UBC Theses and Dissertations

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UBC Theses and Dissertations

Properties of an interacting one-dimensional fermion system Friesen, Waldemar Isebrand 1981

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PROPERTIES OF AN INTERACTING ONE-DIMENSIONAL FERMION SYSTEM by WALDEMAR ISEBRAND FRIESEN B . S c , Brock U n i v e r s i t y , 1973 M . S c , U n i v e r s i t y of B r i t i s h Columbia, 1975 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES (Department of P h y s i c s ) We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA APRIL 1981 (c) Waldemar I s e b r a n d F r i e s e n , 1981 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r a n a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l m a k e i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s m a y b e g r a n t e d b y t h e h e a d o f my d e p a r t m e n t o r b y h i s o r h e r r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t b e a l l o w e d w i t h o u t m y w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f PMVS ICS  T h e U n i v e r s i t y o f B r i t i s h C o l u m b i a 2 0 7 5 W e s b r o o k P l a c e V a n c o u v e r , C a n a d a V 6 T 1W5 Date A$a.zo m i n F - f i 17/19) i i A b s t r a c t For n e a r l y a decade, q u a s i - o n e - d i m e n s i o n a l c o n d u c t o r s have been the s u b j e c t of i n t e n s i v e s t u d y . T h e o r e t i c a l l y , much a t t e n t i o n has been devoted t o the development of o n e - d i m e n s i o n a l Fermi gas models, some which may be s o l v e d e x a c t l y , and t o the c a l c u l a t i o n of t h e i r response f u n c t i o n s . A f t e r a review of t h i s t h e o r y , a d i f f e r e n t approach i s adopted i n the i n v e s t i g a t i o n of two models. The d i e l e c t r i c response t h e o r y of the t h r e e -d i m e n s i o n a l Coulomb gas has been a p p l i e d t o an a n i s o t r o p i c system i n which the p a r t i c l e s i n t e r a c t w i t h an e f f e c t i v e one-d i m e n s i o n a l l o n g - r a n g e p o t e n t i a l . W i t h i n the framework of the a p p r o x i m a t i o n of S i n g w i , T o s i , Land, and S j o l a n d e r , the d i e l e c t r i c p r o p e r t i e s of the model are examined i n o r d e r t o det e r m i n e the c o n d i t i o n s under which i t i s u n s t a b l e w i t h r e s p e c t t o f o r m a t i o n of a charge d e n s i t y wave s t a t e . I t i s found t h a t the p o s i t i v e n e u t r a l i z i n g background must be p o l a r i z a b l e i n o r d e r f o r such an i n s t a b i l i t y t o o c c u r . The same a p p r o x i m a t i o n method, when a p p l i e d t o a one-d i m e n s i o n a l f e r m i o n gas w i t h a 8-function i n t e r a c t i o n may be compared w i t h the e x a c t s o l u t i o n of Yang. T h i s s o l u t i o n , which e x i s t s i n the form of c o u p l e d i n t e g r a l e q u a t i o n s , has been c a l c u l a t e d n u m e r i c a l l y , and, as p r e d i c t e d by the L i e b - M a t t i s theorem, the ground s t a t e i s found t o be non-magnetic. The a p p r o x i m a t i o n of Si n g w i et a l . proves t o g i v e b e t t e r c o r r e l a t i o n e n e r g i e s than o t h e r i n e x a c t methods, p a r t i c u l a r l y a t h i g h e r d e n s i t i e s . i i i T a b l e of C o n t e n t s Page L i s t of F i g u r e s v Acknowledgements v i i i ' C hapter 1 Quasi-One-Dimensional C o n d u c t o r s 1.1 I n t r o d u c t i o n 1 1.2 P e i e r l s I n s t a b i l i t y 5 1.3 One-Dimensional T h e o r i e s 9 A Tomonoga-Luttinger Model 11 B E x t e n s i o n s of the Tomonaga-Luttinger Model 15 C Hubbard Model 21 1.4 Response F u n c t i o n s 26 1.5 Summary ' 34 2 D i e l e c t r i c Response of a One-dimensional E l e c t r o n Gas 2.1 I n t r o d u c t i o n 36 2.2 Quasi-One-Dimensional Model 37 2.3 D i e l e c t r i c P r o p e r t i e s of a ID Fermi Gas 41 A Sum Rule s 45 B C o m p r e s s i b i l i t y 49 2.4 A p p r o x i m a t i o n Methods 51 A STLS A p p r o x i m a t i o n 53 B P r o p e r t i e s of and 56 C CDW I n s t a b i l i t y i n the STLSA 59 D P a i r C o r r e l a t i o n F u n c t i o n 61 E C o l l e c t i v e E x c i t a t i o n s 62 F H a r t r e e - F o c k and Random Phase A p p r o x i m a t i o n s 64 2.5 Model With a T r a n s v e r s e G a u s s i a n D e n s i t y 65 2.6 N u m e r i c a l Method 72 2.7 R e s u l t s 74 2.8 D i s c u s s i o n 94 3 The One-dimensional Fermion Gas w i t h a &--function I n t e r a c t i o n 3.1 I n t r o d u c t i o n 98 3.2 The Exact S o l u t i o n 99 A L i e b - M a t t i s Theorem 103 3.3 The Ground S t a t e 105 A N u m e r i c a l Method 108 B R e s u l t s 110 3.4 Magnetic S t a t e s 112 A S p i n S u s c e p t i b i l i t y 114 B N u m e r i c a l Method 115 C R e s u l t s 116 i v 3.5 Approximate Methods 123 A STLS A p p r o x i m a t i o n 124 B H a r t r e e - F o c k and Random Phase A p p r o x i m a t i o n s 126 C G e n e r a l i z e d Random Phase A p p r o x i m a t i o n 127 D N u m e r i c a l Method 130 E R e s u l t s 131 3.6 D i s c u s s i o n 137 4 D i s c u s s i o n and C o n c l u s i o n s 4.1 Comparison w i t h Other Models 139 A 'g-ology' 139 B Overhauser S p i n D e n s i t y Wave 141 4.2 M u l t i p l e - C h a i n Models 146 A T h r e e - D i m e n s i o n a l 'g-ology' 146 B D i e l e c t r i c Response Theory 147 4.3 Summary and S u g g e s t i o n s f o r F u t u r e Work 150 R e f e r e n c e s 154 V L i s t of F i g u r e s Page F i g u r e 1.1. The o n e - d i m e n s i o n a l f r e e - p a r t i c l e s u s c e p t i b i l i t y . 6 1.2. Form of the momentum d i s t r i b u t i o n i n the Tomonoga model. 13 1.3. K i n e t i c energy d i s p e r s i o n in. the L u t t i n g e r model. 13 1.4. S c a t t e r i n g p r o c e s s e s i n c l u d e d i n the H a m i l t o n i a n ( 1 . 9 ) . 17 1.5. P o s s i b l e s t a t e s of the H a m i l t o n i a n ( 1 . 1 2 ) . 24 1.6. B e t h e - S a l p e t e r e q u a t i o n f o r the v e r t e x p a r t . 29 1.7. F i r s t - o r d e r c o n t r i b u t i o n s t o the v e r t e x p a r t . 30 1.8. V e r t e x diagrams g i v i n g l o w e r - o r d e r l o g a r i t h m i c c o r r e c t i o n s . 31 1.9. Phase diagram of the Fermi gas f o r the c o u p l i n g c o n s t a n t s g,, g^. 33 2.1. S t a t i c d i e l e c t r i c f u n c t i o n 6 s _ , s ( q , 0 ) f o r a s t a b l e ID system. 60 •2.2. (a) The p o t e n t i a l v(q) d e f i n e d by Eq. (2.57) compared w i t h a l / q x p o t e n t i a l , (b) The r e a l space F o u r i e r t r a n s f o r m of v(q) compared w i t h the Coulomb e^/r p o t e n t i a l . 67 2.3. (a) The s t r u c t u r e f a c t o r S(q) f o r UA = 1 (Q.=a 0/2). 76 (b) S(q) f o r U = 5 ( x & = 3.93). 77 2.4. (a) The p a i r c o r r e l a t i o n f u n c t i o n g(x) f o r UA = 1. 78 (b) g(x) f o r U = 5. 79 2.5. (a) The f u n c t i o n G(q) f o r UA = 1. 80 (b) G(q) f o r U = 5. 81 2.6. G(q) f o r U/A « 1 and U/A = 0 . 5 . 83 v i 2.7. (a) Plasma f r e q u e n c i e s Wp(q) f o r UA = 1. - 8 4 2.7. (b) CUp(q) f o r U = 5.- 85 2.8. The e l e c t r o s t a t i c p o t e n t i a l v/€(q,0) between two t e s t charges f o r UA = 1. 87 2.9. The e f f e c t i v e i n t e r a c t i o n ver.e = v / [ l - ( l - G ) v X ] f o r UA = 1. T 89 2.10. The system response t o an e x t e r n a l p e r t u r b a t i o n f^fe.x. = v ^ ( c 3 ' 0 ) f o r u = 3 ' A = 2 / 3 - 9 0 2.11. The c o m p r e s s i b i l i t y r a t i o X©/X- 9 1 2.12. Plasma f r e q u e n c i e s f o r the p o t e n t i a l w(q) = W/q . 93 2.13. G(q) f o r the p o t e n t i a l w(q). 95 3.1. The k e r n e l H(x) d e f i n e d by Eq. ( 3 . 8 b ) . 109 3.2. j>(y) f o r q = 0.1, 1, 3, 10, 20. 110 3.3. The r a t i o kp/Q as a f u n c t i o n of q. I l l 3.4. The s o l u t i o n s <r(x) of Eq. (3.12) f o r v a r i o u s v a l u e s of b and q. 117 3.5. The f u n c t i o n p ( y ) o b t a i n e d from Eq. (3.5b) f o r v a r i o u s b and q. 119 3.6. The f r a c t i o n a l spin-down d e n s i t y d e t e r m i n e d from Eq. (3.6a) f o r v a r i o u s b and q. 120 3.7. The r a t i o k p/Q f o r v a r i o u s b. 121 3.8. The energy d e n s i t y as a f u n c t i o n of the c o u p l i n g s t r e n g t h C. 122 3.9. Second o r d e r energy terms. 128 3.10. The f u n c t i o n s G(C) and G,(C) used t o c a l c u l a t e the e n e r g i e s ( 3 . 1 1 ) , ( 3 . 1 9 ) , and ( 3 . 2 0 ) . 132 3.11. The approximate energy d e n s i t i e s compared w i t h the e x a c t r e s u l t . 133 3.12. The p a i r c o r r e l a t i o n f u n c t i o n f o r v a r i o u s v a l u e s of C. 135 3.13. Comparison of the RPA and GRPA e n e r g i e s t o second and t o i n f i n i t e o r d e r . 137 Energy spectrum of the H a r t r e e - F o c k s t a t e s d e f i n e d by Eqs. (4.4) and ( 4 . 5 b ) . H a r t r e e - F o c k e n e r g i e s f o r the SDW,.ferromagnet and paramagnetic s t a t e s compared t o the e x a c t . energy as a f u n c t i o n c o u p l i n g s t r e n g t h . Plasmon e x c i t a t i o n spectrum o b t a i n e d by G i u l i a n i et a l . (1979). v i i i Acknowledgements I am i n d e b t e d t o P r o f . B i r g e r B ergersen f o r h i s p a t i e n t and knowledgeable guidance d u r i n g the c o u r s e of t h i s work. Thanks a r e due t o Dr. A. N o b i l e f o r some h e l p f u l c o r r e s p o n d e n c e and f o r p o i n t i n g out a n u m e r i c a l e r r o r . F i n a n c i a l s u p p o r t from the N a t i o n a l S c i e n c e s and E n g i n e e r i n g Research C o u n c i l i s a l s o g r a t e f u l l y acknowledged. 1 Chapter 1 Quasi-One-Dimensional Conductors 1.1 I n t r o d u c t i o n ... i n the one d i m e n s i o n a l f r e e - e l e c t r o n model the i n t e r a c t i o n w i t h l a t t i c e d i s p l a c e m e n t s can be t r e a t e d i n a f a i r l y s a t i s f a c t o r y way, and ... such a model has p r o p e r t i e s as might be e x p e c t e d from a s u p e r c o n d u c t o r i f e x t r a p o l a t e d t o one d i m e n s i o n . F r o h l i c h (1954) I t f o l l o w s t h a t f o r a o n e - d i m e n s i o n a l m e t a l w i t h a p a r t l y f i l l e d band the r e g u l a r c h a i n s t r u c t u r e w i l l never be s t a b l e , s i n c e one can always f i n d a d i s t o r t i o n ... f o r which a break w i l l o c cur a t or near the edge of the Fermi d i s t r i b u t i o n . I t i s t h e r e f o r e l i k e l y t h a t a o n e - d i m e n s i o n a l model c o u l d never have m e t a l l i c p r o p e r t i e s . P e i e r l s (1955) The q u e s t i o n of the p o s s i b i l i t y of e x i s t e n c e of d i f f e r e n t phases i n o n e - d i m e n s i o n a l l i n e a r systems p r e s e n t s some t h e o r e t i c a l i n t e r e s t . I t t u r n s out t h a t the answer t o t h i s q u e s t i o n i s n e g a t i v e ... Landau and L i f s h i t z (1958) The s u p e r c o n d u c t i n g s t a t e ... can occur even i n our s t r u c t u r e which i s e s s e n t i a l l y a o n e - d i m e n s i o n a l c h a i n ... L i t t l e (1964) We i n t e r p r e t t h e s e d a t a as a r i s i n g from s u p e r c o n d u c t i n g f l u c t u a t i o n s ... a s s o c i a t e d w i t h a tendency toward h i g h t e m p e r a t u r e s u p e r c o n d u c t i v i t y i n t h e s e pseudo-one-d i m e n s i o n a l s o l i d s . The ground s t a t e of the compounds thus f a r s t u d i e d i s , however, t h a t of a P e i e r l s i n s u l a t o r ... Coleman, M.J. Cohen et a l . (1973) These q u o t a t i o n s are from r e f e r e n c e s c i t e d w i t h a h i g h e r than average f r e q u e n c y i n the s o l i d s t a t e p h y s i c s l i t e r a t u r e of the p a s t decade, and have been s e l e c t e d t o p r o v i d e an i n d i c a t i o n of the m o t i v a t i o n s and e x p e c t a t i o n s a s s o c i a t e d w i t h r e s e a r c h i n t o the s u b j e c t of " q u a s i - o n e - d i m e n s i o n a l " c o n d u c t o r s . I n t e r e s t i n g l y , the t h e o r e t i c a l s t a t e m e n t s p r e d a t e the e x p e r i m e n t a l r e p o r t by about twenty y e a r s . 2 One-dimensional (ID) models have l o n g been a p h y s i c i s t ' s whetstone on which he can sharpen, h i s m a t h e m a t i c a l t o o l s and hone h i s i n t u i t i o n f o r the s h a p i n g of p h y s i c a l t h e o r y t o d e s c r i b e r e a l t h r e e - d i m e n s i o n a l (3D) systems. However, w i t h the advent of c e r t a i n c l a s s e s of s y n t h e t i c m a t e r i a l s which approach, i n some sense, a ID i d e a l , the study of ID models has e v o l v e d from a p l a y g r o u n d i n t o a t e s t i n g ground i n which t h e o r y can a t l a s t be compared w i t h e x p e r i m e n t . Of c o u r s e , r e a l systems a r e always 3D, but not n e c e s s a r i l y i s o t r o p i c , and i f the degree of a n i s o t r o p y becomes v e r y l a r g e , then a ID v i e w p o i n t may prove t o be more f r u i t f u l . M e t a l s f o r which t h i s i s the case have been dubbed " q u a s i - o n e - d i m e n s i o n a l c o n d u c t o r s " . A c c o r d i n g t o P e i e r l s (1955), a s t r i c t l y ID s o l i d would never have a m e t a l l i c ground s t a t e s i n c e a l i n e a r l a t t i c e of atoms or m o l e c u l e s w i l l always d i s t o r t i n o r d e r t o c r e a t e an •energy gap a t the Fermi l e v e l . T h i s statement i s s u p p o r t e d by e m p i r i c a l o b s e r v a t i o n , but i t s a p p l i c a b l i t y i s l i m i t e d s i n c e i t does not t a k e f u l l account of e l e c t r o n dynamics. F r o h l i c h (1954) demonstrated t h a t the gap due t o a P e i e r l s - t y p e l a t t i c e d i s t o r t i o n c o u l d l e a d t o a ID s u p e r c o n d u c t i n g s t a t e i n a model i n which the p o s i t i v e charge background i s t r a n s l a t i o n a l l y i n v a r i a n t . An a l t e r n a t i v e mechanism f o r s u p e r c o n d u c t i v i t y was proposed by L i t t l e (1964) f o r a s t r u c t u r e c o n s i s t i n g of a c e n t r a l m o l e c u l a r " s p i n e " , surrounded by s i d e c h a i n s of p o l a r i z a b l e o r g a n i c dye m o l e c u l e s . The i d e a was t h a t e x c i t o n s p r o p a g a t i n g a l o n g the s i d e c h a i n s would c r e a t e an e f f e c t i v e a t t r a c t i v e p o t e n t i a l between e l e c t r o n p a i r s on the s p i n e . S i g n i f i c a n t l y , 3 the r e s u l t i n g s u p e r c o n d u c t i n g s t a t e was e x p e c t e d t o have a t r a n s i t i o n temperature much h i g h e r even than room t e m p e r a t u r e . Some time l a t e r , Coleman, M.J. Cohen, e t a l . (1973) r e p o r t e d c o n d u c t i v i t y data f o r the o r g a n i c charge t r a n s f e r s a l t TTF-TCNQ which were i n t e r p r e t e d as s u p e r c o n d u c t i n g f l u c t u a t i o n s above 54K, the temperature a t which the compound underwent a m e t a l - i n s u l a t o r t r a n s i t i o n . . The anomalously (or perhaps s p u r i o u s l y ) h i g h c o n d u c t i v i t y was not c o n f i r m e d by the measurements of a l a r g e number of o t h e r workers (Thomas e t a l . 1976), but the c o m b i n a t i o n of the phase t r a n s i t i o n , s u g g e s t i v e c r y s t a l s t r u c t u r e , and a n i s o t r o p i c p h y s i c a l p r o p e r t i e s i d e n t i f i e d TTF-TCNQ as a q u a s i - o n e - d i m e n s i o n a l m e t a l , t h e r e b y b r i n g i n g the i d e a s of P e i e r l s , F r S h l i c h , and L i t t l e t o the f o r e . The l u r e of h i g h t e m p e r a t u r e ID s u p e r c o n d u c t i v i t y has been a s t r o n g one, m o t i v a t i n g much of the r e c e n t p r o g r e s s i n the f i e l d . P a r t i c u l a r emphasis has been on the c h e m i c a l e n g i n e e r i n g of new compounds, m a i n l y o r g a n i c , t o meet t h e o r e t i c a l r e q u i r e m e n t s . In the p r o c e s s , the s y s t e m a t i c study of t h e i r p h y s i c a l p r o p e r t i e s has l e a d t o a more d e t a i l e d p i c t u r e of a s o l i d as an e n t i t y which i s n e a r l y ID. Reviews of the p h y s i c s and c h e m i s t r y of the s e m a t e r i a l s may be found i n the books e d i t e d by K e l l e r (1975, 1977), S c h u s t e r (1975), Devreese, E v r a r d , and Van Doren (1979), and the a r t i c l e by Andre, B i e b e r , and G a u t i e r (1976). Mention of the m e t a l - i n s u l a t o r t r a n s i t i o n i n TTF-TCNQ b r i n g s us t o the well-known theorem of Landau and L i f s c h i t z ( 1958), which s t a t e s t h a t a ID system w i l l never e x p e r i e n c e a phase t r a n s i t i o n , except a t OK,- i f the f o r c e s a r e s h o r t - r a n g e . 4 The essence of the argument i s t h a t under the above c o n d i t i o n , the e n t r o p y term w i l l always dominate the energy i n the f r e e energy of an i n f i n i t e c hain,- w i t h the consequence t h a t thermodynamic f l u c t u a t i o n s w i l l p r e v e n t the e s t a b l i s h m e n t of any long-range o r d e r a t f i n i t e t e m p e r a t u r e s . I t f o l l o w s t h a t 3D e f f e c t s must be taken i n t o account t o e x p l a i n the o b s e r v e d changes of phase. N e v e r t h e l e s s , ID models remain i n v a l u a b l e as a b a s i s f o r our u n d e r s t a n d i n g of q u a s i - o n e - d i m e n s i o n a l c o n d u c t o r s . T h e r e f o r e , a n e c e s s a r i l y r e s t r i c t e d o v e r v i e w of c e r t a i n a s p e c t s of ID t h e o r y f o l l o w s i n the remainder of the c h a p t e r . D i s c u s s e d i n the s u c c e e d i n g s e c t i o n i s the P e i e r l s i n s t a b i l i t y , a r e c u r r i n g f e a t u r e i n the d e s c r i p t i o n of these m a t e r i a l s . A summary of developments i n the h i g h l y a b s t r a c t Fermi gas models i s p r e s e n t e d i n Sec. 1.3, w h i l e the c o r r e s p o n d i n g response f u n c t i o n s , i m p o r t a n t i n the t h e o r y of phase t r a n s i t i o n s , a r e d i s c u s s e d i n Sec. 1.4. In c o n c l u s i o n , some d i f f i c u l t i e s a s s o c i a t e d w i t h the t h e o r y a r e i n d i c a t e d i n Sec. 1.5. F o l l o w i n g t h i s r e v i e w , the main c o n t r i b u t i o n of the t h e s i s i s p r e s e n t e d i n Chaps. 2 and 3. An a n i s o t r o p i c 3D model which i s e q u i v a l e n t t o a ID gas of f e r m i o n s i n t e r a c t i n g w i t h a long-range p o t e n t i a l i s i n t r o d u c e d i n Chap. 2. The d i e l e c t r i c p r o p e r t i e s of t h i s model are then i n v e s t i g a t e d i n a c e r t a i n (which we term STLS) a p p r o x i m a t i o n . E x t e n s i o n of the STLS method t o the case of a S - f u n c t i o n i n t e r a c t i o n p e r m i t s a comparison of approximate r e s u l t s w i t h the known e x a c t s o l u t i o n . T h i s i s the s u b j e c t of Chap. 3, i n which the magnetic s t a t e s of the ^ - f u n c t i o n gas are a l s o c o n s i d e r e d . 5 F i n a l l y , i n the c o n c l u d i n g c h a p t e r we r e l a t e our r e s u l t s t o those p r e s e n t e d i n Sec. 1.4 and t o s i m i l a r 3D s t u d i e s by o t h e r a u t h o r s . S u g g e s t i o n s a r e a l s o made f o r f u r t h e r i n g the work p r e s e n t e d i n t h i s t h e s i s . 1.2 P e i e r l s I n s t a b i l i t y As the P e i e r l s t r a n s i t i o n i s a p h y s i c a l m a n i f e s t a t i o n of a b a s i c i n s t a b i l i t y of a ID f e r m i o n system, a t l e a s t a b r i e f d i s c u s s i o n of t h i s phenomenon i s w a r r a n t e d ( f o r a r e v i e w , see Toombs 1978). L e t us c o n s i d e r a n o n - i n t e r a c t i n g ID gas of e l e c t r o n s c o n f i n e d t o a l e n g t h L, w i t h Fermi wavevector kp and energy € F . The l i n e a r response of the e l e c t r o n d e n s i t y t o a s t a t i c e x t e r n a l p o t e n t i a l w i t h F o u r i e r t r a n s f o r m ^ ( q ) i s g i v e n by O . 6 < f t * k n (I.D where 6^ and n£ a r e the energy and z e r o - t e m p e r a t u r e Fermi d i s t r i b u t i o n f u n c t i o n , r e s p e c t i v e l y , of the s i n g l e - p a r t i c l e k-s t a t e s . The s u s c e p t i b i l i t y i X 0 ( q ) , d e f i n e d as the c o e f f i c i e n t of <p(q) i n Eq. ( 1 . 1 ) , i s shown i n F i g . 1.1 f o r f r e e p a r t i c l e s . Absent i n h i g h e r d i m e n s i o n s i s the ( l o g a r i t h m i c ) s i n g u l a r i t y a t q = 2kj_, which i n d i c a t e s the ID system's i n h e r e n t i n s t a b i l i t y t o an i n f i n i t e s i m a l p e r t u r b a t i o n of the same wavevector. T h i s f e a t u r e i s a consequence of the l i n e a r i t y of the denominator, € k+zkf'^k ~ 2n*kpk f o r k - k f, and remains i n the presence of a l a t t i c e , s i n c e the e l e c t r o n i c band s t r u c t u r e can always be l i n e a r i z e d a t the Fermi l e v e l . The above d i s c u s s i o n s u g gests t h a t the d r a m a t i c 2kp response w i l l have a 6 0 2 q / k F F i g u r e 1.1. The o n e - d i m e n s i o n a l f r e e - p a r t i c l e s u s c e p t i b i l i t y . s i g n i f i c a n t s c r e e n i n g e f f e c t on the e l e c t r o n - p h o n o n i n t e r a c t i o n , r e s u l t i n g i n a s o f t e n i n g of the 2k^ phonon f r e q u e n c y (the g i a n t Kohn anomaly). In the event t h a t the f r e q u e n c y i s d e p r e s s e d t o z e r o , the phonon condenses t o a s t a t i c p e r i o d i c d i s t o r t i o n superposed on the u n d e r l y i n g l a t t i c e . R i c e and S t r a s s l e r (1973) c o n s i d e r e d the i n t e r a c t i o n of a h a l f - f i l l e d t i g h t - b i n d i n g band of e l e c t r o n s w i t h a l a t t i c e . For the F r o h l i c h H a m i l t o n i a n , they demonstrated t h a t w i t h i n a m e a n - f i e l d a p p r o x i m a t i o n the bare phonon f r e q u e n c y , 6U 2 L , i s r e n p r m a l i z e d t o where b i s a c o n s t a n t . Thus, as T d e c r e a s e s toward T c , ffijkp 0. Below the t r a n s i t i o n t e m p e r a t u r e , the system i s a P e i e r l s 7 i n s u l a t o r w i t h a s i n g l e - p a r t i c l e energy gap, due t o the added 2kp p o t e n t i a l , which grows w i t h d e c r e a s i n g t emperature t o a v a l u e a t T = 0 g i v e n by A( 0) = 3.5kpT c. The o c c u p i e d s t a t e s near the new energy gap w i l l combine t o c r e a t e an e x t r a m o d u l a t i o n of the e l e c t r o n i c charge d e n s i t y < f l k f > a c c o r d i n g t o Eq. ( 1 . 1 ) . Coulomb f o r c e s b i n d the charge d e n s i t y wave (CDW) t o the l a t t i c e d i s t o r t i o n i f the w a velength, 27T/kp, i s commensurate w i t h (an i n t e g r a l m u l t i p l e o f ) the o r i g i n a l l a t t i c e p e r i o d . However, i n the incommensurate s i t u a t i o n where the l e n g t h of the u n i t c e l l approaches L, t h e r e i s no p r e f e r r e d e q u i l i b r i u m p o s i t i o n of the CDW so t h a t i t i s f r e e t o move through the c r y s t a l . T h i s " s l i d i n g " CDW i s e s s e n t i a l l y the s u p e r c o n d u c t i n g mechanism of F r o h l i c h (1954). The c r i t i c a l t emperature T £ i s f i n i t e , d e s p i t e the s h o r t range of the e l e c t r o n - i o n i n t e r a c t i o n , because R i c e and S t r a s s l e r took o n l y the 2kp. phonon i n t o a c c o u n t . I f o t h e r l a t t i c e v i b r a t i o n a l modes are i n c l u d e d , the r e s u l t a n t f l u c t u a t i o n s cause T c t o v a n i s h . The e x p e r i m e n t a l l y v e r i f i e d o c c u r r e n c e of the P e i e r l s t r a n s i t i o n i n l i n e a r c h a i n s o l i d s must be a t t r i b u t e d t o 3D i n t e r a c t i o n s . As the temperature i s l o w e r e d , the l e n g t h over which p a r t i c l e s are c o r r e l a t e d on a c h a i n i n c r e a s e s . The Coulomb p o t e n t i a l due t o the d e v e l o p i n g charge o r d e r on n e i g h b o u r i n g c h a i n s w i l l t end t o suppress f l u c t u a t i o n s and c o r r e l a t e the o r d e r e d ' s e c t i o n s u n t i l a s t a t e of 3D l o n g - r a n g e o r d e r i s a c h i e v e d a t a f i n i t e t e m perature (Saub et a l . 1976). Thus, a l t h o u g h the P e i e r l s t r a n s i t i o n i s b a s i c a l l y a ID phenomenon, i t can o n l y o c c u r i n a 3D environment. 