ) [o,>^| S = £*J { 2TM $ X ] (JT 2.21) i s any r e a l number The i r r e d u c i b l e representations of • C7 are the d i r e c t products of the i r r e d u c i b l e representations of the three groups and hence + 4 = = ® A ^ t ^ t , , Xl A ( i y j ) { -2TT<C ( K i ' K , + K j [ ^ , Xl ® A ) i*J {aT^X^ Introducing a two dimensional vector ^ = we can r e w r i t e Q i r ( K, ( I . 2,22) A ^ R ' ° + O x ] l s ) K "^ ) 3 tUI.2.22) defined by ( 0^ K, , K 3 < I) (JT. 2 . 2 3 ) as [ M ^ ^ , X] For the same reason as i n the l a s t case we s h a l l "consider only those i r r e d u c i b l e representations of From the product CT and 0~ with S- I 69 we obtain Thus the representations of \J are mutually conjugate. • i s i r r a t i o n a l -jj^ I) ( K l*i £ s a member contains an i n f i n i t e number of ^ I and From to. itself". ' ' = ; ^ I t also follows that the o r b i t of< which representation A } Due to the assumption that i n e q u i v a l e n t i r r e d u c i b l e representations. as* ' + cannot be an i n t e g e r , therefore except-for— 0 ^(k"i,K3, and 0 K| ' ^ > 0 '> I Wi -v KV (lC2.-2fc) An irreducible belongs to the same o r b i t i f and only i f |. f o r some integer ^ - K. ^ 7) 2 2 3 i t follows that the l i t t l e group r e l a t i v e l ^ " . ^ ' ') { f ° 5 , 7 and CJ° i s the group 5"° Again as a t r i v i a l consequence of the d e f i n i t i o n every i r r e d u c i b l e representation of the l i t t l e group"is an allowable representation. Therefore we can construct the ^physical' i r r e d u c i b l e " representations of CT i n a manner • 70 analogous to the previous case making use of the r e s u l t s i n Appendix T... Let of cf f>i£+ ,X] +,^3? be an a r b i t r a r y element then V Hence the i r r e d u c i b l e representation A by = W j ^ With 2 * induced i s given by K,7i,+ Ki* ) >j £*J> j ^ [ Jj 2^'0 - £ 7 1 , 7 ^ - * , ^ ) J (H.2.2^) 0 , ± 1, ± 2, • • • • . If we use only one l i t t l e group f o r each o r b i t of the p h y s i c a l i r r e d u c i b l e representations of T" 5 we get a l l the p h y s i c a l i r r e d u c i b l e representations of J" as i n (IT, 2. 2 <j ) I t i s e s s e n t i a l to note that the p h y s i c a l i r r e d u c i b l e representations •• of i- n t h i s case are a l l i n f i n i t e l y dimensional.. Assume now the magnetic f i e l d i s p a r a l l e l to a l a t t i c e plane but not along any l a t t i c e vector. Without any loss of g e n e r a l i t y we may take the magnetic f i e l d to be p a r a l l e l to the plane/determined by p r i m i t i v e l a t t i c e vectors # 2 . and- 0$ , Referring to Table 1 we see that we have two cases to be considered separately. H - Case (bi):. number and ^, ^ "FT |\i where ^ are r e l a t i v e prime integers { [ % 4 t Let us denote the Abelian subgroup J" by ^/ _fj v The p h y s i c a l i r r e d u c i b l e representations A ( K i ) - -£*J> [ where K a ; 0 k» , then and = i s an i r r a t i o n a l K3 [^,0' / -2TTC + Ki^ ) { 3 |^ i t follows that J u are given by c ^ ; o ] . A o>i ( j 2TTC 1) X^j (31.2.52) are r e a l number s a t i s f y i n g the i n e q u a l i t i e s < I. [ - ^5), ( k i ) of From the f o l l o w i n g e q u a l i t y 0 ] [ t- ^ ^ , X ] JT^ , o ] 72 A *"°{ [-f£, O o] [ w**s Lfr*., ,>i cl ] The conditions f o r the equation to be v a l i d are and k 3 = ( K + 2. a s ) 3 The two conditions i n s a t i s f i e d when can only be simultaneously K = 0. group r e l a t i v e to A This implies that the l i t t l e ' ' [ J" J T ; and J i s J , Thus we can again construct t h e ^ p h y s i c a l " i r r e d u c i b l e representations of [J from those of Let Of J , J" by the method of induction. f ' H ^ + Oiafla +'Mj , X ] 0 ince be an a r b i t r a r y element 73 the physical irreducible induced where by the ^ •J^J are one ( b i i ): f o r two n by hence the dimensional. representation It is physical irreducible ^K', d also,noted representa- to belong to the / that , We group per H can of J" orbit j ^ " = obtain °* + a l l the without i n the ^ ^ 7 above physical repetition if we construction. where b o t h ^ z y J 3 A R E numbers In t h i s Case and infinite s representations little irrational i s given - •-• a some i n t e g e r irreducible Case i conditions same o r b i t use Uj *• 1/ ~j - A^.'^'O tions for '^ { J J Ki ^, -j^l Ki, Ks/1) that /I ^ ' { J J representation case the ( b i ) f o r we can s i t u a t i o n , i s very choose the A b e l i a n Its similar group physical irreducible J to t h a t to in be representations 74 are given by \ Now we have (ir.2.40) Again because of the f a c t that (X2.4-1) except for ^ = 0 one comes to the conclusions that the v l i t t l e group of A ° { f°) J * and two representations ; r e l a t i v e to ^ ^I, 3 , 0 f k J and and 7° is 75 ^ tki', Kj , I) I J°j belong to the same o r b i t i f y From the f o l l o w i n g r e l a t i o n i t follows that the p h y s i c a l i r r e d u c i b l e representation (.Ki, K3 0 ^ j - j y A^ ' induced by ^ i s given by: £ Tl.A, 4 H A 4 O^flj , X J 1)^/^ 4 Z V Again the representation j J J is infinite dimensional. From Table 1. we see that there are two d i s t i n c t "to be considered cases i n f i n d i n g the p h y s i c a l i r r e d u c i b l e rv' representations of —i ' J~ i f . the magnetic f i e l d H i s not,.', p a r a l l e l to any l a t t i c e plane. — i Case ( c i ) : Zjrsj H = ^ I i , + ^ — i + tyr\ — ^ N ^0 V where are i r r a t i o n a l such that °/i H a i s also an i r r a t i o n a l number 76 while frvyj , ^ a r e r e l a t i v e l y prime L e t us d e n o t e by 0~° w h i c h i s one o f t h e m a x i m a l *A oi.i=-<* Because integers t h e subgroup Abelian . { [_^\ &\ + ^iUa ,X] subgroups of- J" and ^ IN/— ^= c v -J"** c a n be w r i t t e n as t h e d i r e c t > ol.\ t h r e e A b e l i a n groups JQO/^lj \ z product of the [ [oi j\j^ , ( z the^physical irreducible \ and representations of T a r e g i v e n by = where Making -2TV ( 6 ^ u i t KiVii) jj jt^Jj ^ 2 i r i K\ , kY (X + ^ 1 ^ , 0 ^ ) ^ . ) f use of the r e l a t i o n = [ondl 4 o n N a 2 3 y ( X - ^ ' K I - ^ N ^ , •+ ^ i ^ ] we d e d u c e t h e r e s u l t ^(KoKi.Oj ^£_^ ) ? f j [ « : +^N^ X] Lj£«-v£,<f] j % / .(1,2.4-7) 77 M'"2 Since ^'' ^ 2 a r e i r r a t i o n a l and ^ * 0/I, 2, - - • therefore the two q u a n t i t i e s /N" ^-^ - cannot, be non-zero integers.. exceptor ^ * 0 From A and (JT. .2.4-<p ( * > K » > ) { i J 0 >d 2 ar N-i ; ^M^i Thus ^ = 0 . i t follows that the l i t t l e group of \ relative to and the conditions f o r 7 A *"* fj'J ( and 7° j,) and i s C7° itself, *' ftf'jto 0 belong to the same o r b i t are: f o r some integers that 0 4 j ^3 <N , and Let ^ where [^ + i^N-V^)^ a r b i t r a r y element of 7 i ^'j r e s t r i c t i o n that 04 , ^' < /\| integers. Now > must be such be an °e integers with the and , X-^ are a r b i t r a r y 78 and hence t h e ^ p h y s i c a l i n d u c e d by where £ irreducible ' '^ { J~ °.!) i s g i v e n Kl i s an i n t e g e r i s defined representation 2) ^ £ J" j by: i n the range 0 < ^ ' ^ N and by 01'Kevi/Vi's & More e x p l i c i t l y JLtj we c a n w r i t e [ 2-rrC [ A + ^W*,*, - X ( ^ f ^ ) ( ^ ' ^ - .-' £ 111 4 - ^ 3 ) ( l ^ N Y ) ^ - ^7,) ^72)^1 (HT,2.SS) 79 It V i s clear that Case are fj * (cii): a i lirrational j^lj and | j J and KJ ) dimensional. where "j, ,^ numbers s u c h t h a t t h e r a t i o s and ^3 ^ are a l s o i r r a t i o n a l , the maximal Abelian subgroup such Xlj ^ [ ^ i ^ j , i t s physical irreducible representations ^ ( K s ' ^ j r j ^ where ^st l|i ^ L e t us d e n o t e by is Infinitely by that satisfies the i n e q u a l i t i e s . 0 ^ < ' Now 04 and f~ 0 from = we ^ , (X- '^.'p obtain the f o l l o w i n g relation C o n s i d e r t h e s e t o f numbers o f t h e f o r m % (i) , ^ a - Oi *^ / ± 2, • • •• , T h e r e t h e s e t does n o t c o n t a i n set contains non-zero (1.2.^) "] Oa, ^a - ^ 2 ^ 1 a r e two p o s s i b i l i t i e s : any n o n - z e r o integers. where either i n t e g e r or ( i i ) the 80 In t h e f i r s t irreducible exactly little representations as b e f o r e . . group itself. case t h e c o n s t r u c t i o n { fj Then m a k i n g JJ from those of ('3D\ 2 . 5 7 ) F o r from of <J of of p h y s i c a l r e l a t i v e to is we s e e t h a t t h e J and cf C i s 5°' use of the r e l a t i o n 2 we o b t a i n the'physical 4 i n d u c e d by 3> , . I (k irreducible " '° f (f J 3 * **** 0 + < M representation ^'^'^fCTj , ^ > JL' , I The integers^ representation = 0, ±-1, * 2 . , ' " 1)'"' -{ T i <3 and the is infinitely dimensional. On the other hand, i f the set of numbers of the form. 7v, ^j.-01 , £ 0 r ox,, oia » 0 ^ 1 * 2 , - •• • contains non-zero integers then the l i t t l e group [_ to vT , and J* i s no longer of A 1 ( J" J r e l a t i v e J I f a given integer belongs to the set then a l l the i n t e g r a l m u l t i p l e s of the given i n t e g e r also belong' to the set. the f a c t that This together with i s a proper normal subgroup of i n d i c a t e s that the quotient group LJG i s an L infinite group. I t would be easy to work out the elements of L e x p l i c i t l y , but we do not do t h a t , because we do not know how to f i n d the p h y s i c a l i r r e d u c i b l e representations of L Moreover even i f we knew these representations we could not 0~ claim that the representation of obtained from them by induction are i r r e d u c i b l e and c o n s t i t u t e the complete set of 'physical i r r e d u c i b l e representations of (7. . I n other words the r e s u l t s of Appendix I do not apply i n t h i s case. However we can construct a probably complete and i r r e d u c i b l e set of p h y s i c a l representations of way i n d i c a t e d below. G i n the We say "probably" because the assumptions of Appendix I are not s a t i s f i e d i n t h i s case, and we can only conjecture that we obtain i n t h i s way a l l the p h y s i c a l i r r e d u c i b l e representation of (J. To do t h i s we can f i r s t f i n d a l l the p h y s i c a l i r r e d u c i b l e representations of C l>> =f 4 /Xl \ from 82 those of J" the little to (J by group of the ,!J° and W is representations D \, , W where $ ^ We relative induction.- This of [ itself and thus obtained ^ < and A^'^fj" j T the " p h y s i c a l ' i r r e d u c i b l e are g i v e n by • ^ is CT little itself group of and method o f i n d u c t i o n t o o b t a i n t h e f o l l o w i n g of relative ^ J a l s o show t h a t t h e 7" to J"" ° D, i l ± 2 , '<f ~ can + straightforeward-'because representation J ^*2 is we I repeat J J the representations a However we are unable to prove these representations are irreducible. In t h i s irreducible s e c t i o n we representations and-orientations the have c o n s t r u c t e d of for arbitrary of t h e m a g n e t i c f i e l d w i t h case j u s t d e s c r i b e d . In a l l the ^physical'' i r r e d u c i b l e r e p r e s e n t a t i o n s any r e s t r i c t i o n s v.on t h e crystal a l l the p h y s i c a l the magnitudes exception above c a s e s the are o b t a i n e d lattice. I t can be without readily of seen that a l l the representations obtained are u n i t a r y , ; 84 I I I 3-. PERIODIC BOUNDARY CONDITION In the i n v e s t i g a t i o n s of the bulk properties of c r y s t a l l i n e s o l i d s i t i s often convenient p e r i o d i c boundary c o n d i t i o n . dimensions Li tt, , L &2 2 and to introduce the I f a given c r y s t a l i s of Li % ( L t t L , L$(> o) axe 2 i n t e g e r s ) we i d e n t i f y the opposite sides of the c r y s t a l or we imagine an i n f i n i t e number of c r y s t a l samples i d e n t i c a l to the given one are stacked together. When one uses the group; t h e o r e t i c a l language the a p p l i c a t i o n of the p e r i o d i c boundary condition means p i c k i n g out from a l l the i r r e d u c i b l e representations of the group of p r i m i t i v e t r a n s l a t i o n s those which have the property that the representation of the element (E I t ) The vector is. equal to the representation of (fr|t,+ LC^") £ i s an a r b i t r a r y l a t t i c e vector and the s u f f i x i stands f o r 1,2 or 3. I t i s w e l l known that the i r r e d u c i b l e representations of the group of p r i m i t i v e t r a n s l a t i o n s can be l a b e l l e d by vectors of the f i r s t B r i l l o u i n zone and the correspondence between the set of a l l vectors of the f i r s t B r i l l o u i n zone and the set of a l l the i r r e d u c i b l e representat i o n s i s one to one. The p e r i o d i c boundary condition makes us pick out a f i n i t e subset of i r r e d u c i b l e representations corresponding to a f i n i t e subset of vectors i n the f i r s t B r i l l o u i n zone and r e j e c t a l l the others. I t i s generally acceptd that when the r a t i o of the number of atoms on the surfaces of the c r y s t a l to the t o t a l number of atoms i s v a n i s h i n g l y small and when long range i n t e r a c t i o n s connecting surface atoms to i n t e r i o r atoms are not of primary importance* the a p p l i c a t i o n of p e r i o d i c boundary condition does not a l t e r the conclusions of p h y s i c a l importance. Brown pointed out that even though i t i s l e g i t i m a t e to use the p e r i o d i c boundary condition to study the behaviour of an electron i n a c r y s t a l t h i s need not be the case when the c r y s t a l i s s i t u a t e d i n an., external uniform magnetic f i e l d . In the f o l l o w i n g we wish to show that p e r i o d i c boundary- c o n d i t i o n i s incompatible with any of the i n f i n i t e dimensional " p h y s i c a l " i r r e d u c i b l e representations magnetic - t r a n s l a t i o n group. i r r e d u c i b l e representation of the In the case when the p h y s i c a l of .the.magnetic t r a n s l a t i o n group i s f i n i t e dimensional the p e r i o d i c boundary condition can be applied only when the s i z e of the c r y s t a l i s r e l a t e d to the dimension of the representation be described i n a c e r t a i n s p e c i f i c way to below. The f i r s t case we consider i s when the magnetic f i e l d i s The p h y s i c a l i r r e d u c i b l e of (see Section I E , 2 representations ) are given by ( k ^ K j , i) where and Q ^ £ =0 otherwise. A p p l i c a t i o n o f . p e r i o d i c boundary 86 condition means that we want to pick out those d> I.J nJ = 3> For s a t i s f y i n g the c o n d i t i o n [>•*,+ 1 ^ + ^ - 4 - ^ , /t = 1 condition (JT. S.i) f o r a r b i t r a r y value of inequalities 0 and When = A (UL. 3 . 