CENTRAL RESEARCH INSTITUTE FOR PHYSICSBUDAPEST

K F K I - 1 9 8 0 - 6 2

F. W O Y N A R O V I C H

E X C I T A T I O N S W I T H C O M P L E X W A V E N U M B E R S IN A H U B B A R D C H A I N

II. S T A T E S W I T H S E V E R A L P A I R S O F C O M P L E X W A V E N U M B E R S

^ H u n g a r i a n ° Ä c a d c m y o f S c i e n c e s

CENTRAL RESEARCH

INSTITUTE FOR PHYSICS

BUDAPEST

EXCITATIONS WITH COMPLEX WAVENUMBERS IN A HUBBARD CHAIN II. STATES WITH SEVERAL PAIRS OF COMPLEX WAVENUMBERS

F. Woynarovich

Central Research Institute for Physics H-1525 Budapest 114, P.O.B.49, Hungary

HU ISSN 0368 5330 ISBN 963 371 697 7

ISNB 963 371 698 5 /összkiadás/

wavenumbers are studied. The original set of Lieb-Wu equations is replaced by an equivalent set in which only real wavenumbers appear, the total number of which is equal to the sum of the number of complex wavenumbers and the number of electrons needed to make the band half-filled. In a sense discussed in the text, the new set of equations refers to excitations only. The energy- momentum dispersion is also found. Based on the energy spectrum and the

U+oo limiting form of the wavefunction, the excitations can be identified as interacting quasi-particles.

А Н Н О Т А Ц И Я

Исследуются синглетные состояния одномерных Хаббард-цепей, описываемые многими парами комплексных волновых векторов. Оригинальные уравнения ЛИБ-ВУ замещаются эквивалентной им системой уравнений, в которых появляются уже только действительные волновые векторы, число которых равно сумме чисел ком­

плексных волновых векторов и электронов, нужных для полузаполнения зоны. В данном смысле новая система уравнений относится уже только к возбуждениям.

Определяется энергия возбуждений. По форме энергетического спектра, так же как и по форме волновой функции, действительной в пределе U <*>, возбуждения можно считать взаимодействующими квазичастицами.

KI VONAT

1-d Hubbard láncok több komplex hullámszámpárral leírható singlet álla­

potait vizsgáljuk. Az eredeti Lieb-Wu egyenleteket helyettesitjük egy ekvi­

valens egyenletrendszerrel; ebben már csak valós hullámszámok szerepelnek, ezek száma megegyezik a komplex hullámszámok és a sáv félig töltöttségéhez szükséges elektronok számának összegével. A szövegben tárgyalt értelemben az uj egyenletrendszer már csak a gerjesztésekre vonatkozik. Meghatározzuk a gerjesztések energiáját. Az energia spektrum formája, valamint a hullámfüggvé­

nyek az U-*-“> határesetben felvett alakja alapján a gerjesztések kölcsönható kvázirészecskéknek tekinthetők.

In P a p e r I. ( p r e v i o u s p a per, F. W o y n a r o v i c h 19 8 0 ) t h o s e e i g e n s t a t e s of t h e 1-d H u b b a r d H a m i l t o n i a n

h a v e b e e n s t u d i e d w h i c h c o r r e s p o n d t o s t a t e s in w h i c h t h e a m p l i t u d e o f f i n d i n g e l e c t r o n p a i r s o c c u p y i n g t h e same s i t e d o e s n o t v a n i s h e v e n if U is large. It has b e e n e s t a b l i s h e d , th a t t h e s e s t a t e s a r e to be d e s c r i b e d b y s u c h s o l u t i o n s o f the L i e b - W u (1968) e q u a t i o n s

in w h i c h some o f t h e w a v e n u m b e r s к a r e c o m p l e x . S o l u t i o n s w i t h o n e p a i r of c o m p l e x w a v e n u m b e r s w e r e d i s c u s s e d in P a p e r I.

for , a n d for s i n g l e t s t a t e s , w i t h A/^/Velectrons.

In the l a t t e r c a s e w e h a v e f o u n d t h a t t o the w a v e n u m b e r p a i r

(

1

1

(1 .2 )

K t l x t h e r e is o n e л c o u p l e d b y t h e e q u a t i o n

J

-

(This e q u a t i o n is c o r r e c t u p to t e r m s e x p o n e n t i a l l y s m a l l in /V ) . Л is d e t e r m i n e d b y an e q u a t i o n of t h e type

2 art tg i. ( л - Sin ke) + l a r c t y k . ( A - s i n k m) * U j ' (1.4)

w h e r e k t a n d are t h e w a v e n u m b e r s d e f i n e d b y Eq. (1.2) to c o r r e s p o n d to t h e I -s l e f t o u t o f the g r o u n d s t ate 1 set.

T h e y h a v e to s a t i s f y the e q u a t i o n s

*

/

U ' h v i ' H »

(1.5)

° ° ~(0

W i t h kt j kM a n d Л , a l l the o t h e r u n k n o w n s c o u l d b e e x p r e s s e d ; in p a r t i c u l a r , t h e d e n s i t i e s o f t h e rea l к -s, and " n o r n a l "

Л -s, c o u l d be g i v e n as

p ( k j IT (o s (u jth k )d u j + ^{2. COS к}

2* N

U/tf

( % ) ^ r ( s in k - A f "

(1 .6 )

__

2TN ( cos (o ( sin k - sinkt )J -h cos(u(fin к ^- sinkm))J d u

and

№ •

UH

cos új A du

í H U \ {ch[\-iinkt)Íft cA(Ъ -И п к „ )Ц

& )

w i t h t h e f i r s t t e r m s in c u r l y b r a c k e t s b e i n g t h e g r o u n d s t a t e d e n s i t i e s pg(k) a n d S0(\J