8 A number of s o l i d s , namely the t e t r a c y a n o p l a t i n a t e s a l t , KCP, and the TCNQ s a l t s , TTF-TCNQ and TSeF-TCNQ, e x h i b i t a P e i e r l s d i s t o r t i o n (comprehensive t r e a t m e n t s of the p h y s i c s of th e s e m a t e r i a l s can be found i n the r e v i e w s by Heeger, Comes and S h i r a n e , and S h u l t z and Craven i n Devreese et a l . ( 1 9 7 9 ) ) . In p a r t i c u l a r , TTF-TCNQ d i s p l a y s an assortment of phase t r a n s i t i o n s which we s h a l l b r i e f l y summarize. The c r y s t a l s t r u c t u r e c o n s i s t s of s e g r e g a t e d , p a r a l l e l s t a c k s of p l a n a r TTF and TCNQ m o l e c u l e s , p a r t i a l l y i o n i z e d by a charge t r a n s f e r of 0.59 e l e c t r o n s / m o l e c u l e from the TTF t o the TCNQ. Anomalies i n the c o n d u c t i v i t y and o t h e r p r o p e r t i e s have been d e t e c t e d at 54K and 38R. X-ray and n e u t r o n d i f f r a c t i o n e x p e r i m e n t s have been i n t e r p r e t e d , a c c o r d i n g t o a t h e o r y by Bak and Emery (1976), as i n d i c a t i n g a t r a n s i t i o n t o the P e i e r l s s t a t e on the TCNQ c h a i n s a t the h i g h e r t e m p e r a t u r e , f o l l o w e d by a s i m i l a r d i s t o r t i o n on the TTF c h a i n s a t 49K. The measured CDW p e r i o d i c i t y i s c o m p a t i b l e w i t h a kp c o r r e s p o n d i n g t o the obs e r v e d charge t r a n s f e r . F i n a l l y , o r d e r i n g i n the t r a n s v e r s e d i r e c t i o n t o mi n i m i z e the Coulomb energy between the CDWs o c c u r s a t 38K. I t s h o u l d be n o t e d , however, t h a t the b e h a v i o u r of TTF-TCNQ i n t h i s temperature r e g i o n i s not c o m p l e t e l y u n d e r s t o o d . The p o i n t of t h i s s u p e r f i c i a l g l a n c e a t the P e i e r l s i n s t a b i l i t y has been t o i l l u s t r a t e the c o n n e c t i o n between the e m p i r i c a l b e h a v i o u r of a r e a l 3D m a t e r i a l and the o v e r s i m p l i f i e d ID model t h a t was i n t r o d u c e d a t the b e g i n n i n g of t h i s s e c t i o n . U n d e r l y i n g c o m p l e x i t i e s such as the d e l i c a t e b a l a n c e between 3D i n t e r a c t i o n s and ID f l u c t u a t i o n s , and the e l e c t r o n - p h o n o n c o u p l i n g , i s the r o o t cause of the m e t a l - i n s u l a t o r t r a n s i t i o n : 9 the d i v e r g e n t 2kp response of a ID system of n o n - i n t e r a c t i n g e l e c t r o n s . One element which has been m i s s i n g from the d i s c u s s i o n i s the e l e c t r o n - e l e c t r o n i n t e r a c t i o n . C l e a r l y , the e f f e c t of i n c l u d i n g i n t e r p a r t i c l e f o r c e s on the s u s c e p t i b i l i t y (1.1) needs t o be a d d r e s s e d . The f o l l o w i n g two s e c t i o n s o u t l i n e some of the s i g n i f i c a n t t r e a t m e n t s of the problem i n the c u r r e n t l i t e r a t u r e . 1.3 One-Dimensional T h e o r i e s The advantage of e l u c i d a t i n g the p r o p e r t i e s of a s t r i c t l y ID e l e c t r o n gas was r e v e a l e d i n the p r e v i o u s s e c t i o n by the a s s o c i a t o n of the P e i e r l s i n s t a b i l i l t y i n a 3D s o l i d w i t h the 2kp s i n g u l a r i t y i n P C 0 ( q ) , the most n a i v e a p p r o x i m a t i o n t o the charge d e n s i t y s u s c e p t i b i l i t y one can make. As the p h y s i c s of a s i n g l e c h a i n seems t o be b a s i c t o compounds such as TTF-TCNQ, t h e r e f o l l o w s the n e c e s s i t y of o b t a i n i n g the most a c c u r a t e d e s c r i p t i o n p o s s i b l e of a ID system of c o u p l e d e l e c t r o n s . The P a u l i s p i n s u s c e p t i b i l i t y f o r a n o n - i n t e r a c t i n g f e r m i o n system i s a l s o g i v e n by PCe(q), i m p l y i n g an i n s t a b i l i t y w i t h r e s p e c t t o a 2k^ s p i n d e n s i t y wave (SDW). When i n t e r a c t i o n s a re more f u l l y a ccounted f o r , the CDW and SDW responses t o the a p p r o p r i a t e e x t e r n a l s t i m u l i w i l l , i n g e n e r a l , t a k e d i f f e r e n t f u n c t i o n a l forms. P a i r e d - e l e c t r o n s t a t e s , i n d i c a t i v e of B C S - l i k e s u p e r c o n d u c t i v i t y , a re a l s o p o s s i b l e i n p r i n c i p l e . A p r i m a r y o b j e c t i v e , t h e r e f o r e , i n the development of a c o r r e c t ID t h e o r y i s t he p r e d i c t i o n of the type of o r d e r which w i l l m a n i f e s t i t s e l f i n the ground s t a t e . R e p r e s e n t a t i o n of a m o l e c u l a r c h a i n , which i s i t s e l f 3D, by 10 a ID model, n e c e s s i t a t e s some a p p r o x i m a t i o n of the s i n g l e -p a r t i c l e b a s i s s t a t e s . O f t e n t h i s w i l l t ake the form of a s i n g l e band a p p r o x i m a t i o n , i n which case an e f f e c t i v e ID p o t e n t i a l may be c a l c u l a t e d . A l t e r n a t i v e l y , model p o t e n t i a l s which have no d i r e c t r e l a t i o n s h i p t o r e a l systems a r e f r e q u e n t l y employed f o r the sake of c o m p u t a t i o n a l c o n v e n i e n c e . An example i s the two-p a r t i c l e c o n t a c t i n t e r a c t i o n , v ( x ) = £(x). One e x t e n s i v e l y s t u d i e d model i s d e f i n e d by the Hubbard H a m i l t o n i a n : i r r L ' (1.2) . . . . . where c. c r e a t e s an e l e c t r o n w i t h s p i n & = ±1 a t l a t t i c e s i t e i , and the number o p e r a t o r n-t0. = c ^ c . ^ . The e n e r g i e s U and t ar e the i n t e r a c t i o n of two p a r t i c l e s a t the same s i t e and the hopping i n t e g r a l between n e i g h b o u r i n g m o l e c u l e s . A f e e l i n g f o r v a r i o u s e l e c t r o n i c phenomena can be g a i n e d from t h i s model and a d i s c u s s i o n f o l l o w s i n Sec. 1.3C. E q u a t i o n (1.2) i s a s p e c i f i c case of the g e n e r a l H a m i l t o n i a n where € ^ i s the k i n e t i c energy and v ^ ( q ) i s a ( p o s s i b l y ) s p i n -dependent i n t e r a c t i o n . T h i s H a m i l t o n i a n has been w i d e l y i n v e s t i g a t e d f o r d i f f e r e n t forms of 6L. and v ^ / q ) and i n v a r i o u s a p p r o x i m a t i o n s . We proceed now t o d e s c r i b e and summarize the f i n d i n g s of a number of im p o r t a n t s t u d i e s . 11 1.3A Tomonoga-Luttinger Model L e t us c o n s i d e r a gas of f e r m i o n s of mass m i n a p e r i o d i c box of l e n g t h L, i n t e r a c t i n g w i t h a p o t e n t i a l which i s a r b i t r a r y w i t h i n c e r t a i n r e s t r i c t i o n s . The a p p r o p r i a t e H a m i l t o n i a n i s (1.3) w i t h f^. = -R 2k 2/2m and v<rfl./(q) = v ( q ) . For the non-i n t e r a c t i n g system a l l s t a t e s below kp a r e f i l l e d ; when v(q) i s added, p a r t i c l e - h o l e p a i r s a r e c r e a t e d . I f the p o t e n t i a l i s not too s t r o n g , and of s u f f i c i e n t l y l o n g range such t h a t o n l y s m a l l momentum s c a t t e r i n g p r o c e s s e s a re i m p o r t a n t , then o n l y p a i r s near kp a r e e x c i t e d . Now, d e f i n e the two o p e r a t o r s it**6*** w i t h j>^+ + j>£ = j)^, the d e n s i t y f l u c t u a t i o n o p e r a t o r . We observe t h a t f o r q << ^ F ' f ^ [ f ^ d e s t r o y s ( c r e a t e s ) a p a r t i c l e - h o l e p a i r near + ( - ) k p . Tomonoga (1950) showed t h a t i n a subspace S of a l l the e i g e n s t a t e s of the n o n - i n t e r a c t i n g system, the f o l l o w i n g commutation r e l a t i o n s h o l d * S ' (1.4) S c o n s i s t s of those s t a t e s i n which p a r t i c l e s and h o l e s a r e c o n t a i n e d i n the r e g i o n K: kp - k 0 < |k| < k p + kg, kg « kp. N o t i n g t h a t (j9* )* = j>*, we see t h a t (1.4) i s j u s t a statement of boson commutation r e l a t i o n s f o r the o p e r a t o r s (2T/Lk) V 2 y ^ . An a d d i t i o n a l s i m p l i f i c a t i o n i s o b t a i n e d by l i n e a r i z i n g 6^ f o r k i n K, € k - ( 2 | k | / k F - 1)£ F. 12 The l i n e a r spectrum a l l o w e d Tomonaga t o e x p r e s s Eq. (1.3) e n t i r e l y i n terms of boson o p e r a t o r s : «' S * 2 (fil< * A 7 - i> + t ?- *^ + constant T h i s r e l a t i o n may be d i a g o n a l i z e d by a c a n o n i c a l t r a n s f o r m a t i o n . Hence, the low-energy e x c i t a t i o n s of the system a re b o s o n - l i k e c o l l e c t i v e modes, w i t h f r e q u e n c i e s upty - Ifl fi*^TT~T ( 1 - 5 ) G u t f r e u n d and S c h i c k (1968) d e t e r m i n e d the momentum d i s t r i b u t i o n , n^, f o r the ground s t a t e and the c o n d i t i o n on the p o t e n t i a l f o r the assumed commutation r e l a t i o n s (1.4) t o h o l d . As shown i n F i g . 1.2, the s t e p f u n c t i o n d i s c o n t i n u i t y which c h a r a c t e r i z e s n^, the n o n - i n t e r a c t i n g d i s t r i b u t i o n , d i s a p p e a r s , a l t h o u g h n^ p o s s e s s e s an i n f i n i t e s l o p e at k p f o r s m a l l v ( 0 ) . For y ( 0 ) l a r g e enough, ( d n ^ / d k ^ i s l i n e a r and any t r a c e of the Fermi s u r f a c e has v a n i s h e d . I f the assumption t h a t the p r o p e r t i e s of a Fermi gas are de t e r m i n e d by a s m a l l i n t e r v a l around k = k F h o l d s , then the a c t u a l shape of the s i n g l e - p a r t i c l e spectrum s h o u l d be u n i m p o r t a n t . The l i n e a r i z a t i o n of 6/< near £ p has the consequence of p e r m i t t i n g the d i a g o n a l i z a t i o n of (1.3) i n terms of boson o p e r a t o r s , and suggests the m o d i f i c a t i o n of 6^ from a p a r a b o l i c t o a l i n e a r d i s p e r s i o n r e l a t i o n . L u t t i n g e r (1963) proposed a model i n which 6^ cc k and which extends the energy spectrum t o 13 k p - k 0 k F k p + k 0 F i g u r e 1.2. Form of the momentum d i s t r i b u t i o n i n the Tomonoga model ( a f t e r G u t f r e u n d and S c h i c k 1968). i n f i n i t e n e g a t i v e e n e r g i e s as p i c t u r e d i n F i g . 1.3. F i g u r e 1.3. K i n e t i c energy d i s p e r s i o n i n the L u t t i n g e r model. T r e a t i n g r i g h t - and l e f t - g o i n g p a r t i c l e s as d i f f e r e n t e n t i t i e s ( l a b e l l e d 1 and 2 ) , but a l l o w i n g i n t e r a c t i o n s between them, means t h a t s c a t t e r i n g e v e n t s l e a v e p a r t i c l e s on the same b r a n c h , a s i t u a t i o n which i s e q u i v a l e n t t o the s m a l l momentum t r a n s f e r r e s t r i c t i o n of the Tomonoga model. The H a m i l t o n i a n i s 14 w r i t t e n as ("h = 1) H = H 0 + r l , where i t may be obse r v e d t h a t the k i n e t i c term r e p r e s e n t s the energy of m a s s l e s s p a r t i c l e s t r a v e l l i n g w i t h speed V p . A f u r t h e r f e a t u r e of the model i s the f i l l i n g of the n e g a t i v e energy l e v e l s and the o c c u p a t i o n of p o s i t i v e energy s t a t e s up t o |k| = k p when v = 0. For the d e n s i t y f l u c t u a t i o n o p e r a t o r s d e f i n e d by ftW £ c ; + f c * t < v * > t = , ' z one f i n d s boson commutation r e l a t i o n s ( M a t t i s and L i e b 1965) i d e n t i c a l t o Tomonoga's, but which a r e now e x a c t on account of the l i n e a r spectrum and the f i l l e d Fermi sea: £ t o = ; «V*'<Z ( 1 . 7 ) S c h i c k (1968) proved t h a t the k i n e t i c energy H 0 can be e x p r e s s e d i n terms of the J>-operators as w e l l , whence H becomes H^ZlfMW-l)+fJ-lW^^*(^WikM U.B> For the sake of c l a r i t y we l e t (q > 0) 15. and r e w r i t e H i n terms of the new o p e r a t o r s , The e q u i v a l e n c e of the f e r m i o n and boson r e p r e s e n t a t i o n s of H 0 i m p l i e s t h a t the e i g e n s t a t e s of H 0, which may be i d e n t i f i e d by t h e i r d i s t r i b u t i o n of p a r t i c l e s and h o l e s , c a r r y the a d d i t i o n a l quantum l a b e l of the number of plasmons of A- and B-type. S i n c e H i s now q u a d r a t i c , i t may be d i a g o n a l i z e d t o y i e l d where tanh2^>(q) = -v(q)/irvp ( t h i s i m p l i e s the r e s t r i c t i o n v(q) < TTvp) and Wj i s the vacuum r e n o r m a l i z a t i o n energy. The o b v i o u s f e a t u r e of the s o l u t i o n t o note i s t h a t a l l e i g e n s t a t e s are g e n e r a t e d by the boson o p e r a t o r s Afl and B. which c r e a t e plasmons of energy q - s e c h 2 ^ ( q ) . 1.3B E x t e n s i o n s of the Tomonaqa-Luttinger Model The Tomonaga-Luttinger (TL) model i s d e f i c i e n t i n a c o u p l e of r e s p e c t s . F i r s t , i t i n c l u d e s o n l y f o r w a r d - s c a t t e r i n g p r o c e s s e s , i g n o r i n g b a c k - s c a t t e r i n g a l t o g e t h e r , and s e c o n d l y , no account i s taken of s p i n . I n c o r p o r a t i o n of t h e s e c o n s i d e r a t i o n s i n t o the model c o m p l i c a t e s m a t t e r s and p r e c l u d e s , i n g e n e r a l , e x a c t s o l u t i o n s . Numerous papers have d e a l t w i t h v a r i o u s a s p e c t s of the extended TL model, and what has emerged i s a f a i r l y c o h e r e n t p i c t u r e , sometimes termed " g - o l o g y " . To s e t the stage f o r a d i s c u s s i o n of the s a l i e n t r e s u l t s , l e t us i n t r o d u c e some t e r m i n o l o g y . R e c a l l Eq. ( 1 . 3 ) , 16 tro-' ( 1 , 3 ) A d o p t i n g the Tomonoga a s s u m p t i o n , o n l y the s t a t e s near k p a r e deemed t o be i m p o r t a n t , and a c c o r d i n g l y 6^ i s l i n e a r i z e d around the Fermi surfa c e , . £ ^ = v p ( | k | - k p) . Some s o r t of c u t o f f p r o c e d u r e must be employed i n or d e r t h a t t h i s assumption remain v a l i d . Then, as i n the TL model, our system c o n s i s t s of two ty p e s of p a r t i c l e s , t hose w i t h momentum near + k p and _kp, which we l a b e l 1 and 2. However, the p o s s i b l e i n t e r a c t i o n s a r e g e n e r a l i z e d t o a l l o w s p i n - f l i p p r o c e s s e s as w e l l as branch t o branch s c a t t e r i n g . The H a m i l t o n i a n i s o f t e n w r i t t e n i n the form (Solyom 1979) Kl<0 k,<o iki<r,ikjtrL Hkp-hk^foi ikt-zkr-f+Go-, kz<o -1* Ki>0 +l^o ^ ^ i ^ A ^ ^ A ^ ^ (1.9) kz<o The c o u p l i n g s t r e n g t h s , q-, which c a r r y an a d d i t i o n a l 17 s u b s c r i p t // or 1 depending on whether the i n t e r a c t i n g e l e c t r o n s have p a r a l l e l or a n t i p a r a l l e l s p i n s , l a b e l the p r o c e s s e s p i c t u r e d i n F i g . 1.4. The f o r w a r d s c a t t e r i n g e v e n t s g 2 and g^ k r 2 k F - q + |A^A/vVWVV\AAAA ^ 1 1 + k 2 + 2 k F + q k 1 - 2 k F - q 91 t /VWWvVVWVVM g 3 k 2 + 2 k p + q - G k-k r q k 2 + q g 2 jVVVvVvM /vWvv] k 2 + q k 2 9 4 F i g u r e 1.4. S c a t t e r i n g p r o c e s s e s i n c l u d e d i n the H a m i l t o n i a n ( 1 . 9 ) . S i g n s + and - r e f e r t o p a r t i c l e s near + k F and -kp. a r e d i s t i n g u i s h a b l e i n t h a t p a r t i c l e s from d i f f e r e n t branches ( g 2 ) and from the same branch (g^) i n t e r a c t , but i n both cases the momentum t r a n s f e r q ~ 0. B a c k - s c a t t e r i n g (q - 2kp) p r o c e s s e s a r e denoted g, and g^, the former i n v o l v i n g p a r t i c l e s from d i f f e r e n t b r a n c h e s , and the l a t t e r p a r t i c l e s near the same Fermi p o i n t . T h i s l a s t p r o c e s s a l l o w s f o r the pre s e n c e of a p e r i o d i c p o t e n t i a l and a band s t r u c t u r e i n €|<, and n e c e s s i t a t e s the i n c l u s i o n of a r e c i p r o c a l l a t t i c e v e c t o r , G. However, g i v e s a 18 s i g n i f i c a n t c o n t r i b u t i o n o n l y when G = 4kp, t h a t i s when the band i s h a l f - f i l l e d . The n a t u r e of a c u t o f f may be handle d i n two ways. One i s the Tomonoga method of r e s t r i c t i n g the p a r t i c l e momenta t o an i n t e r v a l around ±k p, and i s known as the bandwidth c u t o f f . In t h i s c a s e , the c o u p l i n g s a r e taken t o be c o n s t a n t s , w i t h the consequence t h a t t h e r e a re o n l y t h r e e independent v a l u e s among g ( / / , g | > L , , and qlL. The f a c t o r s g,^ and q%n then o c c u r o n l y i n the c o m b i n a t i o n (g,// ~ g^ //) , which i m p l i e s t h a t the r e l a t i v e v a l u e s may be a d j u s t e d so t h a t g ^ = g ^ = g^. A second approach i s t h e use of a c u t o f f i n the momentum t r a n s f e r q. The q-dependence of the g^ - i s then r e t a i n e d and the above statement r e g a r d i n g t h e i r r e l a t i o n s h i p does not h o l d . I f gxu and g,^ remain non-zero, then a band c u t o f f of some s o r t must s t i l l be made i n or d e r t o a v o i d the n o n - l i n e a r p a r t of 6^ . Numerous c a l c u l a t i o n s have been made f o r the H a m i l t o n i a n (1.9) w i t h v a r i o u s assumptions f o r the c o u p l i n g c o n s t a n t s . The TL model r e t a i n s o n l y the s m a l l q i n t e r a c t i o n and thus c o r r e s p o n d s t o s e t t i n g a l l g(- = 0 except f o r g x and g^. For a more d i v e r s e s e t of c o n s t a n t s , the concept of boson r e p r e s e n t a t i o n s of fe r m i o n o p e r a t o r s has been employed t o enable the a n a l y s i s of such problems. L u t h e r and P e s c h e l (1974) and M a t t i s (1974) i n t r o d u c e d an o p e r a t o r d e f i n e d i n terms of the d e n s i t y o p e r a t o r />/(k), (1.10) w i t h 19 As a r e p r e s e n t a t i o n of the f e r m i o n f i e l d o p e r a t o r 7 & k JK 0;(x) s a t i s f i e s the r e q u i r e d commutation r e l a t i o n i n a d d i t i o n t o m a i n t a i n i n g the form of the e q u a t i o n of motion The s i g n i f i c a n c e of the parameter « i s r e a l i z e d by a comparison of the c o r r e l a t i o n f u n c t i o n s <0 | Oj (x , t ) 6>j | 0> and <0 | ^ - ( x , t ) j / / ; 10>, where 10> denotes the ground s t a t e of H. E q u a l i t y of the two q u a n t i t i e s i s a c h i e v e d , f o r a c o n s t a n t d e n s i t y of k - s t a t e s , i n the l i m i t oc —» 0, whereas f o r a n o n l i n e a r d e n s i t y o< may remain f i n i t e . In the l a t t e r s i t u a t i o n , « i s i n t e r p r e t e d as a bandwidth. L u t h e r and Emery (1974) c o n s i d e r e d the H a m i l t o n i a n (1.9) f o r f i n i t e v a l u e s of qff , g ^ , and g^. They found i t u s e f u l t o d e f i n e s p i n - d e n s i t y o p e r a t o r s as w e l l as d e n s i t y o p e r a t o r s by where <r= ±1 and C ; i s the p a r t i c l e o p e r a t o r f o r the j branch of 20 the L u t t i n g e r spectrum. Both o p e r a t o r s s a t i s f y the a l g e b r a ( 1 . 7 ) : In terms of d e n s i t y o p e r a t o r s , the Luther-Emery H a m i l t o n i a n may be w r i t t e n as + 1 / * < W C^K-/*) *2r ^  (1.u) The l a s t term i s not d i r e c t l y e x p r e s s i b l e i n terms of d e n s i t y o p e r a t o r s so use i s made of the boson r e p r e s e n t a t i o n (1.10) w i t h an added s p i n i n d e x . H may then be s e p a r a t e d i n t o a sum of d e n s i t y and s p i n - d e n s i t y p a r t s , H = H^> + H^ ., where and 21 Hy, i s r e c o g n i z e d as b e i n g of the L u t t i n g e r form (1.6) f o r which the s o l u t i o n has a l r e a d y been d i s c u s s e d . L u t h e r and Emery observ e d t h a t f o r the p a r t i c u l a r v a l u e g ^ = -6irvp/5, H e may be d i a g o n a l i z e d t o g i v e The f i r s t term d e s c r i b e s plasmon e x c i t a t i o n s w i t h a r e n o r m a l i z e d v e l o c i t y , Vp = VF Seek If j iauk 2 f - - ($///- 2 j J/277i/p In the second term the boson s p i n o p e r a t o r s have been r e p l a c e d by s p i n l e s s f e r m i o n o p e r a t o r s w i t h a spectrum where vjl = 4 v p / 5 and A = |g |^|/2iroc. Thus, the e x c i t a t i o n spectrum i n t h i s s p e c i a l case c o n s i s t s of a c o n t i n u o u s band f o r the d e n s i t y f l u c t u a t i o n s and a s i n g l e - p a r t i c l e spectrum w i t h a gap 2L a t kp f o r the s p i n e x c i t a t i o n s . 1.3C Hubbard Model• Hubbard (1957) i n t r o d u c e d the H a m i l t o n i a n (1.2) t y t (1.2) t o d e s c r i b e 3D narrow band m a t e r i a l s . I t has s i n c e been used t o model ID systems f o r which -the Coulomb r e p u l s i o n between e l e c t r o n s i s thought t o be i m p o r t a n t . Coleman, J . A. Cohen, et 22 a l . (1973) in v o k e d the model t o e x p l a i n the low temperature c o n d u c t i v i t y of the f i r s t known h i g h l y c o n d u c t i n g TCNQ s a l t , NMP-TCNQ. T o r r a n c e (1977) argued t h a t the r e s u l t s of neut r o n and X-ray s c a t t e r i n g and o p t i c a l a b s o r p t i o n e x p e r i m e n t s , among o t h e r s , p r o v i d e e v i d e n c e t h a t the e l e c t r o n - e l e c t r o n i n t e r a c t i o n i s l a r g e i n TTF-TCNQ and can best be e x p l a i n e d by Eq. (1.2) w i t h U/t -2-3. The Hubbard H a m i l t o n i a n i n one dimensio n i s amenable t o ex a c t m a t h e m a t i c a l a n a l y s i s , and the ground s t a t e has, i n f a c t , been c a l c u l a t e d by L i e b and Wu (1968) f o r a r b i t r a r y v a l u e s of the two parameters s p e c i f y i n g the model, U/t and the d e n s i t y n. T h e i r s o l u t i o n i s based on the Bethe a n s a t z , which was used by Yang (1967) t o a n a l y z e a system of f r e e f e r m i o n s w i t h a c o n t a c t i n t e r a c t i o n ; t h i s model i s examined f u r t h e r i n Chap. 3. Both s o l u t i o n s e x i s t i n the form of s i m i l a r s e t s of c o u p l e d i n t e g r a l e q u a t i o n s w h ich, u n f o r t u n a t e l y , do not l e n d t hemselves r e a d i l y t o p h y s i c a l i n t e r p r e t a t i o n . Other e x a c t r e s u l t s have been o b t a i n e d f o r z e r o - t e m p e r a t u r e and thermodynamic p r o p e r t i e s i n the l i m i t s U/t << 1 and U/t >> 1, and f o r the case of the h a l f -f i l l e d band (n = 1 e l e c t r o n / l a t t i c e s i t e ) . An i d e a of the m o d i f i c a t i o n s t h a t can be induced i n a system of n o n - i n t e r a c t i n g e l e c t r o n s by the c o u p l i n g between the p a r t i c l e s may be o b t a i n e d from an e x a m i n a t i o n of the Hubbard model f o r l a r g e U. We f o l l o w the tr e a t m e n t of Emery (1977) who extends Eq. (1.2) t o i n c l u d e a n e a r e s t - n e i g h b o u r i n t e r a c t i o n : H • -<2 Kw<cv * H-c) * u l n : t „ i i r * w ( 1 . 1 2 ) By w r i t i n g H i n the.momentum r e p r e s e n t a t i o n , 23 H « - 2 * £ <u« U \ v * uZ < £ ; f t k , 4 ekt 4 V Z cos % A C*kiirSk,-t'<r'c*V C*«r era* and comparing w i t h Eq. ( 1 . 9 ) , we see t h a t t h i s H a m i l t o n i a n has a l l g^ d i f f e r e n t i n v a l u e . I f V = 0, the model c o r r e s p o n d s t o s e t t i n g t h e p e r p e n d i c u l a r components of the gt- e q u a l and the p a r a l l e l components t o z e r o . The terms i n v o l v i n g the parameters t and V w i l l be taken t o be p e r t u r b a t i o n s on t h e ground s t a t e which i s d e t e r m i n e d s o l e l y by the o n - s i t e i n t e r a c t i o n |U| _>> <o . I f the f r a c t i o n a l s i t e o c c u p a t i o n number f < 1 and U > 0, the ground s t a t e has a t most one p a r t i c l e on each s i t e and a m u l t i p l e degeneracy i n the s p i n and s i t e c o n f i g u r a t i o n ( F i g . 5 . 1 ( a ) ) . When U i s a t t r a c t i v e , the s t a t e of lo w e s t energy has a l l e l e c t r o n s p a i r e d f o r any f 5 2. We c o n s i d e r the f o l l o w i n g examples: (1) U > 0, f = 1/2 As i l l u s t r a t e d i n F i g . 1 . 5 ( b ) , the degeneracy i n the ground s t a t e i s l i f t e d when V > 0, s i n c e e l e c t r o n s p r e f e r t o s i t on a l t e r n a t e s i t e s . The a m p l i t u d e of the r e s u l t i n g p e r i o d i c charge d e n s i t y w i t h wavelength 2d (d i s the l a t t i c e s p a c i n g ) , or w i t h wavevector 4kp, i s reduced by a d e l o c a l i z a t i o n of the e l e c t r o n s when t i s f i n i t e . T h i s s i t u a t i o n may be termed a CDW, but i t s o r i g i n l i e s i n the m i n i m i z a t i o n of the i n t e r a c t i o n energy w i t h i n the e l e c t r o n system, r a t h e r than i n the r e d u c t i o n of the k i n e t i c energy t h r o u g h the c o u p l i n g t o an e x t e r n a l p e r t u r b a t i o n as d i s c u s s e d i n Sec. 1.2. P e r m i t t i n g V to become n e g a t i v e 24 encourages both t r i p l e t and s i n g l e t p a i r i n g of p a r t i c l e s on n e i g h b o u r i n g s i t e s ( F i g . 1.5(c)) and hence a s u p e r c o n d u c t i n g s t a t e i f t i s non-zero. (a) X X X • • • X X • X • ( b ) X « X ' X ' X ' X (0 -—t—*—•—t—I—•—i— ( d ) t I t I t J, t I t 1 (e) 11 ' 11 • 11 1 tl • tl F i g u r e 1.5. P o s s i b l e s t a t e s of the H a m i l t o n i a n ( 1 . 