2 ) where Li Li = -C - 2 (11,3.1) implies and any yu, ^ < isl . s a t i s f y condition Ki representations Since o subject to the K, <C 1 ^ to i t i s necessary that X, = 0, 1, 2, • — , ( *- 0 a p o s i t i v e integer x 2-N condition CH,3.3) (31.3.5') ( J . 3.1) requires that and f o r a r b i t r a r y value of iv, , and" When L /C - 3 A = These imply that a p o s i t i v e integer x (2N) i t i s r e a d i l y seen that c o n d i t i o n (1,3,4') (3T. 3-1) leads to Thus applying p e r i o d i c boundary condition means, according to (nr. 3 . s ) , tjr, a. and p i c k i n g out a subset, of the* p h y s i c a l i r r e d u c i b l e representation of with K <; = Conditions (i=t-2,S) where (J i t = 0; i , 2, • v • £Lc-i) s i m i l a r to these are also obtained when one applies the p e r i o d i c boundary c o n d i t i o n to the i r r e d u c i b l e represent a t i o n s of the group of l a t t i c e translations.. However, while i n that case there i s no r e l a t i o n between the s i z e of c r y s t a l and the dimension of the i r r e d u c i b l e representations, i n t h e . present case the s i z e of the c r y s t a l i s r e l a t e d to the dimension of • t h e p h y s i c a l " i r r e d u c i b l e representations. Along >v t h e - d i r e c t i o n s , of (£, and A £ the lengths of the c r y s t a l must be even, m u l t i p l e s of the dimension of the p h y s i c a l ... i r r e d u c i b l e representation when measured i n u n i t s of I and l J-l a I respectively. This means that imposing p e r i o d i c boundary condition i n t h i s case leads t o A r e s t r i c t i o n ; ori the magnetic f i e l d . To attach any p h y s i c a l meaning to the r e s t r i c t i o n would require a j u s t i f i c a t i o n of p e r i o d i c boundary c o n d i t i o n . In view of t h i s we do not think Zak's statement (Zak (1964 I I ) ) that "the imposition of p e r i o d i c boundary c o n d i t i o n leads to q u a n t i z a t i o n of the magnetic f i e l d " needs to have any deep p h y s i c a l meaning In f a c t , i n the case of i r r a t i o n a l magnetic f i e l d i t 88 turns out that p e r i o d i c boundary condition cannot be applied at a l l except along the d i r e c t i o n of the f i e l d . let M. 2 in where M M * : — * A A, To see t h a t , i s an i r r a t i o n a l number as of 0~ the p h y s i c a l i r r e d u c i b l e representations are given by where ft , ft = 0, ± I, -• • 1 condition The p e r i o d i c boundary requires 3" ^ ) When ' -C = | , implies and ^2, f o r a r b i t r a r y i n t e g r a l values offt - ^2-0 i.e. k-, = where be s a t i s f i e d i f L\ i=2 £ = 1 k= 3 h £>^>, ( 0 JW.^ {-2ir\.U (^^ + £ t v ^ ) ^) =• | cannot i s a non-zero integer. ^4.^2, ^ ( l , S. 0 where ( =r , 1, 2/ • • • • , L, - I . we can see at once that s a t i s f i e d as When j - i i K L K JJ » I the c o n d i t i o n i s s i m p l i f i e d to However the c o n d i t i o n When Thus i f we put ^ ^ |_ + + 2 (HT. 2,4) except cannot be |_ =- 0 2 can be s a t i s f i e d provided that Jl » 3 0, l,2 ( , (t-rl). - 89 This means that i n the present case we may only apply p e r i o d i c boundary c o n d i t i o n i n the d i r e c t i o n of the magnetic f i e l d . In a s i m i l a r way i t i s not d i f f i c u l t to v e r i f y that p e r i o d i c boundary condition cannot be applied to other cases discussed i n Section I I I . 2 when the " p h y s i c a l " i r r e d u c i b l e representations of J a r e infinite dimensional. 90 THE GROUP Jfl III.4. AND ITS CO-REPRESENTATIONS We h a v e shown i n S e c t i o n ,—• group CJ of the Hamiltonian magnetic f i e l d contains I I . 3 t h a t t h e symmetry of a f r e e e l e c t r o n i n a uniform £R(2)|0'&3' the a n t i u n i t a r y operator t—• Thus, (It t h e group IfO- may a l s o c o n t a i n a n t i u n i t a r y e l e m e n t s . i s e a s y t o s e e that. W- contains a n t i u n i t a r y elements o n l y when t h e p o i n t g r o u p of the l a t t i c e 6-fold to the magnetic axis perpendicular has a o f $Tl *Xl i s a p r o p e r subgroup be 2 ( f o r a proof o f ''ffll see Opechowski \l by rv If or field.) L e t us d e n o t e t h e u n i t a r y s u b g r o u p rJ A , 2, the index Ii of and G u c c i o n e must (1965)). I—' have seen i n C h a p t e r I I t h a t t h e o r d e r is a t most 21, Hence t h e o r d e r , rv of t h e group o f ^hiKj j We rv ^ffi I [J i s a t most 12 , rv Consider matrices a space which i s i n v a r i a n t under which transform ^Tl The t h e b a s i s f u n c t i o n s of t h e space rv under the operators "^TV i n the usual operators. called general fiTi r do n o t f o r m a r e p r e s e n t a t i o n o f s e n s e i f Tf^ contains a n t i u n i t a r y The s y s t e m o f m a t r i c e s a co-representation o f Iftft • f o r m s what W i g n e r co-representation containing a n t i u n i t a r y operators dimensional (1959) W i g n e r h a s shown i n how orte c a n ' c o n s t r u c t i r r e d u c i b l e of a group finite of from irreducible representations given of i t s u n i t a r y subgroup. In the previous s e c t i o n we h a v e c o n s t r u c t e d p h y s i c a l ' ' irreducible representations A , o f (J f o r arbitrary and o r i e n t a t i o n s |0f t h e m a g n e t i c f i e l d . not constructed magnitudes However we h a v e any r e p r e s e n t a t i o n o r c o - r e p r e s e n t a t i o n o f the group ^71, , This i s due to the f o l l o w i n g three (l) The group ^V- of the l a t t i c e reasons: i s u n s p e c i f i e d unless the point group and the o r i e n t a t i o n of the magnetic f i e l d r e l a t i v e to the c r y s t a l axes are given. This means i n d i v i d u a l cases must be considered separately. (2) Except when the f i e l d i s r a t i o n a l the " p h y s i c a l " i r r e d u c i b l e representations of (J are i n f i n i t e dimensional hence we are not sure that the representations constructed from those of 7 by i n d u c t i o n w i l l be i r r e d u c i b l e . i r r e d u c i b l e representations of 7 (3) If the"physical" are i n f i n i t e dimensional then t h e ' p h y s i c a l i r r e d u c i b l e of f[K must also b e i n f i h i t e dimensional. In t h i s case we are not sure that Wigner's 1 method i s a p p l i c a b l e to construct the p h y s i c a l i r r e d u c i b l e co-representations of W i f i t contains a n t i u n i t a r y elements. In the case of r a t i o n a l f i e l d we do not have the d i f f i c u l t i e s . mentioned i n (2) and (3). We can construct the physicial . i r r e d u c i b l e co-representations of r ffl ifit contains a n t i u n i t a r y elements: using Wigner's method Once the 'physical i r r e d u c i b l e representations of rfA are known. Hence the problem i s reduced to f i n d i n g the p h y s i c a l i r r e d u c i b l e /\, representations of AX : This we can do by making use of the f a c t that the order of the group MJG i s at most 12. Since any group of order l e s s than 60 i s solvable i n p a r t i c u l a r M/7 i s solvable. normal subgroups of Kj. • Ai^Mz, 7 C U, C U z C That means we can f i n d \ ' ' (say) such that y * ^ C <U . 92 i s a f i n i t e composition s e r i e s . of !J i s at most 12 i n U most four groups.) ( i n f a c t , since the index the sequence consists of at Because the sequence of groups forms a composition s e r i e s the index of each group i n the succeeding group of the sequence must be prime. Now we can use of a method of induction and extension s i m i l a r to that described by Raghavacharyulu (1961) f o r the case of space groups, to o b t a i n p h y s i c a l i r r e d u c i b l e representations of lk v those of (7. t from Repeating t h i s process, i f necessary, we can construct the ^'physical i r r e d u c i b l e representation of ^ . i s c l e a r that the representations of AA It obtained i n t h i s way must be f i n i t e dimensional. We can conclude that i n the case of r a t i o n a l f i e l d , ,the co-representations of lpf\, can be obtained i n a s t r a i g h t f o r w a r d way. 93 IV.la. THE LANDAU FUNCTIONS A l t h o u g h we c a n n o t s o l v e an t h e S c h r ' d d i n g e r e q u a t i o n o'f e l e c t r o n i n t h e p r e s e n c e o f a u n i f o r m m a g n e t i c f i e l d and a periodic electric field, the solutions f o r the simpler p r o b l e m o f an e l e c t r o n i n a u n i f o r m m a g n e t i c f i e l d known. Landau solutions. given ( 1 9 3 0 ) was t h e f i r s t t o o b t a i n these We s h a l l u s e one o f t h e c o m p l e t e s e t o f s o l u t i o n s by Johns:'on and L i p p m a n n them b a s i s functions representations only are w e l l ( 1 9 4 9 ) and c o n s t r u c t generating o f J". out of the physical i r r e d u c i b l e They d i f f e r f r o m L a n d a u ' s solutions by p h a s e f a c t o r s . As we h a v e shown t h a t f o r d i f f e r e n t c h o i c e s o f the corresponding magnetic t r a n s l a t i o n groups a r e we s h a l l w r i t e f o r c o n v e n i e n c e t h e v e c t o r symmetric gauge. form using To a v o i d an o b l i q u e coordinate " c r y s t a l l a t t i c e we. a r e F u r t h e r m o r e we s h a l l magnetic f i e l d First of complications i s simple to a l a t t i c e o f a l l we s h a l l briefly the Schrbdinger equation uniform magnetic f i e l d . symbols potential i nthe cubic. vector. summarise t h e p r o p e r t i e s by J o h n s o n and L i p p m a n n i n f o ra free electron i n a Thus t h e f o l l o w i n g i s j u s t a resume r e s u l t s and a c o l l e c t i o n of d e f i n i t i o n s . o f used. L e t us d e n o t e t h e k i n e t i c momentum •by arising assume t h a t t h e d i r e c t i o n o f t h e is parallel o f some o f t h e i r isomorphic, s y s t e m we assume t h a t t h e dealing with the s e t of solutions obtained solving the algebraic gauge : I t i s easy t o v e r i f y that operator 7 [\v. c I f we choose our c o o r d i n a t e system i n such z-axis i s p a r a l l e l can readily to the d i r e c t i o n a way that LI) the of the magnetic f i e l d check t h a t the f o l l o w i n g commutation we relations hold: where [ ^ o / - [ TT* + ] ~ -f^^)] (IV. I. 2) 0 - " 2^ ~ and n: 60 i s the )e| H c y c l o t r o n frequency the magnitude of t h e f i e l d . which are W = d e f i n e d by ^ ^ Introduce H and is *c, the operators defined'by: B e c a u s e of t h e commutation relations (II/. f.2) Si it n cf eo l l o w s t h a t b o t h *c and ^ c (IV. 1-3) and are constants of motion. r "^c and ^-o are not simultaneously measurable. When t h e a b o v e i s c o m p a r e d w i t h c l a s s i c a l of m o t i o n t h e o p e r a t o r s are i d e n t i f i e d c o o r d i n a t e s o f t h e c e n t r e of t h e o r b i t f o r t h e equations as the transverse 95 motion of t h e e l e c t r o n i n a plane magnetic f i e l d . can Using the fact perpendicular to the that write The o p e r a t o r - 2 ^ (" * 7r c o m m u t a t i o n r e l a t i o n . (IV. (• 0 ) + a n d the d e f i n e an e i g e n v a l u e problem w h i c h c a n be shown t o be i d e n t i c a l w i t h t h a t o f one dimensional harmonic o s c i l l a t o r . are given where i n t e g e r o r zero. eigenfunctions operators Consider fat, i> j> \ \, ^, L> 2 where and and X p h y s i c a l dimension. f? the s e t of a l l the such o f ;J4t i s any p o s i t i v e of the three mutually \^i^,L)> I a n d Hence t h e e i g e n v a l u e s simultaneous commuting that JL> (IVM.ID) = X |*.|&,.JL> UV.MO = p l^v, ^ c a n be any r e a l numbers o f t h e a p p r o p r i a t e Explicitly the eigenf unction \^>^,^ in 96 coordinate representation i s hk |U>- where' Aw Ivfe)*! - W polynomial o£ degree Let , f ^2 ^ *-!] , * a n \ x ^ d Hermite l s . " be the three components of any vector ^ then Xt Hence toj, {-j i f y l ^ + JA^) ] K « , i > M a k i n g u s e o f (IV. 1. lb) I .1 — IJ ^ - | 1 I ' ft . -••> and 11 [ f Li (IV, LI') IJ V X — |St-«j, l > we o b t a i n l/yv I I T— • I J and -r Let F 1 - ^ 1 0 1 + ^ ^ + ^ 0 3 b e a n y l a t t i c e vector (llAl.li) then t h e . o p e r a t o r when /^f(v) , XJ [ has t h e f o l l o w i n g i s i n the symmetric |>K,«f) gauge , (IV, M 4 ) From e q u a t i o n s | ^ , p > .