T h e e n e r g y o f these s t a t e s , e v a l u a t e d b y t h e f o r m u l a

£ * E > ( -2.c o s k ) - b c o s K c h x real к

is

(1.8)

£ ~ E0 4- 6( k e) 4 €{k^) t U

w i t h E0 b e i n g t h e g r o u n d s t a t e e n e r g y and

oo

€(k) - 2 COS к + 2 Г — --- cos(u)3ink) Clio

о chuU OJ

4

w h i l e t h e m o m e n t u m e v a l u a t e d b y

(1.9)

(

1

10

(1.1 1)

is

P * Po - p ( k i ) - р ( км )

w i t h

p ( k)

* 1о ( ^ ) sen (из s in k ) из • eh «о М-

Ч

d u

(1.12)

(1.13)

In t h i s p a p e r we i n t e n d to g e n e r a l i s e o u r r e s u l t s in two d i r e c t i o n s : w e look f o r s t a t e s w i t h s e v e r a l (L) p a i r s of c o m p l e x w a v e n u m b e r s a n d at the s a m e time w e do not f i x the b a n d f i l l i n g w h i c h can be l e s s t h a n h a l f . We d e n o t e the n u m b e r of e l e c t r o n s n e e d e d to m a k e the b a n d h a l f f i l l e d b y W ; H * N - N 6 .

To s e p a r a t e c h a r g e and s p i n r e d i s t r i b u t i o n e f f e c t s , w e w i l l loo k for s t a t e s in w h i c h the s p i n p a r t is in its g r o u n d state. T o m a k e sure, t h a t t h e s t ate c a n b e s i n g l e t , we t a k e Ne even.

T h e s t a t e s w i t h s e v e r a l p a i r s o f c o m p l e x к - s a r e e x p e c t e d to h a v e m a n y p a r a m e t e r s , t h u s w e w i l l not b e a b l e to s o l v e the L i e b - W u e q u a t i o n s c o m p l e t e l y . W h a t we w a n t t o s h o w is t h a t e v e n

if t h e n u m b e r o f c o m p l e x к -s is large, t h e y c a n be s e p a r a t e d f r o m the r e a l к -s a n d " n o r m a l " Л -s a n d a s y s t e m of e q u a t i o n s a n a l o g o u s to (1.4) (1.5) c a n be d e r i v e d , w h i c h c o n t a i n s o n l y the p a r a m e t e r s o f t h e e x c i t a t i o n s .

W e n o t e t h a t a l l o w i n g for Ne *- N , t h e t r e a t m e n t b e c o m e s g e n e r a l e n o u g h t o i n v o l v e b o t h k i n d s of c h a r g e - e x c i t a t i o n s

( Paper I. P o i n t 2.3).

In C h a p t e r 2. w e w i l l d e r i v e the s y s t e m of e q u a t i o n s d e t e r ­ m i n i n g the p a r a m e t e r s o f the e x c i t a t i o n s . In C h a p t e r 3. the

s y m m e t r y of the e q u a t i o n s f o u n d is e x a m i n e d w h i l e C h a p t e r 4. is d e v o t e d to the d i s c u s s i o n of t h e n a t u r e o f the s t a t e s in q u e s t i o n .

2. E Q U A T I O N S F O R T H E S T A T E S W I T H S E V E R A L C O M P L E X W A V E N U M B E R S

2.1 E l i m i n a t i o n of the c o m p l e x w a v e n u m b e r s f r o m the L l e b - W u e q u a t i o n s

W e s u p p o s e that, s i m i l a r l y to the c a s e of o n e c o m p l e x к - - p a i r , to e a c h c o m p l e x к - p a i r a A is c o u p l e d b y the

e q u a t i o n s :

sin (k„ + ix n) m Л „ - < - £ v-

0

Stn(Kh- i x „ ) - Л л + i ! L -t 0( t ~ ^ N)

(2.1)

T h e s e e q u a t i o n s are t h e g e n e r a l i s a t i o n s o f (1.3) a l l o w i n g f o r t h e p o s s i b i l i t y of Л b e i n g c o m p l e x . T h e К -s a n d x - s s a t i s f y i n g (2.1) (up t o e x p o n e n t i a l l y s m a l l c o r r e c t i o n s ) a r e

{ { / ( f - » ■ A f ' i b ' l s t f * / ( { [ ■ (2.2a)

0 , Sign ((OSKn) - - sign ( j j - - J * , \ )

H ^ a n s i n j . ( / Щ + ( РеЛщ + +

a n c h j - j j f ä, ЗтЛщ)х+ ( Ш * V (ГЫЛь- l f j

K „ > 0 , s ig n ( cot£ „ ) - - Sign ( j L + j „ / \ HJ

(2.2b)

N o t e t h a t a l s o in th i s ca s e t h e s e t of c o m p l e x k - s c o n s i s t s o f c o m p l e x c o n j u g a t e pairs p r o v i d e d the Л s e t c o n s i s t s of r e a l

Л -s a n d c o m p l e x c o n j u g a t e p a i r s . In the f o l l o w i n g w e w i l l s u p p o s e t h i s , b u t l a t e r w e w i l l see t h a t the e q u a t i o n s d e t e r ­ m i n i n g the Л -s i n d e e d d e f i n e s u c h A -sets.

N o w t h e e q u a t i o n s for t h e c o m p l e x к -s are

н ( к я+ ix„)

-

-

-/L ) -

f i ’ f u '

L

- E l a n t q i - ( s c n ( K ht i x h) - A M)

(-2.3.а)

ze~L

N ( K „ - i K h) - Z T l h - £ la rc tc jiS C ti(K h- i x h) - Ä J ~

/3*1 Г (2.3.b)

L

— E lA r c iC j( s i n f a - i x J - A * , ) P7)»1

It is n o t h a r d to v e r i f y t h a t (2.1) a n d so (2.2.a-b) are the s o l u t i o n s o f the i m a g i n a r y p a r t s o f (2.3.a-b) if t h e c o n d i t i o n s

+ ц H E I a n u / ( sin(K^tCx*)

У-

' ft гч+н u fj

• V Í

(2.4)

a r e fu l f i l l e d . T h e s e i n e q u a l i t i e s are t o b e c h e c k e d at t h e e n d w h e n the X -s a n d Л -s are k n o w n .