1 2 ) . i n d i c a t e s a s i n g l y o c c u p i e d s i t e of e i t h e r s p i n and unoc c u p i e d s i t e , (a) U > 0, V = 0, t = 0, f <•1/2 (b) U > 0 V > 0, f = 1/2 (c) U > 0, V < 0, f = 1/2 (d) U > 0, V > 0, t *0, f = 1 (e) U < 0, V > 0, f < 2 (2) U > 0, f = 1 T h i s s p i n - d e g e n e r a t e s t a t e has each l a t t i c e s i t e s i n g l y o c c u p i e d when t = 0. I n t e r s i t e hopping l e a d s t o an a n t i f e r r o m a g n e t i c exchange c o u p l i n g between n e i g h b o u r i n g s p i n s and c o n s e q u e n t l y a SDW as p i c t u r e d i n F i g . 1 . 5 ( d ) . an 25 (3) U < 0, f = 1/2 The s t a t e w i t h V > 0 i s the same as i n case ( 1 ) , but w i t h a p a i r of e l e c t r o n s l o c a t e d a t e v e r y o t h e r s i t e ( F i g 1 . 5 ( e ) ) . A c o m p e t i t i o n between the CDW and s i n g l e t s u p e r c o n d u c t i v i t y modes a r i s e s i f the p a i r s a r e a l l o w e d t o propagate a l o n g the l a t t i c e . Emery (1979) has a l s o demonstrated how the L u t t i n g e r model may be d e r i v e d by t a k i n g the continuum l i m i t of Eq. (1.12) f o r s p i n l e s s f e r m i o n s : Here the term i n U has d i s a p p e a r e d s i n c e each l a t t i c e s i t e may accommodate o n l y one p a r t i c l e w i t h o u t s p i n . In the l i m i t of d -> 0, the s i t e o p e r a t o r s , c^, upon a p p r o p r i a t e t r a n s f o r m a t i o n , go over i n t o the f i e l d o p e r a t o r s ^ ; ( x ) f o r fe r m i o n s w i t h v e l o c i t i e s ±v P = ±lim 2 t d , and which T j V A / JLWJ. 1. 1. Ill ± \J 11 & TY i U 11 s a t i s f y the D i r a c e q u a t i o n , The energy band, £^ = _ 2 t c o s k d becomes the i n f i n i t e l i n e a r spectrum 6 ^ = ±v_k. F i n a l l y , t he H a m i l t o n i a n which r e s u l t s i s o + 26 Note t h a t t h e - f i r s t two terms i n the second i n t e g r a l s i g n i f y the i n t e r a c t i o n of two p a r t i c l e s on the same branch; t h i s r e p r e s e n t s an e x t e n s i o n of the the o r i g i n a l L u t t i n g e r model ( 1 . 6 ) . 1.4 Response F u n c t i o n s Up t o t h i s p o i n t a number of H a m i l t o n i a n s f o r a ID i n t e r a c t i n g e l e c t r o n gas have been d i s c u s s e d , but t h e i r s i g n i f i c a n c e has not been d e a l t w i t h . The b a s i c aim of these models, as mentioned i n Sec. 1.2, i s t o g i v e some i n s i g h t i n t o the o c c u r r e n c e of phase t r a n s i t i o n s i n the w e a k l y - c o u p l e d l i n e a r c h a i n systems. The f e r m i o n gases t h a t have been c o n s i d e r e d a re t r a n s l a t i o n a l l y i n v a r i a n t , and thus cannot m a n i f e s t any s p a t i a l o r d e r . In r e a l i t y however, the e l e c t r o n gas e x i s t s i n the presence of a l a t t i c e which b r e a k s the t r a n s l a t i o n a l symmetry of the gas and induces a response i n i t . A common method of d e t e r m i n i n g phase i n s t a b i l i t i e s i s t o c a l c u l a t e the response of the f e r m i o n system t o an i n f i n i t e s i m a l e x t e r n a l f i e l d . D i v e r g e n t b e h a v i o u r i s i n d i c a t i v e of an i n s t a b i l i t y i n the q u a n t i t y which c o u p l e s t o the p e r t u r b a t i o n . The low temperature phase of the the system w i l l g e n e r a l l y e x h i b i t a l o n g range o r d e r as measured by an o r d e r parameter. For an i n v a r i a n t system the o r d e r parameter v a n i s h e s ; one must then l o o k a t the s u s c e p t i b i l i t y f u n c t i o n which g i v e s the l i n e a r response of the system t o a weak f i e l d , or a t the c o r r e l a t i o n f u n c t i o n . We d e f i n e the f o l l o w i n g q u a n t i t i e s (Solyom 1979): 1) the F o u r i e r t r a n s f o r m of the charge d e n s i t y 27 2) the F o u r i e r t r a n s f o r m of the z-component of the s p i n d e n s i t y SDW M ko- fe+«tr 3) the c r e a t i o n o p e r a t o r f o r t r i p l e t p a i r s w i t h s p i n component 2or a l o n g the z - a x i s 4) the s i n g l e t p a i r c r e a t i o n o p e r a t o r A d d i t i o n a l o p e r a t o r s c o u l d have been d e f i n e d f o r the x- and y-components of the s p i n d e n s i t y and f o r t r i p l e t p a i r s w i t h a z e r o s p i n p r o j e c t i o n . The o r d e r parameters f o r the CDW, SDW, t r i p l e t s u p e r c o n d u c t i v i t y ( T S), and s i n g l e t s u p e r c o n d u c t i v i t y (SS) phases are then g i v e n by the ground s t a t e averages <0^(q)> of the c o r r e s p o n d i n g o p e r a t o r s . . The frequency-dependent l i n e a r response of the Fermi system t o a g e n e r a l i z e d e x t e r n a l p o t e n t i a l V^(q,w) i s measured by <0^(q,u>)>, the average of 0^(q,w) i n the presence of . The g e n e r a l i z e d s u s c e p t i b i l i t y d e f i n e d by <0^(q,W)> = ^ ( q , ( d ) V ^ ( g , t f ) i s the F o u r i e r t r a n s f o r m of a t w o - p a r t i c l e r e t a r d e d Green f u n c t i o n A l t e r n a t i v e l y , 0 ^ ^ may be e x p r e s s e d i n terms of the t i m e - o r d e r e d Green f u n c t i o n or r e l a t e d t o the c o r r e l a t i o n f u n c t i o n < 0 ^ (q,t)0^(q,0)> by the f l u c t u a t i o n - d i s s i p a t i o n theorem. The i n t e n t of many s t u d i e s • has been t o determine the dependence on the c o u p l i n g c o n s t a n t s of the cu 0 be h a v i o u r 28 of ^^(q,to) . A b r i e f o u t l i n e of what has been found f o l l o w s and w i l l be summarized i n the form of a phase diagram. I t was s t a t e d above t h a t the t w o - p a r t i c l e r e t a r d e d Green f u n c t i o n i s r e l a t e d t o the t i m e - o r d e r e d f u n c t i o n d e f i n e d by 1 *• 5 T The advantage i n c a l c u l a t i n g K i s the ready a p p l i c a t i o n of p e r t u r b a t i o n t h e o r y ( N o z i e r e s 1964), which l e a d s t o the e x p r e s s i o n of K i n terms of a v e r t e x f u n c t i o n , P: where p^ - denotes the v a r i a b l e s , Wj , ^ and G(p^) i s . the s i n g l e - p a r t i c l e Green f u n c t i o n . The B e t h e - S a l p e t e r e q u a t i o n i s an i n t e g r a l e q u a t i o n f o r P; d i a g r a m a t i c a l l y i t reads as i n F i g . 1.6. J i s the i r r e d u c i b l e v e r t e x p a r t , dashed ( s o l i d ) double l i n e s i n d i c a t e e x t e r n a l ( i n t e r n a l ) Green f u n c t i o n s G, and arrows p o i n t i n g up (down) r e p r e s e n t p a r t i c l e s ( h o l e s ) . S u b s c r i p t s pp and ph a r e a t t a c h e d t o P and J t o denote p a r t i c l e - p a r t i c l e and p a r t i c l e - h o l e s c a t t e r i n g . Let us examine the v e r t e x Vpp i n more d e t a i l . The z e r o t h o r d e r v e r t e x i s j u s t the bare i n t e r a c t i o n w h i l e a number of f i r s t - o r d e r diagrams a r e d e p i c t e d i n F i g . 1.7. The dashed ( s o l i d ) l i n e s r e p r e s e n t e x t e r n a l ( i n t e r n a l ) non-29 F i g u r e 1.6 B e t h e - S a l p e t e r e q u a t i o n f o r the v e r t e x p a r t . + i n t e r a c t i n g o n e - p a r t i c l e Green f u n c t i o n s G^(k,cu) r where ± i n d i c a t e s a p a r t i c l e near the Fermi p o i n t ±kp., and the wavy l i n e denotes the bare v e r t e x . For the c h o i c e k = -kp, k' = +k F, the v e r t e x p a r t of F i g . 1.7(a) i s p r o p o r t i o n a l t o f ^ dftu)' C^F-P.w)^7'/<F + P,w-Co0 -(1.16a) I f p = 0, the p o l e s of the two Green f u n c t i o n s approach each o t h e r as W -5> 0, and the i n t e g r a l d i v e r g e s as ln(oJ/u)0) , where t*J0 i s a bandwidth c u t o f f . The same d i v e r g e n c e , but w i t h o p p o s i t e s i g n o c c u r s i n F i g . 1.7(b) i f we w r i t e the i n t e g r a l as 30 k +q;-A + ik-q k+qi - •+1 k' i ! * p f A kVvWvUM k+p |M\AVWVVVW| A | + k'-p P + q k|- + |k' ( c ) k I" + |k' (a) F i g u r e 1.7. F i r s t - o r d e r c o n t r i b u t i o n s t o the v e r t e x p a r t , (a) p a r t i c l e - p a r t i c l e s c a t t e r i n g (Cooper c h a n n e l ) (b) p a r t i c l e - h o l e s c a t t e r i n g ( zero-sound c h a n n e l ) (c) p a r t i c l e - h o l e s c a t t e r i n g i n which the p a r t i c l e and h o l e a r e on the same branch. but no such l o g a r t h m i c term appears i n the diagram ( c ) . Diagrams (a) and (b) a r e examples of the s o - c a l l e d Cooper p a i r and z e r o -sound s c a t t e r i n g c h a n n e l s . C l e a r l y , r e p e a t e d i n s e r t i o n of th e s e elements i n t o a v e r t e x w i l l g e n e r a t e any power of ln(w/w 0) and hence the l e a d i n g terms i n each o r d e r . For the c h o i c e of c o u p l i n g c o n s t a n t s g ( ^ = g,^ = g Z = 9 ' 93 = g# = °' Bychkov e t a l . (1966) summed these ( p a r q u e t ) diagrams e x a c t l y , and o b t a i n e d a r e s u l t f o r V^p i n v o l v i n g the f a c t o r A s i n g u l a r i t y o c c u r s when g < 0; a c c o r d i n g t o Bychkov et a l . , t h i s i m p l i e s an i n s t a b i l i t y of the ground s t a t e w i t h r e s p e c t t o the f o r m a t i o n of bound p a i r s a t a t r a n s i t i o n temperature ( o b t a i n e d by s u b s t i t u t i n g T f o r to) The non-zero v a l u e of T. i n d i c a t e s the inadequacy of the parquet r (1.16b) / Tc = tu" 0exp(Trvp/g). 31 a p p r o x i m a t i o n s i n c e i t c o n t r a d i c t s the theorem r e g a r d i n g ID phase t r a n s i t i o n s . T h i s i s an example of the well-known tendency of m e a n - f i e l d t h e o r y t o o v e r e s t i m a t e the t r a n s i t i o n t e m p e r a t u r e . Menyhard and Solyom (1973) went beyond the p a r q u e t summation and i n c l u d e d f l u c t u a t i o n e f f e c t s by t r e a t i n g l o w e r -o r d e r l o g a r i t h m i c c o r r e c t i o n s i n a c a l c u l a t i o n u s i n g the r e n o r m a l i z a t i o n group method. Now t a k e n i n t o account are terms (such as those shown i n F i g . 1.8) i n which the p r o p a g a t o r s are d r e s s e d w i t h s e l f - e n e r g y i n s e r t i o n s , as w e l l as diagrams b e l o n g i n g t o the e l e c t r o n - h o l e s c a t t e r i n g c h a n n e l , F i g . 1 . 7 ( c ) . W i t h t h e s e improvements Ppp obeys the power law fpp ~ (W0/<y)3/2 F i g u r e 1.8. V e r t e x diagrams g i v i n g l o w e r - o r d e r l o g a r i t h m i c c o r r e c t i o n s . f o r s m a l l f r e q u e n c i e s . Thus we see t h a t t h e e f f e c t of the f l u c t u a t i o n s has been t o d e p r e s s T^ down t o z e r o . W h i l e the above r e s u l t s were d e r i v e d by Bychkov et a l . and Menyhard and Solyom f o r the s u p e r c o n d u c t i n g i n s t a b i l i t y , the same c o n s i d e r a t i o n s a r e a p p l i c a b l e t o the r e m a i n i n g s u s c e p t i b i l i t i e s s i n c e the e x p a n s i o n s of the c o r r e s p o n d i n g v e r t e x f u n c t i o n s w i l l c o n t a i n the same l o g a r i t h m i c terms. A l l 32 responses must t h e r e f o r e be d e t e r m i n e d s i m u l t a n e o u s l y i n o r d e r t o p r e d i c t which i n s t a b i l i t y w i l l be dominant. L u t h e r and P e s c h e l (1974) used t h e i r t e c h n i q u e of b o s o n i z i n g f e r m i o n o p e r a t o r s t o c a l c u l a t e the d e n s i t y and p a i r i n g response i n the L u t t i n g e r model. S i n c e s p i n i s absent i n t h i s model, t h e r e i s no d i s t i n c t i o n between the SDW and CDW and between the TS and SS s u s c e p t i b i l i t i e s . F u r t h e r m o r e , both v (q,a>) and X (q,uo) d i s p l a y the same g^-dependent power law d i v e r g e n c e as W-*0, the former f o r q = 2k F and g ^ > 0, and the l a t t e r f o r q = 0 and g^ < 0. When the b a c k s c a t t e r i n g c o n s t a n t s g,^ and g ( ^ are f i n i t e , the s p i n degree of freedom becomes s i g n i f i c a n t . G u t f r e u n d and Klemm (1976) have c a l c u l a t e d the response f u n c t i o n s f o r the L u t h e r and Emery (1974) s o l u t i o n which e x i s t s f o r q{^ = - 6 T T v p / 5 , a g a i n by means of the boson r e p r e s e n t a t i o n . In the s t a t i c l i m i t , # t o | j 2 k p , t o ) and 0^ss(O,J«J) d i v e r g e as i n the L u t t i n g e r model but w i t h a d i f f e r e n t exponent. However, because of the gap i n the' spectrum of s p i n e x c i t a t i o n s , Xs£>^ ( 2kp ,u)) a n d ^ f T S ( 0 , t o ) remain f i n i t e as u) -> 0. Along the l i n e g ( ^ = -6TTV f/5, CDW b e h a v i o u r i s p r e d i c t e d f o r g^ > -3TTv F/5, w h i l e a SS s t a t e i s e x p e c t e d f o r more n e g a t i v e v a l u e s of g^. In f u r t h e r developments, the TL model has been extended by Solyom (1979) t o i n c l u d e the. small-momentum p r o c e s s , g^, and s p i n , w h i l e Chui and Lee (1975) have mapped the problem onto the c l a s s i c a l 2D Coulomb gas. G u t f r e u n d and Klemm (1976) have i n v e s t i g a t e d the e f f e c t of g^ i n the Luther-Emery model. The p r e s e n t s t a t e of the ' g - o l o g i c a l ' p i c t u r e of the ID Fermi gas may be summarized i n a phase diagram, F i g . 1.9, which 33 s h o w s t h e e x p e c t e d i n s t a b i l i t y a s a f u n c t i o n o f g , (= gt/f = g , j _ ) a n d g z . L e s s d i v e r g e n t r e s p o n s e s a r e i n d i c a t e d i n b r a c k e t s . S S C D W (CDW) ( S S ) S D W (CDW) F i g u r e 1 . 9 . P h a s e d i a g r a m o f t h e F e r m i g a s f o r t h e c o u p l i n g c o n s t a n t s g , , g ^ . ( A f t e r G u t f r e u n d a n d K l e m m 1 9 7 6 . ) A n , i m p o r t a n t d e t a i l t o n o t e i s t h e d i v i s i o n o f t h e p l a n e b y t h e l i n e gj = 2g^_ i n t o t w o r e g i o n s i n w h i c h s u p e r c o n d u c t i v i t y a n d d e n s i t y w a v e s a r e t h e d o m i n a n t r e s p o n s e s . T h i s i s a f e a t u r e w h i c h e x i s t s a l r e a d y i n m e a n f i e l d t h e o r y , a n d c a n , i n f a c t , b e d e r i v e d f r o m a c o n s i d e r a t i o n o f t h e s y m m e t r y o f t h e H a m i l t o n i a n ( 1 . 9 ) w i t h r e s p e c t t o t h e c o u p l i n g c o n s t a n t s ( B a e r i s w y l a n d F o r n e y 1 9 8 0 ) . 34 1.5 Summary In s p i t e of the r e l a t i v e abundance of r e s u l t s , both e x a c t and a p p r o x i m a t e , f o r the Fermi gas model, t h e r e remains a b a s i c problem i n e s t a b l i s h i n g a corre s p o n d e n c e between the c o u p l i n g c o n s t a n t s , g-, and the e f f e c t i v e p o t e n t i a l between e l e c t r o n s i n a r e a l s o l i d . For i n s t a n c e , G u t f r e u n d ( i n K e l l e r 1977) "would r o u g h l y put KCP somewhere i n the second quadrant and TTF-TCNQ i n the f i r s t q u a d r a n t " of the (g ( , g ^ ) - p l a n e i n F i g . 1.9. ( I t a l i c s mine . ) 0 ' I f one b e g i n s w i t h a bare i n t e r a c t i o n v(q) such as the Coulomb p o t e n t i a l , which i s of l o n g range and t h e r e f o r e d i v e r g e n t as q 0, one i m m e d i a t e l y e n c o u n t e r s d i f f i c u l t i e s w i t h p e r t u r b a t i o n t h e o r e t i c t r e a t m e n t s such as the parquet summation and the r e n o r m a l i z a t i o n group method. R e f e r r i n g back t o the i n t e g r a l s (1.16a,b), we note t h a t they now i n c l u d e an i n t e g r a t i o n over the s i n g u l a r i t y i n v ( p ) , i n a d d i t i o n t o the one a l r e a d y p r e s e n t i n the v a r i a b l e to'. T h i s c o m p l i c a t i o n may w e l l i n v a l i d a t e the a f o r e m e n t i o n e d p e r t u r b a t i o n approaches. The assumption of a screen e d i n t e r a c t i o n t o f i t i n t o the 'g-ology' p i c t u r e i s a l s o q u e s t i o n a b l e , because of the problem i n d e t e r m i n i n g which diagrams i n the e x p a n s i o n a re a l r e a d y a c c o u n t e d f o r i n the s c r e e n i n g p r o c e s s . An a l t e r n a t i v e scheme, which bypasses the d i f f i c u l t i e s i n h e r e n t i n p e r t u r b a t i o n t h e o r y , i s t o make an a p p r o x i m a t i o n f o r a q u a n t i t y r e l a t e d t o the response f u n c t i o n . In Chap. 2, the d i e l e c t r i c response of a ID gas of e l e c t r o n s w i t h a model i n t e r a c t i o n i s c a l c u l a t e d by a p p r o x i m a t i n g the e q u a t i o n of motion f o r the d e n s i t y f l u c t u a t i o n o p e r a t o r . The s u s c e p t i b i l i t y , 35 X (q,w), so o b t a i n e d i s then i n v e s t i g a t e d f o r the purpose of &DIA/ d e t e r m i n i n g under what c o n d i t i o n s the system w i l l p o s s e s s a CDW i n s t a b i l i t y . 36 Chapter 2 D i e l e c t r i c Response of a One-dimensional E l e c t r o n Gas 2.1 I n t r o d u c t i o n H aving o u t l i n e d the common c o n c e p t u a l approach t o the problem of the i n t e r a c t i n g o n e - d i m e n s i o n a l e l e c t r o n gas i n the opening c h a p t e r , we pr o c e e d now t o d e f i n e and examine a somewhat d i f f e r e n t model. In Sec. 2.2, the s t a r t i n g p o i n t i s an a n i s o t r o p i c 3D system which b e a r s some resemblance t o the e l e c t r o n i c environment of the m o l e c u l a r c h a i n s t r u c t u r e d i s c u s s e d p r e v i o u s l y . W i t h a s u i t a b l e d e s c r i p t i o n of the non-i n t e r a c t i n g gas, i t i s p o s s i b l e t o t r a n s f o r m the 3D H a m i l t o n i a n i n t o a ID one w i t h a m o d i f i e d i n t e r a c t i o n . Whether or not a CDW s t a t e i s the ground s t a t e of the system can be d e t e r m i n e d from the b e h a v i o u r of the d i e l e c t r i c s u s c e p t i b i l i t y . C o n s e q u e n t l y , the a p p r o p r i a t e f o r m a l i s m f o r the ID case i s de v e l o p e d i n Sec. 2.3 by a d a p t i n g the tr e a t m e n t of the 3D gas by P i n e s (1963). S i n c e the s u s c e p t i b i l i t y cannot be c a l c u l a t e d e x a c t l y , an a p p r o x i m a t i o n . scheme i s i n t r o d u c e d i n Sec. 2.4; the one we have chosen i s the method a p p l i e d w i t h some su c c e s s by S i n g w i , T o s i , Land, and S j o l a n d e r (1968) t o the 3D e l e c t r o n gas. The c a l c u l a t i o n s f o r a p a r t i c u l a r v e r s i o n of the model are o u t l i n e d i n Sees. 2.5 and 2.6. F i n a l l y , t he r e s u l t s a re p r e s e n t e d and a n a l y z e d i n Sec. 2.7, w i t h a d i s c u s s i o n f o l l o w i n g i n the l a s t s e c t i o n of the c h a p t e r . 37 2.2 A Quasi-One-Dimensional Model The model t h a t w i l l be the b a s i s of our approach t o the d i s c u s s i o n of ID Fermi gas i n s t a b i l i t i e s i s d e f i n e d by the f o l l o w i n g f e a t u r e s . F i r s t , N e l e c t r o n s a re p l a c e d i n a 'tube' of l e n g t h L ( l i n e a r d e n s i t y n=N/L) w i t h a c y l i n d r i c a l l y symmetric background p o t e n t i a l which i s t r a n s l a t i o n a l l y i n v a r i a n t a l o n g the x - a x i s . P e r p e n d i c u l a r t o the x - a x i s , the p a r t i c l e s a r e d e s c r i b e d by the e i g e n f u n c t i o n s of a t r a n s v e r s e H a m i l t o n i a n , Hj_. Supposing the d i f f e r e n c e between the two lowest energy e i g e n v a l u e s of H t o be much l a r g e r than the bandwidth, 6p , we take the b a s i s f u n c t i o n s of the n o n - i n t e r a c t i n g e l e c t r o n s t o be g i v e n by where i s the n o r m a l i z e d l o w e s t energy e i g e n s t a t e of H_j_ and k i s the wavenumber i n the x - d i r e c t i o n . We employ h e r e , and f o r the remainder of the c h a p t e r , the n o t a t i o n r = (x,rj.) and k = (k' f ° r r e a l - and r e c i p r o c a l - s p a c e v e c t o r s w i t h magnitudes r and K. S e c o n d l y , the p a r t i c l e s i n t e r a c t w i t h the Coulomb p o t e n t i a l v & ( r ) = e2/r. Then, t o ensure charge n e u t r a l i t y , the e l e c t r o n s a r e assumed t o move i n a smeared-out p o s i t i v e background which j u s t b a l a n c e s the k = 0 component of the e l e c t r o n d e n s i t y . The S c h r o d i n g e r e q u a t i o n f o r the model i s w r i t t e n as where and V gp a r e the e l e c t r o n - e l e c t r o n and e l e c t r o n -background i n t e r a c t i o n s and 38 In terms of the c r e a t i o n and a n n i h i l a t i o n o p e r a t o r s of p a r t i c l e s of s p i n cr i n the s t a t e P^, Eq. (2.1) i s e x p r e s s e d i n the form cr,<rz (2.2) where C = S i n c e V^ -^ i s s e p a r a b l e , the dependence may be i n t e g r a t e d out of the m a t r i x elements, 3 °ier¥ where and t 2 "(t)S T j ^ W (2.3) Here (^<3j_) i s the F o u r i e r t r a n s f o r m of the t r a n s v e r s e charge 39 d e n s i t y , J (2.4) With t h e s e d e f i n i t i o n s one a r r i v e s a t the r e s u l t whence the e l e c t r o n - e l e c t r o n term i n Eq. (2.2) becomes (2.5) C o n s i d e r now the i n t e r a c t i o n of the system w i t h an e x t e r n a l charge d i s t r i b u t i o n zf^(v). The a p p r o p r i a t e H a m i l t o n i a n i s « t e - 2 v < ^ i v w ( r ) i k V ' > c ; < p W where the p o t e n t i a l due t o zyj g J <(£) i s V M -- [dLr! -«y«sfeo I f y 6 x(£,) i s s e p a r a b l e i n t o the form then whence D e f i n i n g 40 H. becomes ex ^ex S L'£ A* K^^Cl«r Ck-^ (2.8) Is We w i l l have o c c a s i o n t o make use of Eqs. (2.6) - (2.8) l a t e r i n the c o u r s e of the development of the d i e l e c t r i c f o r m a l i s m . The background p o s i t i v e charge d e n s i t y i s a charge d i s t r i b u t i o n e x t e r n a l t o the e l e c t r o n system, w i t h the p a r t i c u l a r parameters z = +e and S u b s t i t u t i o n i n t o Eqs. (2.7) and (2.8) y i e l d s and The s e l f - e n e r g y of the background i s and the q = 0 term of H ^ i s g i v e n by S i n c e the above t h r e e terms c a n c e l t o o r d e r N, we r e w r i t e , a f t e r d e f i n i n g the d e n s i t y f l u c t u a t i o n o p e r a t o r the H a m i l t o n i a n a p p e a r i n g i n Eq. (2.2) as 41 « "£ ekC*W + il£ "^/& - n ] <2;9» As i t stands t h i s i s j u s t the H a m i l t o n i a n f o r f e r m i o n s (not e l e c t r o n s ) i n a ID box which i n t e r a c t w i t h the p o t e n t i a l v ( q) d e f i n e d by Eq. ( 2 . 3 ) . 2.3 D i e l e c t r i c P r o p e r t i e s of_ a ID Fermi Gas Le t us now t u r n our a t t e n t i o n t o the problem of the d i e l e c t r i c response of a system d e s c r i b e d by Eq. ( 2 . 9 ) . The tr e a t m e n t of the 3D e l e c t r o n gas (3DEG) by P i n e s (1963, h e r e a f t e r r e f e r r e d t o as P) w i l l be the b a s i s of the e n s u i n g f o r m u l a t i o n . Begin by s u b j e c t i n g the p a r t i c l e s t o the i n f l u e n c e of a t i m e - v a r y i n g e x t e r n a l charge d e n s i t y * / e / r , ^ = */ex k . ) U j + (du> p ^ k ^ e ^ t * S u b s t i t u t i n g i n t o Eq. (2.8) g i v e s the p e r t u r b i n g H a m i l t o n i a n W e, = l£+ f A - A x ^ w ) ^ V"* r f' r/'t (2.10) Note t h a t no time dependence has been a s c r i b e d t o the t r a n s v e r s e d e n s i t y , ^JCi) ' * n o r d e r t o keep the ID p o t e n t i a l v f i s < (q) s t a t i c . E q u a t i o n s (2.8) and (2.10) a r e i n the same form as Eqs. ( 3 -109a,b) i n P so t h a t we may i m m e d i a t e l y make use of h i s r e s u l t f o r the induced charge d e n s i t y c a l c u l a t e d i n l i n e a r r e s p o n s e , P = tutx.d J l(f+)K, lZ [ w . J ^ a - ' oo+l^Ls) (2.11) where (/<j^no * s fc^e m a t E " i x element of the o p e r a t o r p* between 42 the ground s t a t e , t£ , and the e x c i t e d s t a t e , *£ n , of the un p e r t u r b e d system, and W A O = E n - E 0 . In o r d e r t o proceed t o the i n t r o d u c t i o n of a d i e l e c t r i c f u n c t i o n , i t i s n e c e s s a r y t o b a c k t r a c k a l i t t l e t o the tube of e l e c t r o n s t h a t we began w i t h . Imagine an e x t e r n a l charge d i s t r i b u t i o n zf&^(%,u) i n d u c i n g a d e n s i t y J>(^,bJ) i n the tube. The p o t e n t i a l s due t o p and JJ^ are o b t a i n e d by P o i s s o n ' s e q u a t i o n : 1/ ( ) - * » f \ (2.12) The t o t a l p o t e n t i a l seen by a t e s t charge i n the system i s V T ( ^ ) « V ^ ^ . J + VAit By d e f i n i t i o n , the f i e l d s due t o vy and V^^, are r e l a t e d by the d i e l e c t r i c t e n s o r : Combining Eqs. (2.14) and ( 2 . 1 5 ) , we have D e f i n e a d i e l e c t r i c f u n c t i o n rrf^w) by (2.13) (2.14) (2.15) 43 (2.16) i n o r d e r t o w r i t e v e x ( ^ ) = e ( ^ , J v r ( ^ o j ) From Eqs. ( 2 . 1 2 ) , ( 2 . 1 3 ) , (2.16) we have ep(g ,w) = -? (T T - ^ - T - ') A * ' ' ft* V' (2.17) S i n c e our model assumes t h a t A x ^ ) = A x t y ^ A x t y . ) an e x p r e s s i o n i n v o l v i n g o n l y the x-component of the d e n s i t y i s o b t a i n e d by p u t t i n g p&)C(<\L) = f^j.)' Thus, v € x ( q ) = v(q) and f<fu]' - ' h * ' ^ ( 2 . i e ) Comparison of Eq. (2.17) w i t h Eq. (2.11) p r o v i d e s the r e l a t i o n - t — \ ( * + ) !