* lV^"3KjU> = e x p l i c i t form IV and ( l I M - t f ) Jl > we find (IV. I. I &')' • 1 and where a = I From and (IV- \- I f ) 1^,^,1^ generates t a t i o n ' of t h e A b e l i a n subgroup magnetic functions (il/.M^ ( J \ ' \ 4 . I - I °* (1^.1.17) the .functions t i M . ^ |^,p L> translation l^/j^X^ group. we c a n s e e t h a t each o f an i r r e d u c i b l e £ > X] represen0 f the From now on we s h a l l c a l l , t h e t h e Landau f u n c t i o n s . IV.lb. BASIS FUNCTIONS OF THE "PHYSICAL" REPRESENTATIONS OF J FOR THE CASE OF/\RATIONAL MAGNETIC FIELD We turn now to the question of constructing basis functions f o r the p h y s i c a l i r r e d u c i b l e representations of the magnetic t r a n s l a t i o n group from the set of Landau f u n c t i o n s . be any u n i t a r y i r r e d u c i b l e representation of (7* X> Let p any s t a t e vector^ then the ^^ .2 . basis f u n c t i o n which belongs to column of the representation can be .obtained by applying to F the p r o j e c t i o n operator with some f i x e d ^. the ft'^ - 2 1 Thus to the j ^^'Y ^ ) G F column of D ^f,i^> .2 ~&(-0*) ^~ The summation i n the p r o j e c t i o n i s c a r r i e d over a l l the elements see that and \J of the group ^J, operator To i s a basis, f u n c t i o n belonging to we apply to i t an operator ^j' °f K'(T'7)0"aF i s '.a ^ basis f u n c t i o n belonging « 'I column of the representation 2> provided the 99 7 sum over elements of i s convergent. Consider f i r s t the case where the magnetic f i e l d i s given by H ~ ]ejll "j\7 ^3 as i n Section III..£:. " p h y s i c a l " i r r e d u c i b l e representations of J" The are then (\j -dimensional. Since any r e a l number can be w r i t t e n as ^2. N 11^ + and p>2. put ^ =• = 0 , ± 1, * 2 , • • • ^ = o, I. ' •N-1 where ; i s a p o s i t i v e . r e a l number such that +^ ) y a n p». <^ I i n the expression For s i m i l a r reasons we can w r i t e l r = c>, ± i ± 2; 0 ( j o ^ p 3 JL as for O^+f )^ 3 ^ ^ . ; we )'K,|J Z> , / where Then an a r b i t r a r y JL Landau f u n c t i o n w i l l be denoted by Let element of ' (J~ = (^N^ )^+ / J ( ^ 4 ^ 4 ^ ) 0 - , vX ] b e a n O^^^X arbitrary then A y J . [ - a T r c [ ^ ( ^ u + j . a ) ^ + (u-^O^UJ affair* (X-|J(^IV+^3^)] x 100 where, and yU is a positive integer 1 + j' yu' - To construct l e t us apply to the function /h, =, 2 Writing H V fc'j , ^2, I. ^) 2 \ z 0, ^3 = of to f 3 \ 0 y = element of + o^'a) » 2 that of the i r r e d u c i b l e (see +^ + j>*) d (^N (^j>^ 1 X = M a k i n g use (IV- . 1 functions , X . Z. q) ) J 4, > (.jr+ j> ) ^ 3 k-, (7 (ill ^ o projection operator a general [ V , ^ + ('xiN + O a Here [K yU-o, D/ A the b a s i s X> '> * ' *' the such . N) representations N l e s s than where and ^ as ~] we have ' N , -0, ± I, ±2, (lty\ I. 2|) 2^ - •- and , and 2 . i = D, ( J H \ I. < " j ^ ' N 1,2,-- we can , /Sf- I, simplify 101 M «*]> [ - 2 * * v { ( X- I <N ^ „ 2 The summand i n the above expression no longer depends on SK\ and ^ and hence the sum must diverge. X However the . f o l l o w i n g i n f i n i t e ' sum converges f o r any given values of x, A ^ , 2 ( f o r a proof of t h i s statement see Appendix I I . )..and, i n f a c t , i t i s the basis f u n c t i o n we are looking f o r . To v e r i f y t h i s d i r e c t l y we. apply an a r b i t r a r y element of J we have to the f u n c t i o n [W<£ + (/HJW U V \ 124) ., + f) ^ "+ , X] Making use of [W.i.2\) 102 4 [%lM f j " ) a + ~ .2 Here A" ^} ( 2 i r £ i s a positive integer = i'* ft" last [ 2- Ks^') j K a ^ a ^ ^J> [ - 2 T C ( K i ^ i ' t l^^N) I ' The 2 £ , X ] z 0 expression i n a n l e s s than' d i s y N defined and by otherwise ( j V \ 1. 2 5") c a n be w r i t t e n as 103 Comparing function o (,1V- 1 - 2 4 ) [\]/. 1. 24.) transforms j f J wh e r e ^ [ a T i K i - ^ J j according to the ft = 0, I, .2, j ( ,N-t 0 ft'^ column of f u n c t i o n s ^ t j + ; f the. g r o u p the functions i n (IV. I. 2 7 ) For e a c h p a i r o f numbers (IV, )> i s s o l u t i o n of the equation ^a(Ki) O^'M we c o n c l u d e t h a t t h e j Hence t h e , s e t ^ (ir.2.17) with ( l^ - )f 2 ^1 and generates the representation 7 . F o r b r e v i t y we s h a l l simply )j" and by | 'k, i f . / j , ^l^O denote * 'j ) k we h a v e one s u c h s e t of f u n c t i o n s . We should add t h a t t h e c o n t e n t of t h i s 12-7) subsection i s a more t h o r o u g h d i s c u s s i o n o f w h a t Zak ( 1 9 6 4 I I ) d i d b e f o r e . 104 IV.lc. BASIS FUNCTIONS OF THE "PHYSICAL" IRREDUCIBLE REPRESENTATIONS OF FOR THE CASE OF ...IRRATIONAL; MAGNETIC FIELD Let the magnetic f i e l d ' be given by where ^ H 3 i s i r r a t i o n a l •;(: s:'ee Section I I I . £.) The " p h y s i c a l " . i r r e d u c i b l e representations of countably i n f i n i t e dimensional. Landau f u n c t i o n by and , LT = ^^ r |^, 0, * 1, * 2, • • numbers such that 0 ^ CT J" ^e Let us denote an a r b i t r a r y ( ^ +• ^) (X v - ( ^ ^ |>3) ^ ^> and , ^ where ^are .real ps < < • . y^a, + Consider the e f f e c t of an element of the group a on the Landau f u n c t i o n \K we have Let us apply t h e . p r o j e c t i o n operator r ) ^'i^+'h-i&s i ^ . + p )o. Itf^OTC ,)> 2 J to t h e Landau f u n c t i o n ~ ^ Q ^ / y Kt s C ^ | ^ f , ^ Lp+f>i)<*- , |H 5 I *l *> + n a 4i + n d *J K s I U~ + |<>i) 4 )> f = 3 2 v 3 ' " [ *K>£ 4 ^ 4 + *j K ^ f ,Xl i s independent of be t h e b a s i s X, W sum does n o t c o n v e r g e . . However r e s u l t makes i t p l a u s i b l e t h a t will n ' ' The summand . i n (IV. I. 2^) and h e n c e t h e t r i p l e e (ir4-^)^> 2 v, v h ,X J 4 ^ £ + S ? , X ] I -K, ^ a . A t where | (^'-t j),^ a. ) f u n c t i o n w h i c h we a r e l o o k i n g by d i r e c t c o m p u t a t i o n , as i n (lV, I. 2 8 ) we , 'Kj this (A* t Ks) "lb ^ for. obtain In' f a c t ( Comparing | r (;'+ V + ^ ) « . , l ^ ^ x ) (I V. I.&o) ->( K/ ^) CL/ with (l^. • ? ) 2 2 • (IV.f, we see that has the required property., , {v+*i)'^)> Hence we conclude that the set of functions where 'yU « . 0 , * \, ± 2 , * • * • j of the equation C^.f '^! representation ^(.K,, Kj,0 numbers IT. and 2 = ^' ( ^) > 0 i s a solution generates the f j p. or each p a i r of, (kl) we have one such set of f u n c t i o n s . Every Landau f u n c t i o n belongs to one and only one of such set. 107 IV.Id COMPARISON.OF THE BASIS FUNCTION OF THE IRREDUCIBLE REPRESENTATIONS OF (J "PHYSICAL" FOR THE CASES OF ' RATIONAL AND IRRATIONAL MAGNETIC F I E L D S . -In t h i s the s e c t i o n we s h a l l compare t h e b a s i s "physical" i r r e d u c i b l e representations TJ = shall <** a. call H and with fH - p - ^ ^ Ms a = a a r a J of H = 4rk t h o s e when f u n c t i o n s of when 1^ . " t i o n a l magnetic W e field i r r a t i o n a l magnetic f i e l d . n In \el-U ( Sections field I I I . 2 and I V . 1 we h a v e s e e n t h a t when t h e m a g n e t i c i s r a t i o n a l the "physical" i r r e d u c i b l e representations are finite representations are dimensional. The b a s i s functions of these e x p r e s s e d i n terms of t h e L a n d a u . f u n c t i o n s l i n e a r c o m b i n a t i o n s o f an i n f i n i t e number o f them. the magnetic f i e l d is irrational C7 of basis a r e j u s t s e t s .pf L a n d a u f u n c t i o n s . . . a r e When, the "physical" i r r e d u c i b l e representations functions of infinite dimensional and t h e I n t h e f o l l o w i n g we w a n t t o show t h a t a c o r r e s p o n d e n c e between t h e b a s i s representation and functions of (7 f o r the case of r a t i o n a l magnetic a subset of the b a s i s representations field of CT absolute can be made a r b i t r a r i l y of a " p h y s i c a l " i r r e d u c i b l e F o r any g i v e n values of d i f f e r e n c e between c o r r e s p o n d i n g b a s i s to approach Let functions ^ ^ expansion w i l l field f o r t h e case of i r r a t i o n a l magnetic c a n be e s t a b l i s h e d . the of a " p h y s i c a l " i r r e d u c i b l e small as t h e q u a n t i t y i n a manner we now be a g i v e n irrational be w r i t t e n as f o l l o w s ^/H ^/ ? functions i s made describe. number, whose • decimal, 108 7 Here are Js i n t e g e r s such t h a t sequence , 0 ^ are r e l a t i v e l y /hi ( i r , JJ, j r - • &c ) x 10 _ 6 be a member o f t h e a b o v e s e q u e n c e From S e c t i o n k} = |^ K?fc K ,<f.) J 2> e ^ ' ^ = i ( ^ (Of a n f d = 1 of ' • J R that the magnetic f i e l d We s h a l l of f u n c t i o n s now c h o o s e o « of | *s J V , K"*, K , j- ) 3 \J « N;-l Kj^> ; I /R indicates ^ f u n c t i o n s which will e q u i v a l e n t t o t h e one ( ^ = 0, 1, 2, d e n o t e d by t h e s y m b o l f 4 ^ ) ^ ^ t ^ / K:, Ks, i ) i n t h e symbol a s e t of basis d e f i n e d by if j <K,. (OvN + ^ i s rational. generate a representation g e n e r a t e d by the set 0, 1,2, representation The s u b s c r i p t of r a t i o n a l H •= — - I V . 1, when generates the i r r e d u c i b l e , A a c ^ y where |o ' + i of f u n c t i o n s where ) We d e f i n e a N| , ' " ' K i / " " b 1,-2, 3,-•• prime i n t e g e r s , thus Nc Let ^ ^ ~}J , o f r a t i o n a l numbers Nt numbers. C ^~ i s any n o n - n e g a t i v e i n t e g e r and 0 ) , The new s e t K3 )) are R 109 and '. - 2 e 2 T ^ l K f , ) U , l W N * + j ' 4 f , ) < i , ^ > g on if 0 > +' The representation generated by them i s s t i l l u n i t a r y . When the magnetic i s given by i n f ini.te. set of Landau functions where ft = 0, X \, ±2, - " vv J-l = )% ; / ^ A a n d ^ the 3 ^^/^^x ( ^ 4. p ) a. , V ~ ft e subscript i n d i c a t e s that the magnetic f i e l d i s i r r a t i o n a l , generates D the representation /K3/I ^ of ( ^, ^, ^ , ^3, ft )) where Now we make correspond . = (j{\fc)f J. (. f + pO to a subset of ^ > x in the f o l l o w i n g way |% wh-ere - K,*, K ; i( ^ ^ K j ,ft]) 4 given values of x, ^ , 2 > J * s (^'+p )A., ^ ^ A > a 3 (IV. 1.32) We wish to show that f o r any the d i f f e r e n c e D given by (lV,|,33) 110 approaches to zero as just ( | v , /, S l ) | K, Denote From i n the manner we described. From and approaches to ^ | e l^'+pO^ ^'^ (IV, |, 12) |^ (llA l.3l') and ^/a. K > R - K Nc^(>0v we can w r i t e H>z we can see that U/^ U / O | R as follows | b B Y y 3D, ^' Ill where V * P = ^ The Hermite polynomial odd powers of * The p'roof that l ) contains only even or x according to whether Tba. —* 0 as ^ ^ i s even or odd. —* i s almost the same f o r both cases. • . / • We s h a l l f i r s t consider the case.that where 0. 2L i s the c o e f f i c i e n t of in ^ Hn(*). i s even,, then Let us consider the sum 2t / fH p t' -'^ v From and the i n e q u a l i t i e s i t f o l l o w s that the f u n c t i o n jt^j ( ' l * ) decreasing f o r <" - \j~2i- , p 2 ^ ^ ^ 2/- i s monotonic and monotonic increasing f o r , Consequently • f o r f i x e d values of the f u n c t i o n • (H^ff^ ")*] ^ 0 ^, -ft' and 2l Y> 112 i s monotonic increasing f o r and monotonic decreasing f o r As ^ both OM and ^ e x i s t s a p o s i t i v e integer where (N") sequence (_IV, 1-37) then Wu — * °*. c i t Therefore there such that f o r * > i s the-first'non-z-ero r a t i o n a l number i n the b ~ a n ft- ~ - ' — r •• - • From the i n e q u a l i t i e s v d , (lv, I.37') w e c a n s e e that when (KN.-^^p^yj I ^ ^ regarded as a f u n c t i o n of x > <C + f+(M*^ 0 W hen i s monotonic increasing f o r ^ negative, and monotonic increasing f o r 0 i z positive. Hence 2 113 -I ( I V . I. Pc where (^(x) a continuous I f we ^ = i and 1 variable. d e n o t e by To Jwj> { * i " * ] 8 By -via, i s t r e a t e d as c h a n g i n g v a r i a b l e s we,have. t h e s m a l l e r o f t h e two q u a n t i t i e s H'*^*-** then o3 - H 1 4 (IV. As //Vi ~ 7 ixs , < Consequently a p p r o a c h e s t o z e r o as L e t us now I t ' i s e a s y t o see that 6 /M .consider ^ 4 approaches to ID, . This I A* I 2 1 2.i| 5 a (*) ^• is just the absolute l.4o) 114 d i f f e r e n c e of two Landau functions corresponding to two i d e n t i c a l set of quantum numbers but d i f f e r e n t magnetic f i e l d s and For given values of 11 ^ i 2- lp\Q.\°'* respectively. the Landau functions are continuous when treated as functions of the magnitude of the magnetic f i e l d . Thus Since both i t follows from as ^tJUi ID, When and U>2. (j|A 1,54) > ^ -as ^Jf^c 0 lb,—> f o r even values of ^^.y HUU) - 2J s Ku\ ^ M ^ } ( - i ( JL=o v ( " 7 ' . ^ T l * (iv. I, 49 2 §'(I) by Again since As. 0 case 2l+l We define ^ approaches to zero I 1*2 ^ • tend to zero as ."^fai—> that also i s odd, l e t i n the previous —* |Vc exists a positive integer there as C t such t h a t f o r <C > 115 If the r a t i o n a l number ^ %>o 2 i s chosen with i > then v Po ^ ^ I M i i i ^ K J j / 2 fr-^W+f^j[ t^+fr'+fo*-^ ^ 'K^D Now f o r No Pi, / s u f f i c i e n t large both summands i n the l a s t expression f o r S ^ ) when regarded as f u n c t i o n of monotonic decreasing f o r p o s i t i v e values of o\ _ % are Hence i n exactly the same way as we showed 5 ( f ) —> 0 when • ^ / N i —"> ^ . ^'(i) — > 0 when ^ ^ ^) —> ( we can show that I t i s also clear' that i n showing thati i as 0 ^J}ii — * ^ f o r any value of "W *]£) — » 0 even or odd. Hence we have, shown when "^'/Nc Thus the set of functions - as fyt ~ 1 ^Jtfi > ^ / <:—(VJLZL —* ^ the argument used i s v a l i d > . | k,° ^ k*, J , / ' ^ converges to the s e t i n the sense that k I >x K 3 where X 116 | |< KP, 'j-' )) - f o r any given values of | <*c, , k ,, 4 ^/Z . 1 Ki^ K 2 K3 "J-')) | — > O The sequence of " p h y s i c a l " i r r e d u c i b l e representations of, ! 