The e q u a t i o n s d e f i n i n g t h e r e a l к -s are

N

A /k , - 2Ж1 - - f E Z t r o + a ^ -(s itb k f—Я а ) * £ Z arc ~ A* ) J

' 1 Л-1 ы mxf kándisc out.

%-L

- j Z la rtit) Jt-(ft'*, k j - h p ) + Z Z&rc,frj ^ (s ű b k j - Л * ) ] coni'.

(2.5)

w h e r e we u n d e r s t a n d

_ Л л L / ■ 1 я \ ) , SCn,k-ReA , stn,k~ReA

Re jla r c t q - &rc+() ^— --- * arctcf

( J con,*. U. +

% - 7suA

u J l M b , i ( s b k - * ) ]- X ы

1 / , m < 2 ‘ + ( s , « k- * * 4

Re jlA rtiX jb ; (Stlbk-A)J^^

Xscyn, (sin,k-RcA^ lf-

distort. и

4 /&,Л/<Т-

]м j l a r c H ^ ( s c n k - A ) J = 0

' u Jdiscont.

(2.6. a)

(2.6 . b)

T h e I set in (2.5) c o n s i s t s o f i n t e g e r s if A/e/i is e v e n a n d h a l f - o d d - i n t e g e r s if /Vf/z. is odd. N o t e t h a t the I f s e t d e f i n e d as

I - = 1' - £^./Ь l a r c t f £ - ( s t k í ; -% f l) - E 2arc.+tj -jLfciksk, ) 7

( ß ** J

(2.7)

c o n s i s t s of i n t e g e r s (or h a l f o d d - i n t e g e r s ) if t h e / - s are i n t e g e r s (or h a l f o d d - i n t e g e r s ) . Thus, d e p e n d i n g on the

p a r i t y 0 f (al - H)/2 w e have t o c h o o s e N - 2 L - H d i f f e r e n t I ' -s fro m o n e o f the s e t s

- { ( N - i ) , - ^ A J -3 ) , 4 ^ - 3 ) , £ ( » - * ) (2.8.a)

- f (/9-г; , ... { ( * - * ) / { и (2.8.b)

E q u a t i o n (2.5) d e f i n e s к - s a l s o t o t h e l ' -s lef t o u t fro m (2.8.a) o r (2.8.b). W e w i l l d e n o t e th i s k -s b y t h e i n d e x к (for " h o l e " ) . T h e d e n s i t y o f the k - s, s a t i s f y i n g (2.5) is

( % ) ( u/k )

p (k ) = jL + JL— -Leos к E ,, , - , + -|r- ZcoskEl

у Z T tTTN ( % ) + (§ с ц к -Л р )г l T N M ( % ) L s-Csikk-Au,)1-

(2.9)

As this p ( k ) c o n t a i n s a l s o the c o n t r i b u t i o n o f t h e v a r i a b l e s kK j t o r e p l a c e - s u m s , w e h a v e to use

H+IL

/ W

f <k> -

T h e e q u a t i o n s for the "nor m a l "

Я

*.

£ Z a rt+ zj (Я ^-SOt k j J + Z Arc, t ( j +1** ,)J +

//4 m-1

+ z l a r c i r C j L ^ - s ^ ^ ) ] - 2- T f c + E l a r c l g l ( ^ - ^ ) +

hief fvef (2.11)

+

E

N o w u s i n g (2.1) and t h e i d e n t i t y

L a r c t < ] £ ( * K - A * , - < ? ) -h l a r c t g E f a - A ^ + i f i ) =

(2.12)

• l a r c t q j j ( ÁK ~ Л М) + Я s ig n ( f c ( \ - л * ) )

O n e h a s (up to e x p o n e n t i a l l y s m a l l terms)

Í N

E 2. a r c t y J L ( * t - s ( n k r ) - Z 7 ^ + Z lo rc it) (2.13).

/Уд. u f i ‘ i

w i t h

j J = 4 - E i s i g n ( t e ( К - л * ) ) ft?

•}. is i n t e g e r if ( / J - H ~ 2 L ) / 2. is o d d a n d h a l f o d d - i n t e g e r if (AJ-H-ZL )/2. is e v e n * N o t e tha t (2.13) is f o r m a l y the same as t h e c o r r e s p o n d i n g e q u a t i o n for a s y s t e m w i t h N - Z L —H

e l e c t r o n s , all d e s c r i b e d b y r e a l k - s .

T h e A -s c o u p l e d to the c o m p l e x к - s are d e t e r m i n e d b y the e q u a t i o n s :

L

Г L an+q ± ( / \ h - s in k }) + £ Z « r c i c j ^ ( A h - s ík(km+í x* , ) ) + ft*1

L ie'L

£ z^ f9 £ ( Л „ - $ С ^ - Я п , ) ) - 2 X £ + 1 2-arc-tr) ь (/1*-vL) +

Mui (Ь*1

L

+ E

2

A*J

If in the f i r s t t e r m of the l.h.s. w e r e p l a c e the c o n t i n u o u s p a r t of the s u m o v e r the rea l к - s b y t h e c o r r e s p o n d i n g

i n t e gral, t h e n w e have, up to t e r m s p r o p o r t i o n a l t o (c a l c u ­ l a t i n g a l s o the s u m o f the d i s c o n t i n u o u s p a r t s b y m e a n s of p ( £ ) w o u l d i n t r o d u c e an e r r o r o f t h e o r d e r o f % / )