- ] L K J%*» Lto-oj^+tf a+u^+CSJ ( 2 . 1 9 ) w i t h v(q) g i v e n by ( 2 . 3 ) . D e f i n i n g the f u n c t i o n Eq. (2.19) may be r e w r i t t e n as / . ,. „ p J *»*o \(0*) lZ - iirV(l> S(, u) €itM) ' ^ « 1 - < " A ^ ' L M J ' W ' ( 2. 2 1 ) where P denotes the p r i n c i p a l p a r t . E q u a t i n g i m a g i n a r y p a r t s i n t h i s l a s t e q u a t i o n , one a r r i v e s a t the sum r u l e 44 *3 L^^e^. --v»v(i)s(i) ( 2 . 2 2 ) where Sly) - jjj J Si^u) du) From the d e f i n i t i o n of S ( q ) , i t may be shown (see P, pp. 72-, 129-) t h a t S(q) i s the F o u r i e r t r a n s f o r m of the ti m e -independent d e n s i t y - d e n s i t y c o r r e l a t i o n f u n c t i o n , and c o n s e q u e n t l y can be r e l a t e d t o the p a i r d i s t r i b u t i o n f u n c t i o n , g ( x ) , by f r . , 1 _/<,)< (2.23) The f u n c t i o n g(x) i s the p r o b a b i l i t y of l o c a t i n g a p a r t i c l e a d i s t a n c e x from another one, and i s t h e r e f o r e of p h y s i c a l i m p o r t a n c e . The requirement t h a t g ( x) be no n - n e g a t i v e i s one c r i t e r i o n of the v a l i d i t y of an a p p r o x i m a t i o n scheme. Another q u a n t i t y of i n t e r e s t i s the d i e l e c t r i c s u s c e p t i b i l i t y which measures the d e n s i t y response of the e l e c t r o n gas t o an e x t e r n a l p e r t u r b a t i o n and i s d e f i n e d by On comparison w i t h Eq. (2 . 1 8 ) , we make the o b s e r v a t i o n t h a t -/ - vtyXCl.lti) ( 2.24) and c o n s e q u e n t l y t h a t 45 *H'^~tL ^l(fpno^lU)-ulK0+iSr to+J^+if}. (2.25) S i n c e the summand i n t h i s e q u a t i o n r e q u i r e s a knowledge of the m a n y - p a r t i c l e e i g e n s t a t e s and e n e r g i e s , 'X (Q fW) may be e v a l u a t e d e x a c t l y o n l y i n the n o n - i n t e r a c t i n g case i n which {%^X9) (2.26) 2.3A Sum R u l e s The d i e l e c t r i c f u n c t i o n , or e q u i v a l e n t l y the s u s c e p t i b i l i t y , of the 3DEG obeys a number of sum r u l e s which d e r i v e from p h y s i c a l c o n s i d e r a t i o n s and the a n a l y t i c b e h a v i o u r of 6 and X. These r u l e s (see P i n e s and N o z i e r e s 1966, h e r e a f t e r r e f e r r e d t o as PN, pp. 205-), w i t h the Coulomb p o t e n t i a l vc(g) = 47Te 2/Q 2 e x p l i c i t l y - i n d i c a t e d , a r e : IT Q Z OOT^X du) = - ZY^ L (2.27a) o 00 I (2.27b) 46 I I IT lim w T y v i X (a .60 j du) = o /a j |->o J° * (2.27c) li'w j ToTm 6(g,0o) du) - ""Vto\Vi~'K\/c_(q) $-*>° 0 " (2.27d) We pr o c e e d now t o determine what the ID analogues of the above e x p r e s s i o n s a r e i n o r d e r t o p r o v i d e c o n s i s t e n c y checks on the a p p r o x i m a t i o n s t h a t w i l l be made. The f-sum r u l e (2.27a) i s a consequence of p a r t i c l e c o n s e r v a t i o n and (2.25) ; d i m e n s i o n a l i t y i s nowhere a c o n s i d e r a t i o n i n the d e r i v a t i o n so t h a t the ID v e r s i o n i s The r e m a i n i n g t h r e e r u l e s h o l d it 6 and % s a t i s f y the Kramers-K r o n i g r e l a t i o n s r oo 7T Vo'-OO (2.29a) p r« T 7 7 — 7 1 — 00-00 ~* (2.29b) where P denotes the p r i n c i p a l p a r t and the wavevector dependence has been s u p p r e s s e d . These r e l a t i o n s a r e themselves dependent on OC and 6 b e i n g a n a l y t i c i n the upper h a l f of the complex - 6 J p l a n e . By d e f i n i t i o n , OC i s a c a u s a l response f u n c t i o n and t h e r e f o r e e x h i b i t s the r e q u i r e d a n a l y t i c i t y except p o s s i b l y f o r a p o l e a t 00 = 0 ( i n which case (2.29b) must be m o d i f i e d by a d d i n g a term i n v o l v i n g the r e s i d u e of the p o l e ) . Then Eq. (2.29b), t o g e t h e r 47 w i t h the p e r f e c t s c r e e n i n g c o n d i t i o n f o r the 3DEG, / = 0 (2.30) l e a d s t o Eq. (2.27b). In the ID c a s e , p e r f e c t s c r e e n i n g o c c u r s i f v ( q ) d i v e r g e s i n the s m a l l q l i m i t s i n c e , a c c o r d i n g t o Eq. (2.37) / i m € ( Q O ) = / •+ nzv(q)K €, however, i s not a c a u s a l response f u n c t i o n ( a l t h o u g h l / € i s ) so t h a t i t s f u n c t i o n a l form must be i n v e s t i g a t e d t o determine i t s a n a l y t i c p r o p e r t i e s . Because any s i n g u l a r i t i e s i n 6 w i l l appear as z e r o e s i n 1/f, we r e c a l l the d e f i n i t i o n ( 2 . 1 9 ) : Ttfifi - / * "Z^ 2 K f ^ ) n o l { u-w^+tf ~ w+«no+tf J For complex U) = + iu)^, l/£ i s a l s o complex, and thus can assume r e a l v a l u e s o n l y on the r e a l and i m a g i n a r y axes. When LJ = which i s a f u n c t i o n i n c r e a s i n g m o n o t o n i c a l l y from t o 1 as u)z-2>*>. Thus, i f l / € ( q , 0 ) < 0, t h e r e w i l l be a p o l e i n €{q,ui) on the i m a g i n a r y a x i s . For r e a l CJ = cJt , 1/f w i l l have an i m a g i n a r y p a r t i n the r e g i o n of e x c i t a t i o n f r e q u e n c i e s tdM • O t h e r w i s e , 48 R e f e r r i n g t o t h e s i n g l e - p a r t i c l e e x c i t a t i o n s p e c t r u m i n F i g . 2.7, we n o t e t h e s i g n i f i c a n c e o f t h e ' i n s t e p ' , d e m a r c a t e d by vJ < W_(q). In a d d i t i o n t o t h e a b s e n c e of s i n g l e - p a r t i c l e e x c i t a t i o n s , t h e r e w i l l be no s t a t e s i n t h i s r e g i o n i n w h i c h m u l t i p l e p a i r s a r e e x c i t e d . I t may be c o n c l u d e d , t h e r e f o r e , t h a t u n l e s s a c o l l e c t i v e mode a p p e a r s w i t h a f r e q u e n c y l e s s t h a n 6 u _ ( q ) , l / t 5(q,6J) i s r e a l f o r t*) < £\)_(q) ( t h i s h o l d s f o r a l l q ± 2mkp, m i n t e g e r , s i n c e t h e i n s t e p r e p e a t s p e r i o d i c a l l y w i t h p e r i o d 2 k p ) . S i n c e i t d i v e r g e s n e g a t i v e l y a s 60, -> t>0_(q) , l/£ w i l l p a s s t h r o u g h z e r o i f l / f ( q , 0 ) > 0. In summary t h e n , c?(q , W ) has a p o l e on t h e r e a l ( i m a g i n a r y ) a x i s i f f ( q , 0 ) i s p o s i t i v e ( n e g a t i v e ) . T h i s n o n - a n a l y t i c i t y of € has t h e i m p o r t a n t c o n s e q u e n c e t h a t t h e K r a m e r s - K r o n i g r e l a t i o n (2.29b) i s not t r u e and hence t h a t t h e r e e x i s t no ID v e r s i o n s of t h e sum r u l e s ( 2 . 2 7 b , d ) . An e x c e p t i o n t o t h e b e h a v i o u r j u s t o u t l i n e d may o c c u r , however, when q i s an i n t e g r a l m u l t i p l e of 2kp, i n w h i c h c a s e 6^(2mkp.) = 0 and t h e p o s s i b i l i t y o f a z e r o of l / £ f o r r e a l id i s n e g a t e d . Then t5(2mkp,w) w i l l be a n a l y t i c i n t h e upper W-plane p r o v i d e d c?(2mkp,0) > 0. I f t h i s i s s o , t h e c o n d u c t i v i t y sum r u l e i s d e r i v e d as f o r t h e 3DEG and has t h e form J o €(itu>) ^ 1 I J ^ L - V ( c i ) • <i = Z^kf ( 2 > 3 1 ) By t h e above d i s c u s s i o n , t h e r e i s no c o m p r e s s i b i l i t y sum r u l e , but a u s e f u l r e l a t i o n i n v o l v i n g t h e c o m p r e s s i b i l i t y may s t i l l be o b t a i n e d . 49 2.3B C o m p r e s s i b i l i t y The c o m p r e s s i b i l i t y i s d e f i n e d i n terms of the ground s t a t e energy E 0 by *K (2.32) E 0 can be c a l c u l a t e d w i t h the a i d of the well-known theorem (see PN, p. 297). 0 J * A ( 2 . 3 3 ) . where E i i A ^ ^ e 2 ^ * s t^ i e t o t a l i n t e r a c t i o n energy of the ground s t a t e w i t h v a r i a b l e c o u p l i n g c o n s t a n t Xe2. £ 0 i s the energy of the n o n - i n t e r a c t i n g ground s t a t e (X = 0 ) , which i n ID i s U6p/3. By Eq. ( 2 . 3 2 ) , 1 . , t l - . whence Xo I , *— - r — = / + T7TZ- 0/2. x (2.34) E^^.(K) may be e x p r e s s e d i n terms of the X-dependent s t r u c t u r e f a c t o r S(q,X) i n the f o l l o w i n g way: I t w i l l prove u s e f u l t o d e f i n e the f u n c t i o n * '\ (2.35) so t h a t i n the thermodynamic l i m i t , Eq. (2.34) may be r e w r i t t e n as 50 (2.36) On the o t h e r hand, f o r a charged Fermi gas w i t h a compensating b a c k g r o u n d , X i s r e l a t e d t o £ by (see PN, pp. 20,209) l + Z € F *• (2.37) Thus, the c o n s i s t e n c y r e q u i r e m e n t on the c o m p r e s s i b i l i t y i s t h a t the r a t i o X»/K o b t a i n e d from X '2-€F £->o t(f,0)-l (2.38) be the same as t h a t p r e d i c t e d by Eq. (2.36) We remark a l s o t h a t the c o r r e l a t i o n energy e - e - E f COrr ° i s , by v i r t u e of Eq. (2.33) and the form of the HF ground s t a t e energy, ^ e x p r e s s a b l e as (lkf Kl f F f (2.39) 51 2.4 A p p r o x i m a t i o n Methods When i n t e r p a r t i c l e f o r c e s a r e i n c l u d e d , a p p r o x i m a t i o n s must be made, the two most common of which a r e t h e H a r t r e e - F o c k a p p r o x i m a t i o n (HFA) and the random phase a p p r o x i m a t i o n (RPA). In the former scheme an e l e c t r o n moves i n the averaged p o t e n t i a l due t o a l l the o t h e r e l e c t r o n s so t h a t i t responds t o VexfAx as a f r e e p a r t i c l e , i . e . ^ ^p-%0. In the RPA, the response i s t h a t of a f r e e p a r t i c l e t o the t o t a l p o t e n t i a l vy: />/p>)= x0(pco)vr(pu)) = kJi,(*)v(i)Lf(pu)-+ftx(p<A))] which i m p l i e s t h a t ^ (n n) = X o ( j ^ ) * * P A l J ' w ' l-vti)X0(pul) (2'40) Flaws i n h e r e n t i n each of t h e s e a p p r o x i m a t i o n s a r i s e from the n e g l e c t of c e r t a i n p a r t i c l e c o r r e l a t i o n s . In the HFA, the o n l y ones taken i n t o account a r e those due t o the P a u l i p r i n c i p l e between p a r a l l e l s p i n e l e c t r o n s and c o n s e q u e n t l y c o r r e l a t i o n s between a n t i p a r a l l e l s p i n p a r t i c l e s are i g n o r e d c o m p l e t e l y . On the o t h e r hand, the RPA i s a time-dependent H a r t r e e t h e o r y and thus t a k e s inadequate account of exchange e f f e c t s . An e f f o r t t o remedy t h i s d e f e c t f o r the 3DEG was f i r s t made by Hubbard who suggested an approximate X(q,od) of the form x ( n ,,, = X o i i f * )  l-Ll'&^)lvt(piC0C^^) (2.41) where G(q) = Q 2/4(Q 2+K P 2) ' and v. (q) = 4TTe2/Q2 i s the F o u r i e r t r a n s f o r m of the Coulomb i n t e r a c t i o n . S i n c e t h e n , a number of a u t h o r s have adopted Eq. (2.41) as 52 a s t a r t i n g p o i n t w i t h the i n t e n t i o n of o b t a i n i n g a G(cj) which more a c c u r a t e l y t a k e s i n t o account l o c a l f i e l d e f f e c t s . The p h y s i c a l s i g n i f i c a n c e of G(q)° may be deduced by f i r s t of a l l d e f i n i n g an e f f e c t i v e p o t e n t i a l t o which an e l e c t r o n responds by I t then f o l l o w s from the d e f i n i t i o n and from the t o t a l p o t e n t i a l t h a t vr.'veff = v c & f We see, t h e r e f o r e , t h a t G(q) i s r e l a t e d t o the d i f f e r e n c e i n t h e p o t e n t i a l s seen by an e l e c t r o n and a t e s t c h a r g e ; t h a t i s , the l o c a l f i e l d . In g e n e r a l , G(q) s h o u l d a l s o be a f u n c t i o n of W; however, the (^-dependence i s almost u n i v e r a l l y n e g l e c t e d because of the c o m p l e x i t y i t i n t r o d u c e s i n t o a c a l c u l a t i o n . K u g l e r (1975) has i n v e s t i g a t e d the p r o p e r t i e s of G(q,W) = G'(q,w) + iG"(q,60), and o b t a i n s the Kr a m e r s - K r o n i g r e l a t i o n T a k i n g the l i m i t of 60 0 from which we see t h a t G(q) i s a c t u a l l y d e t e r m i n e d by the 53 f u n c t i o n a l form of G(cj,W) a t a l l f r e q u e n c i e s . In a d d i t i o n t o m o d i f y i n g the s t a t i c b e h a v i o u r of % and €, G(cj,W) w i l l a l t e r the plasmon spectrum of the . e l e c t r o n gas, s i n c e i t appears i n the denominator i n Eq. ( 2 . 4 1 ) . One d i f f i c u l t y a r i s i n g from the replacement of G(q,^) by i t s s t a t i c l i m i t has been p o i n t e d out by V a i s h y a and Gupta (1973). These a u t h o r s showed t h a t the c o m p r e s s i b i l i t y sum r u l e (2.27d) and the sum r u l e f o r the t h i r d f r e q u e n c y moment of 0C(q,u) cannot be s a t i s f i e d s i m u l t a n e o u s l y i f G i s f r e q u e n c y -independent. N e v e r t h e l e s s , i n the f o l l o w i n g work we w i l l f o l l o w the a c c e p t e d p r a c t i c e of s e t t i n g G(g,£«>) = G ( g ) . 2.4A STLS A p p r o x i m a t i o n One of the most s u c c e s s f u l methods f o r c a l c u l a t i n g G(q) was proposed by Si n g w i et a l . (1968, h e r e a f t e r r e f e r r e d t o as STLS) and i n v o l v e s the r e t e n t i o n of a term d i s c a r d e d i n the RPA. R e v e r t i n g t o f i r s t q u a n t i z e d n o t a t i o n , the e q u a t i o n of motion of the d e n s i t y f l u c t u a t i o n /• = u t ~* i s g i v e n , f o r the J %, l H a m i l t o n i a n i n the absence of an e x t e r n a l f i e l d 54 I n s t e a d of d r o p p i n g the l a s t term as i s done i n the RPA, r e w r i t e i t as and make the a p p r o x i m a t i o n J • V - */<-/ /) (2.44) Eq. (2.43) may then be w r i t t e n i n the form Thus the bare Coulomb p o t e n t i a l which appears i n the RPA i s r e p l a c e d by an e f f e c t i v e p o t e n t i a l R e p l a c i n g v^(q) by v ^ ^ ( q ) i n the RPA s u s c e p t i b i l i t y g i v e s /-Xoty.^Wty) (2.46) whence comparison w i t h Eq. (2.41) r e v e a l s t h a t i n the STLS a p p r o x i m a t i o n (STLSA) Q}v(%) t * (2.47a) Eqs. (2.41) and (2 . 4 7 a ) , a l o n g w i t h the 3D v e r s i o n of t h e sum r u l e , (2.22), 55 f *° . rod c o n s t i t u t e a s e l f - c o n s i s t e n t scheme which i s the b a s i s of the STLS t r e a t m e n t of l o c a l f i e l d c o r r e c t i o n s . The j u s t i f i c a t i o n f o r the STLSA d e r i v e s from the f a c t t h a t i t g i v e s b e t t e r r e s u l t s f o r the p a i r c o r r e l a t i o n f u n c t i o n g(£) at s m a l l r than o t h e r methods. In f a c t , the RPA and the Hubbard a p p r o x i m a t i o n g i v e n e g a t i v e - and t h e r e f o r e u n p h y s i c a l - v a l u e s f o r g ( r ) as r -> 0 i n the range of m e t a l l i c d e n s i t i e s . T h i s i s because, as was mentioned p r e v i o u s l y , they n e g l e c t s h o r t - r a n g e c o r r e l a t i o n s , t h e r e b y g e t t i n g wrong the s i z e of the c o r r e l a t i o n h o l e s u r r o u n d i n g each p a r t i c l e . One f a i l i n g of the STLSA s h o u l d be mentioned, namely t h a t the two v a l u e s of the c o m p r e s s i b i l i t y c a l c u l a t e d from the 3D analogues of Eqs. (2.36) and (2.38) do not a g r e e . The c o r r e c t i o n of t h i s inadequacy has been the s u b j e c t of much f u r t h e r work by Si n g w i and c o w o r k e r s , c u l m i n a t i n g w i t h the t r e a t m e n t of V a s h i s h t a and S i n g w i (1972) who manage t o o b t a i n good agreement f o r 7 < 0 A : . The STLSA i s the approach w e , s h a l l adopt i n t a c k l i n g the ID model under c o n s i d e r a t i o n . In a d a p t i n g the o r i g i n a l 3D method t o the ID c a s e , we note t h a t the t h e o r y l e a d i n g t o the s e l f -c o n s i s t e n t e q u a t i o n s i s e s s e n t i a l l y independent of d i m e n s i o n a l i t y , t h e r e b y making the ID f o r m u l a t i o n a s t r a i g h t f o r w a r d p r o c e s s . Of c o u r s e , the Coulomb p o t e n t i a l v c ( q ) must be r e p l a c e d by the p o t e n t i a l v ( q ) d e f i n e d by ( 2 . 3 ) . We are f r e e t o d e f i n e a f u n c t i o n G(q) as b e f o r e : 56 /- ll-Sty] vtyKJl,*) (2.48a) where O^ o (QftO) i s g i v e n by ( 2 . 2 6 ) . The a p p r o x i m a t i o n l e a d i n g t o Eq. (2.47a) i s j u s t as a p p l i c a b l e i n one d i m e n s i o n as i n t h r e e , so t h a t To complete the s e l f - c o n s i s t e n t method, the sum r u l e d e r i v e d e a r l i e r i s r e s t a t e d f o r c o n v e n i e n c e : r to For a p a r t i c u l a r v (q) t o be s p e c i f i e d l a t e r , Eqs. (2.48) are the b a s i s of our t r e a t m e n t of the d i e l e c t r i c response of a ID gas. 2.4B P r o p e r t i e s of % and £ The m a t h e m a t i c a l p r o p e r t i e s t h a t the e x a c t % and £ sh o u l d p o s s e s s have been o u t l i n e d i n Sec. 2.3B. We now examine the STLS a p p r o x i m a t i o n t o these f u n c t i o n s i n o r d e r t o a s c e r t a i n any i n c o n s i s t e n c i e s w i t h the r e q u i r e m e n t s , and t o e s t a b l i s h a d d i t i o n a l f e a t u r e s due t o the e x p l i c i t form p r o v i d e d by the STLSA. C o n s i d e r f i r s t the s u s c e p t i b i l i t y ^ ^ - r ^ s d e f i n e d by Eq. (2 . 4 8 a ) . The e x c i t a t i o n spectrum f o r which I m ( ^ S T ^ ^ ) v a n i s h e s w i l l c o n s i s t of the n o n - i n t e r a c t i n g s i n g l e - p a r t i c l e continuum of F i g . 2.7 and a c o l l e c t i v e e x c i t a t i o n branch d e t e r m i n e d by the p o l e s of XiTLS. T h i s d i f f e r s from the e x a c t spectrum which would have I m ( 0 O ^ 0 f o r 00 > U)+(q) because of m u l t i p a i r e x c i t a t i o n s . Thus, any p o l e s which appear i n X^rLS w i t h f r e q u e n c i e s to (q) > uJAq) would be damped and pushed o f f the r e a l a x i s i n an e x a c t (2.48b) (2.48c) 57 c a l c u l a t i o n . More e x p l i c i t l y , 0CC (to) b e i n g a c a u s a l response f u n c t i o n i s r e a l o n l y f o r im a g i n a r y oo and f o r r e a l 00 o u t s i d e the r e g i o n 6v>_(q) < W < oJ>+(q). C o n s e q u e n t l y , ^C^^ w i l l have s i n g u l a r i t i e s o c c u r r i n g under the f o l l o w i n g c o n d i t i o n s : 1) f o r G(q) < 1, a p o l e on the r e a l a x i s a t CO (q) > U)+iq) 2) f o r G(q) > 1, a p o l e on the r e a l a x i s a t AJ (q) <60_(q) i f |0£.(q,0)| < [ (G(q)-Dv(q) | ] " 1 = % C ( q ) , or a p o l e on the im a g i n a r y a x i s i f |OC,(q,0)| > 'X f(q) • 3) f o r G(2kp) = 1, a s i n g u l a r i t y a t 00 = 0. We note t h a t the req u i r e m e n t t h a t 0C^rL^(qrO) be a n a l y t i c f o r Im(w) > 0 i s v i o l a t e d i n the case where G(q) > 1 and | ^ 0 ( q » n ) | > ^ ( q ) . The plasmon mode g i v e s a c o n t r i b u t i o n t o the i n t e g r a l i n Eq. (2.48c) which i s d e t e r m i n e d as f o l l o w s . From i t s d e f i n i t i o n i n ( 2 . 2 5 ) , X(«o) near the p o l e i s g i v e n by 2 whence s , , ( t ) = J I M X M A . = - i r | r ^ p o The term |(y>^)p 0| 2 i s o b t a i n e d by o b s e r v i n g t h a t which becomes, i n the STLSA / 2. 58 W • # I, Hence Eq. (2.48c) may be r e w r i t t e n as rOJfC<j) ^ K M o / TTw J V We now show t h a t the f-sum r u l e (2.28) i s s a t i s f i e d i d e n t i c a l l y i n the STLSA f o r G(q) < 1. S i n c e ^ J T L S ^ ' 4 0 ^ I S a n a l y t i c o f f the r e a l a x i s , i t s a t i s f i e s the K r a m e r s - K r o n i g r e l a t i o n where, i t must be remembered, the i n t e g r a l i n c l u d e s a c o n t r i b u t i o n from the plasmon p o l e . In the l i m i t of u) oo , 0CST<_s(q,W) —> nq 2/mu) 2 so t h a t the e x p r e s s i o n becomes •oO PI P f which i s i d e n t i c a l t o Eq. ( 2 . 2 8 ) . A l t h o u g h the a n a l y t i c b e h a v i o u r of £ S T L s ( q ^ ) i s of l i t t l e r e l e v a n c e by v i r t u e of the d i s c u s s i o n i n Sec. 2.3B, the s t a t i c d i e l e c t r i c f u n c t i o n i s of some i n t e r e s t . E x p l i c i t l y , w i t h ( 0) . , In p a r t i c u l a r , the d i v e r g e n c e of 0Co(2kp) l e a d s t o 59 W^F.O) " ' - oik) which w i l l be n e g a t i v e f o r G(q) < 1, i n d i c a t i n g t h a t t h e r e w i l l be an i n t e r v a l of q - v a l u e s around 2kp- f o r which £5-7-^ 5 (Q* 0) < 0. The p h y s i c a l s i g n i f i c a n c e of a n e g a t i v e 6(cjf0) w i l l be e l a b o r a t e d upon i n Sec. 2.7, as the erroneous b e l i e f t h a t t h i s f e a t u r e i m p l i e s an i n s t a b i l i t y i n the e l e c t r o n gas has been put f o r w a r d i n the l i t e r a t u r e . Eq. (2.37) r e q u i r e s t h a t £(q,0) > 1 as q 0 t o ensure t h a t the c o m p r e s s i b i l i t y , \ , i s p o s i t i v e , and hence t h a t the system remain s t a b l e . In the STLSA, Eq. (2.37) becomes where | % f l ( q , 0) | 2m/-»r'n2 k F as q -*> 0. From the a n s a t z (2.48b) f o r G ( q ) , i t f o l l o w s t h a t X > 0 i f J0 %v^} 9 ^ H < zir€f (2.50) T h i s i n e q u a l i t y p r o v i d e s one c r i t e r i o n f o r the c r e d i b i l i t y of the r e s u l t of any c a l c u l a t i o n . The f u n c t i o n € ( q , 0 ) i s s k e t c h e d i n F i g . 2.1 f o r the s i t u a t i o n where the s t a b i l i t y c o n d i t i o n i s s a t i s f i e d . 2.4C CDW I n s t a b i l i t y i n the STLSA I t was p o i n t e d out i n Chap. 1 t h a t the the phenomenon of the CDW i n s t a b i l i t y m a n i f e s t s i t s e l f i n the " d i v e r g e n c e of the s t a t i c s u s c e p t i b i l i t y , which n e c e s s i t a t e s , t h e r e f o r e , a c l o s e r s c r u t i n y of the s i n g u l a r b e h a v i o u r of £(q,0) i n the STLSA. R e c a l l i n g E q .(2.48a), 60 O CJ F i g u r e 2.1. S t a t i c d i e l e c t r i c f u n c t i o n € J T X S ( q , 0 ) f o r a s t a b l e ID system. X(o) = r Xo(f}  we observe t h a t t h e r e e x i s t two p o s s i b i l i t i e s f o r i n f i n i t i e s i n O ^ C j r O ) . One i s the v a n i s h i n g of the denominator when / By d e f i n i t i o n , % 0 i s n e g a t i v e ; t h u s any s o l u t i o n must be one f o r which G(q f i£ W) > 1. A d i v e r g e n c e may a l s o r e s u l t a t q = 2 k p s i n c e 2 k p ) | oo . In t h i s c a s e , xUkF) -i l-GUkF)v(zkF) which blows up i f G ( 2 k p ) = 1. There i s , however, a p h y s i c a l c o n s i d e r a t i o n which i n h i b i t s the p r e d i c t i o n of a CDW i n the p r e s e n t t h e o r y . I n the d i s c u s s i o n of the p a i r c o r r e l a t i o n f u n c t i o n , g ( x ) , which f o l l o w s , i t - w i l l be shown t h a t the a s y m p t o t i c l i m i t of G(q) i s 61 A p h y s i c a l l y m e a n i n g f u l g ( x) i s p o s i t i v e d e f i n i t e which a c c o r d i n g l y r e q u i r e s G(q) ^ 1 as q oo . We must c o n c l u d e , t h e r e f o r e , t h a t i f the v a l i d i t y of the • STLSA i s not t o be c h a l l e n g e d by a n e g a t i v e g ( 0 ) , G(q) must e x h i b i t some s o r t of peak e x t e n d i n g a t l e a s t t o one a t q = 2kp i n o r d e r f o r the two c o n d i t i o n s d e t a i l e d above t o be s a t i s f i e d . 2.4D P a i r C o r r e l a t i o n F u n c t i o n The p a i r c o r r e l a t i o n f u n c t i o n i s r e l a t e d t o the s t a t i c form f a c t o r by (2.51) - oo R e c a l l the d e f i n i t i o n of G(q) i n the STLSA: We c a l c u l a t e now the a s y m p t o t i c l i m i t of G(q) assuming t h a t f o r l a r g e q, v(q) ^  l / q * . -oo V f 00 - oo Thus as q o o , 1 - G(q)-»g(0). 