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UBC Theses and Dissertations

Magnetic operator groups of an electron in a crystal Tam, Wing Gay 1967

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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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