3 | ^ ^ generated by converges to the i n f i n i t e l y dimensional representation generated by ) % K,' K, •) > t 2 117 IV.2 ANALOGUES. OF BLOCH THEOREM In the study of the p r o p e r t i e s of conduction electrons i n c r y s t a l l i n e s o l i d s , the Bloch theorem plays an important role* This theorem i s a consequence of the t r a n s l a t i o n a l symmetry of the Hamiltonian of the system. I f the c r y s t a l i s placed i n a uniform magnetic f i e l d t h i s symmetry i s s t i l l present though we. must now consider the group of magnetic t r a n s l a t i o n s . Consequently one can obtain statements s i m i l a r to the Bloch theorem f o r electrons moving i n a p e r i o d i c p o t e n t i a l and a . uniform magnetic f i e l d * In the f o l l o w i n g we s h a l l discuss two analogues of the Bloch theorem. As i s w e l l known, the Bloch theorem states that eigenfunctions of the Schrodinger equation can be w r i t t e n i n the form e^ r k£ where JJL$ [y ) ) i n v a r i a n t under a l l l a t t i c e t r a n s l a t i o n s . proved i n the f o l l o w i n g way, Let J) is This theorem can be be an i r r e d u c i b l e representation of the group of l a t t i c e t r a n s l a t i o n s given by 4) ( e A, 4 %^a. t O-ijO, ) _c£'(*'A.+-'M;+'>i.2b) We construct the basis f u n c t i o n (jV.a.a) (r*) which generates I) by applying to any s o l u t i o n of (IV.2..I) corresponding to 3) the p r o j e c t i o n operator , Then the f u n c t i o n <U£(r) definded to be the product -e" 1 tytl*) which i s can be seen to be i n v a r i a n t under l a t t i c e t r a n s l a t i o n s * The f a c t o r & in a Bloch f u n c t i o n i s a s o l u t i o n of the Schrodinger equation f o r a f r e e e l e c t r o n . Since generates the i d e n t i c a l representa- A£(.v) t i o n of the group of l a t t i c e t r a n s l a t i o n s the f u n c t i o n l i k e the f u n c t i o n U|Q(0 ^ . generates the r e p r e s e n t a t i o n of that group. We s h a l l now obtain an analogue (due to Zak (1965 )) of the Bloch theorem i n the presence of uniform magnetic f i e l d by g e n e r a l i s i n g i n a s t r a i g h t f o r w a r d manner the proof of Bloch theorem j u s t sketched. ck v„ - J -rf Assume that the magnetic f i e l d i s r a t i o n a l , *"* — |e.|sQ. 77 ^ The i r r e d u c i b l e representations of the A b e l i a n subgroup |[4^ 2' + OI NX.+ ( l 2 i Xl \ of the magnetic t r a n s l a t i o n group are given by where £ = - 2.TT [ K i S, + /h, 5L, + O v N ^ + ^ 3 5 K + KJ G . ) T h e and L a n d a u functions are s o l u t i o n s of the Schrodinger equation f o r a f r e e e l e c t r o n i n a uniform magnetic f i e l d when /C(^) i s chosen i n the symmetric gauge but, as we have seen i n S e c t i o n IV. 1 , they do not generate i r r e d u c i b l e representations of the group { [ t>j , X l \ 119 •If to a Landau f u n c t i o n we apply the p r e j e c t i o n operator 2 "2 e N 4 -ty} (-TC^^^'^O [ ^1 (here the symbol i s an a b b r e v i a t i o n f o r the t r i p l e sum and denote the r e s u l t by representation 4. ( <fis£(r) k| <^f(y) then ' / -s, <) f [ T ki K N X Jtl ) generates the , X"] ^ . Consider now the s o l u t i o n s of the Schrodinger equation. As the Hamiltonian ^3 commutes with operators of the magnetic t r a n s l a t i o n group, i t , i n p a r t i c u l a r , commutes with operators of the group ^[ , X] \ Hence iHfi and the operators of the Abelian subgroup [[£N>XJ form a s e t of commutative operators and there e x i s t s a set of simultaneous eigenfunctions. Let ^ ( v ) be a s o l u t i o n of (l\/*2,3) which generates the representation function h/£ A of ] [.ttf , A J J . Introduce a defined by the f o l l o w i n g equation By d e f i n i t i o n and from the way £^£[r) I t i s easy to see that i s constructed we also have 120 Comparing equations Therefore i n the (IV. 2-5) and (y ) function e.* a rb'le s i m i l a r t o t h a t o f as we equation it <p£ (O have s e e n , 2>7) 1 obtain w e (see (IV. 2.4)) i n the Bloch .relation: <££(Y*) p l a y s function for, i s a s o l u t i o n of the f o r a free electron i n a uniform the Schrodinger magnetic f i e l d generates a "physical" i r r e d u c i b l e representations of and the Abelian subgroup of the magnetic t r a n s l a t i o n group. L i k e the p e r i o d i c p a r t ^£ [?) the second f a c t o r representation interesting ble W^t l v ) in representation lattice t r a n s l a t i o n s ' i n the [ g r o u p and t + w$ * 3 identical translations. representation Hence t h e y p l a y which i n , Xl i s i r r a t i o n a l we of t h e A b e l i a n ) the group electron. can repeat the subgroup of t h e m a g n e t i c t r a n s l a t i o n its.irreducible representations. Landau f u n c t i o n s It is g e n e r a t e an i r r e d u c i b , X~] j case of B l o c h the magnetic f i e l d -Ui & generates the p l a y a rb'le s i m i l a r t o t h a t o f t h e a b o v e p r o c e s s by m a k i n g u s e £ function, does n o t of t h e g r o u p case should Bloch lattice W£(*) present If 1/^(r) of the group of to note t h a t of the In t h i s case the themselves generate p h y s i c a l i r r e d u c i b l e of t h e A b e l i a n the group r o l e s i m i l a r to ^ £ ^ fl, 4 ^ 4j , XI £ l r i n the j Bloch 121 function Now we turn to another analogue of the Bloch theorem. Again we s h a l l s t a r t by assuming that the magnetic f i e l d i s rational, If ^f(v) given f u n c t i by on %# [r) i s any s o l u t i o n of tlV.2.3) ( £ - I T ( Ki t, + generates the i r r e d u c i b l e representation of the Abelian group /ftO then the )) Ki A l^' **'**' ) 1 ] ^ the vector p o t e n t i a l £L£/M, i s w r i t t e n again i n the symmetric gauge we have Let us introduce a f u n c t i o n F^(.^ ) < which i s defined by the. equation (JV.2.10) More e x p l i c i t l y i t i s given by 122 If a, + V^N £ - e A Z EN e + ^3 ^ then x*j> (-TC A ) 0 M ^ V ) y a x*k ( - T T I ^ />,W) V I'M-**); x 123 A l t e r n a t i v e l y we may w r i t e (IV. 2, |2) as or This demonstrates that remains i n v a r i a n t up to a phase f a c t o r under the group {[ , X"]j • If the magnetic f i e l d i s made to vanish by p u t t i n g o-vv =• D and N - I , i t i s easy to see that the Abelian subgroup £ f^N v ^0 ) °^ ^he magnetic t r a n s l a t i o n group becomes the group of ordinary l a t t i c e t r a n s l a t i o n operators. representation & ^> The i r r e d u c i b l e becomes an i r r e d u c i b l e representation of the, group of l a t t i c e t r a n s l a t i o n s , and the vector j£ equation i s defined in-the whole B r i l l o u i n zone. Now (J\A2,|2) becomes and hence we have the Bloch theorem as a s p e c i a l case of the above when the magnetic f i e l d i s zero. In appearance the second analogue of the Bloch theorem we have j u s t discussed resembles that introduced by Harper (1955) and discussed i n greater d e t a i l by Jannussis They looked f o r s o l u t i o n s of (IV. 2.)) Bj£(^\> However the a r b i t r a r y vector and not the vector (1964). having the form vector they used i s an / defined above. From the transformation property they obtained f o r the f u n c t i o n Q]£ w e can see that i t does not generate an i r r e d u c i b l e 124 representation of the Abelian-'subgroup £ [ magnetic t r a n s l a t i o n group, e'^" &fir) to , X lj of the Hence the resemblance of tyt I?) i s only s u p e r f i c i a l * For the case of i r r a t i o n a l magnetic f i e l d we can make use of the A b e l i a n subgroup of the magnetic t r a n s l a t i o n group and obtain s o l u t i o n s of the Schrodinger equation. 2 - 3 ) which generate ' p h y s i c a l " i r r e d u c i b l e representations of £ £ oa, 2?i + i n e x a c t l y the same manner as above. As discussed i n S e c t i o n E L 2. the p h y s i c a l i r r e d u c i b l e representations of = where M<J>{-2-n-C ) J ( KlOa, + 2 i r (^ -iTj = = are given by 5t + i ^ ^ , f 2-rra X + k "g ) 3 • a n d Using the p r o j e c t i o n operator we can obtain from any s o l u t i o n a basis f u n c t i o n ^ij-() ^ generating Of ^ r - far) of lit?) ^ (." ^> ) I< 0^ -3) 2 1 ; ' ' tW-^lS) 125 If we define the function ) by the f o l l o w i n g equation then i t has the property or where [ ^ , X'] - - W j l ^ X O Rj^tr) tlV.2,17) = V , oJ + ^ 3 ^ 3 I t i s i n t e r e s t i n g to, note that i n the case of i r r a t i o n a l magnetic f i e l d we do not get the Bloch theorem, by putting 8 or the Abelian subgroup L i x ,^Oj j-j = 0 , °^ the magnetic t r a n s l a - t i o n group becomes a subgroup of the group of l a t t i c e translations. 126 IV.3. PERIODICITY OF THE ENERGY SPECTRUM I t i s w e l l known t h a t , as a consequence of the invariance of the Hamiltonian under the group of l a t t i c e t r a n s l a t i o n , the energy spectrum of a Bloch e l e c t r o n has the p e r i o d i c i t y of the r e c i p r o c a l l a t t i c e . More•precisely, i t means that i f Bit) i s the energy eigenvalue corresponding to the Bloch f u n c t i o n e*^ ^ ( . v ) and energy eigenvalue corresponding to i ^c{£+0) r the s ug g + where O then = 2ir ( U.tfi t B (£ ) = B functions + + e'^UQir) %£ ) and s ir, , 1T 2 v t 3 =. 0, ±. I , ±2, • • • • ) , ' We also know that the twb Bloch and e ^ + ^ ' y H£+£(0 generate the same i r r e d u c i b l e r e p r e s e n t a t i o n of the group of l a t t i c e t r a n s l a t i o n s and because of the r e l a t i o n can r e s t r i c t ourselves to £ we vectors i n the f i r s t Brillouin zone f o r the c h a r a c t e r i s a t i o n of the energy values. In other words the energy eigenvalue of a Bloch f u n c t i o n can be l a b e l l e d by the same vector ^ which l a b e l s the i r r e d u c i b l e representation of the group of l a t t i c e t r a n s l a t i o n s generated by the Bloch f u n c t i o n . We have seen that the Hamiltonian ^f = ^ ( ^ 8 2 i s i n v a r i a n t under the group of magnetic t r a n s l a t i o n s W ) , If we l a b e l the energy eigenvalues i n a s u i t a b l e way we can obtain statement about the p e r i o d i c i t y of the energy spectrum s i m i l a r to that f o r the Bloch e l e c t r o n . Fischbeck ( 19631) i n v e s t i g a t e d the p e r i o d i c i t y of the energy spectrum by studying the transformation p r o p e r t i e s of a system of eigenf unctions of under a s e t of which d i f f e r very l i t t l e 127 from'the operators of the magnetic t r a n s l a t i o n group but, j u s t because of t h i s d i f f e r e n c e , they do not form a group.. He considered both the cases of r a t i o n a l and i r r a t i o n a l magnetic f i e l d s . However some of h i s r e s u l t s about the branches of the energy spectrum are i n c o r r e c t . For the case of r a t i o n a l magnetic f i e l d , Zak (1964 I I I ) obtained the same p e r i o d i c i t y r e l a t i o n s of the energy spectrum as Fischbeck by using group t h e o r e c t i c a l argument. In t h i s s e c t i o n we s h a l l discuss the p e r i o d i c i t y of the energy spectrum f o r both r a t i o n a l and i r r a t i o n a l magnetic f i e l d s making use of the-' corresponding magnetic t r a n s l a t i o n groups and t h e i r p h y s i c a l i r r e d u c i b l e representations we have obtained. Let H us again s t a r t by assuming that the magnetic f i e l d i s equal to j^j&'JJ &3 a n d consider the s o l u t i o n s of the Schrbdinger equation We have seen i n S e c t i o n IV. 2. that s o l u t i o n s of (IV-3,1) be w r i t t e n i n the form, ^= 2tr ( K i • + K»"S) can' where -tytl?) such that the f u n c t i o n generates the ^physical" i r r e d u c i b l e representation of the Abelian subgroup {[t , X ]] H (_ tn - ^ of the' magnetic t r a n s l a t i o n group. + + ) By the construction analogous to the one used i n obtaining the p e r i o d i c part <^a(.^) of the Bloch f u n c t i o n which we described i n the l a s t s e c t i o n we have obtained 128 where 2; SLTT 7 CO i s a s o l u t i o n of ( i r X •+ i t (?) a n d We shall by &(£) l a b e l the denote the . Thus as - strictly i n the by the that (i^,4) corresponding of the group If/^tr) ( l V . 3.3) {[t"N,X]^[ may may which generate. Hence, within by ° * ^ we e l e c t r o n we only f o r zone d e f i n e d , to l a b e l s of the ''physical" i s defined Tiff rela'tion t o see c a s e of the B l o c h £ p o r t i o n of the B r i l l o u i n vector outside Y WIO eigenf unctions speaking j However b e c a u s e of e a s energy eigenvalue energy eigenvalues corresponding i s e~" irreducible representations the Let define of t h a t p a r t of the B r i l l o u i n < ? for zone by the the 129 Since , where t.ft a%>0~\ •jj- - 0 , \, i 2, - N- commutes , / / then \_^K>:^~] ^ ( . O with the' Hamiltonian a s o l u t i o n of (IV.3. l) i s also with energy eigenvalue equal to E(£) , 0~] Consider , the transformation of the'/f unction under an element [t , K\ u of the group have discussed i n Section I I I . 2 | i f representation A ( J ^ As we ^ ( v ) generates the of the group { [ £N , A ] j must generate the representation to A'*"' '. {*) conjugate Therefore 1 Thus the f u n c t i o n [^zjDl apart from being a '^(.Y) s o l u t i o n to the same energy value of ^physical' i r r e d u c i b l e representation group generates the A '"^ 4 ^" \ of the { , Consequently we have Combining (IV. S . S ) and (lV-2.7) we have the f o l l o w i n g p e r i o d i c i t y of the energy spectrum where Is, , U , i % = 0, ± 1, ± 2, • • • • , a When the magnetic f i e l d i s i r r a t i o n a l i.e.where ^ H = "j^JQ^^ i s an i r r a t i o n a l quantity we can use the p h y s i c a l i r r e d u c i b l e representations of the Abelian subgroup 130 ^ [ ^ , # 1 ^ J Xl + of the magnetic t r a n s l a t i o n group to - parametrise the energy eigenvalues as before. %(y) If function i s any s o l u t i o n of (IV- S. then the obtained by using the p r o j e c t i o n operator %q^lr) corresponding to the i r r e d u c i b l e representation where ^ Hence + the group j_ given by s has the p r o p e r t i e s that i t i s a s o l u t i o n of ^^.