HrlL

s £ A ^s

* “**(% -***•)an un- z f№ - * > * &

1 ^J ( ² . ¹⁶ )

* « К ? * * A > « W * Í „ + # /

-

V e u s e d h e r e the i d e n t i t i e s ( 2 . 6 . a) and T

dk - ore«« -Jf + (A-t)v j ( 2 . 17)

a r e eh. i_

z ( Л +1) * - + ( 2 . 1 8 . b)

T h u s sununing up Eqs. (2.3.a), (2.3.b) a n d (2.15), a n d u s i n g a l s o (2.1) a n d (2.12), one o b t a i n s

2 U H L

Z l a n t y £ - ( / „ - smtcA) - I T + Z U r c t y ^ f o i - A u , ) (2.19)

A*/ 14,1

w i t h

- AJ ssjn>(%e,AHJ ~ % +■ Z

+

1 7

—

2 2

aU к 2. ' 2_

+ 4~ Z (X e ( A ) ) fi

5^ b e i n g i n t e g e r if N - L - H is o d d a n d h a l f o d d - i n t e g e r if N - L - H is even.

T h e a c t u a l s y s t e m to be s o l v e d is t h e s y s t e m o f (2.5), (2.13) a n d (2.19). K n o w i n g all the k i-s and к , -s,

Я -s,

/*-s , the c o m p l e x w a v e n u m b e r s c a n be c a l c u l a t e d , the o r i g i n a l I , a n d } q u a n t u m n u m b e r s can be d e t e r m i n e d and t h r o u g h

(2.3.a-b) also t h e e x p o n e n t i a l y s m a l l c o r r e c t i o n s to the к-s and of (2.2.a-b) can b e o b t a i n e d .

It is i n t e r e s t i n g to n o t e , t h a t t h e s y s t e m (2.5) , (2.13) ,

(2.19) is e n t i r e l y s y m m e t r i c (as far as the s t r u c t u r e is concerned) in the v a r i a b l e s k j , Д ^ a n d к^ , A „ . The o n l y a s y m m e t r y is that the n u m b e r o f -s is t h e h a l f o f the n u m b e r of k j -s w h i l e the n u m b e r o f /1*. -s can be les s than h a l f the n u m b e r of -s (if H is n o t zero) . B u t t h i s is o n l y due to t h e fact tha t w e are i n v e s t i g a t i n g S * * О sta t e s . It is n o t h a r d to see, th a t if w e w e r e c a l c u l a t i n g s t a t e s w i t h 5%t 0 , t h e n e v e n thi s a s y m m e t r y w o u l d d i s a p p e a r . (In the g e n e r a l case, w i t h N e e l e c t r o n s , Л^-s and M fc L ) Л^- s ( ■£ A/e — (M, + M b) ) the p r e s c r i p t i o n f o r the Z ' , a n d Д/ p a r a m e t e r s w o u l d be tha t j ' -s are i n t e g e r n u m b e r s if is even, ^ -s are i n t e g e r s if A/e - M, is odd, a n d -s are i n t e g e r s if

Ne - Л7г is odd) .

2.2 E l i m i n a t i o n of the n o r m a l ____X - s .

E q u a t i o n ( 2.13) m a y h a v e m a n y s o l u t i o n s d e p e n d i n g on t h e c h o i c e of the p a r a m e t e r s ^ . As w e s t a t e d in o u r p r o g r a m , w e w o u l d

like to d e s c r i b e s u c h s t a t e s in w h i c h t h e spi n d e g r e e s o f f r e e ­ d o m are n o t e x c i t e d . To do this, b a s e d on the a n a l o g y of (2.13) a n d the c o r r e s p o n d i n g e q u a t i o n o f a s y s t e m w i t h N - H - L L

fy I

e l e c t r o n s , we h a v e to c h o o s e t h a t - s e t w h i c h is c h a r a c t e r i s -

t

tic o f the g r o u n d s t a t e of a s y s t e m o f N - H - L L e l e c t r o n s , i.e.

t h e set

/ / N — H - Z L / f N - H - L L ,1

- г (— - - - V j ~z(— — V / •••

W i t h this c h o i c e of the -s, t h e d e n s i t y o f the

A

M )

i % ) x + ( Ь - я * к) г.

oo

С*л-)

p ( k ) dk , 2 > T W ^ 2 / --- - M '

Г J ( U/ J L + ( A - * T

(2.22)

T h e s o l u t i o n o f (2.22) is e a s i l y o b t a i n e d b y F o u r i e r t r a n s f o r m a ­ t i o n :

oo

д а ; - £ f л» - _ L Г * 4

w f

Л/«

(2.23)

T h i s О Д a l l o w s us to e l i m i n a t e t h e Д -s f r o m y j ^ o f (2.9) w i t h the r e s u l t

-43Ü

g ( k ) = 1 + cos k J -e ~ (os(tosink) d u )

0 ^ ‘i '

o? -loZ M L

- J.— COS к / J?-- — Г 1 COs(m ( s in к - S í n к А) о!О U-N о с К и Я .

+ .j! — . Zcosk )

(Ч)

IX-H m-1

у-

(2 .24)

B y m e a n s o f f ( A ) f r o m (2.5) e q u a t i o n s for the v a r i a b l e s c a n be o b t a i n e d

N

о « w i и '

U Z .