62 2.4E C o l l e c t i v e E x c i t a t i o n s The f r e q u e n c i e s Wp(q) of the c o l l e c t i v e e x c i t a t i o n s a r e de t e r m i n e d by the z e r o e s of 6(q,w) or a l t e r n a t i v e l y by the p o l e s of y ( q , ( J ) . For w<W_(q) or w > w + ( q ) , ImX(q,«0) = Im^„(q,co) = 0, so t h a t Wp(q) i s the s o l u t i o n of U s i n g the d e f i n i t i o n of R e % 0 ( q , w ) , the r o o t s of the e q u a t i o n a r e g i v e n by where z , 2. t-OCe^ (2.52) For G(q) < 1, oc^ > 1 and 6Up(q) > W + . Thus f o r G(q) < 1, t h e r e always e x i s t s an e x c i t a t i o n of frequ e n c y 6Vp(q) and the plasmon d i s p e r s i o n c u r v e may j o i n t he p a r t i c l e - h o l e continuum o n l y when l*>+(q) = W_(q) = 0 (at q = 0) p r o v i d e d t h a t i n the l i m i t q — 0 , v ( q ) —» <*» slower than 1/q 2 . T h i s r e s u l t , t h a t the l o n g w avelength o s c i l l a t i o n s of the ID gas even w i t h a long-range f o r c e have z e r o f r e q u e n c y , i s t o be ex p e c t e d by the f o l l o w i n g argument. A q = 0 e x c i t a t i o n i m p l i e s p o s i t i v e and n e g a t i v e charge imbalances a t i n f i n i t e s e p a r a t i o n . In 3D, the r e s t o r i n g f o r c e i s f i n i t e s i n c e the e l e c t r i c f i e l d due t o an i n f i n i t e sheet of u n i f o r m l y d i s t r i b u t e d charge i s c o n s t a n t , independent of d i s t a n c e ; hence ft>p = 4TThe 2/m. The s i t u a t i o n i s d i f f e r e n t i n ID because the t r a n s v e r s e charge d e n s i t y i s f i n i t e and l o c a l i z e d ; 63 the f o r c e t e n d i n g t o r e s t o r e charge d i s p l a c e m e n t s a t i n f i n i t e s e p a r a t i o n v a n i s h e s . C o n s e q u e n t l y , q = 0 c o l l e c t i v e o s c i l l a t i o n s can be e x c i t e d a t no c o s t i n energy. To i n v e s t i g a t e the d i s p e r s i o n of the l o n g w avelength modes the s m a l l - q dependence of G(q) i s r e q u i r e d . From Eq. (2.48b) / (2.53) where the l a s t l i n e r e s u l t s from the evenness of S ( q ) . I f v(q) d i v e r g e s as q 0 , then G ( q ) — * 0 and from Eq. (2.52) whence As G ( q ) — > 1 " [ 1 + ] , => oo [ 0 ] , whence OJp(q)—^ w+(q) [ i A ) _ ( q ) ] . Thus t h e r e i s a d i s c o n t i n u i t y i n Wp a t G = 1, a p p e a r i n g i n the i n s t e p of the p a r t i c l e - h o l e spectrum when G(q) > 1. The c o l l e c t i v e mode s o f t e n s toward z e r o f r e q u e n c y as G i n c r e a s e s , and Wp(q) becomes p u r e l y i m a g i n a r y i f G(q) i s such t h a t o t ^ < t J _ ( q ) / w + ( q ) . T h i s , t h e n , i s the c o n d i t i o n f o r the system t o s t a t i c a l l y d i s t o r t i n t o a CDW s t a t e . From the 64 d i s c u s s i o n i n Sec. 2.4C such an i n s t a b i l i t y i m p l i e s the e x i s t e n c e of a peak i n G(q) and hence a range of q f o r which the system i s u n s t a b l e . The a s y m p t o t i c l i m i t of o)p(q) i s W +(q) as l o n g as G(q-*<*>) < 1. S i n c e t h i s i s a l s o the requi r e m e n t t h a t g(0) > 0, i t i s e v i d e n t t h a t the plasmon spectrum always extends t o l a r g e q a l o n g the upper edge of the s i n g l e - p a r t i c l e spectrum whenever the STLSA g i v e s a p h y s i c a l l y a c c e p t a b l e g ( x ) . 2.4F H a r t r e e - F o c k and Random Phase A p p r o x i m a t i o n s I f G(q) = 1 i n Eq. ( 2 . 4 2 ) , we f i n d , by u s i n g the d e f i n i t i o n of V T(q , e o ) , t h a t Veff = V e 5 c , which i s j u s t the HF statement t h a t the p a r t i c l e s respond t o the e x t e r n a l f i e l d . Eq. (2.48a) y i e l d s , as i t s h o u l d , the r e s u l t OCiq,^) = Xoiq,^) which, when s u b s t i t u t e d i n t o Eq. ( 2 . 4 8 c ) , shows the s t r u c t u r e f a c t o r t o be: SH p( t) = ( ' ^ F • Lkr (2.55) The p a i r c o r r e l a t i o n f u n c t i o n may then e a s i l y be deduced from Eq. ( 2 . 5 1 ) : 3Cx) '- 1 " T ^ r (2.56) Note t h a t g(0) = 0.5 s i n c e o n l y p a r a l l e l s p i n p a r t i c l e s a r e c o r r e l a t e d , and c o n s e q u e n t l y t h a t 1 - g(0) * G(<* ) = 1. T h i s i n e q u a l i t y a r i s e s because of the n o n - s e l f - c o n s i s t e n c y of the HFA i n t h e STLSA sense. By p u t t i n g G(q) = 0 i n t o Eqs. (2.42) and (2.48a), one o b t a i n s V&ff = V T and *X = %0/[ ( l - v ( q ) ) & ] , or the RPA. The 65 e x p r e s s i o n f o r % i s s u f f i c i e n t l y complex t h a t a n a l y t i c a l forms f o r S(q) and g(x) cannot be d e r i v e d , but a g a i n i t may be p o i n t e d out t h a t 1 - g ( 0 ) * G(oo ) = 0. F i n a l l y , we remark t h a t the HF r e s u l t s (2.55) and (2.56) ar e u s e f u l s i n c e they w i l l be c o r r e c t i n the l i m i t of h i g h d e n s i t y as the i n t e r a c t i o n s t r e n g t h v ( q ) — > 0 . 2.5 Model W i t h a T r a n s v e r s e G a u s s i a n D e n s i t y U n t i l now the development of our q u a s i - l D model has been f o r a g e n e r a l t r a n s v e r s e w a v e f u n c t i o n , 9 ( r ^ ) . In o r d e r t o p e r f o r m n u m e r i c a l c a l c u l a t i o n s we choose @ (£ L) t o be a n o r m a l i z e d G a u s s i a n As a reminder of the model under c o n s i d e r a t i o n , we v i s u a l i z e a system of N e l e c t r o n s , e s s e n t i a l l y l o c a l i z e d t o a c y l i n d e r of l e n g t h L and r a d i u s 2d, and h a v i n g o n l y a s i n g l e degree of freedom, a l o n g the c y l i n d e r a x i s . (2.57) A c c o r d i n g t o Eqs. (2.3) and ( 2 . 4 ) , ©(r^) d e t e r m i n e s the ID i n t e r a c t i o n p o t e n t i a l t o be: (2.58) where the e x p o n e n t i a l i n t e g r a l 66 ' <LK IK The q — * 0 and q—>•* l i m i t s of v ( q ) a r e : ^ , o (2.59) From the b e h a v i o u r e x p r e s s e d i n (2 . 5 9 ) , i t i s p o s s i b l e t o show t h a t the r e a l space p o t e n t i a l v ( x ) has the l i m i t i n g forms Thus, a t l a r g e d i s t a n c e s the i n t e r p a r t i c l e p o t e n t i a l i s t h a t due t o p o i n t c h a r g e s , w h i l e a t s m a l l s e p a r a t i o n i t i s t h a t of p a r a l l e l p l a n a r charge d i s t r i b u t i o n s . The f u n c t i o n s v (q) and v ( x ) a r e shown i n comparison t o the Coulomb p o t e n t i a l i n F i g . 2.2. We f i n d i t c o n v e n i e n t t o e x p r e s s a l l f u n c t i o n s and v a r i a b l e s as d i m e n s i o n l e s s q u a n t i t i e s by h e n c e f o r t h t a k i n g as s t a n d a r d u n i t s the Fermi parameters 6F and kp-. For c l a r i t y , d i m e n s i o n l e s s v a r i a b l e s U = e 2 k p / 6 F and A = a k p a r e d e f i n e d ; the p o t e n t i a l i n u n i t s of £ p / k p i s then w r i t t e n as 1.2. (2.61) I t i s a l s o customary t o d e f i n e a d i m e n s i o n l e s s measure of the i n t e r p a r t i c l e s p a c i n g , x s by 1/n = x^a^ , aa b e i n g the Bohr r a d i u s . We note t h a t the c o u p l i n g c o n s t a n t U = 4x s/7T i s i n v e r s e l y p r o p o r t i o n a l t o the d e n s i t y , w h i l e the pr o d u c t UA = 2fl/a g i v e s the r a d i a l e x t e n t of the charge d i s t r i b u t i o n . From 67 F i g u r e 2.2. (a) The p o t e n t i a l v ( q ) d e f i n e d by Eq. (2.57) compared w i t h a 1/q 2 p o t e n t i a l . (b) The r e a l space F o u r i e r t r a n s f o r m of v ( q j compared w i t h the Coulomb e 2 / r p o t e n t i a l . 68 now on, the parameters U and A w i l l s e r v e t o c h a r a c t e r i z e the p h y s i c a l system. For r e f e r e n c e , the d i m e n s i o n l e s s e q u a t i o n s d e f i n i n g the STLSA a r e : _ i _ 1" _J i ] = Y(Q 0)) = X o ( l ' " }  Vi}) L G(pOJ) V l-Ll-GtylvtyXoCpw) (2.62a) (2.62b) (2.62c) ° 27T0 | U)z-0)}(o) T -Y< „ ) - « ' \ ' U-C%)<U<U*(i> ( 2 . 6 2 d ) 0>±(.%)- I f ± z % \ As they s t a n d , the e q u a t i o n s ( 2 . 3 5 ) , (2.36) and (2.38) a re not v e r y u s e f u l f o r the purpose of c a l c u l a t i n g the c o m p r e s s i b i l i t y f o r the model j u s t o u t l i n e d . For a system w i t h parameters U and A, the f u n c t i o n g i v e n by (2.35) i s where S(q ; A u,A) i s the s e l f - c o n s i s t e n t s t r u c t u r e f a c t o r f o r the system c h a r a c t e r i z e d b y X u and A, and 69 A p p l y i n g the d e f i n i t i o n kp =TTN/2L, and making the change of v a r i a b l e \U \, Eq. (2.36) becomes: . i £ . r k i f : + ± - ] f - u f 6* /"OO ^ *\\ e*1 Et(Azf)Ls(p\,l\)-llt D e f i n i n g the l a s t e x p r e s s i o n may, upon u s i n g the d e r i v a t i v e s M JL 5A_ A. djf iff be r e c a s t as Xo f u In the STLSA, Eq. (2.38) assumes the form (2.63) Xo _ j _ l + G(i)v(i)Xo(i,o) I n v o k i n g Eq. (2. 5 3 ) , and s u b s t i t u t i n g the v a l u e % (q -»0,0) = -1/ir y i e l d s = / 70 F i n a l l y , i n t e g r a t e by p a r t s and use the f a c t t h a t t o a r r i v e a t the r e s u l t (2.64) Once the p a r a m e t r i c dependence of S(q) on U and A i s known, i t i s a s t r a i g h t f o r w a r d m atter t o c a l c u l a t e and compare 7C0/K from the two r e l a t i o n s (2.63) and ( 2 . 6 4 ) . The plasma f r e q u e n c y g i v e n by Eq. (2.54) i s , i n Fermi u n i t s , S i n c e v(q) ^ - l n q a t s m a l l q, we see t h a t the plasmon d i s p e r s i o n c u r v e goes t o z e r o as q\/-lnq' ; i . e . w i t h i n f i n i t e s l o p e . T h i s b e h a v i o u r agrees w i t h t h a t p r e d i c t e d by the Tomonoga model i f the p r e s e n t form of v(q) i s s u b s t i t u t e d i n t o Eq. ( 1 . 5 ) . The h i g h d e n s i t y l i m i t of the model i s o b t a i n e d by l e t t i n g U -»s>0, A—± o o as the p r o d u c t UA remains c o n s t a n t . Then *X—> % 0 and S(q) —3> S,I^.(Q)« However, the s e l f - c o n s i s t e n t G(q) does not t a k e H F the c o n s t a n t HF v a l u e of 1, but r a t h e r must be s o l v e d from Eq. ( 2 .62b). T h i s i s done by s u b s t i t u t i n g f o r l a r g e A the p o t e n t i a l (2.59b) and S H F ( q ) ( g i v e n by (2.55)) i n t o (2.62b) which then reads (2.65) 71 (2.66) We remark t h a t Eq. (2.66) i s the p r e v i o u s l y mentioned Hubbard a p p r o x i m a t i o n a p p l i e d t o our ID model. The l o c a l f i e l d d e s c r i b e d by t h i s G(q) has no i n f l u e n c e on the s t r u c t u r e f a c t o r s i n c e a t the h i g h k i n e t i c e n e r g i e s i m p l i e d by U—>0 the i n t e r p a r t i c l e p o t e n t i a l i s of n e g l i g b l e importance compared t o the P a u l i p r i n c i p l e i n k e e p i n g the p a r t i c l e s a p a r t . I t i s w o r t h w h i l e t o c o n s i d e r the case i n which UA —> «o i n such a way t h a t the r a t i o U/A2 remains f i n i t e . A g a i n v(q) = U/A 2q 2 = W/q2 s w(q), and G(q) i s g i v e n by but now the f i n i t e n e s s of w(q) means t h a t "X does not tend to ^ and the f u l l STLS scheme must be employed t o a c h i e v e s e l f -c o n s i s t e n c y . The p h y s i c a l s i g n i f i c a n c e of t h i s form of the p o t e n t i a l i s r e a l i z e d by n o t i n g t h a t the F o u r i e r t r a n s f o r m of 1/q 2 i s |x|, the ID Coulomb p o t e n t i a l d e f i n e d by d 2W/dx 2 = -wi*(x). Because the f i e l d due t o t h i s p o t e n t i a l i s c o n s t a n t f o r a l l d i s t a n c e s , we expect the q = 0 plasmon f r e q u e n c y t o be s h i f t e d t o a f i n i t e v a l u e , and indeed one f i n d s , from Eq. ( 2 . 6 5 ) , (2.67) 72 2.6 N u m e r i c a l Method The t a s k a t hand i s t o e x t r a c t from Eqs. (2.62a,b,c) s e l f -c o n s i s t e n t s o l u t i o n s of G(q) and S ( q ) . Our model has been f o r m u l a t e d so t h a t two parameters U and A a r e r e q u i r e d t o s p e c i f y the d i m e n s i o n l e s s p o t e n t i a l a p p e a r i n g i n these e q u a t i o n s . V a r i a t i o n of U and A such t h a t UA i s c o n s t a n t e n a b l e s us t o study the b e h a v i o u r of the system as a f u n c t i o n of d e n s i t y , w h i l e the e f f e c t of the t r a n s v e r s e d i m e n s i o n , CL, can be examined by h o l d i n g U c o n s t a n t and cha n g i n g A. The s o l u t i o n of the model, denoted M W ^ ( U ) , f o r p a r t i c u l a r v a l u e s of U and A b e g i n s w i t h the com p u t a t i o n of the p o t e n t i a l v(q) d e f i n e d by (2.61) f o r a s e t of q - v a l u e s ( q ) ^ t o be det e r m i n e d by the a c c u r a c y d e s i r e d f o r S ( q ) . The s o l u t i o n of Eqs. (2.62) then proceeds w i t h the c h o i c e of an i n i t i a l S ( q ) , a c h o i c e which i s s i m p l i f i e d by c a l c u l a t i n g s y s t e m a t i c a l l y a sequence of M W ^ ( U ) . In the h i g h d e n s i t y or s m a l l U l i m i t , S(q) approaches the HF e x p r e s s i o n ( 2 . 5 5 ) . T h e r e f o r e , f o r the chosen UA, ta k e U s m a l l and commence the i t e r a t i o n p rocedure w i t h S u e ( q ) . F o r subsequent h i g h e r v a l u e s of U (lower d e n s i t i e s ) , the bes t i n i t i a l s t r u c t u r e f a c t o r i s the s o l u t i o n f o r M,.(U) w i t h the n e a r e s t s m a l l e r v a l u e of U. The i t e r a t i o n scheme f o r M M ^ ( U ) i s q u i t e s t r a i g h t f o r w a r d . W i t h the a p p r o p r i a t e S(q) f o r {q}, G(q) i s computed a c c o r d i n g t o (2.62b). These v a l u e s a r e then used i n (2.62a) t o get ?C(q,fa>) i n the i n t e g r a l i n ( 2 . 6 2 c ) ; the newly c a l c u l a t e d S(q) then g e n e r a t e s a new i t e r a t i o n . R e p e t i t i o n of t h i s p rocedure c o n t i n u e s u n t i l S(q) has converged t o a s u f f i c i e n t degree of a c c u r a c y . 73 A few comments on the n u m e r i c a l p r o c e d u r e s adopted are i n o r d e r . The i n t e g r a l i n Eq. (2.62b) i s e v a l u a t e d by means of Simpson's method and t h e r e f o r e the {q} chosen a r e i m p o r t a n t . L e t us r e w r i t e t h i s e x p r e s s i o n by c hanging v a r i a b l e s For l a r g e q S(q) — v l , so t h a t the i n t e g r a t i o n must extend out f a r enough t h a t the p a r t of the i n t e g r a l n e g l e c t e d i s s m a l l . In a d d i t i o n , the s p a c i n g Aq of the {q} must be chosen s m a l l enough t o g a i n the d e s i r e d a c c u r a c y . Of c o u r s e , the number of p o i n t s i s l i m i t e d t o some e x t e n t by p r a c t i c a l c o n s i d e r a t i o n s . A p r a c t i c a l p o i n t worth making i s t h a t the speed of convergence was g r e a t l y improved by t a k i n g the average of the i n p u t and the r e s u l t i n g S(q) f o r a p a r t i c u l a r i t e r a t i o n t o g e n e r a t e a s u c c e s s i v e one. I t i s e a s i l y shown t h a t i f the output f u n c t i o n s S converge, the convergent s o l u t i o n i s a s e l f -c o n s i s t e n t one. The r e m a i n i n g i n t e g r a l t o be e v a l u a t e d i s the f r e q u e n c y i n t e g r a l i n Eq. ( 2 . 6 2 c ) . Because of the f i n i t e l i m i t s and the e x i s t e n c e of a f u n c t i o n a l form f o r OC(io), the 1 6 - p o i n t G a u s s i a n q u a d r a t u r e method was employed. A s i m i l a r i n t e g r a l i n the f-sum r u l e , ( 2 . 2 8 ) , was c a l c u l a t e d i n the same way. Once the s e l f - c o n s i s t e n t S(q) i s d e t e r m i n e d , the p a i r c o r r e l a t i o n f u n c t i o n i s o b t a i n e d by F o u r i e r t r a n s f o r m a t i o n as p r e s c r i b e d by Eq. ( 2 . 5 1 ) . N o t i n g t h a t S(q) i s an even f u n c t i o n , we must compute 0 74 where ^ m ( t y i s the maximum q f o r which S i s c a l c u l a t e d . T h i s i n t e g r a l l e n d s i t s e l f r e a d i l y t o n u m e r i c a l c o m p u t a t i o n by the F i l o n method, w i t h the r e s t r i c t i o n t h a t x may not become too l a r g e compared t o the s p a c i n g of the q - v a l u e s . F i n a l l y , we comment on the e v a l u a t i o n of the c o m p r e s s i b i l i t y from Eqs. (2.62) and (2.63). The l a t t e r r e l a t i o n i n c l u d e s an i n t e g r a t i o n of S(q) which i s e a s i l y a c c o m p l i s h e d w i t h Simpson's method and g i v e s Xo /7\ d i r e c t l y . Less s t r a i g h t f o r w a r d i s the thermodynamic which i n v o l v e s d e r i v a t i v e s w i t h r e s p e c t t o U and A of the terms P(U,A) and ju(U,/\) - Ptt.AleA For a s p e c i f i e d M W ^ ( U ) the p r o c e d u r e i s as f o l l o w s : 1) C a l c u l a t e S(q) s e l f - c o n s i s t e n t l y and then p{\,oc) f o r a s e t of M < X e t(\) where \= 0,AU, 2Au, . . . , U; = A, A±&A, A±2AA; and AU and Ah a r e numbers whose magnitudes are s u i t a b l e f o r the n u m e r i c a l i n t e g r a t i o n and d i f f e r e n t i a t i o n i n s t e p s 2) and 3 ) . 2) Compute the q u a n t i t i e s yu.(U,A) by the Simpson's and/or t r a p e z o i d a l method. 3) C a l c u l a t e the d e r i v a t i v e s such as 9A 2AA 4) Sum a l l terms i n Eq. (2.63) t o get X,/% 2.7 R e s u l t s The q u a n t i t i e s which come out d i r e c t l y from the s e l f -c o n s i s t e n t s e t of e q u a t i o n s (2.62) a r e the f u n c t i o n s G ( q ) , S ( q ) , and X(q,oJ) or c"(q,W), as w e l l as the plasma f r e q u e n c i e s , uJpiq). 75 C a l c u l a t i o n s were performed f o r v a r i o u s d e n s i t i e s a t a number of tube d i a m e t e r s . F i g . 2.3a shows the U-dependence of S(q) when UA = .5 ( c o r r e s p o n d i n g t o OL = a 0 / 2 ) . As U i n c r e a s e s , the k i n k which appears a t q = 2 f o r U = 0 becomes smoothed out and the t a i l of the d i f f e r e n c e , 1 - S ( q ) , extends out t o l a r g e r q. The p h y s i c a l s i g n i f i c a n c e i s r e a l i z e d from the p a i r c o r r e l a t i o n f u n c t i o n p l o t t e d i n F i g . 2.4. In the n o n - i n t e r a c t i n g l i m i t when U — ^ 0 , the o n l y o p e r a t i v e c o r r e l a t i o n i s t h a t due t o the P a u l i p r i n c i p l e i n v o l v i n g i d e n t i c a l - s p i n p a r t i c l e s , so t h a t g(0) = 0.5. The i n c r e a s e d c o u p l i n g between o p p o s i t e - s p i n e l e c t r o n s and, t o a l e s s e r e x t e n t , between those of l i k e s p i n t h a t r e s u l t s as U becomes l a r g e r enhances the i n t e r p a r t i c l e c o r r e l a t i o n s , t h e r e b y deepening the c o r r e l a t i o n h o l e s u r r o u n d i n g each p a r t i c l e . E v e n t u a l l y g(0) d e c r e a s e s t o a n e g a t i v e v a l u e f o r U i 10, i n d i c a t i n g the breakdown of the STLSA. A l t e r n a t i v e l y , U can be h e l d c o n s t a n t w h i l e A i s v a r i e d ; the r e s u l t s f o r U = 5 (x s=3.93) a r e d i s p l a y e d i n F i g . 2.3b. For d e c r e a s i n g A, the o b s e r v e d b e h a v i o u r i s s i m i l a r t o t h a t when UA i s f i x e d and U i s r a i s e d . T h i s must be so s i n c e a s h r i n k i n g tube r a d i u s has the e f f e c t of l o c a l i z i n g charge and hence of i n c r e a s i n g the i n t e r p a r t i c l e i n t e r a c t i o n . The f u n c t i o n G(q) i s p l o t t e d i n F i g . 2.5a f o r M ( ( U ) . Most s i g n i f i c a n t i s the f e a t u r e l e s s s t r u c t u r e of G when U ± 0; no h i n t i s r e t a i n e d of the w e l l - d e f i n e d peak near q = 2 which o c c u r s f o r U = 0. As e x p e c t e d , s i m i l a r o b s e r v a t i o n s a r e made f o r M|5-(5) i n F i g . 2.5b where the l i m i t A —><*» i s e q u i v a l e n t t o MU^(U -»0). T h i s s u g g e s t s t h a t a b e t t e r measure of the c o u p l i n g than U i s the r a t i o U/A = e 2 / d . k i , which i s the s i m p l e s t d i m e n s i o n l e s s 76 F i g u r e 2.3. (a) The s t r u c t u r e f a c t o r S(q) f o r UA = 1 ( f l.=a D/2). 77 U = 1 0 -J L 0 X k r F i g u r e 2.4. (a) The p a i r c o r r e l a t i o n f u n c t i o n g(x) f o r UA 79 F i g u r e 2.4. (b) g(x) f o r U = 5. A = . 5 A = oo - L q / k . F i g u r e 2.5. (b) G(q) f o r U = 82 c o m b i n a t i o n of the c o n s t a n t e and the parameters a and kp. In o r d e r t o check t h a t the d i s a p p e a r a n c e of the peak i s not an a r t i f a c t of the c o m p u t a t i o n a l method, the b e h a v i o u r of G(q) i s t r a c e d f o r s m a l l v a l u e s of U/A f o r M i 5.(U). What we f i n d , i n F i g . 2.6 i s the d i m i n u t i o n and b r o a d e n i n g of the peak as G(q) smoothens out w i t h i n c r e a s i n g U/A. The c r i t e r i o n f o r a CDW which was e s t a b l i s h e d i n Sec. 2.4C, i s t h a t G(q) > 1 f o r some q. A c c o r d i n g t o F i g . 2.5, t h i s w i l l happen o n l y f o r those M ^ ( U ) f o r which G(*>) > l ; i . e . f o r systems f o r which the STLSA i s not r e a s o n a b l e . From the above d i s c u s s i o n , no u nusual f e a t u r e s i n the plasmon spectrum a r e a n t i c i p a t e d . In F i g . 2.7, cop(q) remains above the p a r t i c l e - h o l e continuum and i s c o n s i s t e n t w i t h the s m a l l q, q^ - l n q 1 , b e h a v i o u r p r e d i c t e d i n Sec. 2.5. The appearance of f r e q u e n c i e s below o)_(q) i s dependent on G(q) b e i n g g r e a t e r than 1, a s i t u a t i o n which we have seen above t o be u n p h y s i c a l i n the STLSA. In Sec. 2.3B i t was c o n c l u d e d t h a t the f-sum r u l e , (2.28) i s the o n l y sum r u l e a p p l i c a b l e t o the ID e l e c t r o n gas, and moreover, i t was shown i n Sec. 2.4B t o h o l d i d e n t i c a l l y i n the STLSA. As a check on our n u m e r i c a l p r o c e d u r e , the r u l e was c a l c u l a t e d and found t o be v e r y n e a r l y s a t i s f i e d i n a l l c a s e s . F u r t h e r - i n s i g h t i n t o the p r o p e r t i e s of our model i n the STLSA may be g a i n e d by c o n s i d e r a t i o n of the two q u a n t i t i e s v(q)/6(q,0) and (2.68) For an e x t e r n a l charge d i s t r i b u t i o n (q,0) i n t e r a c t i n g w i t h F i g u r e 2.6. G(q) f o r U/A << 1 and U/A = 0.5. F i g u r e 2.7. (a) Plasma f r e q u e n c i e s cup(q) f o r UA = 1. The dashed l i n e s i n d i c a t e the bounds of the p a r t i c l e - h o l e continuum. 86 the ID e l e c t r o n gas, the s e l f - e n e r g y of A* p l u s the change i n energy of the system, t o second o r d e r i n p e r t u r b a t i o n t h e o r y , due t o the i n f l u e n c e of p i s g i v e n by Thus f o l l o w s the i d e n t i f i c a t i o n of v/e as the e f f e c t i v e p o t e n t i a l - between two charges i n the presence of the system. In the STLSA, €(q,0) i s always n e g a t i v e around q = 2 f o r G(q) < 1. T h i s i s c o n t r a r y t o the a s s e r t i o n of Landau and L i f s h i t z (1960) t h a t the d i e l e c t r i c c o n s t a n t must always be g r e a t e r than u n i t y i n a s t a b l e system. As has been p o i n t e d out by K h i r z n i t s (1976), the c r i t e r i o n f o r s t a b i l i t y must be f o r m u l a t e d i n terms of the response f u n c t i o n l/£(q,0) f o r q = 0, i n which case the c o n d i t i o n l / f ( q , 0 ) < 1 a l l o w s n e g a t i v e v a l u e s of € as w e l l . For q—>0 r however, i t i s t r u e t h a t £(q,0) > 0 i n o r d e r t h a t the system not c o l l a p s e . T h i s i s not t o say t h a t t h e r e i s no consequence of the f a c t t h a t 6(q,0) < 0 f o r some q. I f the c o n s t r a i n t of r i g i d i t y of the background i s r e l a x e d , t h e n the t o t a l system of background + e l e c t r o n s w i l l f i n d i t e n e r g e t i c a l l y f a v o u r a b l e t o d i s t o r t w i t h a wavevector, or range of w a v e v e c t o r s , near q = 2. F i g . 2.8 shows v(q)/£(q,0) f o r some v a l u e s of U when UA = 1. The v i o l a t i o n of the s m a l l - q s t a b i l i t y r e q u i r e m e n t f o r U £. 3 i s not an i n h e r e n t p r o p e r t y of the model but must r a t h e r be a t t r i b u t e d t o the STLSA, as w i l l be d i s c u s s e d s h o r t l y . The e f f e c t i v e i n t e r a c t i o n between two p a r t i c l e s i n the system i s g i v e n by Eq. ( 2 . 6 8 ) . We note t h a t v g # (q) i s n e c e s s a r i l y r e p u l s i v e as l o n g as G(q) remains l e s s than one. In 87 F i g u r e 2.8. The e l e c t r o s t a t i c p o t e n t i a l v/€(q,0) between two t e s t c h a r g e s f o r UA = 1. 88 the e v e n t u a l i t y of G(q) p e a k i n g t o a v a l u e g r e a t e r than 1 + [ v ( q ) | % ( q , 0 ) | ] " 1 , something which we do not o b s e r v e , v ^ would become a t t r a c t i v e f o r some s p r e a d of wavevectors and, as a r e s u l t , the e l e c t r o n gas would d i s t o r t even w i t h a non-p o l a r i z a b l e background. The m a n i f e s t a t i o n of such an event would be the appearance of the plasmon d i s p e r s i o n c u r v e a t the lower edge of the p a r t i c l e - h o l e continuum and. the subsequent s o f t e n i n g of tOp(q) down t o z e r o f r e q u e n c y . Sample c u r v e s f o r v f i ^ (q) are shown i n F i g . 2.9. In the RPA, the e f f e c t i v e i n t e r a c t i o n between two p a r t i c l e s i n the gas i s the same as t h a t f o r charges e x t e r n a l t o the system so t h a t C l e a r l y Vgf^C can never be a t t r a c t i v e so t h a t i n the RPA t h e r e i s no p o s s i b i l i t y of an i n s t a b i l i t y whether the background i s r i g i d or n o t . For a s l i g h t l y d i f f e r e n t p o i n t of view, we c o n s i d e r the d e n s i t y response of the system t o an e x t e r n a l p e r t u r b a t i o n , namely the q u a n t i t y f/j>t)( = v% shown i n F i g . 2.10. In the STLSA, the i n d u ced d e n s i t y a t q = 2 i s always g r e a t e r than j ) ^ , and d i v e r g e s as G(2) —>1, s i g n i f y i n g the f o r m a t i o n of a CDW. In c o n t r a s t , f/ft*. - 1 f o r a 1 1 3 a t anY d e n s i t y i n the RPA, c o n f i r m i n g the c o n c l u s i o n of the p r e c e d i n g p a r a g r a p h . I t has been mentioned above t h a t l i m e"(q,0) = - o o f o r U Jt 3 and UA = 1. By Eq. ( 2 . 3 8 ) , t h i s i m p l i e s a n e g a t i v e c o m r e s s i b i l i t y and hence an i n s t a b i l i t y . On the o t h e r hand,7Co/X c a l c u l a t e d a c c o r d i n g t o Eq. (2.36) remains p o s i t i v e f o r l a r g e r 89 90 91 F i g u r e 2.11. The c o m p r e s s i b i l i t y r a t i o K9/K. Curve I i s c a l c u l a t e d from the thermodynamic r e l a t i o n ( 2 . 3 6 ) , and I I from the d i e l e c t r i c e x p r e s s i o n ( 2 . 