(r) (|V> S. f) "fyC* - 2.TT (_ K^>i Kj 1$ ) and i t generates the representation { [^4, A of ,Xl] , We denote the energy eigenvalue of by Since i s the l a b e l of the'physical''. i r r e d u c i b l e representations i t s a t i s f i e s the f o l l o w i n g : - J?j = Sir (kX + ki"Z and 0 From (IV.3,^) £ K. , ^ - * i t i s c l e a r that ) I tlV.S.ID) = + <J where CTJ can 2TT (Ir,^ + 0» £ ) = extend the region given definition of 0^3,10) by by A g a i n because J^a&i, o~] t r ) [-KJ^JJ Writing use the to (IV. of = 3, II) we any value of Ef^j) ( + (W- l) value , of of >O (J* ^ (?) conjugate to with generates 0 ^ of K K l) 3 and making + and 1 3. 16) hence £ )) (|V,3.I3) ^ is irrational then I k*i + ^ ^ )^ is t h e s e t o f numbers i s a continuous d e f i n e d by Hg But everywhere dense i n t h e . u n i t i n t e r v a l that the relation i n terms of 2, - - • K outside we have ( Kt, - 0j ± ), Since Hence ^] j • Hence I ( IT to vectors eigenf unction ^^j^X, (2TC((K- +^)^ £ the E(*x). above e q u a t i o n £ for equal + %K, { = 0,± 1, ±2, - • • . commutes w i t h representation the group 3 £• Htn ) i s a l s o an energy eigenvalue t h e Ui, LT and then For from ( 0 , 4) , f u n c t i o n of £(£s ) (l\A3.t3) I f we assume i n the region i s i n d e p e n d e n t of we have the 132 Suppose there e x i s t s a value K, 0 As the set o£ numbers ^ ( ^ J . ^ ) ^ such that k\° < \ 0 and i s everywhere dense i n the u n i t i n t e r v a l no matter how small a neighbourhood one chooses around K, there always e x i s t s an integer ^ 0 i^i^j such that i s i n s i d e the neighbourhood f o r which This then c o n t r a d i c t s the assumption that Thu.s i t i s s u f f i c i e n t to use K j energy eigenvalue of; i s continuous to c h a r a t e r i s e the f o r the case of i r r a t i o n a l magnetic f i e l d . We can obtain the above r e s u l t from (|V 3. #) ^vvi considering a sequence of r a t i o n a l numbers Ni-*^ such that ( j V . 3^) T>t} ^ "J^ TTj / v " ' For then r e l a t i o n would imply the value of £•(£) or £ ( IT(K|^I + k i ^ + values of but " j ^ " ~ . OfA^ ' by kl K3 and ^ ) ) w a s independent of the p a r t i c u l a r Ki , Fischbeck has obtained p e r i o d i c i t y r e l a t i o n s with (|V. 3-7) anc * (lV.3.12) i r r a t i o n a l magnetic f i e l d s . identical when specialise<* to r a t i o n a l and He concluded that the d i f f e r e n c e between the cases of r a t i o n a l and i r r a t i o n a l f i e l d i s that' from one branch of the energy spectrum ( i . e . dealing only with the v a r i a t i o n of along ^ ) one can generate m d i f f e r e n t branches f o r the case of r a t i o n a l f i e l d and an i n f i n i t e number of branches f o r the case of i r r a t i o n a l field 133. w i t h i n the range. Q £ k\ < I , These statements are not correct because from r e l a t i o n s (IV, 3,5") and (IV, 3.n) . i t follows that the energy eigenvalues w i l l not change i f we add to the vector £ (or ^ ) any m u l t i p l e of 2irt^ . I t i s perhaps not e n t i r e l y t r i v i a l to remark that p e r i o d i c i t y conditions l i k e (IV, 3. t) and (IV, 2, i3) are meaningful only when a corresponding scheme of l a b e l l i n g of the energy eigenvalues i s given. For a d i f f e r e n t scheme d i f f e r e n t statements can be made of the energy eigenvalues of the eigenfunctions of the same p h y s i c a l system (see f o r example Harper (1955) and Jannussis (1964)). In S e c t i o n I I . 5 we have derived two s u f f i c i e n t U~i > Ji. conditions f o r two magnetic t r a n s l a t i o n groups corresponding to magnetic f i e l d s H i , Hz to be isomorphic.. Let us f i r s t r e c a l l these conditions i n the case when both Hi ; H2. (1) are r a t i o n a l f i e l d s : If S § | ^ ^ ^ = and i T i - H S where. M and (2,^) N an i n t e g e r , such that M is are <-' <*- r e l a t i v e l y prime then the groups <7i , 01 are M isomorphic. ^ MlZ^ I f = a n d K ^ + 2 | f e M ^ where rr '-f M i s an i n t e g e r then , 3s are isomorphic. We now wish to show that when e i t h e r of the above two conditions i s s a t i s f i e d the two corresponding energy spectra w i l l have the same p e r i o d i c i t y i n the r e c i p r o c a l space. When c o n d i t i o n ( l ) . i s s a t i s f i e d , M, N are r e l a t i v e l y prime, therefore the°physical"irreducible representations !J of corresponding to H, = Z must be N-dimerisional j u s t as f o r (ft Since the energy p e r i o d i c i t y depends only on N Similarly fit condition (IV. 3. fr) i t w i l l be the same f o r both cases. i f the second s u f f i c i e n t c o n d i t i o n f o r isomorphism i s s a t i s f i e d , that i s & - ^ + *m * Ha t — a — ) * * = then t h e ' p h y s i c a l i r r e d u c i b l e are both N dimensional representations of Q~i and J l and hence the corresponding energy- spectra again s a t i s f y the same r e l a t i o n (lV,3.<S) As we have seen, the second s u f f i c i e n t condition f o r isomorphism i s true without any m o d i f i c a t i o n i f we replace by an i r r a t i o n a l f i e l d i . e . Hi = J^Q J s , Then a W Z - Mi + previous M ^ a 1 5 also i r r a t i o n a l , and from our d i s c u s s i o n we can conclude that the energy spectra corresponding to the magnetic f i e l d s characterised by of Hi ' KJ Hi and Hz are but are both independent of the value k"i , The f i r s t s u f f i c i e n t condition f o r isomorphism, on the other hand, i s more g e n e r a l l y v a l i d i f Hi Ha = M Hi i n the sense because i s i r r a t i o n a l and fJ\ can now be any integer. Hence the two energy spectra w i l l again be characterised by K} but independent of the value of Ki . Thus f o r the case of i r r a t i o n a l f i e l d we can j u s t conclude th&t the two conditions f o r isomorphism do not give anything new about the energy as a f u n c t i o n of Ki and Ki - 135 The above statements involve any approximation* H> - H, y^T + are q u i t e general and does not The r e l a t i o n M ^ has p r e v i o u s l y been derived by Azbel (1963,1964) as a c o n d i t i o n f o r the energy as a f u n c t i o n of the Bloch vector to be the same i n the very s p e c i a l case of an e f f e c t i v e oneband Hamiltonian. two values of f-j In p a r t i c u l a r , the energy would be f o r the the same p e r i o d i c f u n c t i o n of £ • If we take M * ' , then we obtain a ( l a t t i c e constant) - ID~ cm. • - M <~ 10^ c 2 gauss. I t i s d i f f i c u l t to imagine a s i t u a t i o n where interband t r a n s i t i o n s can be a neglected i n the presence of a f i e l d of the order of 10' guass and the e f f e c t i v e one-band Hamiltonian s t i l l gives a v a l i d d e s c r i p t i o n of the p h y s i c a l system. 136 V.l. PERTURBATION CALCULATIONS FOR THE CASE OF IRRATIONAL FIELD The e f f e c t of a weak p e r i o d i c p o t e n t i a l on the motion of a free electron i n a uniform magnetic f i e l d has been studied by Zak (1964III), f o r a very s p e c i a l case of a r a t i o n a l magnetic f i e l d , making use of the basis functions of the "physical''irreducible representations of the magnetic t r a n s l a t i o n group. Since he was i n t e r e s t e d s p e c i f i c a l l y i n the e f f e c t of the p e r i o d i c p o t e n t i a l on the motion of the electron i n a plane normal to the magnetic f i e l d he assumed that the c r y s t a l l a t t i c e was a two dimensional square The lattice. d i r e c t i o n of the magnetic f i e l d was assumed to be perpendicular to the two dimensional l a t t i c e and i t s magnitude given by perturbation -L He then proceeded to c a l c u l a t e the energy of a s i n g l e Landau l e v e l f o r t h i s choice of the magnitude of the magnetic f i e l d . In t h i s chapter we extend Zak's treatment of pertubation c a l c u l a t i o n i n three d i r e c t i o n s : (1) We consider a l l r a t i o n a l (not j u s t ) d a n i r r a t i o n a l magnetic f i e l d s . (2) We derive s e l e c t i o n rules f o r a r b i t r a r y p e r i o d i c p o t e n t i a l and a r b i t r a r y r a t i o n a l and i r r a t i o n a l f i e l d s (3) .Despite the f l u c t u a t i o n i n the dimensions of " p h y s i c a l " i r r e d u c i b l e representations of CT when a f i e l d i s changed from r a t i o n a l to i r r a t i o n a l we can show that i f a sequence of r a t i o n a l f i e l d s tend to 1 an i r r a t i o n a l f i e l d the corresponding perturbation 137 matrices of the p e r i o d i c p o t e n t i a l a l s o tend to the p p e r t u r b a t i o n m a t r i x f o r the i r r a t i o n a l field. In t h i s chapter we s h a l l put main emphasis on the case of i r r a t i o n a l f i e l d . Since the set of r a t i o n a l numbers has zero measure i t i s reasonable to regard the i r r a t i o n a l f i e l d as the normal case and r a t i o n a l f i e l d as an e x c e p t i o n . However, the p e r t u r b a t i o n problem w i l l be discussed f o r both r a t i o n a l and i r r a t i o n a l f i e l d s on equal f o o t i n g . As we s h a l l see l a t e r , to f i n d the p e r t u r b a t i o n energy we have to solve an i n f i n i t e or f i n i t e set of l i n e a r equations according as the magnetic f i e l d i s i r r a t i o n a l or rational. U n f o r t u n a t e l y we are unable to solve these equations unless the number of l i n e a r equations i n the set concerned i s small. Because the p h y s i c a l i r r e d u c i b l e r e p r e s e n t a t i o n s of the magnetic t r a n s l a t i o n group are u n i t a r y we can o b t a i n s e l e c t i o n r u l e s without r e s o r t i n g to d i r e c t computation matrix elements of the p e r i o d i c p o t e n t i a l i n a way to the case when the group i s f i n i t e . of similar This we s h a l l do f o r a general p e r i o d i c p o t e n t i a l . To see how the p e r t u r b a t i o n m a t r i x of the p e r i o d i c p o t e n t i a l tends to a l i m i t as the r a t i o n a l f i e l d i s made to approach an i r r a t i o n a l f i e l d we assume a very simple p e r i o d i c potential. However the l i m i t i n g behaviour of the m a t r i x elements does not depend on the p e r t i c u l a r form of the 138 p o t e n t i a l chosen. As f a r as mathematical r i g o u r i s concerned, our treatment has some questionable aspects, i n the sense that we deal with c e r t a i n i n f i n i t i e s ,'in a way which we have not been able to j u s t i f y exactly. In the remainder of t h i s s e c t i o n we s h a l l ; first consider the d i r e c t computation of matrix elements of perturbation using a simple p e r i o d i c p o t e n t i a l f o r the case / of i r r a t i o n a l field.. Let us now enumerate the assumptions we make about the l a t t i c e and the magnetic f i e l d f o r i n the f o l l o w i n g discussions : (1) The l a t t i c e of the " c r y s t a l " i s a two dimensional square l a t t i c e and the p e r i o d i c p o t e n t i a l i s given by V(».fc) where V V. ( " * ^ + i s a p o s i t i v e i n t e g e r and CX i s the l a t t i c e constant. (2) The vector p o t e n t i a l ' X(^) I w r i t t e n i n the s symmetric gauge. (3) The magnetic f i e l d i s perpendicular t o the * y ^ plane. Since we are not i n t e r e s t e d i n the motion of the e l e c t r o n i n the. z - d i r e c t i o n we imagine that i t i s constrained to move on the X , ^ plane. Hence we w i l l drop the z-dependent p a r t ' i n a l l the Landau functions and consider the subgroup of the magnetic t r a n s l a t i o n s group f o r 139 which the t r a n s l a t i o n s , are r e s t r i c t e d ' to those on the' *, ^ plane. Let the magnitude of the f i e l d be given by — — ri ( | i s an i r r a t i o n a l number) then the p h y s i c a l i r r e d u c i b l e representations c a n (Jx K3 = 0 , ^3 = 0 by putting" nvt^o where of X>^''^ - r ^/ ^ • -> (TJE.2,2^) so that -1 - <o, ± I, ± 1, - — y be obtained from and The L a n d a u f u n c t i o n s 0 4 k, <£ I . ( a f t e r dropping 2-dependent parts ) A^n - P l <rf>*>] . ^ ^ ^ ) V ( 4 ^ ) f t ( 1 / ^-(Ai-y;"-\ 1 = 0, ±. 1. * 2, where and X. = <l ; . 0£ ; ^>< i "J^I i 2 r r -^t _ generate the representation. * J)^"'^ °f . We the ' »» - , shall assume i n t h i s periodic potential neglect i s weak so t h a t we t r a n s i t i o n between s t a t e s function's w i t h by corresponding d i f f e r e n t q u a n t u m number L e t us c o n s i d e r defined V^*' ^) and t h e f o l l o w i n g c a s e the matrix that may t o Landau Yi . element Vt^',^') j*<^) (v ,,.. where e ' ^ H v J ' J ) . <SU^V= I n t e r p r e t i n g the integral- over x as a d e l t a f u n c t i o n we see, that the matrix element vanishes unless one of the f o l l o w i n g condition i s s a t i s f i e d : —13" 4 ^ (yL+f- ^'-f') = 0 (U (.. l/. K 4."), To s a t i s f y ^ respectively. • ^V |AK| 0 2 j^'and ^' must be equal to I t i s easy to see that ^("r ) ^ ^(iirfO / 0,1 7 i /.4."; and ¥e s h a l l denote the expression 2irV |/U J V (.^u, p , yL,p) =. 