(2.25)

Г

m l ch.uSL <u

£/ л

A s by the A4-s and /l„-s all the o t h e r u n k n o w n s a r e d e t e r m i n e d the p r o b l e m is r e d u c e d to the s o l v i n g o f Eqs. (2.19) and (2.25)

2.3 E n e r g y and m o m e n t u m

The e n e r g y is c a l c u l a t e d b y the f o r m u l a T

E = - Л/ ’ l l cos к p ~ ( k ) d k - £ l ( c o s ( k h i-iX hl) + C O S f a - c x * ,) ) (2.26)

J trt‘1

-1Г

w h i c h g i v e s

H U L

£ = E 0 -t- Z t ( kk ) (2.27)

Ar1

w h e r e 6(k ) is g i v e n b y (1.10). T h e m o m e n t u m e v a l u a t e d by m e a n s o f the f o r m u l a s (1.11) , (2.7) , (2.8) , (2.14) , (2.20) ,

(2.21) a n d (2.25), t u r n s o u t t o be (up to n l T ) H+Zl-

P - E - p ( b ) + f ( 2 -28)

A -/

w h e r e p (к ) is g i v e n by (1.13) , a n d is ze r o if is odd a n d 7Г if ( N - H ) / 2. is e v e n (in the g e n e r a l case

w h e n N - H can be odd, 1 +■ (N ~ H )/l) mod I X ) . T h e a p p e a r e n c e of this is n o t c o n n e c t e d w i t h the p r e s e n c e o f c o m p l e x w a v e - numb e r s . I t r a t h e r r e s e m b l e s t h e fact, tha t e v e n the g r o u n d state m o m e n t u m of a H e i s e n b e r g c h a i n c a n be 0t T /z . or T

d e p e n d i n g on the p a r i t y of t h e n u m b e r o f sites, a n d tha t eve n the g r o u n d s t a t e m o m e n t u m o f a h a l f f i l l e d H u b b a r d c h a i n can be

0 or T d e p e n d i n g on the p a r i t y o f N /u .

2 . 4 A s p e c i a l s o l u t i o n f o r t h e - s

The e q u a t i o n s (2.19) a n d (2.25) a r e h i g h l y n o n l i n e a r b u t t h e r e is o n e case w h e n they c a n be r e p l a c e d by a l i n e a r i n t e g r a l e q u a t i o n . T h i s is t h e case, w h e n H * 0 , L is m a c r o s c o p i c

( c o m p a r a b l e to N ) a n d we c h o o s e for the %' s e t the n u m b e r s

- Г Л - О , - { ( L - 3 ) , ■ ■ ■ { ( l - f )

W i t h t h i s c h o i c e o f Jh' -s a l l Ah w i l l b e real, a n d the n u m b e r o f A -s b e t w e e n A a n d A +dA c a n be g i v e n as ( i L ) • f ( A ) c(A , w h e r e

ZL

F (A) "(ZL) ' Z IhCA-Sihk^ (2

C o m b i n i n g t h i s w i t h (2.23) o n e finds t h a t the d e n s i t y of all Л -s a n d A -s is t h e same a s the d e n s i t y o f A -s in the g r o u n d s t a t e

OC

6( b ) + т у - f ( b ) -

I ^

4

u s i n g t h e £ ( л) t o e v a l u a t e t h e sum o v e r the /)л -s in (2.25) , we find th a t

!>«)■$ К ( 2 ' 32)

i.e. in th i s s p e c i a l case t h e q u a s i p a r t i c l e s are n o t i n teracting.

W e s h o u l d e m p h a s i z e , that t h i s h o l d s o n l y if L is large, and the e r r o r i n t r o d u c e d by u s i n g f ( A ) in the s u m m a t i o n s (for w h i c h 4

Vl is an upper limit) is s m a l l e n o u g h , and if w e c h o o s e (2.29) to c h a r a c t e r i z e t h e system.

2.5 On the c o n d i t i o n s (2.4)

A n y s o l u t i o n o f the s y s t e m (2.5), (2.13) and (2.19) is m e a n i n g f u l only if s u b s t i t u t i n g the A -s into (2.4), the

i n e q u a l i t i e s are s a t i s f i e d . W e w e r e n o t able to s h o w in g e n e r a l that t h e y hold, o n l y in two c ases: o n e in w h i c h t h e n u m b e r of e x c i t a t i o n s is s m a l l c o m p a r e d t o N (i.e. the n u m b e r of Л -s is not m a c r o s c o p i c ) and at t h e same t i m e the s p i n p a r t is n e a r to its g r o u n d state. T h e o t h e r case is w h e n a l t h o u g h the n u m b e r of Л -s is m a c r o s c o p i c , t h e s y s t e m is n e a r to t h e s t ate d e s ­ c r i b e d in Point 2.4. In b o t h c a s e s t h e sums in (2.4) c a n be

e s t i m a t e d b y i n t e g r a t i n g o v e r the Л -s u s i n g t h e i r g r o u n d state

* dens i t y . T h i s e s t i m a t i o n s h o w s t h a t b o t h a n d are

d e f i n i t e l y p o s i t i v e . It is a l s o true, t h a t for a s m a l l n u m b e r of c o m p l e x к -pairs, t o eac h p a i r t h e r e m u s t e x i s t а Л s a t i s f y i n g

( 2 . 2 . a - b ) . (If t h e r e w h e r e c o m p l e x к p a i r w i t h o u t Л , then to that p a i r and *jm s h o u l d be zero.)

2.6 T h e n u m b e r of s o l u t i o n s

Eq. (2.19) in the large U limit g o e s ov e r i n t o the s e c u l a r e q u a t i o n of a H e i s e n b e r g c h a i n w i t h l e n g t h L L + H a n d w i t h L + H s p i n s p o i n t i n g in one d i r e c t i o n and L spins p o i n t i n g in the o ther. If B e t h e ' s h y p o t h e s i s h olds, (i.e. all e i g e n s t a t e s of a H e i s e n b e r g c h a i n can be d e s c r i b e d b y the B e t h e

( Z L + H)

eq u a t i o n s ) the n t h e s e e q u a t i o n s m u s t h a v e ( и / sol u t i o n s .