3 8 ) . 92 v a l u e s of U as can be seen from F i g . 2.11. T h i s i n c o n s i s t e n c y i n d i c a t e s a weakness i n the STLSA f i r s t p o i n t e d out by STLS f o r the 3DEG. A l a r g e improvement was e f f e c t e d by V a s h i s h t a and S i n g w i (1972) by a d d i n g a d e n s i t y - d e p e n d e n t term t o G ( q ) : M i ) = W i } where the f a c t o r 1/2 < ot < 1 i s chosen t o o p t i m i z e the agreement between the thermodynamic and d i e l e c t r i c v a l u e s of % . The s u c c e s s of t h i s m o d i f i c a t i o n s u g g e s t s i t would be a n a t u r a l next s t e p i n an attempt t o b e t t e r the t r e a t m e n t p r e s e n t e d i n . t h i s c h a p t e r . P h y s i c a l l y , i n c l u s i o n of the new term amounts t o t a k i n g three-body c o r r e l a t i o n s i n t o a c c o u n t . These c o r r e l a t i o n s become im p o r t a n t a t l a r g e s e p a r a t i o n s , i . e . as q — 0 . Hence t h e i r n e g l e c t i n the c a l c u l a t i o n of lim£(q,0) becomes i n c r e a s i n g l y s i g n i f i c a n t as the c o u p l i n g s t r e n g t h r i s e s w i t h d e c r e a s i n g d e n s i t y , and l e a d s t o the breakdown of the c o m p r e s s i b i l i t y sum r u l e i n the STLSA. D e s p i t e t h i s inadequacy, the f a c t t h a t the thermodynamic c o m p r e s s i b i l i t y remains p o s i t i v e t o lower d e n s i t i e s l e a d s us t o b e l i e v e t h a t q u a n t i t i e s such as g ( x ) , which are s i m i l a r l y dependent on G(q) or S(q) f o r a l l q, are r e l i a b l y c a l c u l a t e d i n the STLSA. The p o t e n t i a l w(q) = W/q2 was p r e v i o u s l y i n t r o d u c e d as a l i m i t i n g case of v ( q ) . Because of the s t r o n g e r s i n g u l a r i t y i n w a t q = 0, q u a l i t a t i v e l y d i f f e r e n t r e s u l t s a r e t o be e x p e c t e d when v(q) i s r e p l a c e d by w i n Eqs. ( 2 . 6 2 ) . One such change i s i n the b e h a v i o u r of the plasma f r e q u e n c y , p l o t t e d i n F i g . 2.12. As p r e d i c t e d by Eq. ( 2 . 6 7 ) , the l o n g - w a v e l e n g t h f r e q u e n c y i s 93 Figure 2.12. Plasma frequencies for the potential w(q) = W/q2 94 s h i f t e d t o a f i n i t e v a l u e , as f o r the 3DEG. In a d d i t i o n , Wp (q) a c q u i r e s a t W — 6 a minimum, which deepens and moves t o h i g h e r q as W i n c r e a s e s . M o d i f i e d as w e l l i s the f u n c t i o n G ( q ) . The peak around q = 2 when W = 0 does not d i s a p p e a r w i t h s t r o n g e r c o u p l i n g as i n F i g . 2.5, but r a t h e r r i s e s toward one. As d i s c u s s e d i n Sec. 2.4C, the b e h a v i o u r o b s e r v e d i n F i g . 2.13 i s what one would expect i f the system were p r o c e e d i n g t o a t r a n s i t i o n i n t o a CDW s t a t e a t some c r i t i c a l v a l u e of W. B e f o r e such a t r a n s f o r m a t i o n o c c u r s however, t h e r e seems t o be an i n s t a b i l i t y i n the s o l u t i o n s of Eqs. ( 2 . 6 2 ) , s e l f - c o n s i s t e n t f u n c t i o n s G(q) and S(q) b e i n g i m p o s s i b l e t o o b t a i n ' f o r W £ 20. The n a t u r e of t h i s d i f f i c u l t y has not been i n v e s t i g a t e d . Campos e t a l . (1977) have a l s o performed c a l c u l a t i o n s w i t h a q " 2 p o t e n t i a l . They use the i n t e r p a r t i c l e s p a c i n g x ? as a c o u p l i n g parameter, but the c o n n e c t i o n between W and x s i s not apparent t o us. In any c a s e , Campos et a l . o b t a i n r e s u l t s f o r u>p(q) which a r e s i m i l a r t o o u r s ; the cur v e f o r t h e i r maximum v a l u e of * s = 11 c o r r e s p o n d s a p p r o x i m a t e l y t o the one f o r W = 20 i n F i g . 2.13. However, no mention i s made of any unu s u a l b e h a v i o u r f o r l a r g e r v a l u e s of x . 2.8 D i s c u s s i o n T h i s c h a p t e r has seen the i n t r o d u c t i o n of a model f o r an a n i s o t r o p i c system of i n t e r a c t i n g e l e c t r o n s . By making s u i t a b l e a s s u m p t i o n s , the H a m i l t o n i a n may be t r a n s f o r m e d i n t o a p u r e l y ID one w i t h a p a r t i c l e - p a r t i c l e i n t e r a c t i o n g i v e n by Eq. ( 2 . 5 8 ) . We have i n v e s t i g a t e d the d i e l e c t r i c response of the system by 96 a d a p t i n g the f o r m a l i s m f o r the 3D e l e c t r o n gas t o the ID case and then a p p l y i n g the STLS a p p r o x i m a t i o n . P r o p e r t i e s t h a t have been c a l c u l a t e d u s i n g t h i s approach i n c l u d e the p a i r c o r r e l a t i o n f u n c t i o n , the spectrum of c o l l e c t i v e e x c i t a t i o n s , the c o m p r e s s i b i l i t y , and the d i e l e c t r i c s u s c e p t i b i l i t y . T h i s l a s t q u a n t i t y i s an i n d i c a t o r of any charge d e n s i t y i n s t a b i l i t i e s i n h e r e n t i n the system. We have found t h a t no such i n s t a b i l i t i e s e x i s t i n our model, a t l e a s t f o r t h a t range of parameters f o r which the STLSA i s v a l i d . Such a r e s u l t i s c o n t r a r y t o t h a t o b t a i n e d by G i u l i a n i et a l . (1979) who c o n s i d e r a l a t t i c e of c h a i n s . By making the STLSA f o r S(q) and by t r e a t i n g i n t e r c h a i n i n t e r a c t i o n s i n the RPA, the model r e a p p e a r s as a ID one w i t h the i n t e r a c t i o n e s s e n t i a l l y the same as W(q). G i u l i a n i e t a l . do not s o l v e the problem s e l f -c o n s i s t e n t l y i n the STLS sense, making i n s t e a d the a n s a t z t h a t the p a r a l l e l and a n t i - p a r a l l e l s p i n c o r r e l a t i o n h o l e s a r e square w e l l s . They f i n d a plasmon spectrum which behaves i n much the same way t h a t was h y p o t h e s i z e d i n Sec. 2.4D f o r the o c c u r r e n c e of a CDW s t a t e . At some wavevector the plasmon branch merges w i t h the s i n g l e - p a r t i c l e continuum; C0p(q) then f a l l s i n t o the i n s t e p and goes t o z e r o , i n d i c a t i n g an ' e x c i t o n i c ' or CDW i n s t a b i l i t y . As a s i m i l a r tendency was o b s e r v e d i n our t r e a t m e n t f o r the p o t e n t i a l w(q), we f e e l t h a t the r e s u l t s of G i u l i a n i et a l . a r e p r i m a r i l y a consequence of the form of t h e i r p o t e n t i a l , r a t h e r than the l a c k of s e l f - c o n s i s t e n c y or the s p e c i f i c shape of the c o r r e l a t i o n h o l e s . One of the assumptions made i n f o r m u l a t i n g the model i n Sec. 2.2 was t h a t 6p be much l e s s than the energy d i f f e r e n c e 97 between the two l o w e s t e i g e n s t a t e s of Hj_. What c o n d i t i o n does t h i s impose on the a l l o w e d v a l u e s of k F and a.? A bound on the parameter A = a k p i s o b t a i n e d by f i r s t n o t i n g t h a t the t r a n s v e r s e G a u s s i a n w a v e f u n c t i o n (2.57) i s , i n f a c t , j u s t the ground s t a t e of the 2D harmonic o s c i l l a t o r w i t h f r e q u e n c y &>Wo = ft/2ma. O b v i o u s l y t h e n , the f i r s t e x c i t e d s t a t e has an energy h i g h e r by an amount fift)^, so t h a t the r e q u i r e d c o n d i t i o n must be £p/na)j{0 << 1. For our model t o be v a l i d , A must be r e s t r i c t e d t o v a l u e s much l e s s than one. The compound KCP ( K z [ P t ( C N ) ^ ]Br f l • 3(H^O)), which was mentioned i n Sec. 1.2 i n c o n n e c t i o n w i t h the P e i e r l s d i s t o r t i o n , i s a p o s s i b l e p h y s i c a l r e a l i z a t i o n of the model p r e s e n t e d i n t h i s c h a p t e r f o r two r e a s o n s . F i r s t , KCP has n e u t r a l c h a i n s , u n l i k e TTF-TCNQ i n which t h e r e i s a charge t r a n s f e r between c h a i n s . S e c o n d l y , Z e l l e r and Bruesch (1974) have i n t e r p r e t e d t h e i r o p t i c a l measurements as showing n e a r l y f r e e - e l e c t r o n ID be h a v i o u r i n the c o n d u c t i o n c h a n n e l s p r o v i d e d by the c l o s e l y -spaced Pt atoms. 98 Chapter 3 The O ne-dimensional Fermion Gas w i t h a £-function I n t e r a c t i o n 3 .1 I n t r o d u c t i o n In the p r e v i o u s c h a p t e r a q u a s i - o n e - d i m e n s i o n a l e l e c t r o n gas has been m o d e l l e d w i t h an i n t e r p a r t i c l e p o t e n t i a l t h a t i s hoped t o be r e a l i s t i c f o r a t l e a s t some p h y s i c a l systems. The model has been a n a l y z e d i n the STLS a p p r o x i m a t i o n which i s ex p e c t e d t o y i e l d r e a s o n a b l e r e s u l t s , but which can a t best be compared o n l y t o o t h e r i n e x a c t methods such as the RPA and HFA. I d e a l l y , one wants t o get a f e e l i n g f o r the v a l i d i t y of the STLSA i n ID by s e e i n g how i t measures up t o an ex a c t s o l u t i o n . At our d i s p o s a l i s the o n l y a n a l y t i c a l l y t r a c t a b l e model of a n o n - i d e a l gas of f e r m i o n s e x t a n t , namely one i n which the p a r t i c l e s i n t e r a c t w i t h a S - f u n c t i o n p o t e n t i a l . U n f o r t u n a t e l y , any c o n c l u s i o n s t h a t we may draw from a comparison between the e x a c t t h e o r y and the STLSA w i l l be somewhat d i f f i c u l t t o e x t e n d t o the p o t e n t i a l (2.58) because of the s h o r t range of the c o n t a c t i n t e r a c t i o n . C o n s e q u e n t l y we w i l l have t o be s a t i s f i e d w i t h some q u a l i t a t i v e s t a t e m e n t s about the r e s u l t s of Chap. 2. The e s s e n t i a l s of the Yang (1967) a n a l y s i s of the C-f u n c t i o n problem a re p r e s e n t e d i n the f o l l o w i n g s e c t i o n , a l o n g w i t h a d i s c u s s i o n of the theorem due t o L i e b and M a t t i s (1962) which proves t h a t under c e r t a i n c o n d i t i o n s the ground s t a t e of a ID system i s unmagnetized. Yang o b t a i n e d a s o l u t i o n i n the form of c o u p l e d i n t e g r a l e q u a t i o n s . These a r e s o l v e d i n Sec. 3.3 f o r 99 •the g r o u n d s t a t e and i n S e c . 3.4 f o r s t a t e s w i t h n o n - z e r o m a g n e t i z a t i o n . Then i n S e c . 3.5, we a p p l y t h e a p p r o x i m a t i o n methods o f t h e p r e c e d i n g c h a p t e r t o t h e p r e s e n t model and compare t h e r e s u l t s w i t h t h o s e o f t h e e x a c t t r e a t m e n t . F i n a l l y , t h e c h a p t e r i s c o n c l u d e d w i t h a d i s c u s s i o n of t h e i m p l i c a t i o n s o f t h e r e s u l t s o b t a i n e d . 3.2 The E x a c t S o l u t i o n An a n a l y t i c a l s o l u t i o n t o t h e ID ^ - f u n c t i o n model was f i r s t s k e t c h e d i n a p a p e r by Yang ( 1 9 6 7 ) . F o l l o w i n g h i s n o t a t i o n , we w r i t e t h e H a m i l t o n i a n f o r N s p i n - 1 / 2 p a r t i c l e s c o n f i n e d t o a l i n e of l e n g t h L ( d e n s i t y n=N/L) as H - - ! f e + ^ r ^ - * j > O.I) where c > 0 and "ft 2/2m = 1. As H c o n t a i n s no e x p l i c i t s p i n d e p e ndence, any w a v e f u n c t i o n s a t i s f y i n g t h e S c h r o d i n g e r e q u a t i o n i s s e p a r a b l e i n t o s p a t i a l and s p i n f u n c t i o n s ; ^ ( x { x ^ ) , an e i g e n f u n c t i o n o f ( 3 . 1 ) ; and X( «r, , . . ^ ) , an e i g e n f u n c t i o n of t h e s p i n o p e r a t o r s , A H. A A A , M L e t % • d e n o t e a s p i n f u n c t i o n o f a p a r t i c u l a r c o n f i g u r a t i o n o f J s p i n s w i t h t o t a l z-component of t h e s p i n , M. Then t h e t o t a l w a v e f u n c t i o n $ may be w r i t t e n as M S i n c e t h e OCj t r a n s f o r m i n t o e a c h o t h e r under a p e r m u t a t i o n of t h e 0*£ , t h e V';(xi I - M ^ ) must be r e l a t e d t o e a c h o t h e r by a 100 permutation of the xj in such a way that £ i s antisymmetric. In can be expressed in terms of single s p a t i a l and spin functions, ^ and % say, as follows: Here P- i s an element of the permutation group, and the factor f>. ( - ) 1 is +(-) depending on whether P{ is composed of an even (odd) number of transpositions. Because of the invariance of the Hamiltonian under the permutation of p a r t i c l e indices, the function *K need only be determined in the domain R, : 0 < x , < ... < x w < L of the N-dimensional coordinate space. In the domain Rj obtained from the permutation Pj of the indices in the above inequality, the t o t a l wavefunction becomes In his solution, Yang imposed periodic boundary conditions and assumed the so-called Bethe ansatz for t/> . Let Q {Q|,...,Qlsj} and T = {T, , .. . ,TN} label permutations of the set of integers {1,...,N}. Then, in the domain V 0 < x<2, <•••< X Q L N one writes where the p^ _ are a set of unequal numbers. The expansion c o e f f i c i e n t s a. are determined by the symmetry of under permutations and by the boundary conditions. These constraints 101 a r e t h e c o n t i n u i t y o f a t x^. = x^. ^ , a n d t h e c o n d i t i o n o b t a i n e d f r o m t h e ^ - f u n c t i o n p o t e n t i a l i n E q . ( 3 . 1 ) t h a t 0* 9fc He L (,+ 1 W i t h o u t f u r t h e r e l a b o r a t i o n o f t h e d e t a i l s , we p r o c e e d t o t h e s o l u t i o n w h i c h i n v o l v e s t h e d e t e r m i n a t i o n o f t h e numbers pt- , and an a d d i t i o n a l s e t o f u n e q u a l n u mbers, Aj , w h i c h d e t e r m i n e t h e s p i n o f t h e w a v e f u n c t i o n t . I n t h e t h e r m o d y n a m i c l i m i t o f N —Vo© f L—>oo , and M ( t h e number o f down s p i n s , s a y ) — s u c h t h a t M/L, N/L a r e f i n i t e , t h e p a r a m e t e r s p and A become c o n t i n u o u s v a r i a b l e s w i t h d i s t r i b u t i o n f u n c t i o n s j) (p) a n d 0"(A) t o be d e t e r m i n e d f r o m t h e c o u p l e d i n t e g r a l e q u a t i o n s : ZTo-(k) = -f 8 a. A ~—~—77~7 + r Q J-8 ( 3 . 2 a ) tic ftp) - l + \_<Lh -TzT^fW ( 3 . 2 b ) The i n t e g r a t i o n l i m i t s B a n d Q a r e s e t by t h e n o r m a l i z a t i o n c o n d i t i o n s , ( 3 . 3 a ) g ( 3 . 3 b ) F i n a l l y , t h e e n e r g y d e n s i t y o f t h e s y s t e m i s j u s t t h e t o t a l k i n e t i c e n e r g y of f r e e p a r t i c l e s w i t h a momentum d i s t r i b u t i o n 102 * f P ( ? h (3.4) We note t h a t the i n t e r a c t i o n s t r e n g t h , c, does not appear e x p l i c i t l y i n the e x p r e s s i o n f o r the energy, but r a t h e r i n f l u e n c e s £ through J) ( p ) . From the e x p r e s s i o n s (3.3a) and (3.4) i t i s t e m p t i n g t o t h i n k of y ) (p) as the Fermi d i s t r i b u t i o n f u n c t i o n , np, which g i v e s the p r o b a b i l i t y of o c c u p a t i o n of the f r e e e l e c t r o n plane-wave s t a t e w i t h momentum p, and of Q as the Fermi momentum. T h i s view i s c o r r e c t o n l y i n c e r t a i n l i m i t s when J>(p) tends t o a constant" v a l u e ; o t h e r w i s e i t i s the d e n s i t y of quantum numbers p, each of which has an o c c u p a t i o n p r o b a b i l i t y of u n i t y . In g e n e r a l , n^, which i s a s t e p f u n c t i o n i n the non-i n t e r a c t i n g regime, i s e x p e c t e d t o be smeared out around kp by the i n t e r p a r t i c l e p o t e n t i a l , a l t h o u g h some v e s t i g e of the Fermi •surface might be r e t a i n e d . S e t t i n g B = oo and i n t e g r a t i n g Eq. ( 3 . 2 a ) , 2 j r 00 -00 rto r& f>4 +H^Ml^ y i e l d s m = n/2. Thus, i n t h i s s p e c i a l case J?(p) d e s c r i b e s the non-magnetic s t a t e , which Yang c l a i m s t o be the ground s t a t e by v i r t u e of a theorem proved b y - L i e b and M a t t i s (1962). P r i o r t o a d i s c u s s i o n of t h i s theorem, we r e c a s t the e q u a t i o n s (3.2) - (3.4) i n t o a more f u n c t i o n a l form f o r f u r t h e r r e f e r e n c e . D e f i n i n g d i m e n s i o n l e s s v a r i a b l e s x = A/c,. y = p/c, q = Q / c , b = B/c, and making the re p l a c e m e n t s 103 2-Rf(p) -±J)(x) , 27T«r(A)-><r(y) one o b t a i n s : -b ~? (3.5a) - / f 27T (3.5b) n _ / 1 t " Z.TT r d.x <r(x) ( 3.6a ) (3.6b) e 3 " ITT (3.7) 3.2A L i e b - M a t t i s Theorem Let us c o n s i d e r the H a m i l t o n i a n ( 3 . 1 ) , w i t h the i n t e r a c t i o n term r e p l a c e d by a r e a l s p i n - and v e l o c i t y - i n d e p e n d e n t p o t e n t i a l , V ( x / , . . . , ) , which i s symmetric i n the N p a r t i c l e c o o r d i n a t e s . I f N = 2, the e i g e n s t a t e of minimum energy i s a s i n g l e t , h a v i n g a symmetric s p a t i a l w a v e f u n c t i o n which i s n o d e l e s s i n the i n t e r i o r of the box, i . e . f o r 0 < X| ,x^ < L. L i e b and M a t t i s (1962) extended t h i s r e s u l t t o an a r b i t r a r y even number of p a r t i c l e s , p r o v i n g t h a t the s t a t e s of the system a r e o r d e r e d a c c o r d i n g t o t h e i r t o t a l s p i n S. In e f f e c t , the theorem 104 says t h a t the s t a t e h a v i n g S = 0 i s the ground s t a t e of the system u n l e s s V i s p a t h o l o g i c a l , i n which case t h e r e e x i s t s a degeneracy. R e c a l l the e x p a n s i o n f o r the t o t a l w a v e f u n c t i o n <£ ; denote the s p a t i a l p a r t by x^ +, , . . . , x w ) , where JLt = N/2 - M and the bar s i g n i f i e s the antisymmetry of <A w i t h r e s p e c t t o p e r m u t a t i o n s of the {xj,...,^,,} and of the {x^ +( , . . . , x ^ } . The essence of t h e ' p r o o f by L i e b and M a t t i s c o n s i s t s of showing t h a t i n t he domain d e f i n e d by R M: 0<x, <...<y<L ; 0<x^+l < . . . <x^ <L the f u n c t i o n & h a v i n g the lo w e s t energy i n R^ i s the one which belongs t o S = M. Fu r t h e r m o r e , t h e r e i s no degeneracy i f & i s n o d e l e s s i n s i d e R ; t h u s , a p a t h o l o g i c a l p o t e n t i a l i s d e f i n e d as one which causes 4* t o v a n i s h w i t h i n t h i s domain. S i n c e the e i g e n s t a t e s a r e degenerate w i t h r e s p e c t t o M, i t f o l l o w s t h a t the one w i t h S = M = 0 i s the ground s t a t e . In a p p l y i n g t h i s theorem t o the Yang model ( 3 . 1 ) , we note two t h i n g s . F i r s t , the ^ - f u n c t i o n p o t e n t i a l i s i l l - b e h a v e d i n the above sense o n l y i n the l i m i t as c—^to. S e c o n d l y , the boundary c o n d i t i o n s imposed on <l> are l o c a l ( e . g . & = 0 when x = 0,L), p r e c l u d i n g p e r i o d i c c o n s t r a i n t s . However, we expect the n a t u r e of the boundary c o n d i t i o n s t o be un i m p o r t a n t when L -»"o, the l i m i t which i s taken i n de d u c i n g the e q u a t i o n s ( 3 . 2 ) - ( 3 . 4 ) , and we w i l l t h e r e f o r e assume the ground s t a t e has S = 0. 105 3.3 The Ground S t a t e As the f i r s t s t e p i n the a n a l y s i s of the s o l u t i o n when b •o, l e t us s i m p l i f y Eqs. (3.5a,b) as f a r as p o s s i b l e . D e f i n i n g fM = I T J i f i l+<4(<j-x)2 and the F o u r i e r t r a n s f o r m s fM~- ^\ dLkftkja "to iky -A3 a l l o w s Eq. (3.5a) t o be r e w r i t t e n as 2 T r 00 LkU-vf) -to 7 T T - +. F(k) Upon use of the r e l a t i o n "ti I e ~ = He Hi -to one o b t a i n s t-te' and s u b s e q u e n t l y S u b s t i t u t i n g f o r F ( k ) cr(x) = (4 dk W7- i* ite~'ki i which e v e n t u a l l y reduces t o 106 R e p l a c i n g <r(x) i n Eq. (3.5b) by t h i s e x p r e s s i o n g i v e s , a f t e r some m a n i p u l a t i o n , (3.8a) where - A* '--to Zl + f(*+!f)lh0*lrf* (3.8b) Thus the c o u p l e d e q u a t i o n s (3.5a,b) have been reduced t o a s i n g l e non-homogeneous Fredholm e q u a t i o n of the second k i n d . An a n a l y t i c s o l u t i o n f o r ^o(y) i s d i s c o u r a g e d by the i n t e g r a l r e p r e s e n t a t i o n of H ( y ) , and the problem must t h e r e f o r e be t a c k l e d n u m e r i c a l l y . B e f o r e p r o c e e d i n g i n t h i s f a s h i o n , i t i s i n s t r u c t i v e t o examine two l i m i t i n g c a s e s of Eq. ( 3 . 8 ) . F i r s t , c o n s i d e r the h i g h d e n s i t y extreme when q—>oo. By F o u r i e r a n a l y z i n g the e x p r e s s i o n -oo f p(y) = / + J p(x)HCx-y) one o b t a i n s ' " W k ) (3.9) where / M and -DO -Ki Eq. (3.9) i m p l i e s t h a t 107 •Mo) where - a . r* * r & Thus f o r q — J ) ( y ) = 2 f o r a l l y, which, from Eqs. (3.6a) and ( 3 . 7 ) , has the consequence t h a t n/c = 2q/tr and £ = n*"rrl/12 = 2 fpkp/3 . The l a t t e r r e s u l t i s , as i t s h o u l d be f o r a h i g h d e n s i t y system, the energy d e n s i t y of a gas of n o n - i n t e r a c t i n g f e r m i o n s . Note a l s o t h a t y>(p) = np and t h a t Q = qc = kp . In the o p p o s i t e extreme of q—>0, Eq. (3.8) shows t h a t J>(y) — v l , which c o r r e s p o n d s t o a d e n s i t y of p - v a l u e s which i s the same as the d e n s i t y of s t a t e s of a c o l l e c t i o n of non-i n t e r a c t i n g s p i n l e s s f e r m i o n s . Hence, the number and energy d e n s i t i e s must, be, as i s e a s i l y v e r i f i e d , n/c = q/TT (Q = 2kp) and £= 8£pkp/37T\ We observe t h a t i n the low d e n s i t y l i m i t £ i s independent of c, a l t h o u g h the p a r t i c l e i n t e r a c t i o n i s the dominant energy. T h i s seeming paradox has i t s r o o t s i n the p e r f e c t c o r r e l a t i o n between p a r t i c l e s due t o the s t r e n g t h of the c o u p l i n g compared t o the k i n e t i c energy. The v a n i s h i n g of the w a v e f u n c t i o n when any two p a r t i c l e s , r e g a r d l e s s of s p i n , t o u c h i s c h a r a c t e r i s t i c of a c o l l e c t i o n of s p i n l e s s f e r m i o n s and r e s u l t s i n a z e r o c o n t r i b u t i o n t o the energy by the f - f u n c t i o n p o t e n t i a l . Such a low d e n s i t y b e h a v i o u r i s c o n t r a r y t o the Wigner c r y s t a l l i z a t i o n one would expect when the i n t e r a c t i o n i s l o n g - r a n g e . E s t a b l i s h i n g the b e h a v i o u r of the s o l u t i o n i n the l i m i t s of 108 q—» 0 and q — » oo i s the best one can do a n a l y t i c a l l y w i t h Eq. ( 3 . 8 ) . For i n t e r m e d i a t e v a l u e s of q, n u m e r i c a l methods must be r e s o r t e d t o . Our aim w i l l be t o c a l c u l a t e £ as a f u n c t i o n of the parameter C = c / k F , s i n c e £ can c o n v e n i e n t l y be e x p r e s s e d i n Fermi u n i t s (Cpkp) as The i n t e g r a l i s a f u n c t i o n of C s i n c e ^>(y) depends o n l y on q which can i t s e l f be r e l a t e d t o C by r e w r i t i n g Eq. (3.5a) as c " H J..*y/(y' < 3- 1 1 ) 3.3A N u m e r i c a l Method S o l u t i o n of Eq. (3.8) and s u b s e q u e n t l y the c a l c u l a t i o n of £(C) i s a s t r a i g h t f o r w a r d p r o c e d u r e as o u t l i n e d i n the f o l l o w i n g s t e p s : (1) Choose a v a l u e of q and then e v a l u a t e , by means of a Simpson i n t e g r a t i o n , the f u n c t i o n H(x) d e f i n e d by Eq. (3.8b) f o r the set of p o i n t s (0=x0,x, ,...,x2W= 2q) w i t h c o n s t a n t s p a c i n g Ax = xt-+/ - X J = q/N. H(x) i s p l o t t e d i n F i g . 3.1. (2) S e l e c t a f u n c t i o n y)((x) as an i n i t i a l guess f o r y(x). S i n c e y>(x) has lower and upper bounds of 1 and 2 as d i s c u s s e d above, a r e a s o n a b l e c h o i c e i s a c o n s t a n t , 1 < j)0 < 2, which w i l l i n c r e a s e w i t h q. (3) For the s e t {xj x^},- c a l c u l a t e y z ( x j ) from Eq. (3.8a) by Simpson's method, and make use of - the f a c t t h a t j> and H a r e even f u n c t i o n s . 109 I I i I I 0 1 2 3 4 x F i g u r e 3.1. The k e r n e l H(x) d e f i n e d by Eq. ( 3 . 8 b ) . (4) A g a i n employ Simpson's method t o compute n/c and 6/c3 by s u b s t i t u t i n g p (x) i n t o Eqs. (3.6a) and ( 3 . 7 ) . (5) Set j>{ (x) = y^(x) and r e p e a t s t e p s (3) and (4) u n t i l l i m i t i n g v a l u e s are a c h i e v e d i n n/c and £/c 3. (6) C a l c u l a t e C and 5(C). S i n c e the c u r v a t u r e of jO(x) i n c r e a s e s 'with q, more i t e r a t i o n s a r e r e q u i r e d t o produce convergence i o r l a r g e r q. Ten i t e r a t i o n s or l e s s a r e s u f f i c i e n t t o o b t a i n an answer con v e r g e n t t o one p a r t i n 1 0 s f o r q £ 20. The r e s u l t s t h u s o b t a i n e d must then be checked t h a t they a r e independent of the p o i n t s p a c i n g Ax. 110 3.3B R e s u l t s Eq. (3.8) has been s o l v e d f o r a wide range of the parameter q. The r e s u l t i n g f u n c t i o n p(y) i s shown i n F i g . 3.2 f o r a number of q - v a l u e s . B e g i n n i n g w i t h the c o n s t a n t j> (y) = 1 f o r q << 1, 2 y/q F i g u r e 3.2. J?