04(i^u)B^ which i s S>' , Hence, by ®*(ty)<^j- 2 c • independent of the value of yU. Let K +f)*> J ^ l ^ f ' ) ^ and | <H, (/*Vf>>i x he the landau f u n c t i o n s f o r which ( _ 1/ Making use of we o b t a i n ^/^;//',p") = = C' (v, 1.7) (say) The expression obtained f o r the matrix elements Vl J+> p; a n d W y u , p ; yu" p") does not depend on the p a r t i c u l a r values of yu, p ; yU', ^ Tlius whatever the values of j^, p i f the c o n d i t i o n s ( V. 1.4) V (- ^•> ^'''f") yu\ p V vip+P£) are non-zero and equal t o /*'+^>f^ f o r a n y are ( p ; jJ,^') p a r t i c u l a r , i f . V ( . yk, p ; y^', ^'J =*• i n t e ; and yu"> p"\ ; ( IA 1 , t ') and s a t i s f i e d the matrix elements ; yi/', p' and ; and C , y In then g e r Prom the d i s c u s s i o n s i n S e c t i o n f• IV, \ we know that the space / f t ^ ) spanned by the set of f u n c t i o n s 142 L OL Dj ±. I, ± 2 , • • • for representation ^>^ { Oj. j where WtJ> the spaces ^ ( k V ) and ^ ('0 {K(A + P')A> 1 t ( which imply a r e {£ j 0 I*? = ^ [ f ^ f - and yu" I, o') 'spanned by the f u n c t i o n s / respectively with (_ and.. J) ^ a n d 0 f X.J ^^./V such that c o n d i t i o n s , I f there ( _ y. <. 4) and that K",' = i n t e g e r + ^ X ( i n t e g e r ) Ki — Ki" = integer- + This means that k\, K"/ , K*| W satisfy (integer) (,IE 2 . 2 ) Consequently the spaces the c o n d i t i o n V f o r two p h y s i c a l i r r e d u c i b l e r e p r e s e n t a t i o n s equivalent. g e n e r a t e J s a t i s f i e d then k", — and Ki =• l ^ ) ^ , S i m i l a r l y { I'V l/-+p>>*.!l , ( u « o * - l ^ > » ) > the r e p r e s e n t a t i o n s exist generate the /<((K/) of and t° be /{(^/'J which are connected by non-zero matrix elements to ^ ( K i ) generate p h y s i c a l i r r e d u c i b l e r e p r e s e n t a t i o n s equivalent t o that generated by yi ( K i ) Let us represent f o r a given ^ the Landau f u n c t i o n s and a l l p o s s i b l e values of y> [ |S (^•+^)&)>x\ and p p o i n t s on a h o r i z o n t a l l i n e i n the f o l l o w i n g way. an o r i g i n on the l i n e and represent by We choose the f u n c t i o n | fo, (ppOx by the point whose distance from the o r i g i n i s given by 143 £y^-vp)a, C o n d i t i o n s any given f u n c t i o n and (\A I > 4-') ( V. I • 4-) ( K\ + j> o) | thus mean that I s connected by non-zero 0 matrix elements t o two Landau f u n c t i o n s whose r e p r e s e n t a t i v e p o i n t s on the h o r i z o n t a l l i n e are separated from ^-^J^ , by d i s t a n c e s Let (j^o-v^o")^ tfc+f!)-^ (/Ao+p.') Y ~ /* f'' + l+ = /^+(>-! , and i n general C ^ = 0, * l, ±2, '). + f^O^i 3The p o i n t s of the set £ are separated by d i s t a n c e s equal t o m u l t i p l e s of one another. I t i s c l e a r that ^) V from has non-zero matrix element between Landau f u n c t i o n s corresponding t o two neighbouring p o i n t s of the set.. Denoting ("[ f>'tf)f by K T the corresponding spaces » » and L ^ - j ) by ^ (K ^ > ) then 1 ^ (^"J* ^ • - - ^ ( k / 1 ( are connected by non-zero matrix elements. f o r any space /£ [x^) h","^ 0 ; More p r e c i s e l y , i n the sequence there are non-zero matrix element of \A*<^) connecting i t t o /L[K^ ) and ) but no other. Prom the above d i s c u s s i o n i t should be c l e a r that the m a t r i x of VL*-J ^) spaces has the f o l l o w i n g form ..^(fc^; .• i n the i n f i n i t e d i r e c t sum of the 144 l-O 8/ c 6' In the above we only show the m a t r i x elements between the spaces /U^"/" ) , A^^) and 0 J> { sequence of spaces j u s t mentioned. f o r an i n f i n i t e m a t r i x . ) among the i n f i n i t e Each block here stands The block l a b e l l e d by {^'j K"/°') J f o r example, gives the m a t r i x elements of Vt^>^) space JlK ) LD> spanned by £| (^+ ) ^> j , ^ * o ± i , ± v ) . x The block l a b e l l e d by I*,® , KJ-°) between the f u n c t i o n s £ I -, (^ + ^ o) X : \ I'S ( M fi') °- \ 01 i n the y gives the m a t r i x elements A which span the spaces and A.(K[ ) 6> a n d 145 (.K^) respectively. The non-zero m a t r i x elements "between d i f f e r e n t subspaces are a l l euqal to C ; and a r e , i n g e n e r a l , o f f - d i a g o n a l i n the corresponding b l o c k . The m a t r i x elements between b a s i s function's of a given space are equal and along the diagonal of the b l o c k . The symbol ^ matrix elements i n the block used to denote i s given by Co4 ( 2 i r i r p , / : ) 8 ' (1/, I t i s easy t o see t h a t . t h e i n f i n i t e p e r t u r b a t i o n m a t r i x ( U, 1.^) can be decomposed i n t o an i n f i n i t e number of i d e n t i c a l submatrices each of which i s given by 0 0 d 0 0 0 d 0 o 0 0 0 0 0 0 0 0 • 0 0 0 > 0 0 0 d 0 0 0 0 0 * d B: c 0 0 0 0 • 0 0 d si d 0 0 0 > 0 6 0 d d 0 0 > 8* d Prom the matrix ( IA I. the p e r t u r b a t i o n energy E ( we can see that we must solve the f o l l o w i n g i n f i n i t e set of homogeneous l i n e a r equations: UO) 146 where ( £ =. o \, ±2, y numbers and Q =• * ) and Ei are the unknown J c', The set of values of E\ obtained by s o l v i n g (I/. gives the spectrum of energy lev-els i n t o which a Landau level with fixed i s broadened under the i n f l u e n c e of the perturbing p o t e n t i a l Vt*/^-). Each of the 3© energy l e v e l s i s s t i l l i n f i n i t e l y degenerate because the same i n f i n i t e set of equations (_V. occurs an i n f i n i t e countable number of times. Although each of the l i n e a r equations of the i n f i n i t e set i n (. l/> I. i s very simple we have not been able to f i n d a method to solve them. From the above d i s c u s s i o n we can see how the energy l e v e l s of a f r e e e l e c t r o n i n a magnetic f i e l d are a f f e c t e d by the presence of a p e r i o d i c p o t e n t i a l . set the p e r i o d i c p o t e n t i a l VC*, ^) To do t h i s we equal to zero. first In t h i s rv case the Hamiltonian commutes w i t h the group (\J± whose elements are [_ VI , X J plane and X where W i s any r e a l number such that For a given value of { IN $ ^ j i s any v e c t o r on the x (~^<^<»>) 0 X < \ . the set of Landau f u n c t i o n s generates an i r r e d u c i b l e r e p r e s e n t a t i o n oi Vj., T h i s r e p r e s e n t a t i o n i s i r r e d u c i b l e , because f o r any two f u n c t i o n |<vt, j^,> an element of Since j§ which transforms and |S,^> | ^, ^ to there i s |<i\., ^ )> 2 can be any r e a l number ( i n some u n i t of length) 147 the dimension of the r e p r e s e n t a t i o n of 4/i. generated by the Landau f u n c t i o n s i s non-countably i n f i n i t e . I f the p e r i o d i c p o t e n t i a l ]/L >fy) i s introduced, x the H a m i l t o n i a n commutes w i t h the elements of the subgroup ^Jj_ but not w i t h a l l the elements of V i The r e p r e s e n t a t i o n space formed by the set of Landau f u n c t i o n s is [,l^ ^>\ l now decomposed i n t o subspaces, according to the i r r e d u c i b l e r e p r e s e n t a t i o n s of C L which are countably i n f i n i t e dimensional. Each subspace i s c h a r a c t e r i s e d by a K to the i r r e d u c i b l e r e p r e s e n t a t i o n ^ ' ^ k { generates. Vl't'ty) corresponding of Gx which i t We have seen that the p e r t u r b a t i o n m a t r i x of connects a countably i n f i n i t e number of subspaces Jl*n (e.g. (. V, I. t as i n /ILK?"), ) , and they a l l generate equivalent >—- r e p r e s e n t a t i o n s of Qj. • The m a t r i x of V(*< Al^) > v v s ' ^) (see (V, l , ^) i n the subspaces J (Ki L t > J )t Ji[^) ) can be decomposed i n t o an i n f i n i t e (countable) number of i d e n t i c a l submatrices each of which i s given by ( V, l . l i ) • By s o l v i n g the i n f i n i t e set of equations ( V. I. a r i s i n g from (V, I, II) we get an i n f i n i t e (countable) number of eigenvalues B\ which give the energy values of the l e v e l s i n t o which the degenerate Landau l e v e l s i n Jlk\ ) , A(^) n s p l i t by the p e r t u r b a t i o n . ,A(^°) > are However each of such l e v e l s s t i l l has a countably i n f i n i t e degeneracy. we o b t a i n the same set of equations ( I / . 1, I I ) T h i s i s because accountably i n f i n i t e number of times from the p e r t u r b a t i o n m a t r i x J 148 CV, of I n other words from ( V, I, °\ ) we o b t a i n a countably i n f i n i t e number of energy l e v e l s each of which i s countably i n f i n i t e degenerate. We can c h a r a c t e r i s e t h i s set of energy l e v e l s by the set of k/s J K, ^ K^, 1 which l a b e l the subspaces i n v o l v e d . Of course, we can repeat e x a c t l y the same procedure f o r another sequence of subspaces c h a r a c t e r i s e d by another set of K * 1 such that the subspaces i n the sequence generate e q u i v a l e n t rv i r r e d u c i b l e r e p r e s e n t a t i o n s of \Ji countably i n f i n i t e number•of l e v e l s . p o s s i b l e v a l u e s of k/s and thus o b t a i n another Because the number of ±Q non-countable we s h a l l have non-countable number of sets of l e v e l s c h a r a c t e r i s e d by sequences of K"' s obtained i n the way we have explained. S i m i l a r l y we can d i s c u s s the e f f e c t of p e r t u r b a t i o n produced by VL*> fy) when the magnetic f i e l d i s r a t i o n a l . However, we s h a l l not repeat the above d i s c u s s i o n . a n t i c i p a t e the r e s u l t s of S e c t i o n V. 3 following irrational We s h a l l and j u s t s t a t e the d i f f e r e n c e s between the cases of r a t i o n a l and fields. For the case of r a t i o n a l f i e l d the p e r t u r b a t i o n m a t r i x of VLx,^) dimensional. corresponding to (. V* 1. i s finite The d i f f e r e n t energy l e v e l s obtained by s o l v i n g a f i n i t e set of l i n e a r equation obtained from the p e r t u r b a t i o n m a t r i x i s now N - f o l d degenerate ( KI i s the dimension of the "physical" i r r e d u c i b l e r e p r e s e n t a t i o n s of rv J± ) However the number of such sets of l e v e l s (each s e t l a b e l l e d by a f i n i t e sequence of (.K,K )'s z ) is still non-countable 150 7.2. SELECTION RULES POR THE CASE OP IRRATIONAL MAGNETIC PI ELD In t h i s s e c t i o n we s h a l l d e r i v e s e l e c t i o n r u l e s f o r the p e r t u r b a t i o n produced by an a r b i t r a r y periodic- p o t e n t i a l V(*) making use of the 'physical" i r r e d u c i b l e r e p r e s e n t a t i o n s of We s h a l l do t h i s only f o r the case of i r r a t i o n a l magnetic f i e l d since e x a c t l y the same procedure a l s o a p p l i e s to the case of r a t i o n a l f i e l d . . Let Vi?) be any three dimensional p e r i o d i c p o t e n t i a l and the f u n c t i o n ^jpf*''* ^ 1 / ^ " " ^ be b a s i s f u n c t i o n s belonging to the 1 and i r r e d u c i b l e representations JLrk ]}t columns of the p h y s i c a l '') K l > K3 a n ^ -^t^/KiJ^ (HI, 2.2^) ) of the magnetic t r a n s l a t i o n group ( J" , s e e (Note we are d e a l i n g now w i t h the whole group and not the subgroup Oi of the magnetic t r a n s l a t i o n group). elements C of ? the f u n c t i o n s f- Thus under any ( K | > K j ) and 7 ^ c k ; , K ° transform as f o l l o w s and - 2 V 2VA (0-J Assuming that: matrix elements -like ( ^ exist; we consider the f o l l o w i n g i n f i n i t e sum L ^ V ; , Vl* )^/ ' - j 151 where the summation Using (V, 2,1) let to and ^ - .S J i s c a r r i e d over a l l elements of CT . (V. 2.1') [o^#i+ ^ ^t + 2 we can w r i t e y ^s^3 )0 then according ll2.2p and (K'', «i) r . .j, !1TTJ(>-^7 ( 1+ T ) ) -2.7U £ C Hence 0~ A ^1 »ij I t then f o l l o w s that 5 —1 " y 152 1 1 A f >_> o / - S where cr i s an i n t e g e r determined by the equation if k". = + I (f (K/ - X ) )^ ^ The above r e s u l t s can be summarized by the f o l l o w i n g two equations: if .p and X) are i n e q u i v a l e n t ; and I if i) and The symbol Old) <D are e q u i v a l e n t . i s used to denote the. order of the group j \ 153 Prom ( V- 2, i') i t i s apparent t h a t the value of the matrix element i s independent of X X i s i n f i n i t e and so i s 0(\7) Hence equation (V, 2.£ j / hence the sum over the order of the group. need not "be i n c o n s i s t e n t ; i t i s f o r m a l l y analogous to t h a t expressing the s e l e c t i o n r u l e s i n the case of f i n i t e groups. We should l i k e to emphasize t h a t the argument l e a d i n g to equation ( 2, L ) ^ (_V. 2. b') as l a c k of r i g o u r i s concerned i s no worse as. f a r than the corresponding equations obtained by d i r e c t computation, given i n S e c t i o n V, I f o r the case of the very s p e c i a l two dimensional p o t e n t i a l . I t i s a l s o c l e a r that the s e l e c t i o n r u l e s ( V. 2 . £ ) (V>2.£ ) / t are a l s o v a l i d between Landau f u n c t i o n s w i t h d i f f e r e n t values of m, 154 V.3. PERTURBATION CALCULATIONS POR THE CASE OP RATIONAL MAGNETIC FIELD Let us now consider the p e r t u r b a t i o n c a l c u l a t i o n f o r the case when the-, magnitude of the magnetic f i e l d i s given by We s h a l l a l s o make the same assumptions -rr-n> -?r- about the l a t t i c e , the d i r e c t i o n of the magnetic f i e l d and the p e r i o d i c p o t e n t i a l as those enumerated i n Section. Vol., K3 =» Q By p u t t i n g 0i and ^physical" i r r e d u c i b l e r e p r e s e n t a t i o n i n (IT. Q.17) =0 3 ~^ ^" -' ) ki > ) 0 f (j^ given by (K, Hi, 1) r where j , ^' -A and , . = 0 , 1 , , 2 -1 , N-l 3 0 * K.yi-ft and In S e c t i o n |V.I I ^ / K ^ K ^ ) , we have seen that the s e t of f u n c t i o n s C^ =- &, 1,1, - • / N-'\) which are given by the i s 155 generates t h e i r r e d u c i b l e r e p r e s e n t a t i o n (V'2,z) In the r e l a t i o n between and the Landau f u n c t i o n p and K K> KiN + y* + )^> g is has the f o l l o w i n g e x p l i c i t form 2 where 1 A-,* /' - 2E a 1 N Since the b a s i s f u n c t i o n s [J*= o,1,2, - ,H-\) are i n f i n i t e sum of Landau f u n c t i o n s , the matrix element of V/tx, ^) between any two of these f u n c t i o n s w i l l i n general diverge. Here we introduce two a r b i t r a r y steps which we do not j u s t i f y . When e v a l u a t i n g matrix elements of we s h a l l consider the f u n c t i o n K,, k"» as f o r some l a r g e p o s i t i v e i n t e g e r JY\ K can be w r i t t e n as 2 - ^ / <>L 4 ^ 2 \Jl%i^) We a l s o assume that where cx i s some i n t e g e r We s h a l l see that the matrix elements of i> are independent of v/Y*, . Consider the matrix element of V(>/^) between the 156 functions and [U |ft,ki, *>. ^ ) 4-) 3, and j ^ ' ) , Using (V, 3,2) we get vY X where V V = lc>k) -2TTC f) f , Prom the i n t e g r a t i o n over -sc ( d e l t a f u n c t i o n ) i n (y, 2.5) we see that except when one of the f o l l o w i n g c o n d i t i o n s i s s a t i s f i e d 157 ( i . e . K", = K/ p = p' If ) only the l a s t c o n d i t i o n can he s a t i s f i e d and i n t h i s case _L V = ^ 2 V a.W (KaV- Ki na.) f Vo r I A^r air S"^' > 06 Hence i n the space spanned by the f u n c t i o n s where yU = o, 1 , 2 , elements of Vt>/^-) ^-l K, ki. yU) there are only diagonal m a t r i x and these elements are equal to I f we denote the highest common f a c t o r of jr and 'K. by [vr,^ where and and ^ / ( r , ^ ) by ^fi , )fr , ^ 04 <1 M , I f we regard we may w r i t e are i n t e g e r s such that b ^ ^ <1 ^ We can now w r i t e c o n d i t i o n ( V . 