A t the same ti m e one h a s ( Z L + H ) p o s s i b i l i t i e s to c h o o s e t h e 1^ * set. S u p p o s i n g c o n t i n u o u s b e h a v i o u r in t h e large U limit, o n e

c a n c o n c l u d e t h a t the e q u a t i o n (2.25) t o g e t h e r w i t h (2.19) m u s t h a v e N ! / { ( N - H - I L ) ! ( L + Н ) ! L ! J s o l u t i o n s for all U 0 and t h i s is e x a c t l y the n u m b e r of s t a t e s in w h i c h t h e r e are H *■ L e m p t y a n d L d o u b l y o c c u p i e d sites, w i t h t h e s p ins b e l o n g i n g t o t h e N - H - Z L s i n g l y o c c u p i e d s i tes b e i n g in g r o u n d state.

T h u s (2.25) a n d (2.19) d e s c r i b e s all of t h e s e s t a t e s (for s m a l l e n o u g h L to be su r e t h a t (2.4) is v a l i d ) . It is i n t e r e s t i n g to note, th a t w i t h t h e same r e a s o n i n g , c o u n t i n g i n t o a c c o u n t t h e n u m b e r of d i f f e r e n t s o l u t i o n s of Eq. (2.13) we h a v e th a t E q s .

(2.5), (2.13) and (2.19) s h o u l d h a v e

N-H

p ( I L + H \ ( N - L L - H

“ ' L * \(n - z l ~ H ) / L (2.33)

I d i f f e r e n t s o l u t i o n s , w h i c h is e x a c t l y t h e n u m b e r of s t a t e s

o f N - H e l e c t r o n s in a c h a i n of l e n g t h N . U n f o r t u n a t e l y to c o n c l u d e t h a t the s y s t e m (2.5), (2.13) a n d (2.19) d e s c r i b e s a l l s o l u t i o n s of the p r o b l e m one s h o u l d h a v e t o s h o w t h a t (2.4) h o l d s for all solutions.

3. S Y M M E T R Y OF T H E S Y S T E M (2.5) (2.13) (2.19)

As we a l r e a d y m e n t i o n e d , t h e s y s t e m (2.5), (2.13) and (2.19) is e n t i r e l y s y m m e t r i c in the v a r i a b l e s k j , and kK , л н

N o w we s h o w t h a t t h i s s y m m e t r y is p r e s e n t in s o m e f o r m a l s o in t h e m o m e n t u m a n d t h e e n e r g y o f t h e system.

Let us d e f i n e t h e c o m p l e x w a v e n u m b e r s ****** and £*_***

for the A 4 v a r i a b l e s a n a l o g o u s l y to (2.1) a n d c a l c u l a t e

4r< n

(3.1)

u s i n g (2.5), (2.9) a n d that

T

(3.2)

- A

- (k<(m) * 1 **(*)) ~ (**(•") ~ iX*(*)) ^{■h n} scq*, f t e * « ( » ) ) (

$ t ( x ) b e i n g the s t e p f u n c t i o n , we f i n d th a t t h e v a l u e o f (3.1) is s i m p l y (up to П - 2 Л )

*

(3.3)

d e p e n d i n g o n w h e t h e r Л/-/7 ( M b e i n g t h e n u m b e r o f d o w n spins) is e v e n ( Я ) o r o d d (0) . T h u s i n t e r c h a n g i n g the r o l e s of the

v a r i a b l e s kj , A* a n d Л*, c h a n g e s t h e sign of the m o m e n t u m of the s y s t e m (up to a t e r m X )

If we c a l c u l a t e the sum

z

*« к * (3.4)

- E 2 (cos (K4 +i% K) v- c o t f a - i x « ) )

w e fin d t h a t thi s is e q u a l t o M -U . T h u s in thi s s e n s e the s t a t e s in w h i c h the r o l e s o f t h e v a r i a b l e s , Д* and к/, / /!„ are i n t e r ­ c h a n g e d are " c o m p l e m e n t e r s " . T h i s c o m p l e m e n t a r i t y c a n be u s e d to c a l c u l a t e the e n e r g y and w a v e f u n c t i o n o f h i g h l y e x c i t e d s t a t e s o r low e n e r g y s t a t e s of a H u b b a r d c h a i n w i t h n e g a t i v e U

(From this, for e x a m p l e , one k n o w s t h a t t h e h i g h e s t e n e r g y s t ate of N e l e c t r o n s is t h e one in w h i c h a l l к -s are c o m p l e x a n d t h e d i s t r i b u t i o n o f A„ -s is the same as the d i s t r i b u t i o n o f A* -s in the g r o u n d s t a t e , b u t t h i s is a l s o the g r o u n d s t a t e of the c h a i n w i t h - U. .)

T h i s c o m p l e m e n t a r i t y m a y be c o n n e c t e d w i t h t h e p r o p e r t y of the H u b b a r d H a m i l t o n i a n , tha t if we i n t r o d u c e h o l e s i n s t e a d of the u p - s p i n e l e c t r o n s , t h e n

H ы «/

U - Z nté - H 1-1

(3.5)

w h e r e H has t h e s a m e s t r u c t u r e as H . T a k i n g i n t o a c c o u n t

the p a r a l l e l i s m t h a t in t h e c o m p l e m e n t a r y s t a t e s the p a r a m e t e r s A< d e s c r i b i n g t h e spin p a r t c h a n g e r o l e w i t h t h e A* p a r a ­

m e t e r s c o n n e c t e d w i t h the c h a r g e d i s t r i b u t i o n , a n d t h a t the

Л A I

t r a n s f o r m a t i o n w h i c h t r a n s f o r m s H i n t o И i n t r o d u c e s d o u b l y o c c u p i e d o r e m p t y s i t e s i n s t e a d of the s i n g l y o c c u p i e d on e s ( u n c o m p e n s a t e d s p i n s ) , and v i c e v ersa, the a b o v e - s u s p e c t e d c o n n e c t i o n s e e m s v e r y p o s s i b l e .