(y) f o r q = 0.1, 1, 3, 10, 20. the i n t e r c e p t f>(0) i n c r e a s e s w i t h q up t o the maximum y(y) = 2 when q —> oo . In a d d i t i o n , the c u r v a t u r e i n p becomes more pronounced, u n t i l a s t e p f u n c t i o n appears a t i n f i n i t e q. D i s c u s s e d a l r e a d y has been the i n t e r p r e t a t i o n of p as the Fermi f u n c t i o n f o r n o n - i n t e r a c t i n g s p i n - z e r o and s p i n - 1 / 2 f e r m i o n s i n the s m a l l - and l a r g e - q l i m i t s . Such meaning cannot be a s c r i b e d I l l t o p f o r i n t e r m e d i a t e v a l u e s of q. Wi t h the c a l c u l a t e d p , the d e n s i t y may be d e t e r m i n e d . In F i g . 3.3, the dependence of n/c on q i s i n d i c a t e d i n a p l o t of kp/Q a g a i n s t q = Q/c. The r a t i o kp/Q i n c r e a s e s from the s t r o n g -1 r F i g u r e 3.3. The r a t i o kp/Q as a f u n c t i o n of q. c o u p l i n g v a l u e of 1/2 t o the w e a k - c o u p l i n g asymptote kp/Q = 1. F i n a l l y , the energy £ i s shown i n F i g . 3.12 as a m o n o t o n i c a l l y i n c r e a s i n g f u n c t i o n of the i n t e r a c t i o n s t r e n g t h C. The range of v a r i a t i o n i s between the a f o r e m e n t i o n e d l i m i t s of 2/3TT and 8/37T. 112 3.4 M a g n e t i c S t a t e s The assumption has been made t h a t the L i e b - M a t t i s theorem h o l d s f o r t h i s model, and hence t h a t the ground s t a t e has a t o t a l s p i n S = 0. S i n c e the e q u a t i o n s (3.5) - (3.7) a l l o w s o l u t i o n s w i t h a r b i t r a r y m a g n e t i z a t i o n , we w i l l check t h i s a s sumption by c a l c u l a t i n g the e n e r g i e s of s t a t e s w i t h non-zero s p i n . I t i s immediately e v i d e n t t h a t the f e r r o m a g n e t i c s t a t e , m = 0, i s o b t a i n e d when b = 0. In t h i s case = l r whence As mentioned p r e v i o u s l y i n the d i s c u s s i o n of the l i m i t b = oo, q ->oo, the parameters Q and £ a r e c h a r a c t e r i s t i c of a non-i n t e r a c t i n g s p i n l e s s f e r m i o n gas. T h i s must be so s i n c e the s p a t i a l w a v e f u n c t i o n a s s o c i a t e d w i t h the s t a t e of c o m p l e t e l y a l i g n e d s p i n s i s t o t a l l y a n t i s y m m e t r i c , as i s the w a v e f u n c t i o n f o r a c o l l e c t i o n of s p i n l e s s f e r m i o n s . P a r t i a l l y m agnetized s t a t e s a r e o b t a i n e d f o r f i n i t e v a l u e s of b. F o u r i e r d e c o m p o s i t i o n of the f u n c t i o n s J>(y) , CT(x) i s then of no use, n e c e s s i t a t i n g the s u b s t i t u t i o n of Eq. (3.5b) i n t o (3.5a) d i r e c t l y . The i n t e g r a l e q u a t i o n t o be s o l v e d f o r C ( x ) may then be reduced t o the form and n/c = q/tT or Q = 2kf, £/c 3 = q 3/3 or £= f T 2 n 3/3. where g i v e n by 113 C a l c u l a t i o n of c r(x) p e r m i t s the d e t e r m i n a t i o n of the q u a n t i t i e s ; .6 When q —>M>, i t i s e a s i l y deduced t h a t CT(x) — > 1, t h e r e b y l e a d i n g i n t h i s l i m i t t o the i d e n t i f i c a t i o n of <r as the Fermi f u n c t i o n f o r s p i n down p a r t i c l e s , n^p . The parameter B = m/TT may then be i n t e r p r e t e d as the Fermi wavevector f o r the gas of d e n s i t y m. These remarks a r e a statement of the o b v i o u s f a c t t h a t f o r h i g h d e n s i t i e s the system c o n s i s t s of two d i s t i n c t non-i n t e r a c t i n g Fermi gases. I t f o l l o w s t h a t (3.13) a r e s u l t which may a l s o be o b t a i n e d from the e q u a t i o n s (3.5). 114 In the l i m i t where q —>• 0 , the c o n d i t i o n t h a t m < n/2 l e a d s t o the r equirement t h a t cr(x) < q/2b, a b e h a v i o u r which has no o b v i o u s p h y s i c a l i n t e r p r e t a t i o n . T h i s i s not s u r p r i s i n g as i t has a l r e a d y been p o i n t e d out i n Sec. 3.3 t h a t the l o w - d e n s i t y or s t r o n g - c o u p l i n g regime i s analogous t o a gas of n o n - i n t e r a c t i n g f e r m i o n s of z e r o s p i n . In o t h e r words, the s p i n degree of freedom becomes i r r e l e v a n t , and a l l p o s s i b l e s p i n s t a t e s a r e d e g e n e r a t e . Such a degeneracy r e p r e s e n t s the breakdown of the L i e b - M a t t i s theorem due t o the p a t h o l o g y of the p o t e n t i a l when c/kp—* oot and m a n i f e s t s i t s e l f i n a magnetic s u s c e p t i b i l i t y w hich d i v e r g e s as the c o u p l i n g s t r e n g t h i n c r e a s e s . 3.4A S p i n S u s c e p t i b i l i t y The b e h a v i o u r of a s p i n system i n a magnetic f i e l d it i s c h a r a c t e r i z e d by the s u s c e p t i b i l i t y d e f i n e d i n terms of the t o t a l magnetic moment, ju r , by I f the f i e l d i s a p p l i e d a l o n g the a x i s of s p i n q u a n t i z a t i o n , then the Zeeman energy d e n s i t y i s g i v e n by where ju, i s the magnetic moment of the p a r t i c l e . For g i v e n d e n s i t i e s n,m, the l o w e s t energy of the system s u b j e c t e d t o M i s where £(n,m,c) i s the s o l u t i o n of the Yang e q u a t i o n s . Now 115 m i n i m i z i n g w i t h r e s p e c t t o m, y i e l d s the spin-down d e n s i t y of the l o w e s t energy s t a t e . The s p i n s u s c e p t i b i l i t y per p a r t i c l e i s which may be r e w r i t t e n w i t h the a i d of Eq. (3.14) as i s e a s i l y c a l c u l a t e d f o r s m a l l magnetic f i e l d s i n the w e a k - c o u p l i n g l i m i t by employing the e x p r e s s i o n (3.13) f o r £(n,m,c). One f i n d s a r e s u l t which i s independent of * (#) = - ^ t 3.4B N u m e r i c a l Method The problem of c a l c u l a t i n g cr(x) from Eq. (3.12) i s e s s e n t i a l l y i d e n t i c a l t o t h a t of d e t e r m i n i n g y ( y ) i n Sec 3.3A, d i f f e r i n g o n l y i n the form of the k e r n e l , which here i s a f u n c t i o n of two v a r i a b l e s , and i n the non-homogeneous p a r t of the e q u a t i o n , which i s no l o n g e r a c o n s t a n t . E x p l o i t i n g the 116 symmetry p r o p e r t i e s ^ i s computed f o r the s e t of p o i n t s {0=x 0 ,x ( , . . .x w =b;-b=x(J ,x(', ...,x^ N=b} w i t h Ax = A x ' . Then, making the i n i t i a l guess (r(x) = f ( ( x ; q ) , the i t e r a t i v e p rocedure o u t l i n e d i n s t e p s (3) - (6) i s f o l l o w e d . By c a l c u l a t i n g n/c, m/c, 6/c 3 f o r v a r i o u s i n t e r v a l s i z e s A x , i t was found t h a t these q u a n t i t i e s were c o n s t a n t t o one p a r t i n 1 0 5 f o r Ax < b/50. About 12 i t e r a t i o n s were needed t o i n s u r e convergence of c ( x ) f o r b/q 10; t h i s number d e c r e a s e d w i t h b/q u n t i l one i t e r a t i o n was s u f f i c i e n t f o r b/q < 0.2. For l a r g e r a t i o s b/q, a v e r a g i n g of the i n i t i a l and c a l c u l a t e d o'(x) f o r one i t e r a t i o n t o p r o v i d e the i n p u t f u n c t i o n f o r the s u c c e e d i n g s t e p h e l p e d t o speed up the convergence. 3.4C R e s u l t s As the p a r t i c l e and s p i n d e n s i t i e s , n and (n-2m)/2, are d e t e r m i n e d o n l y a p o s t e r i o r i f o r a g i v e n parameter p a i r ( b , q ) , i t i s not p r a c t i c a l t o attempt a c a l c u l a t i o n of the energy as a f u n c t i o n of the f r a c t i o n a l spin-down d e n s i t y m/n f o r c o n s t a n t n. I n s t e a d , we c a l c u l a t e £(c) as a f u n c t i o n of b. F i g . 3.4 d i s p l a y s the f u n c t i o n <r(x) which i s the s o l u t i o n of Eq. (3.12) f o r a number of v a l u e s of b and q. T a k i n g i n t o account the c h a n g i n g h o r i z o n t a l s c a l e , we observe t h a t f o r a p a r t i c u l a r v a l u e of q, the . e f f e c t of i n c r e a s i n g b i s t o e xtend <r(x) t o h i g h e r x, l e a v i n g the shape of o" and the i n t e r c e p t <T(0) v i r t u a l l y the same. For f i x e d b, <r tends t o z e r o as q —> 0 and t o 117 F i g u r e 3.4. The s o l u t i o n s <r(x) of Eq. of b and q. (3.12) f o r v a r i o u s v a l u e s 118 one as q —>oo . In the same way j3(y) i s shown i n F i g . 3.5. Here, a change i n b from b << q t o b > q has a c o n s i d e r a b l e e f f e c t on J) , as i s w e l l i l l u s t r a t e d by the c u r v e s f o r q = 5. Not o n l y does yMO) i n c r e a s e , but the o v e r a l l form of J> i s a l t e r e d . These graphs may be compared w i t h the b = oo l i m i t i n F i g . 3.2. An a d d i t i o n a l o b s e r v a t i o n t o be made i s t h a t f o r c o n s t a n t b, which would appear i n F i g . 3.5 as a o " - f u n c t i o n s p i k e a t y = 0 w i t h magnitude I d e a l l y , Eq. (3.11) would be s o l v e d f o r enough v a l u e s (b,q) t o g e n e r a t e the s u r f a c e s m(b,q), n ( b , q ) , and £(b,q), and hence c*(m,n). As our s a m p l i n g of the (b,q)-parameter space i s r a t h e r l i m i t e d , we e x h i b i t the r e s u l t s as i n F i g s . 3.6 - 3.8. In F i g . 3.6, the b e h a v i o u r of the f r a c t i o n a l spin-down d e n s i t y as a f u n c t i o n of b and q i s i n d i c a t e d . For f i n i t e b, the c u r v e s are c o n s t r a i n e d t o l i e between the l i n e s m/n = 0 when b = 0 and m/n = 1/2 when b—*«o. The s m a l l - q v a l u e s of m/n r i s e r a p i d l y w i t h b t o the l i m i t of 1/2, w h i l e f o r l a r g e q, m/n—>0. F i g . 3.6 i s i n s u f f i c i e n t t o d etermine m(b,q) and n(b,q) s e p a r a t e l y , so the q u a n t i t y kp/Q (=TTn/2qc) i s p l o t t e d i n F i g . 3.7. F i n a l l y , the q u a n t i t y of i n t e r e s t , the energy d e n s i t y , i s shown i n F i g . 3.8. As h y p o t h e s i z e d , £(C,b) f o r magnetic s t a t e s remains g r e a t e r than t h a t f o r the assumed ground s t a t e w i t h S = 0. As an a i d i n u n d e r s t a n d i n g the r e s u l t s , we r e c a l l t h a t 119 2 F i g u r e 3.5. The f u n c t i o n p ( y ) o b t a i n e d from Eq. v a r i o u s b and q. ^ (3.5b) f o r 120 F i g u r e 3.6. The f r a c t i o n a l spin-down d e n s i t y d e t e r m i n e d from Eq. (3.6a) f o r v a r i o u s b and q. 121 F i g u r e 3.7. The r a t i o kp/q f o r v a r i o u s b. 122 F i g u r e 3.8. The energy d e n s i t y as a f u n c t i o n of the c o u p l i n g s t r e n g t h C. 123 lU + Q\ lo•>>«>) = ~Y which a r e r e s u l t s f o r n o n - i n t e r a c t i n g s p i n l e s s and s p i n - 1 / 2 f e r m i o n gases. The i n f i n i t e degeneracy a t C = <* i s broken w i t h d e c r e a s i n g C as £(C,b) g r a d u a l l y d e p a r t s from £ 0(C) = £(C,b=o»), reac h e s a b-dependent minimum, and then i n c r e a s e s t o an energy £(c-»0,b) g i v e n by 1) 8/3TT i f b/q—>0, s i n c e i n t h i s case m/n — > 0 even f o r l a r g e b, 2) 2/3TT i f b/q-»oo s i n c e .m/n 1/2 , 3) 2/3TT < £(c-»0;b) < 8/3TT i f b -> oo i n such a way. t h a t b/q i s f i n i t e . W h i l e t h e s e r e s u l t s do n o t , of c o u r s e , prove t h a t the ground s t a t e i s the one w i t h z e r o t o t a l s p i n , t h e y a r e c e r t a i n l y c o n s i s t e n t w i t h such an a s s u m p t i o n . 3.5 Approximate Methods I t w i l l be r e c a l l e d t h a t the o r i g i n a l i n t e n t i o n of the i n v e s t i g a t i o n of the ^ - f u n c t i o n model was t o compare an e x a c t s o l u t i o n w i t h the p r e d i c t i o n of the STLS a p p r o x i m a t i o n . In o r d e r t o do t h i s , we w i l l now r e f o r m u l a t e the STLS scheme d e v e l o p e d i n the p r e v i o u s c h a p t e r f o r the case of a c o n t a c t i n t e r a c t i o n . In a d d i t i o n , the RPA, g e n e r a l i z e d RPA, and HFA w i l l be b r i e f l y d i s c u s s e d . 124 3.5A STLS A p p r o x i m a t i o n The s e l f - c o n s i s t e n t e q u a t i o n s d e f i n i n g the STLSA become somewhat s i m p l i f i e d when the p o t e n t i a l has the form V(x) = 2 c 5(x) w i t h F o u r i e r t r a n s f o r m v ( q ) = 2c. For the sake of conv e n i e n c e we w i l l work i n Fermi u n i t s . Begin w i t h Eq. (2.62b) and s u b s t i t u t e f o r v ( q ) : -oo which can, by change of v a r i a b l e , be reduced t o OrU) = £ = - i f l s M - ildk v U (3.15a) Thus f o r a g i v e n v a l u e of c, G(q) i s a c o n s t a n t . The d i e l e c t r i c s u s c e p t i b i l i t y becomes l-ZtLl-6l%.(p«>) (3.15b) w h i l e the s t a t i c form f a c t o r remains as /•oo / I lm- X, La,001 *u) o (3.15c) Sty = i J l»*%(pti))A<a The f u n c t i o n S(q) which s a t i s f i e s t he e q u a t i o n s (3.15a,c) may be used as i n Sec. 2.3C t o c a l c u l a t e the ground s t a t e energy: (c where 6 0 = N/3 i s the t o t a l ground s t a t e energy of an assemblage of n o n - i n t e r a c t i n g f e r m i o n s , and the i n t e r a c t i o n energy E t - n - f ( A ) i s g i v e n by i »vt TT if Here the f i r s t term on the r i g h t s i d e of the e q u a t i o n r e p r e s e n t s 125 the i n t e r a c t i o n energy of a u n i f o r m d i s t r i b u t i o n of p a r t i c l e s and has been e x p l i c i t l y i n c l u d e d s i n c e we a r e d e a l i n g w i t h n e u t r a l p a r t i c l e s . Making use of E q .(3.15a), one i s a b l e t o w r i t e e ^ ( X ) = IT " (3.16) whence the energy d e n s i t y becomes 0 (3.17) I t i s i n f o r m a t i v e to o b t a i n the l i m i t i n g b e h a v i o u r of £ S T ^ g ( C ) . As C—>0, (3.17) g i v e s the c o r r e c t n o n - i n t e r a c t i n g r e s u l t , 2/3TT. In the o p p o s i t e extreme where C—>eo, we examine the e x p r e s s i o n (3.15b) f o r the s u s c e p t i b i l i t y . I f G(C) does not approach 1 as C — o o , then c l e a r l y X —> 0 and S(q)—»0, whence (3.15a) shows t h a t G d i v e r g e s . F u r t h e r m o r e , f o r S(q) = 0, the p a i r c o r r e l a t i o n f u n c t i o n g(x) = 1 - £(x), an u n p h y s i c a l r e s u l t . T h i s h o l d s t r u e even i f G ( C ) — 1 slow e r than 1/C. On the o t h e r hand, i f G(C) 1 f a s t e r than 1/C, X — t h e f r e e p a r t i c l e s u s c e p t i b i l i t y , and G = 1/2, a r e s u l t which i s i n c o n s i s t e n t w i t h our i n i t i a l a ssumption t h a t G — * 1. I t must be c o n c l u d e d , t h e r e f o r e , t h a t f o r s e l f - c o n s i s t e n c y JIM G-CCJ ~ I - 7j Hence, i t f o l l o w s t h a t fC w i t h the consequence t h a t c^-j-^^C) d i v e r g e s l o g a r i t h m i c a l l y w i t h 126 l a r g e C. By v i r t u e of Eqs. (2.51) and (3. 1 5 a ) , the p a i r c o r r e l a t i o n f o r v a n i s h i n g s e p a r a t i o n i s g(0) = 1 - G(C). T h i s e x p r e s s i o n e n a b l e s us t o n o t e , f i r s t of a l l , t h a t w r i t i n g Eq. (3.16) as merely r e f l e c t s the f a c t t h a t f o r a c o n t a c t p o t e n t i a l . the i n t e r a c t i o n energy i s d e t e r m i n e d s o l e l y by the p r o b a b i l i t y of two p a r t i c l e s b e i n g l o c a t e d a t the same p o i n t . S e c o n d l y , the a s y m p t o t i c b e h a v i o u r of G(C) ensures t h a t g(0) always remains p o s i t i v e , u n l i k e what i s obser v e d f o r the lon g - r a n g e p o t e n t i a l i n Chap. 2. 3.5B H a r t r e e - F o c k and Random Phase A p p r o x i m a t i o n s The well-known HF r e s u l t f o r the t o t a l energy of the paramagnetic ground s t a t e i s er<r' 0 where n p o . = j 1 ; |p| < k F \ o ; |p| > k F W i t h v (q) = 2C, the energy d e n s i t y t a k e s the s i m p l e l i n e a r form 127 I t s h o u l d be p o i n t e d o u t , however, t h a t t h i s i s not the HF ground s t a t e , as a w a v e f u n c t i o n d e s c r i b i n g a s p i n d e n s i t y wave s t a t e w i t h a lower energy can be c o n s t r u c t e d . T h i s p o i n t w i l l be d i s c u s s e d f u r t h e r i n Chap. 4. The RPA energy i s p a r t i c u l a r l y easy t o c a l c u l a t e f o r the f -f u n c t i o n p o t e n t i a l . X^p^, o b t a i n e d when G = 0 i n Eq. (3.15b), y i e l d s a f t e r one i t e r a t i o n of the STLS proce d u r e a new v a l u e , G ((C). Combining Eqs. (3.15a) and ( 3 . 1 7 ) , the energy may be e x p r e s s e d as 3.5C G e n e r a l i z e d Random Phase A p p r o x i m a t i o n I t i s n e c e s s a r y t o p o i n t out t h a t the RPA i s not a c o n s i s t e n t a p p r o x i m a t i o n f o r a s h o r t - r a n g e i n t e r a c t i o n f o r the f o l l o w i n g r e a s o n . The RPA amounts t o a summation of the r i n g diagrams i n a p e r t u r b a t i o n s e r i e s i n v ( q ) . For the 3D Coulomb p o t e n t i a l t h e s e diagrams r e p r e s e n t the most d i v e r g e n t terms i n each o r d e r ; summing them g i v e s the dominant c o n t r i b u t i o n t o the c o r r e l a t i o n energy f o r the 3DEG, a t l e a s t a t v e r y h i g h d e n s i t i e s . When v( q ) i s a c o n s t a n t , however, the d i v e r g e n c e s d i s a p p e a r , and the exchange terms become comparable t o the d i r e c t g raphs. As an example, c o n s i d e r the second o r d e r c o r r e c t i o n s r e p r e s e n t e d d i a g r a m a t i c a l l y i n F i g . 3.9. The r i n g graph (a) denotes an energy per p a r t i c l e o f : 128 k+qqi k ' - q c r k+qcr k ' - q a (a) F i g u r e 3.9. Second o r d e r energy terms, (b) E l «- l - f t2 r r e(i-iti)9(i-ik'\) ^ ^  ,2/ )( dk'otk - ci " 27P-Both i n t e g r a l s a r e c o n v e r g e n t , y i e l d i n g T_ Zt Us T P / , T i n t J h(m-t)-j ( 3 . 2 0 ) The exchange term (b) has one l e s s s p i n summation and t h e r e f o r e d i f f e r s by a f a c t o r of -1/2 , so t h a t the t o t a l second o r d e r energy i s E^/2. 129 For a more c o n s i s t e n t a p p r o x i m a t i o n we c o n s i d e r the e q u a t i o n of motion of the p a r t i c l e - h o l e o p e r a t o r tit" 5 ck+^ <w= ? /v The RPA t a k e s i n t o account o n l y the k' = q term i n the above e x p r e s s i o n , which i s e q u i v a l e n t t o r e p l a c i n g the p a r t i c l e - h o l e o p e r a t o r s by t h e i r ground s t a t e a v e r a g e s ; i . e . The g e n e r a l i z e d RPA (see PN pp. 317-) i s d e r i v e d by e x p r e s s i n g a l l ^ - o p e r a t o r s i n the c u r l y b r a c k e t by s i n g l e - p a r t i c l e o p e r a t o r s and then r e p l a c i n g p a i r s of o p e r a t o r s , one at a t i m e , by t h e i r ground s t a t e a v e r a g e s . For i n s t a n c e , fk+lk'-lPk' = C H c r C ( c +(<V tf+k'v' Cpf<r' 130 In t h i s a p p r o x i m a t i o n t h e n , the e q u a t i o n of motion becomes where the s i n g l e - p a r t i c l e HF e x c i t a t i o n e n e r g i e s a r e HF HF HF In the s p e c i a l case of v(q) c o n s t a n t , the e q u a t i o n of motion i n the GRPA i s the RPA r e s u l t w i t h v r e p l a c e d by v/2. I t f o l l o w s from Eq. (3.19) t h a t The e n e r g i e s £ j ^ p ^ an& ^GRpA s h o u l d a l s o be c a l c u l a b l e by the method of Gell-man and Brueckner (1957) f o r the 3DEG. 3.5D N u m e r i c a l Method The procedure f o r computing the b a s i c q u a n t i t i e s G(q) and S(q) i n the STLSA i s e s s e n t i a l l y the same as t h a t o u t l i n e d i n Sec. 2.6, w i t h the added s i m p l i f i c a t i o n t h a t G(q) i s now independent of q. I t f o l l o w s t h a t s e l f - c o n s i s t e n c y i s a c h i e v e d when the v a l u e of G has converged t o a c o n s t a n t . As b e f o r e , the number of i t e r a t i o n s i n c r e a s e s w i t h c o u p l i n g s t r e n g t h , t y p i c a l f i g u r e s b e i n g 3 f o r C - 1 and 10 f o r C - 10 .Upon the d e t e r m i n a t i o n of G f o r a range of C - v a l u e s , a s t r a i g h t f o r w a r d n u m e r i c a l i n t e g r a t i o n a c c o r d i n g t o Eq. (3.17) g i v e s us the q u a n t i t y of i n t e r e s t , ^ S T L ^ O - Of c o u r s e , the 131 s t r u c t u r e f a c t o r S ( q ) , and c o n s e q u e n t l y the p a i r c o r r e l a t i o n f u n c t i o n , may a l s o be c a l c u l a t e d once G(C) i s known. In the RPA and GRPA, the energy i s e x p r e s s e d by Eqs. (3.19) and ( 3 . 2 1 ) . The q u a n t i t y G,(C) i s o b t a i n e d t r i v i a l l y a f t e r the f i r s t i t e r a t i o n of the STLS scheme i n i t i a l i z e d by s e t t i n g G = 0. 3.5E R e s u l t s Because they determine the i n t e r a c t i o n energy i n the a p p r o x i m a t i o n schemes d i s c u s s e d above, the f u n c t i o n s G(C) and G((C) a r e shown i n F i g . 3.10. U n l i k e G, which has the a s y m p t o t i c l i m i t of one, Gj d i v e r g e s f o r l a r g e C. The e f f e c t of t h i s b e h a v i o u r i s t h a t S^pp, and £-gg P^ must a t t a i n a maximum'and de c r e a s e t o n e g a t i v e v a l u e s as C i n c r e a s e s . From Eqs. (3.19) and (3.21) the maximum can be shown t o occur when Gj(C)= 1 i n the RPA and when G,(C/2) = 1 i n the GRPA. These are j u s t the c o n d i t i o n s f o r which g(o) v a n i s h e s . The d i r e c t consequence of g(o) becoming n e g a t i v e i s t h a t f l u c t u a t i o n s from a u n i f o r m d e n s i t y i n t e r a c t w i t h an a t t r a c t i v e p o t e n t i a l , b r i n g i n g about a decrease i n the t o t a l energy. The c o n t e n t of the s e remarks i s i l l u s t r a t e d i n F i g . 3.11, where the e n e r g i e s c a l c u l a t e d i n the v a r i o u s a p p r o x i m a t i o n s can be compared w i t h the e x a c t r e s u l t . We observe t h a t the GRPA r e p r e s e n t s a d e f i n i t e improvement on the RPA, r e p r o d u c i n g the ex a c t energy f a i r l y w e l l up t o C — 1. For s t r o n g e r c o u p l i n g , f a l l s below £ , s i g n a l l i n g the ' breakdown of the p e r t u r b a t i o n e x p a n s i o n i n powers of C. More i m p o r t a n t i s the b e h a v i o u r of £ S T L £ which i s a v e r y good a p p r o x i m a t i o n below C — 1.5. Above t h i s v a l u e , the 132 F i g u r e 3.10. The f u n c t i o n s G(C) and G,(C) used t o c a l c u l a t e the e n e r g i e s ( 3 . 1 1 ) , ( 3 . 1 9 ) , and ( 3 . 2 0 ) . 133 F i g u r e 3.11. The approximate energy d e n s i t i e s compared w i t h the exact- r e s u l t (dashed c u r v e ) . 134 agreement g r a d u a l l y worsens as £$ri_S d i v e r g e s l o g a r i t h m i c a l l y and £ tends t o the l i m i t 8/3TT. S t i l l , the STLSA i s seen t o be a g r e a t improvement over the GRPA. A l s o shown i n F i g . 3.11 i s the HF energy, Eq. ( 3 . 1 8 ) . Immediately apparent i s the c o r r e l a t i o n energy, d e f i n e d as the d i f f e r e n c e between the t o t a l energy and £^p- We note t h a t the GRPA and the STLSA tend t o o v e r - and u n d e r e s t i m a t e the c o r r e l a t i o n energy , a f a i l i n g which e x i s t s as w e l l i n the 3DEG (see e.g. Hedin and L u n d q v i s t , 1969). The p a i r c o r r e l a t i o n f u n c t i o n has been computed f o r a number of v a l u e s of C, and i s p r e s e n t e d i n F i g . 3.12. Comparison w i t h F i g . 2.4 r e v e a l s a q u a l i t a t i v e d i f f e r e n c e i n g(x) between s h o r t - and long-range p o t e n t i a l s . In the h i g h - d e n s i t y regime the f u n c t i o n s a r e s i m i l a r as they approach the H F " l i m i t . For low d e n s i t i e s , however, they d i f f e r i n shape a t s m a l l x, g(x) r i s i n g l i n e a r l y i n the &-function model, but not as s t e e p l y as f o r the extended i n t e r a c t i o n of Chap. 2. A l t h o u g h the STLSA p r e d i c t s a r a t h e r poor energy a t C = 5, we expect the e x a c t g ( x) t o bear a c l o s e resemblance t o the c u r v e i n F i g . 3.12 i f the a p p r o x i m a t i o n i s r e a s o n a b l e f o r s m a l l C. 3.6 D i s c u s s i o n F i g u r e s 3.8 and 3.11 summarize the i m p o r t a n t r e s u l t s of t h i s c h a p t e r . The former i n d i c a t e s the b e h a v i o u r of the energy as the t o t a l s p i n of the system i s v a r i e d , and c o n f i r m s the assumption t h a t the ground s t a t e has S = 0. In the l a t t e r f i g u r e , we a r e a b l e t o make a comparison between the r e s u l t s f o r the energy c a l c u l a t e d from the e x a c t s o l u t i o n and from s e v e r a l 135 0 1 2 3 x F i g u r e 3.12. The p a i r c o r r e l a t i o n f u n c t i o n f o r v a r i o u s v a l u e s of C. 136 a p p r o x i m a t i o n methods. As such a comparison was the o r i g i n a l m o t i v a t i o n f o r the i n v e s t i g a t i o n of the ^ - f u n c t i o n model, we now attempt t o i n t e r p r e t our f i n d i n g s f o r the purpose of g a i n i n g f u r t h e r i n s i g h t i n t o the STLSA. F i r s t l e t us c o n s i d e r the RPA and GRPA. I t has been p o i n t e d out e a r l i e r t h a t the r i n g diagrams a r e f i n i t e t o any o r d e r ; c o n s e q u e n t l y every term i n the p e r t u r b a t i o n e x p a n s i o n i s f i n i t e , and the s e r i e s can, i n p r i n c i p l e , be summed. To second o r d e r i n C, the RPA and GRPA e n e r g i e s a r e : where £-H(=. and a r e g i v e n by Eqs. (3.18) and ( 3 . 