3,£) -t-^*-+ ^ as; as given then we can see that i n order to s a t i s f y the above c o n d i t i o n we must have V.3, where and 6 = 0 Hence, when (\A I vc*y /A i s s a t i s f i e d we o b t a i n i ' -' KV K V ID") 159 CHr 2- I f the c o n d i t i o n (y^l^o') oV N + ^' -v p + / 2 and i s satisfied i.e, = ^K/ 4 ^ + p A' where 7 >y D and we o b t a i n i n a s i m i l a r way p- frz-sL < o 160 We s h a l l now show t h a t t h e above c o n d i t i o n s f o r t h e non-vanishing of m a t r i x element of V(*>^) ean a l l be summed by s a y i n g t h e m a t r i x element can be d i f f e r e n t from zero o n l y when the s t a t e s c o n s i d e r e d belong t o e q u i v a l e n t representations. From ^ * ( p ± J l) V -ftp* I f we denote ^ )^ Let ^(K^kj.) ^y + ^ then and / f f i c ' / K t ) the s e t s of f u n c t i o n s £ 1^,^,1^ D, i , 2., ^,N-I . Comparing see t h a t ^(^I.KJ.) irreducible i t follows be the spaces j a n d (y. 3 J 2 ) and yi(k/, k,.) r e p r e s e n t a t i o n s of spanned by {j^/*" '' ^ a , ^ ] where 1 (ut.Q. I * ) with generate equivalent ^physical" . Now, as i n t h e ease of i r r a t i o n a l magnetic we want t o f i n d a s e t of spaces space /£( K i ^ , Kj) 1 we (K, K ) ^ a field, of which a. g i v e n i s a member such t h a t f o r each space i n t h e set t h e r e e x i s t non-zero m a t r i x elements of Vi^, ^) 1 c o n n e c t i n g t h i s space w i t h a t l e a s t another member of t h e set. From has non-zero (^V. 3.»l) and m a t r i x elements the c o r r e s p o n d i n g v a l u e s of 3.H') we can see t h a t Vl*'Jf) between two spaces o n l y when are equal. Hence t h e s e t 161 of spaces we want t o f i n d must have the same value of Let us assume that the set of "basis f u n c t i o n s of J± (^^\ K ^ ) where .K^ are given by =• ( . j l From c o n d i t i o n s we have seen that ^(kj- , kl) Di elements of /Al K/ 1 i s connected by non-zero matrix t.o only two other spaces \ft >^) y ^v) where The space ( V, 3. O , (V.2.A'J /LLkP,Kx) and and K^i are given by , K"j.) i s spanned by the f o l l o w i n g set of functions where j>, - ( p « + - ^ 7 . the space J l { , to and Jlk^jK*) ) where I n a s i m i l a r way we can see that i s connected by non-zero matrix element 162 Repeating t h i s process we can o b t a i n a sequence of spaces A*? «0 , A *.), - --^(ir^U) • • • where and such that /dtO') ( V. 3,14) \/Li>^) has non-zero m a t r i x element connecting and ^ ( l ^ + ' p However' not a l l the spaces i n are d i s t i n c t . we put ^ " Ow 1 E h i s i s easy t o see, because i f i n ( V, 3, \S') we o b t a i n =A ^ Prom the f a c t that (otherwise ^ U"!^(J^) "jja and <v' and Ov/ =^ 0 *>•) must be r e l a t i v e l y prime could not be r e l a t i v e l y prime), i s the smallest value of f o r which Hence we have obtained a sequence of rtV d i s t i n c t . A K V o , M*?> such that Vd'ty) and hence A K spaces 0 has non-zero m a t r i x elements between any two neighbouring spaces i n the sequence i f we introduce the convention that the last, space and the f i r s t space i n the sequence are neighbours. In o b t a i n i n g the spaces i n ( I / , 3«'0 made use of (\A S, i3>) from I V 3, we have only and not IV\ 3, 13') , I n a s i m i l a r way, we could a l s o o b t a i n ^ spaces 163 d i s t i n c t from one another and possess.; the same p r o p e r t y as those i n [V, 3,|{) and . The set (V, J, i&) we j u s t d e s c r i b e d . ,p,-4£), Here we have t of spaces i s the same as the s:et [\J,Z>ll') the only d i f f e r e n c e i s that the spaces are arranged i n a d i f f e r e n t Order. and hence I n fact., since Vf/'P - (1/2, i<0 K", '~ 7^ (0u GTonsequently we can say: each space y l i ^ j ^ ) i s connected only to two spaces J± [ Kv) i n (1/ and ^ ( ^ v ^ K L ) - 1 by non-zero m a t r i x elements of Vfx,^) . No space i n the sequence (I/, i n the sequence. L V- 3. i(>) i s connected i n t h i s way to spaces not It:- i s a l s o c l e a r that, the spaces i n a l l generate e q u i v a l e n t ' p h y s i c a l ' i r r e d u c i b l e r e p r e s e n t a t i o n s of (Ti . Prom the above d i s c u s s i o n i t f o l l o w s that the case considered by Zak (1964 I I I ) , ^ - 1 one. i s a very s p e c i a l I n t h i s case a. g i v e n space' J i ^ W ^ a ) i s not connected to any other space by non-aero m a t r i x elements of 164 V C^/^) / / i n c o n t r a s t t o the general ease where (U^'i^) spaces w i l l "be connected i n the way we have d e s c r i b e d . Before we w r i t e down the ^'H * ^ ' N i n the space formed by the d i r e c t sum of the Vi^/ft) spaces m a t r i x of Xl*'* 1 '**) and A * *'' > i ,l >) l e t us t u r n back f o r a moment t o [\A and I (V. S.ll) Prom these equations we can see that the m a t r i x of i n each space i s a scalar matrix. IV.3.IO N W*/^) The d i a g o n a l element depends on the value of K , ^ only through the term co< (2T|>^) j u s t as (V. 1.5") matrix elements of V ( * / ^) i n the i r r a t i o n a l case. The connecting two d i f f e r e n t spaces can only have two values which d i f f e r by a f a c t o r However we can apply u n i t a r y t r a n s f o r m a t i o n t o the b a s i s f u n c t i o n s of ^(K, °, ^) , 1 L{K^,^) y a l l the non-zero elements of , / JL (k,' "?K>) t o make v/ between two d i f f e r e n t ] / L * i ^ ) spaces equal. To i l l u s t r a t e how t h i s can be done l e t us consider the f o l l o w i n g example. ^(KiyKV) as /f(K,) Prom now on we s h a l l simply w r i t e where the dependence on Ka is understood. Example Let ix- 1 y The i n t e g e r s 71 = > V- > Ys s and i n t h i s p a r t i c u l a r case * s f 7 *Ou = y a r e on- ~ defined by the r e l a t i o n 4- t h e r e f o r e we have 165 We. c a l c u l a t e the values of using ( V, and the corresponding K 's t and o b t a i n iff , > Jr K | ^ /J (K^- T h i s means that the subspace ^ is connected through non-zero m a t r i x elements of \A*/ ^-J the subspaces other. S^),/^- | S) and /4 a n The m a t r i x of Vl*^) t h i s case). (the -ralue of i n the space can now be obtained by means of ( V 5 , "]) I f we denote by 2 X 1 / 0 ./ * r and denote by = d no S t a r t i n g fr.om any subspace J LK ) the number of subspaces thus connected must be 4 where to TT By (^ = | A K | ^ ( ^ , , ^ C ) 0,1,2,3-) J ( £j the expression and ( V. the expression &('r) f d in 166 the m a t r i x of has the f o l l o w i n g form: \A*, X, c c" c c c Bp c c c \\ c + c c~ c s, c c c 6, C c C" c c c and c C c 13. c are defined by 2irv Kt e + r c C c C \\ c c Here c B, c c c c C C (-0 _ C ^ = e -2-irvKi . C As we have discussed i n the general case the nonzero matrix elements of by the ordered p a i r [k^, VL*>ft) K, ) 1 4 i n the block c h a r a c t e r i s e d 0 ) a r e not i d e n t i c a l . However the d i f f e r e n c e i s only a phase f a c t o r Now we want t o d e s c r i b e how we can f i n d e ^ 3 u n i t a r y t r a n s f o r m a t i o n s of the b a s i s f u n c t i o n s of 2 ^'-0 167 and / ( O yllvcp) such that non-zero matrix element w i l l be the same i n each o f f - d i a g o n a l block of Consider the f o l l o w i n g 3 (IA3.2I) u n i t a r y matrices ^ defined i n the f o l l o w i n g way Uy * = where ^ , + Por any p o s i t i v e number <: < ; (-fP\ i J & ) g # i s defined as the , (*)j=- X, l a r g e s t i n t e g e r l e s s than ji^ - I , i ^ - 2. By d i r e c t s u b s t i t u t i o n we o b t a i n - 3, i J Applying these u n i t a r y transformations to the b a s i s f u n c t i o n s of ^ [ Ki ' ) l J , JL [V?* ) /L t [j^, V^ > ) respectively ) i s equivalent t o m u l t i p l y i n g some of the b a s i s f u n c t i o n s by the f a c t o r the f u n c t i o n e "- 2irvl<l p o r instance by a p p l y i n g /• K , ° > ) i s transformed to [ ) ^ to /f(^, J w J w h i l e a l l the other four f u n c t i o n s remain unchanged. because the b a s i s f u n c t i o n s of J> { k[ ) D> m a t r i x element [4-, 0) i n the block ^ , D) Now are u n a l t e r e d the i s changed 168 to C from C w h i l e the other elements i n the block do not change. When \'«^\o) e U W and |K^\ I) - lirCUl are changed U l 2 ) K®) both same f a c t o r . block [k^J to the f u n c t i o n s e ' ^ ^ l kf, o) r e s p e c t i v e l y . The matrix element i n the block and i s a p p l i e d to /dCk,^) n d (o, i) w i l l not change because under 0 |^'\D) and ]KP,\) are m u l t i p l i e d by the However the m a t r i x element ( 4 , 0 ) ^ i ^ ) i s changed from C"* to C , way we can see the only change i n the block that the element a ( 4 , o ) becomes C i n the In a s i m i l a r ( _ Kr ' ^ J i n s t e a d of i C . + F i n a l l y i t i s not d i f f i c u l t t o check t h a t a f t e r the c a r r y i n g out the t r a n s f o r m a t i o n s t h e non-zero matrix elements of the ( (k,^ , K^) block a l l become C , I t i s c l e a r that the + diagonal matrix element of ( IAS."* ) 1 are unaffected by the transformations, The above method of c o n s t r u c t i n g u n i t a r y t r a n s f o r m a t i o n s to make a l l the non-zero m a t r i x element of V(*'^) between two subspaces t /4(k:T°J a p p l i c a b l e no matter what are the values of J\J. is u" , ^ and We can a l s o c y c l i c a l l y permute the b a s i s f u n c t i o n s i n the subspaces Al ^)j K /i(^) and ^ (kf") such that non-zero m a t r i x elements appear only along the diagonal i n each b l o c k . Consequently, to f i n d the e f f e c t on a given Landau l e v e l produced by a weak p e r i o d i c p r o t e n t i a l f o r t h i s s p e c i a l case we have to solve the c h a r a c t e r i s t i c equation corresponding to the f o l l o w i n g p e r t u r b a t i o n m a t r i x . S 169 c 0 c~ C e, c 0 Q c a, c 0 c i3 In g e n e r a l we As are 6 have an ^ i n (V. I, i l ) three non-zero matrix where o c c? 0 •Bj r S3 dimensional perturbation matrix. i n e a c h row of the m a t r i x elements. i n the general (\A3.22) The there form of the p e r t u r b a t i o n case i s 0 0 - -- C,~ c, 0 ... o - -- o 0 2ir VD I 0 A*!* 6o4 ^ ( 2 J L & ) (I/. 3.24.) and C = e (In the s p e c i a l Similarly C, + case considered and C, are defined = o by hence C i s real) 17 0 We are unable to solve the c h a r a c t e r i s t i c equation of (1/.3.2-3) e x a c t l y except i n the few cases of lowest order. 171 V.4. COMPARISON OP PERTURBATION MATRICES We s h a l l now c o n s i d e r what happens to the p e r t u r b a - t i o n m a t r i x [ y , 3. 2 3 ) i r r a t i o n a l number ^ when we allow ^/N i n the manner d e s c r i b e d i n S e c t i o n \V We w i s h t o show t h a t i n the l i m i t (V,2,23) to approach to ar. —>^ X the m a t r i x a f t e r making s u i t a b l e u n i t a r y t r a n s f o r m a t i o n s approaches to the m a t r i x ( V. 1. I ) for irrational 1 First field. of a l l l e t us perform the u n i t a r y t r a n s f o r m a t i o n which corresponds to permuting the rows and columns of (IM.23) t o o b t a i n the f o l l o w i n g m a t r i x 0 0 0 0 0 0 c) c, (For - - - c convenience we have assumed t h a t i s not e s s e n t i a l f o r our argument.) column from \^-~ 4 1 ) )k [V. 4,1^ to by the f a c t o r c o r r e s p o n d i n g rows by e ^ T v 1 TV. - ) i s odd, but i t We m u l t i p l y (o^'->)ri (1/4., I) each of the m a t r i x and each of the and o b t a i n the m a t r i x i n (U 4,2) 172 0 +1 B 0 C, C Bi 0 2 / The reason f o r doing t h i s i s to remove the elements C , C, + to the f i r s t row and f i r s t - column of the m a t r i x r e s p e c t i v e l y , According to j = 0, 1 , 2 , where the m a t r i x element 8y and s i n c e (V.A..2.) I K 3,I%) p - r as ~ f ^ P * = fl / . = (p 0 - f Ji£) for , i s given "by (see ( \A 3 . 2 ^ ) we may w r i t e B . , . ^ Because ) ( where Now we can r e w r i t e 173 2 c, s 0 c, (1/4.3) 8., c, 0 c 4- c;I 3p. ^1 Let us r e c a l l that N —^ When large 04 t h e r e f o r e f o r hi ^fj must v a n i s h , and, from ( V. l.iS') sufficiently we o b t a i n C, In the l i m i t of 77 — * ^ where '/p' z= 7? ^ , Ci w i l l approach the value which i s j u s t the o f f - d i a g o n a l 174 element c' the p e r t u r b a t i o n of i r r a t i o n a l f i e l d . A l s o , when ^ — * ^ according to (\/-3,.24-) where 0 Since as o v — > c* , in }r the expression becomes = ( f + ^ )f • value of £^ of (v*. )• \i) f or the case matrix Thus *V approaches the ( ^ i. 11) . is T'hus as fixed therefore —•> ^ ( V. 4, 3) = becomes. . In f a c t the r e s u l t s of t h i s s e c t i o n can be r e a d i l y extended t o the case of a general p e r i o d i c potential. Hence i n c a l c u l a t i n g the e f f e c t of a weak p e r i o d i c potential on the Landau l e v e l s we may regard the case of i r r a t i o n a l magnetic f i e l d as a l i m i t i n g case f o r r a t i o n a l magnetic, fields. 