4. ON THE N A T U R E OF S T A T E S W I T H C O M P L E X W A V E N U M B E R S

The goal of this c h a p t e r is to c o n n e c t o u r r e s u l t s o b t a i n e d t h r o u g h the l e n g t h y a l g e b r a i c w o r k of the p r e v i o u s c h a p t e r s w i t h some less r i g o r o u s b u t m o r e or less t r a n s p o r e n t p i c t u r e s . A full a n a l y s i s of the s t r u c t u r e of the e i g e n s t a t e s w o u l d r e q u i r e the k n o w l e d g e of the d i f f e r e n t c o r r e l a t i o n f u n c t i o n s , b u t this is b e y o n d o u r grasp. Instead, w e h a v e to be s a t i s f i e d w i t h some i n d i r e c t r e a s o n i n g or w i t h the e x a m i n a t i o n of l i m i t i n g cases.

We w i l l b a s e o u r a r g u m e n t s on the e n e r g y - m o m e n t u m r e l a t i o n s h i p , a n d on the a n a l y s i s of the U ~*oc b e h a v i o u r o f the w a v e f u n c t i o n .

4.1 Q u a s i - p a r t i c l e p i c t u r e for the e n e r g y - m o m e n t u m r e l a t i o n ­ ship

To h a v e a c l o s e r look at the e n e r g y a n d the m o m e n t u m , let us c o n s i d e r first a s y s t e m in w h i c h t h e r e a r e N e =Л/-/У e l e c t r o n s in a s t ate d e s c r i b e d by real w a v e n u m b e r s . A c c o r d i n g t o (2.27) and (2.28), the e n e r g y m e a s u r e d f r o m the g r o u n d s t a t e e n e r g y of a h a l f f i l l e d band, a n d the m o m e n t u m are

£ “ E к *1

(4.1) H

P “ £ , - р ( кк ) * T O i ' Ne /z .)

If w e take a s y s t e m w i t h m o r e e l e c t r o n s t h a n N , w i t h Nq ~ N * H ' we have

Ч '

e - L 6(k h) t н 'и A *1

(4 .2) h'

P ° E -p(kj + T( l + N e/i)

k ’ i

(such a s t a t e c a n be o b t a i n e d by t a k i n g a s t a t e w i t h N - H ' electrons

Г ^ У "к

a n d a c t i n g on it by the o p e r a t o r e x p ^ - i + ^nt + C»f cn i) f T h i s i n t r o d u c e s h o l e s i n s t e a d o f p a r t i c l e s , a n d c h a n g e s the e n e r g y by H 'U a n d the m o m e n t u m by T t i ' . N o w l o o k i n g at t h e e n e r g y of a s t a t e w i t h N€ - N - H e l e c t r o n s and w i t h L p a i r s o f c o m p l e x w a v e n u m b e r s

H + IL

в • £ G ( k i ) + L u A=/

(4.3) H+LL

p - E - p ( K ) t t (< +- ^ t / L) A

w e see t h a t it is like the e n e r g y o f a s t a t e w i t h L+H holes in a s u b b a n d w i t h d i s p e r s i o n -€ (k (p )) ((4.1)) a n d L p a r t i c l e s in an o t h e r b a n d w i t h d i s p e r s i o n C(k(p))-t-u ((4.2)) . T h u s t h e form o f the e n e r g y s u g g e s t s t h a t i n t r o d u c i n g p a i r s o f c o m p l e x w a v e n u m b e r s i n s t e a d of real к -s a c t s like e x c i t i n g a n u m b e r of c a r r i e r s f r o m one b a n d to the o ther. Th i s p i c t u r e , h o w e v e r , r e f l e c t s o n l y t h e a p p a r e n t a d d i t i v i t y o f the e n e r g y a n d m o m e n t u m , a n d gives the right c o e f f i c i e n t o f U . O n e s h o u l d not, h o w e v e r , f o r g e t t h a t the s t a t e s u n d e r c o n c i d e r a t i o n are e x c i t e d s t a t e s of a m a n y - b o d y system, a n d e v e n if t h e s e q u a s i p a r t i c l e s a n d h oles c a n be i d e n t i f i e d in s o m e l i m i t i n g c a s e s as some so r t of s p a t i a l c o n f i g u r a t i o n s , t h e i r e n e r g y and m o m e n t u m is c a r r i e d by the s y s t e m

*

S e e p. 33.

as a w h o l e . T h i s is e x p r e s s e d in the f a c t t h a t i n t r o d u c i n g a p a i r of c o m p l e x w a v e n u m b e r s c h a n g e s the v a l u e o f all the o t h e r w a v e n u m b e r s , and c o n s e q u e n t l y the c o n t r i b u t i o n o f the o t h e r e l e c t r o n s to the t o t a l e n e r g y a n d m o m e n t u m . A n o t h e r p o i n t is t h a t t h e s e e x c i t a t i o n s , if w e t r e a t t h e m as q u a s i p a r t i c l e s , s h o u l d b e r e g a r d e d as i n t e r e a c t i n g ones. Th i s is r e f l e c t e d in t h e fact t h a t the m o m e n t a o f the q u a s i p a r t i c l e s a r e n o t free p a r a m e t e r s , they a r e c o n n e c t e d w i t h t h e a c t u a l q u a n t u m n u m b e r s t h r o u g h a s y s t e m o f e q u a t i o n s ((2.19), (2.25)). In this r e s p e c t the p i c t u r e is v e r y s i m i l a r t o the o n e w e can c o n n e c t w i t h the m o t i o n o f the e l e c t r o n s t h e m s e l v e s : w e h a v e a s y s t e m o f p a r t i c l e s w h i c h can p r o p a g a t e a l o n g a chain, a n d can s c a t t e r on e a c h other.