2 0 ) . Comparing the s e q u a n t i t i e s t o Z^p^ and f g g p A i n F i g . 3.13, we see t h a t f o r s m a l l C, the g r e a t e r p a r t of the t o t a l energy i s a l r e a d y c o n t a i n e d i n second o r d e r . A l s o e v i d e n t i s the improvement due t o the second o r d e r exchange term. From a p h y s i c a l p e r s p e c t i v e , the RPA t r e a t s p o o r l y c o r r e l a t i o n s a t s e p a r a t i o n s s m a l l e r than 1/kp, the e x t e n t of the P a u l i exchange h o l e . When the p a r t i c l e s i n t e r a c t v i a a 8-f u n c t i o n p o t e n t i a l , i t i s the p r o b a b i l i t y of p a r t i c l e s b e i n g s i m u l t a n e o u s l y a t the same p o s i t i o n which d e t e r m i n e s the energy and i m p a r t s p a r t i c u l a r importance t o the exchange i n t e r a c t i o n : hence the n e c e s s i t y of i n c l u d i n g exchange diagrams even f o r s m a l l c o u p l i n g , and the improvement of the GRPA over the RPA. The j u s t i f i c a t i o n f o r the STLSA g i v e n i n Chap. 2 was i t s i n c l u s i o n of l o c a l f i e l d e f f e c t s ; i m p l i e d . t h e n , i s a d e s c r i p t i o n of s h o r t - r a n g e c o r r e l a t i o n s which i s s u p e r i o r t o t h a t of the RPA. G r a p h i c d e m o n s t a t i o n of t h i s p o i n t has a l r e a d y been made i n F i g . 3.10, i n which g(0) remains p o s i t i v e f o r a l l C 137 i n the STLSA, u n l i k e what o c c u r s even i n the GRPA. Gi v e n t h e s e c o n s i d e r a t i o n s , what may we i n f e r about the STLSA as a p p l i e d i n Chap. 2? F i r s t of a l l , t he p o t e n t i a l t h e r e i s of l o n g range, which l e a d s us t o e x p e c t m e a n - f i e l d a p p r o x i m a t i o n s such as the RPA and STLSA t o be b e t t e r than f o r a c o n t a c t i n t e r a c t i o n . L e s s d e l e t e r i o u s w i l l be the e f f e c t of g e t t i n g g(0) i n c o r r e c t l y , which the STLSA does f o r s t r o n g c o u p l i n g i n F i g . 3.10. S e c o n d l y , the STLSA has been shown t o be at l e a s t p a r t i a l l y c o r r e c t i n i t s h a n d l i n g of the l o c a l f i e l d , and w i l l t h e r e f o r e be b e t t e r than the RPA f o r lower d e n s i t i e s . U n f o r t u n a t e l y , t h e r e i s no co r r e s p o n d e n c e between the c o u p l i n g 138 c o n s t a n t s C and U/A i n the two models, so t h a t we cannot s t a t e the range of U/A f o r which the STLSA seems t o be v a l i d , as we c o u l d i n t h i s c h a p t e r . A s i m p l e p h e n o m e n o l o g i c a l improvement t o the STLSA i s suggested by r e c a l l i n g t h a t the GRPA i s e q u i v a l e n t t o m u l t i p l y i n g the p o t e n t i a l i n the RPA by a f a c t o r of 1/2. The net e f f e c t i s t o " p u l l down" the c u r v e G((C) i n F i g . 3.10 t o G,(C/2), t h e r e b y r e d u c i n g the c o r r e l a t i o n energy. In F i g . 3.11, we observe t h a t the STLSA i s c o r r e c t e d by i n c r e a s i n g the c o r r e l a t i o n energy. T h i s o b j e c t i v e may. be a c c o m p l i s h e d by s u b s t i t u t i n g f o r G(X) i n Eq. (3.17) a new f u n c t i o n , Ggff (X), which p o s s e s s e s the a s y m p t o t i c form G^f(X) /v l - l/\2 as . W i t h i n the STLS framework, the p o t e n t i a l appears o n l y i n Eq. (3.15b) which, upon m u l t i p l y i n g C by a s u i t a b l y chosen f u n c t i o n V ( C ) , becomes S e l f - c o n s i s t e n t s o l u t i o n of the e q u a t i o n s (3.15a,c) and (3.22) y i e l d s r e m i n i s c e n t of V a s h i s h t a ' s and Singwi'S (1972) d e n s i t y - d e p e n d e n t c o r r e c t i o n t o G which was mentioned i n Sec. 2.7. (3.22) Geff (c) = G(,|;(c)c) • For s m a l l C, G p ^ ( C ) = G(C), which r e q u i r e s v> (C -> 0) = 1, w h i l e the C — > aa l i m i t , G(vC) = 1 - 1/VC demands t h a t v ( C ) ^  C. We c o n c l u d e by remarking t h a t the r e v i s e d C-dependence of G-rr i s 139 Chapter 4 D i s c u s s i o n and C o n c l u s i o n s 4.1 Comparison w i t h Other Models The p r e v i o u s two c h a p t e r s have d e a l t w i t h the f e r m i o n gas i n which the p a r t i c l e s i n t e r a c t w i t h both a long-range p o t e n t i a l and a £-function i n t e r a c t i o n . In the f o l l o w i n g s e c t i o n we attempt t o l o c a t e t h e s e models w i t h i n the framework of the ' g-o l o g y ' t h e o r y of Chap. 1, w h i l e i n Sec. 4.IB a comparison of the Overhauser SDW c a l c u l a t i o n i s made w i t h the n u m e r i c a l r e s u l t of the e x a c t Yang s o l u t i o n . 4.IA 'g-ology' An assumption b a s i c t o the t h e o r y o u t l i n e d i n Sees. 1.3 and 1.4 i s the e x i s t e n c e of a c u t o f f i n the bandwidth or the momentum t r a n s f e r which a l l o w s the l i n e a r i z a t i o n of the energy d i s p e r s i o n r e l a t i o n around £p. In s e e k i n g t o connect the 'tube' model of Chap. 2 w i t h the 'g-ology' approach, we need t o i d e n t i f y the p o t e n t i a l ( 2 . 5 8 ) , (2.59) (2.58) w i t h the c o u p l i n g constants,, g^, of Eq. ( 1 . 9 ) . S i n c e v(q) i s s p i n - i n d e p e n d e n t , q^/f = g^, and i n the absence of a p e r i o d i c p o t e n t i a l , g 3 = 0. F u r t h e r , g z and q^ both d e s c r i b e q Cc 0 140 p r o c e s s e s , so t h a t qz = g^. Because of the d i v e r g e n c e i n v(q) a t q = 0, l e t us se t the 2kp component of the p o t e n t i a l gj = 0. Then the r e m a i n i n g p a r a m e t e r s , g i and g^, a r e those of the TL model w i t h s p i n . G u t f r e u n d and S c h i c k (1968) d e t e r m i n e d the c o n d i t i o n on the p o t e n t i a l f o r the s p i n l e s s TL model t o be a v a l i d d e s c r i p t i o n of p a r t i c l e s w i t h a n o n - l i n e a r d i s p e r s i o n . I n c l u s i o n of s p i n a l t e r s the p o t e n t i a l by a f a c t o r of two, whence the requi r e m e n t i s « I (4.1) For the l i m i t i n g forms of v(q) i n ( 2 . 5 9 ) , the i n t e g r a n d behaves as q ( - l n q ) and q~3 a t the lower and upper l i m i t s , e n s u r i n g t h a t the i n t e g r a l c o n v e r g e s . Thus f o r kp or l a r g e enough, the i n e q u a l i t y (4.1) i s s a t i s f i e d and the TL model s h o u l d be a r e a s o n a b l e a p p r o x i m a t i o n . A c c o r d i n g t o Solyom (1979), the TL H a m i l t o n i a n may be broken i n t o two commuting p a r t s , Hp and H < r ( c f . Eq. (1.6) which does not i n c l u d e g^ or s p i n ) , w i t h Z and tier = iH JTvp [<r/p)ff-, C-p) + <rz(-p)<rz(p where ^ i ( p ) and 0~;(p) a r e the number- and s p i n - d e n s i t y o p e r a t o r s d e f i n e d as i n Sec. 1.3B. We observe t h a t Hg. i s d i a g o n a l and t h a t H„ i s e q u i v a l e n t t o (1.6) w i t h a r e n o r m a l i z e d v e l o c i t y v F + g„/7T / 1 141 and a p o t e n t i a l 2 g r . For the g e n e r a l i z e d s u s c e p t i b i l i t y behaves as CA)~/L-( G u t f r e u n d and Klemm 1976), where and + (-) r e f e r s t o CDW or SDW (TS or SS) response. Thus f o r a r e p u l s i v e i n t e r a c t i o n , " ^ ^ and ^ s p ^ d i v e r g e , and w i t h e q u a l exponents. As F i g . 1.9 i n d i c a t e s t h a t any amount of p o s i t i v e g ( -c o u p l i n g f a v o u r s the SDW over the CDW i n s t a b i l i t y we might i n f e r the same f o r the p o t e n t i a l v ( q ) . At b e s t , such a c o n c l u s i o n i s r a t h e r t e n t a t i v e s i n c e g^ has not been i n c l u d e d i n deducing. F i g . 1.9. In a d d i t i o n , the g^ are g e n e r a l l y taken t o be c o n s t a n t , or at l e a s t f i n i t e , around q = 0,2kp; the e x t e n s i o n of r e s u l t s t o a s i n g u l a r p o t e n t i a l such as v(q) must then be q u e s t i o n a b l e . As f o r the c o n t a c t p o t e n t i a l employed i n Chap. 3, (4.1) c l e a r l y does not h o l d when v ( q ) i s c o n s t a n t . L i e b and M a t t i s (1965) showed t h a t the momentum d i s t r i b u t i o n i n the L u t t i n g e r model i s n^ = 1/2 f o r a l l k. A l t h o u g h t h i s u n p h y s i c a l r e s u l t i s an a r t i f a c t of the i n f i n i t e n e g a t i v e Fermi sea, we expect the tendency toward a f l a t t e n i n g of n^ t o remain i n the £-function model, t h e r e b y p r e c l u d i n g a comparison w i t h the g - t h e o r y . 4.IB Overhauser S p i n D e n s i t y Wave The knowledge of the e x a c t energy of the fe r m i o n gas w i t h a £-function i n t e r a c t i o n e n a b l e s a comparison t o be made w i t h the Overhauser (1960) SDW s t a t e . Overhauser showed t h a t t h e r e e x i s t s an e x a c t Hartree-Fock' s o l u t i o n of the H a m i l t o n i a n (3.1) which i s lower i n energy than the normal paramagnetic s t a t e w i t h p l a n e 142 wave o r b i t a l s f i l l e d up t o kp, and which has a s p i r a l s p i n d e n s i t y . The main p o i n t s of the a n a l y s i s a r e as f o l l o w s . R e c a l l Eq. (3.1) f o r N p a r t i c l e s c o n f i n e d t o a l i n e of l e n g t h L, w i t h p e r i o d i c boundary c o n d i t i o n s . H = - 2-m• + 2.c E sdi-ft) i * l i * j ° (3.1) Let us suppose the system e x h i b i t s a s p i n d e n s i t y g i v e n by < S> = S L Cos t(jt) x + sen t(j%] y I and hence an exchange p o t e n t i a l Vfe) = 3 L Cos L<p)<tx + s i n ( ^ ) <r^ J where <rx and (Ty are the P a u l i m a t r i c e s f o r s p i n s q u a n t i z e d a l o n g jz. The a m p l i t u d e s S and g are t o be de t e r m i n e d s e l f - c o n s i s t e n t l y by demanding t h a t the HF e q u a t i o n s (4.2) be e q u i v a l e n t t o the S c h r o d i n g e r e q u a t i o n f o r the s i n g l e -p a r t i c l e e i g e n s t a t e s • a n d e n e r g i e s : By c h o o s i n g f o r ^ L the l i n e a r c o m b i n a t i o n of s p i n - u p and s p i n -down p l a n e waves, (4.4) 143 and s u b s t i t u t i n g i n t o Eq. ( 4 . 3 ) , one f i n d s the e x a c t s o l u t i o n f o r the c o e f f i c i e n t s and e n e r g i e s K (W^J^^F " /^Me^wTP (4.5b> The net r e s u l t i s the s p l i t t i n g of the degenerate s p i n - u p and spin-down bands w i t h a gap of magnitude ^f/^+^7'^2 a t k = 0, as shown i n F i g 4.1. F i g u r e 4.1. Energy spectrum of the H a r t r e e - F o c k s t a t e s d e f i n e d by Eqs. (4.4) and (4 . 5 b ) . 144 Having determined the s i n g l e - p a r t i c l e w a v e f u n c t i o n s , we s u b s t i t u t e the N lowes t energy s t a t e s from the £~ band, w i t h |k| ^ 2 k p , i n t o Eq. ( 4 . 2 ) . E v a l u a t i o n of the i n t e g r a l s produces an e q u a t i o n i d e n t i c a l i n form t o (4.3) and l e a d s t o a s e l f -c o n s i s t e n t p o t e n t i a l , V ( x ) , i f g = - 2 k p q / s i n h (7Tq/c) . The energy i s c a l c u l a t e d t o be Computing the s p i n d e n s i t y , one g e t s S(a) = L Tr( dk o((c J (dos(^) x + stn (^)y the e q u a t i o n f o r a r i g h t - h a n d e d s p i r a l . Exchange of the c o e f f i c i e n t s b ( ^ and b'^ i n (4.4) produces a l e f t - h a n d e d h e l i x . M i n i m i z a t i o n of Egov*/ w ^ t n r e s p e c t t o the wavevector q i s the l a s t s t e p i n the s o l u t i o n . The v a l u e of q so o b t a i n e d ranges from a maximum of 2 k p a t c = 0 t o z e r o as c i n c r e a s e s t o c' = 4TTkp/3. Other e n e r g i e s of i n t e r e s t a r e those of the normal and f e r r o m a g n e t i c HF s t a t e s , which a r e g i v e n by FM ~ -3 Comparison w i t h the m i n i m i z e d v a l u e s of Eq. (4.6) r e v e a l s t h a t E, ESDW " EH f o r a 1 1 c ' a n d ESOW ~ f o r c < c'. Thus, of the t h r e e HF s t a t e s , the SDW phase i s the s t a b l e one up t o an i n t e r a c t i o n s t r e n g t h c'. At t h i s v a l u e q = 0, and the wavelength of the SDW . becomes i n f i n i t e . F i n a l l y , we compare E ? 0 ^ as c a l c u l a t e d by Edwards and H i l l e l (1968) w i t h the ex a c t ground 145 s t a t e energy computed i n Chap. 3. In F i g . 4.2 we observe t h a t E„» c o i n c i d e s w i t h e x a c t r e s u l t o n l y i n the n o n - i n t e r a c t i n g SOW . l i m i t , i n which case the SDW a m p l i t u d e v a n i s h e s . I t may be F i g u r e 4.2. H a r t r e e - F o c k e n e r g i e s f o r the SDW ( E S ( > W , taken from Edwards and H i l l e l 1968), f e r r o m a g n e t i c (E F N,) , and paramagnetic ( E N ) s t a t e s compared t o the e x a c t energy (E) as a f u n c t i o n of c o u p l i n g s t r e n g t h . c o n c l u d e d , t h e r e f o r e , t h a t the Overhauser SDW s t a t e i s not a v a l i d d e s c r i p t i o n of the t r u e ground s t a t e . 146 4.2 M u l t i p l e - C h a i n Models The s i g n i f i c a n c e of i n t e r c h a i n c o u p l i n g s i n s u p p r e s s i n g f l u c t u a t i o n s and s t a b i l i z i n g phase changes i n a s i n g l e . c h a i n has been mentioned i n p a s s i n g i n Chap. 1, and i t i s o b v i o u s t h a t a complete t h e o r y of l i n e a r c h a i n c o n d u c t o r s must i n c l u d e 3D e f f e c t s , e s p e c i a l l y i f thermodynamic p r o p e r t i e s a r e t o be p r o p e r l y e x p l a i n e d . Two approaches t o the problem which have been adopted are d i s c u s s e d i n Sees. 4.2A and 4.2B. 4.2A T h r e e - D i m e n s i o n a l 'g-ology' As might be e x p e c t e d , the Fermi gas models reviewed i n Chap. 1 have been extended by a d d i n g i n t e r c h a i n c o u p l i n g terms t o the TL and Luther-Emery H a m i l t o n i a n s . Klemm and G u t f r e u n d (1976) supplement Eq. (1.11) w i t h f o r w a r d and backward s c a t t e r i n g i n t e r a c t i o n s between n e a r e s t - n e i g h b o u r c h a i n s of the form (m i s the c h a i n i n d e x ) c r e r ' Here *v x i s the analogue of the c o u p l i n g c o n s t a n t s g^. and g^, and ^, c o r r e s p o n d s t o gj . I f = 0, the q u a l i t a t i v e b e h a v i o u r of the response f u n c t i o n s remains unchanged a l t h o u g h the exponents i n the power law d i v e r g e n c e s a r e r e n o r m a l i z e d . However when y j i s f i n i t e , the CDW i n a m e a n - f i e l d t h e o r e t i c t r e a t m e n t becomes a 3D c o - o p e r a t i v e phenomenon and the t r a n s i t i o n temperature i s s h i f t e d t o a non-zero v a l u e . The model may be f u r t h e r 147 s o p h i s t i c a t e d by a l l o w i n g i n t e r c h a i n hopping; t h i s f e a t u r e has the e f f e c t of making the s u p e r c o n d u c t i n g t r a n s i t i o n t e mperature f i n i t e . A d i s c u s s i o n of t h i s and o t h e r Fermi gas t r e a t m e n t s of the the c o u p l e d - c h a i n problem may be found i n the review by Solyom (1979). 4.2B D i e l e c t r i c Response Theory A square a r r a y of t h i n c o n d u c t i n g s t r a n d s ( c h a i n s ) was s t u d i e d i n the RPA by W i l l i a m s and B l o c h (1974). They took t h e i r s i n g l e - p a r t i c l e b a s i s s t a t e s t o be of the same form as i n Chap. 2, namely where d ^ i s a l a t t i c e v e c t o r of the square a r r a y . The t r a n s v e r s e f u n c t i o n ® ( j ^ ' d j ^ ) , l o c a l i z e d t o the s t r a n d p o s i t i o n e d a t = d i , i s assumed t o be a G a u s s i a n of l i m i t e d e x t e n t so t h a t t h e r e i s no a p p r e c i a b l e o v e r l a p w i t h n e i g h b o u r i n g c h a i n s , w h i l e the i n t r a c h a i n f u n c t i o n ^>(x) may be e i t h e r a f r e e - p a r t i c l e or a t i g h t - b i n d i n g o r b i t a l . Thus the s t r a n d s are c o u p l e d o n l y t h rough the Coulomb i n t e r a c t i o n , and not by i n t e r s t r a n d h o p p i n g . Confinement of the e l e c t r o n s t o a s i n g l e c h a i n i m p l i e s t h a t t h e r e i s a 3D v e r s i o n of the e f f e c t i v e p o t e n t i a l (2.3) ( f o r q i n the f i r s t B r i l l o u i n z o n e ) : where Gj_ i s a 2D r e c i p r o c a l l a t t i c e v e c t o r i f the f r e e - e l e c t r o n Cf*(x) i s chosen, and ^ ( q ^ ) i s d e f i n e d as i n Eq. ( 2 . 4 ) . The d i e l e c t r i c f u n c t i o n i s g i v e n i n the RPA by 148 e<k£l($^~ li+t'ffa-atyx.Ci,*)] J (4.8> Here 6 i s the background d i e l e c t r i c c o n s t a n t and 0C„(q»w) i s the ID f r e e - p a r t i c l e s u s c e p t i b i l i t y e x p r e s s e d by (2 . 2 6 ) . For the l o n g - w a v e l e n g t h c o l l e c t i v e e x c i t a t i o n s ( t h e z e r o e s of 6, r i ( q , u i ) ) , W i l l i a m s and B l o c h f i n d f r e q u e n c i e s which a r e dependent on Jf, the a n g l e of ^ w i t h r e s p e c t t o the s t r a n d a x i s . In the q 0 l i m i t 60p(q) ranges from the c l a s s i c a l plasma f r e q u e n c y Up2 = 4 ne2/6 m, a t £ = 0 , t o z e r o a t £ = 90 , w h i l e f o r l a r g e q, OJ^(q) tends t o the 3D i s o t r o p i c v a l u e . F u r t h e r m o r e , OJpiq) remains above the ID p a r t i c l e - h o l e continuum f o r a l l q, a r e s u l t ' w h i c h i s the same as t h a t found i n Chap. 2. The W i l l i a m s and B l o c h model has been dev e l o p e d f u r t h e r by G i u l i a n i et a l . (1979) by i n c l u d i n g exchange and s h o r t - r a n g e c o r r e l a t i o n s . R e t a i n i n g the RPA f o r the i n t e r c h a i n Coulomb c o u p l i n g , the STLS a n s a t z i s made f o r the d e n s i t y c o r r e l a t i o n f u n c t i o n , where g(x) i s the c o r r e l a t i o n f u n c t i o n d e f i n e d i n Chap. 2. The consequence of t h i s m o d i f i c a t i o n i s t o add a term t o the RPA e f f e c t i v e p o t e n t i a l ( 4 . 7 ) : where [S(q) - 1] i s the F o u r i e r t r a n s f o r m of [g(x) - 1 ] , However, the STLS s e l f - c o n s i s t e n c y c r i t e r i o n i s not imposed on S(q) i n t h i s c a l c u l a t i o n as G i u l i a n i e t a l . assume r e c t a n g u l a r c o r r e l a t i o n h o l e s f o r e l e c t r o n s w i t h p a r a l l e l and w i t h 149 a n t i p a r a l l e l s p i n s : (4.10) where d i s the mean i n t e r p a r t i c l e s p a c i n g a l o n g a c h a i n , h i s the depth of the h o l e , and Q(x) i s the s t e p f u n c t i o n . The s i g n i f i c a n c e of the the new p o t e n t i a l (4.7) i s t h a t f o r s u i t a b l e c h o i c e s of the model parameters, U(q) becomes a t t r a c t i v e f o r q £ 2kp, w i t h the d r a m a t i c r e s u l t i l l u s t r a t e d i n F i g . 4.3. The plasmon branch merges w i t h the s i n g l e - p a r t i c l e e x c i t a t i o n spectrum, o n l y t o reappear i n the gap below the continuum as an e x c i t o n mode. Imaginary v a l u e s of 6t>p(q) h e r a l d a CDW i n s t a b i l i t y when U(q) becomes s u f f i c i e n t l y n e g a t i v e . In the l i m i t of i n f i n i t e i n t e r c h a i n s e p a r a t i o n , the e f f e c t i v e p o t e n t i a l U(q) reduces t o [1 - G ( q ) ] v ( q ) , the term which appeared i n Eq. (2.62a) i n our c a l c u l a t i o n . I t was shown i n Chap. 2 t h a t i t i s n e c e s s a r y t o have the l o c a l f i e l d c o r r e c t i o n G(q) > 1 i n o r d e r t o o b t a i n the k i n d of i n s t a b i l i t y d e p i c t e d i n F i g . 4.3. G i u l i a n i et a l . s a t i s f y the e q u i v a l e n t of t h i s c o n d i t i o n i n Eq. (4.9) by t a k i n g f o r the p a i r c o r r e l a t i o n f u n c t i o n the e x p r e s s i o n ( 4 . 1 0 ) . ( R e c a l l t h a t G(q) i s a f u n c t i o n a l of g(x) i n the STLSA.) We remark t h a t the r e q u i r e d b e h a v i o u r i n G(q) i s not observed f o r e i t h e r v ( q ) or w(q)oc 1/q 2, a l t h o u g h the tendency i s p r e s e n t i n the case of the l a t t e r p o t e n t i a l (see F i g . 2.13). The i m p l i c a t i o n i s t h a t the form of g(x ) has some b e a r i n g i n d e t e r m i n i n g the e x c i t o n i c i n s t a b i l i t y . 150 F i g u r e 4.3. Plasmon e x c i t a t i o n spectrum o b t a i n e d by G i u l i a n i e t a l . (1979). 4.3 Summary and Suggest i o n s f o r F u t u r e Work One avenue which has been d e v e l o p e d i n the attempt t o u n d e r s t a n d q u a s i - l D - c o n d u c t o r s i s the Fermi gas model o u t l i n e d i n Chap. 1. The main t h r u s t of t h i s t h e o r y i s t o c a l c u l a t e response f u n c t i o n s f o r v a r i o u s t y p e s of o r d e r f o r the purpose of a s c e r t a i n i n g the p o s s i b l e i n s t a b i l i t i e s p o s s e s s e d by a l i n e a r c h a i n m a t e r i a l . For a r e a l i s t i c i n t e r p a r t i c l e p o t e n t i a l , t h e r e i s a d i f f i c u l t y i n i d e n t i f y i n g the c o u p l i n g parameters and p e r t u r b a t i o n t h e o r y seems t o be u n s u i t a b l e . An a l t e r n a t e approach has t h e r e f o r e been p r e s e n t e d i n Chap. 2. Here, the d i e l e c t r i c response of an a n i s o t r o p i c model system w i t h an e f f e c t i v e ID i n t e r a c t i o n has been c a l c u l a t e d by a method p a t t e r n e d a f t e r the STLS a p p r o x i m a t i o n f o r the 3D 151 e l e c t r o n gas. I t was found t h a t the s t a t i c s u s c e p t i b i l i t y does not d i v e r g e f o r the range of model parameters f o r which the a p p r o x i m a t i o n i s deemed t o be r e a s o n a b l e . The STLS a p p r o x i m a t i o n has a l s o been a p p l i e d , i n Chap. 3, t o a ID gas of f e r m i o n s i n t e r a c t i n g v i a a c o n t a c t p o t e n t i a l . Comparison w i t h the e x a c t s o l u t i o n e n a b l e s us t o make some q u a l i t a t i v e s t a t e m e n t s about the v a l i d i t y of the a p p r o x i m a t i o n . In a d d i t i o n , c a l c u l a t i o n of the magnetic s t a t e e n e r g i e s f o r the model c o n f i r m s t h a t the L i e b - M a t t i s theorem h o l d s . The model e x p l o r e d i n Chap. 2 by the STLS method may be improved and extended i n a number of ways. One which has a l r e a d y been mentioned i n Sec. 2.7 i s the m o d i f i c a t i o n by V a s h i s h t a and S i n g w i (1972) of the l o c a l f i e l d c o r r e c t i o n i n o r d e r t o o b t a i n an a c c u r a t e c o m p r e s s i b i l i t y . T h e i r c o r r e c t i o n , would e n t a i l a c o n s i d e r a b l e amount of c o m p u t a t i o n , namely the s i m u l t a n e o u s s e l f - c o n s i s t e n t c a l c u l a t i o n of G"srLS ^ o r t* i e s e v e r a l v a l u e s of the d e n s i t y - d e p e n d e n t parameters U and A needed t o determine the d e r i v a t i v e 2G^TLi(q)/dn. A l t h o u g h computer t i m e - i n t e n s i v e , the p r o c e d u r e i s s t r a i g h t f o r w a r d . More i n t e r e s t i n g would be the a p p l i c a t i o n of the STLS method t o the s p i n s u s c e p t i b i l i t y of a s i n g l e c h a i n . A c a l c u l a t i o n of t h i s s o r t has been performed by Lobo et a l . (1969) f o r the 3D e l e c t r o n gas. B r i e f l y , the ID problem i s c o n c e i v e d as f o l l o w s . The s p i n s u s c e p t i b i l i t y i s g i v e n by 152 X ( o n ) ~ i T K'ttP) I ' ' " ' - ^ ) S ^ ) * . f y « ) (4.11a) where / X 0(q , w j ) i s the f r e e - e l e c t r o n s u s c e p t i b i l i t y ( 2 . 2 6 ) , i s the e l e c t r o n magnetic moment, and G(q) i s the s p i n v e r s i o n of G ( q ) . As f o r G ( q ) , an a n s a t z i s made f o r G ( q ) : -ob r Here [S(q') - 1] i s the F o u r i e r t r a n s f o r m of the f u n c t i o n i' l v ( 4 . i i b ) g (x) = [ g ^ (x) - g ^ ( x ) ] , g^^(x) and g ^ ( x ) be i n g the s p i n c o r r e l a t i o n f u n c t i o n s i n t r o d u c e d i n the p r e v i o u s s e c t i o n . F i n a l l y t o c l o s e the s e l f -c o n s i s t e n t p r o c e d u r e , S(q) must s a t i s f y the sum r u l e f oo (4.11c) The e q u a t i o n s (4.11) a re c l e a r l y s o l v e d i n e x a c t l y the same manner as the d i e l e c t r i c r e l a t i o n s (2.48) and a s i m i l a r a n a l y s i s of the r e s u l t s may be made. T h i s approach has, i n f a c t , been employed by G i u l i a n i e t a l . (1979) i n the study of the c o u p l e d - c h a i n problem d i s c u s s e d above. T h e i r model n e c e s s i t a t e s the replacement of (4.11a) by V I -ItyX.CpU)) (4.12a) and of (4.7b) by % k T ^ i * ( 4 . 1 2 b ) S u b s t i t u t i n g the e x p r e s s i o n (4.10) f o r g ^ and they f i n d an SDW i n s t a b i l i t y f o r s m a l l v a l u e s of h, as opposed t o the CDW i n s t a b i l i t y f a v o u r e d f o r h £ 1. S i n c e t h e s e r e s u l t s depend on 153 the form of S(q) a p p e a r i n g i n Eqs. 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