175 BIBLIOGRAPHY Ashby, N. and Miller, s.C. 1965, Phys. Rev. 1£9, A428. A z b e l , M.Ya. 1963, J.E.T.P. (U.S.S.R.) 17, 665. A z b e l , M.Ya. 1963, J.E.T.P. (U.S.S.R.) 19, 634, Brown, E. 1964, Phys. Rev. 133, A1038. Chambers, W.G. 1965, Phys. Rev. 140, A135. Chambers, W.G. 1966, Phys. Rev. 149, 493. Cohen, M.H. and Palicov, L.M. 1961, Phys. Rev. Lett. 7, 231. Clifford, A.H. 1937, Ann. of Math. 38, 533. P i s c h b e c k , H. J. 1963, Phys. Stat. Sol. 3, 1082. Pischbeck, H.J* 1963, Phys. Stat. sol. 3, 2399. Harper, P.G. 1955, Proc. Phys. Soc. A68, 879. Jannussis, A. 1964, Phys. Stat. Sol. 6, 217. Johnson, M.H. and Lippmann, B.A. 1949, Phys. Rev. 76, 828. Koster, G.P. 1957, Solid State Physics ed. by P. Seitz and D. Turnbull , Vol. 5 (Academic Press, New York) Landau, L. 1930, Zeits. f. Physik 64, 629. Lomont, J.S. 1959, "Application Of Finite Groups", (Academic Press, New York). Mariwalla, K.H. 1966, J. Math. Phys. 7, 114. Opechowski, W. and Guccione, R. 1965, "Magnetism" ed. by S.T. Rado and H. Suhl, Vol. IIA (Academic Press, New York). Peierls, R.E. 1933, Zeits, f. Physik 80, 763. Pippard, A.B. 1962, Proc. Roy. Soc. (London) A270, 1. Pippard, A.B. 1964, Phil. Trans. Roy. Soc. London, A256, 317. Raghavacharyulu, I.V.V. 1961, Can. J. Phys. 39, 830. Tam, W.G. and Opechowski, W. 1966, Phys. Lett. 23, 212. 176 Wannier, G.H. 1962, Rev. Mod. Phys. 34, 645. Weiss, G.H. and Maradudin, A.A. 1962, J . Math. Phys. 3, 771. Wigner, E.P. 1959, "Group Theory And I t s A p p l i c a t i o n To The Quantum Mechanics Of Atomic Spectra" (Academic P r e s s , New York) Chapter 26. Zak, J . 1964, Phys. Rev. 13_4, A1602. Zak, J . 1964, Phys. Rev. 134, A1607. Zak, J . 1964, Phys. Rev. 136, A776. Zak, J . 1965, Phys. Rev. 139, A1159. 177 APPENDIX I ADAPTATION OF CLIFFORD'S RESULTS FOR THE CONSTRUCTION OF THE INFINITE DIMENSIONAL PHYSICAL IRREDUCIBLE REPRESENTATIONS OF C l i f f o r d (1937) 'had shown how one could by s t a r t i n g from the i r r e d u c i b l e representations of an i n v a r i a n t H of an a r b i t r a r y group representations of (l) Or O construct the i r r e d u c i b l e His method depends on two assumptions: the i r r e d u c i b l e representations of J-j dimensional and (2) representation of f i n i t e index i n subgroup are f i n i t e the l i t t l e group of any i r r e d u c i b l e H r e l a t i v e to 0 and fj has a dr ,.• ( C l i f f o r d does not use the term " l i t t l e group", which i s more recent). In constructing p h y s i c a l i r r e d u c i b l e 7 of representations we f i r s t f i n d the one-dimensional p h y s i c a l i r r e d u c i b l e representations of an Abelian i n v a r i a n t 7" of subgroup With the one exception discussed i n Section H . 2 . J", the l i t t l e group of any physical" i r r e d u c i b l e representation v, { r e l a t i v e to Q rv J~ and rv !j is 7 Q itself. However unless the magnetic f i e l d i s r a t i o n a l the index of 0~ infinite. cases. in J is Hence C l i f f o r d ' s theorems do not apply to these In t h i s Appendix we s h a l l prove two theorems which are adaption of some of C l i f f o r d ' s r e s u l t s f o r groups with rv the above p r o p e r t i e s of , To make i t easy for. comparison with C l i f f o r d ' s o r i g i n a l a r t i c l e we s h a l l use h i s notations. 178 Theorem 1 Let <G be a d i s c r e t e i n f i n i t e group and Abelian normal subgroup of 3) O such that the index of /-I i n dimensional i r r e d u c i b l e r e p r e s e n t a t i o n of Of H an Let f j ^ be a countably i n f i n i t e i s countably i n f i n i t e . representation of fl subduced by (JIQ, ( and Then ^f) (J^H the is fully reducible. If ^ i s any i r r e d u c i b l e component of the other i r r e d u c i b l e components of Ql^ then a l l are conjugates of ^/ ^ (L) r e l a t i v e to Q and every such conjugate of must occur i n the decomposition of Proof Let (R. be the r e p r e s e n t a t i o n space of. (% (and 01H) must be reducible under 1%H as a l l the i r r e d u c i b l e representations of H if are one dimensional. <£> i s a subspace of such that i t i s i n v a r i a n t and i r r e d u c i b l e under j-\ then r In other words, must be one dimensional. I f i s any element of <Sj then r(=> must also be i n v a r i a n t under , For i f ^ i s any element of \-\ then I t i s evident that Y"^ i s also one dimensional. the basis of <5? I f e. i s such that where °<-(u) i s the one dimensional r e p r e s e n t a t i o n of the element u £ H generated by u re ^ ot ( r' u then r) re. 179 Hence the representation of generated by r <£> i s f-f conjugate to that generated by , Since Gr i s a discrete r-v i n f i n i t e group and H i s of i n f i n i t e index i n G} then . r V there e x i s t s a sequence of elements 7, , j) /jk>,"'"' in Z O such that G = f H f,h + + - - •• + where ^, ==• i d e n t i t y element. and i n v a r i a n t under an element V <5\ + • Because dimensional r£> f.ff i s one $2^ then there must e x i s t of the sequence ^> -jbj > > > B L "t e T2 k e • • • such that t "the element i n the sequence of 'jb's with the smallest number as s u f f i x s a t i s f y i n g t h i s condition. (5> 0 nr Then = 0 ' 2 £ 3 © ^ <c) and S i m i l a r l y there must be an element ^'s such that T 3 (<£>£> under V 3 of r (5? =-o i s i n v a r i a n t under Yj * t from the sequence of i s not contained i n £> © anc ^ ^ © *I <^ ® Xs €J , Then i s invariant Yji^) n £JIH. Without loss of g e n e r a l i t y we may again assume. 3 i s chosen to be that p a r t i c u l a r element of the sequence -y's with the smallest index and s a t i s f y the above condition. I t i s easy to see that i s a subspac'e of assumption that ^ ® i n v a r i a n t under (Jl^ ® CIQ., By the i s i r r e d u c i b l e then Hence we are able to choose from the sequence of subsequence G>^c€> & ^ * Y, = i d e n t i t y , (fc -• r, £ £> r., y z ( A y - 3 - .. > Y y , © ^ a ' ;>such that ® • ' • 180 with the property © Yl £ © v > • © r^., ^ ) % 6* ( that We thus secceed i n decomposing n 6 (JR. into a sum of mutually 1 «. 0 independent subspaces, i n v a r i a n t and i r r e d u c i b l e , under Oln . (jl^ We have proved that the i r r e d u c i b l e component of each y <5 i s a conjugate r e l a t i v e to O in of that i n ©. That every conjugate must occur i s obvious from the manner we obtain the subsequence . Corollary DIH , the If the l i t t l e group of representation generated by is to \-\ o irreducible r e l a t i v e to i t s e l f then each of the conjugates of O and Ql^ H relative O, appears once i n Proof Since the l i t t l e group of H is H (JLH r e l a t i v e to <3 i t s e l f then the conjugate generated by €> and must /r> <-0 be d i s t i n c t from 6)<!-H if ^ ^ < Y We want next to show that i f Yb + 7 y different. two elements T, ^ ^ and then the corresponding conjugates must be Assume that t h i s i s 'rjiot so, that i s , there are 7 - , Y,* i n the above sequence of V f o r any element jure u. t H = we have o y'i such that 181 then [ Tc~'M. r; ) S « 1 ( rj tire) 1 This would imply that T "' Yy S representation of as • be impossible since each conjugates of (E> - ry - i r A 01^ Y V generates the same 1 c f-f Y7' which we have j u s t proved to does not belong to r e l a t i v e to H , ^ Thus appears once i n Ola. G Theorem 2 Let <j3g subgroup be an. i r r e d u c i b l e of the Abelian of a d i s c r e t e i n f i n i t e group Q rv l i t t l e group of and induced by the rv f-| r-> rv d? and rv r e l a t i v e to Q The representation of invariant be H ""^ of' H (BH itself, i s then irreducible. Proof For the case that the index of the l i t t l e group (which i s H itself) in O proved the theorem. infinite. where We can then w r i t e ^cH ^ ^ \ }^^' ' / if <j>,~ i Because the l i t t l e group of and H" of H is already Let us therefore assume the index i s (=identity) / such that i s f i n i t e , C l i f f o r d has rv- Ji a r e elements of > (JDH r e l a t i v e to (3 j-j i t s e l f , the i r r e d u c i b l e representations rv defined by ^ $^ ^ 182 are. a l l d i s t i n c t conjugates. By i n d u c t i o n , we construct a representation ' (Jl^ O i n the f o l l o w i n g way. For any element A. 10, = 0 I t i s easy to check that ^ If ^ Aia(0 s of i n Or , otkeri^ise , ^LQ i s indeed a representation of i s the representation'space of (fey then the d i r e c t sum i s the.representation space of To see that such that the representation of -tf subduced by (B/f to % £q By Theorem 1 the representation space of £^ t which by assumption contains subspaces is be the i r r e d u c i b l e component of (JIQ irreducible, l e t contains (jig €> |> €> y 4 ; <5> *,ft^/"Thus This shows that contains each of the must be i d e n t i c a l (JIQ. i s i r r e d u c i b l e . 183 APPENDIX I I In t h i s appendix we want to prove that the l i n e a r combination of an i n f i n i t e number of Landau functions denoted by Jflv/ (see ( i\A l . zL K, k ^ K j , ^ ) f u n c t i o n of x , ^, %, We do t h i s by showing that f o r any given values of X, ^, 2s H.( ) i s a w e l l defined the i n f i n i t e s e r i e s (Ax,) ^ J ^ 1 ± £ H ) i s a b s o l u t e l y convergent. Let ck = *1 ( > o) and * = Y -(ih-pO* then °* Since HKI'O i s a polynomial show that 2 we only have to show in ^ / of degree 'H/ , to 184 2 where | -, /|< ( L w o£ K < ^ - i s an Integer such that Consider the f o l l o w i n g i n f i n i t e sum I 2 Since there e x i s t s an i n t e g e r (A) therefore o4 2 2 E 7\ .=.>^(JL) J I t i s easy to see that 2 < -K-^+Y) ' 2 E e ^ ^ + ^ i such that 185 Hence - 4r I- 1 With some s l i g h t m o d i f i c a t i o n o'f the above argument we can r e a d i l y show that Thus, ' which i s the required, r e s u l t .
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Magnetic operator groups of an electron in a crystal Tam, Wing Gay 1967
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Title | Magnetic operator groups of an electron in a crystal |
Creator |
Tam, Wing Gay |
Publisher | University of British Columbia |
Date Issued | 1967 |
Description | In this thesis the problem of an electron in a crystal in the presence of a uniform magnetic field is investigated using group theory method. A group of operators [ symbol omitted ] commuting with the Hamiltonian of an electron in the presence of a uniform magnetic field and a crystal electric potential is constructed. This group is homomorphic to the group [ symbol omitted ] (a magnetic space group) of space time transformations that leave the magnetic field and the crystal electric potential invariant. The property of the subgroup [ symbol omitted ] of [ symbol omitted ] that under the above homomorphism is mapped onto the lattice [ symbol omitted ] of [ symbol omitted ] is studied in detail. It turns out the structure of [ symbol omitted ] depends on the magnitude and the orientation of the magnetic field, so that, in fact one has to deal with an infinite class of groups. In particular, it is useful to divide this infinite class of groups into two subclasses: one subclass is then referred to as corresponding to "rational" magnetic fields, the other as corresponding to "irrational" magnetic field. The group [ symbol omitted ] is a generalisation of the "magnetic translation group" recently introduced by Zak for the special case of a symmetric gauge. He also constructed "physical" irreducible representations of the "magnetic translation group" for the special case of a "rational" magnetic field. In this case a group [ symbol omitted ] always has a maximal Abelian subgroup with a finite index. (The term "physical" representation simply means a representation which can be generated by functions of spatial coordinates.) In this thesis no such restriction is introduced: the "physical" irreducible representations of [ symbol omitted ] are also constructed for the case of irrational magnetic field, in which case the index of a maximal Abelian subgroup is always infinite; the "physical" irreducible representations are then always infinite dimensional. Using a complete set of Landau functions the basis functions generating "physical" irreducible representations of [ symbol omitted ] are found for the special case when the crystal is simple cubic and the magnetic field is parallel to a lattice vector. It turns out when the field is "irrational" the basis functions are countably infinite sets of Landau functions, and the energy spectrum depends only on one of the parameters labelling the "physical" irreducible representations of [ symbol omitted ]. The problem of perturbation produced by a weak periodic potential on the Landau levels for a free electron in a magnetic field is also considered. In this connection we make plausible the validity of certain quite general selection rules for an arbitrary periodic potential. |
Subject |
Electrons Crystals Quantum theory |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-09-15 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0085922 |
URI | http://hdl.handle.net/2429/37354 |
Degree |
Doctor of Philosophy - PhD |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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