In this s c a t t e r i n g p r o c e s s e s t h e y can c h a n g e m o m e n t u m a n d d e p e n ­ d i n g on t h e i r m o m e t a , t h e i r p h a s e is s h i f t e d as w e l l . T o h a v e a s t a t i o n a r y state, w e h a v e to fit the m o m e n t a a n d the p h a s e s h i f t s p r o p e r l y . T h e s e c o n d i t i o n s a r e e x p r e s s e d in t h e L i e b - W u e q u a t i o n s a n d we c a n p u t this p i c t u r e b e h i n d the e q u a t i o n s (2.19), (2.25),too.

T h i s a l s o e x p l a i n s w h y we c a n n o t tell, w h i c h m o m e n t a are to be.

a s s o c i a t e d w i t h the h o l e s a n d w h i c h o n e s w i t h p a r t i c l e s .

T h e a n a l o g y o f Eqs. (2.19), (2.25) w i t h t h e o r i g i n a l L i e b - W u e q u a t i o n s m a k e s p o s s i b l e an a l t e r n a t i v e i n t e r p r e t a t i o n . W e m a y r e g a r d t h e q u a s i p a r t i c l e s as i d e n t i c a l ones w i t h e n e r g y m o m e n t u m d i s p e r s i o n £(k(p)) , b u t c a r r y i n g an " i s o s p i n " ± f/L . T h e n we do n o t h a v e to t h i n k in t e r m s of t w o b a n d s b u t w e h a v e to i n t e r ­ p r e t the U as t h e c r e a t i o n e n e r g y o f a p a i r o f t h e s e q u a s i p a r t i c l e s w i t h isospins + t/t. a n d - >/i. .

4.2 The U— oo l i m i t of the w a v e f u n c t i o n

T h e above d e t a i l e d i n t e r p r e t a t i o n of the s t a t e s w i t h c o m p l e x w a v e n u m b e r s is b a s e d m e r e l y o n the f o r m o f the e n e r g y a n d the s t r u c t u r e o f the e q u a t i o n s d e t e r m i n i n g the p a r a m e t e r s c o n n e c t e d w i t h t h e s e e x c i t a t i o n s . N o w w e try t o find o u t a b o u t the n a t u r e of t h e s e states in the l a r g e U limit, w h e r e a l s o the f o r m of the w a v e f u n c t i o n b e c o m e s m o r e t r a n s p a r e n t .

M a k i n g the d — » oo l i m i t in t h e w a v e f u n c t i o n (see

e x p r e s s i o n s (2.3-B) of P a p e r I.) o n e finds, t h a t so m e o f the a m p l i t u d e s d i v e r g e . As in the n o r m a l i s e d w a v e f u n c t i o n o n l y the t e r m s w i t h t h e s t r o n g e s t d i v e r g e n c e w i l l g i v e f i nite c o n t r i ­ b u t i o n s , p i c k i n g o u t the m o s t d i v e r g e n t t e r m s w e can s e p a r a t e t h o s e c o n f i g u r a t i o n s w h i c h c a n be r e a l i s e d e v e n if U is v e r y large. T h i s way w e g e t the r e s u l t t h a t for l a rge U o n l y t h o s e c o n f i g u r a t i o n s r e m a i n in w h i c h the n u m b e r o f d o u b l y o c c u p i e d

s i tes is e q u a l to t h e n u m b e r o f c o m p l e x к - pairs. In the a m p l i t u d e o f these c o n f i g u r a t i o n s o n l y those p e r m u t a t i o n s P a n d "H gi v e c o n t r i b u t i o n s in w h i c h the + a n d **-i'x*, w a v e n u m b e r

p a i r b e l o n g s to o n e d o u b l y o c c u p i e d s i t e , a n d t h e A ^ b e l o n g s t o the d o w n spin at this site. U s i n g t h e fact, t h a t for l a r g e U a l l the S in k f - S c a n b e n e g l e c t e d c o m p a r e d to t h e Л й -s a n d A„ -s, a n d a l s o u s ing (2.1) the a m p l i t u d e of t h e c o n f i g u r a t i o n s in

q u e s t i o n can be g i v e n as i

(4.4)

H e r e the p e r m u t a t i o n Ö a r r a n g e s t h e c o o r d i n a t e s я, nt ... n „.H int o n o n d e c r e a s i n g o r d e r w i t h the r e s t r i c t i o n tha t f r o m two e q u a l c o o r d i n a t e s tha t of the e l e c t r o n w i t h d o w n s p i n m u s t c o m e

f d

first. T h e n.Qj -s r e f e r to s i n g l y o c c u p i e d sites, t h e to d o u b l y o c c u p i e d ones and P g o e s o v e r all p e r m u t a t i o n s of the r e a l w a v e n u m b e r s . T h e f u n c t i o n s (ff and f L a r e e s s e n t i a l l y t h e H e i s e n b e r g e i g e n f u n c t i o n s :

<f) • H, A ^{( A *v} ■ Axm-l ) (—

T \ I

i I f f t u/q

Ащ - U/tf

) - l i

У + “/ч I Ат л-í - « А ,

(4.5)

А ( . . . A*, А т ы - . ) ^ /(Ьтпн- A jri) ^- u/z.

A (■ Á ru , A r t - - ■ ) ‘ (A n * - A j , ) *■

T h e n u m b e r s у'* are the c o o r d i n a t e s of t h e d o w n s p i n s in the c h a i n of s i n g l y o c c u p i e d s i tes in i n c r e a s i n g order, фy is f o r ­ m a l l y the sam e as (ju w i t h the d i f f e r e n c e t h a t U m u s t be r e p l a c e d b y - U , the b y t h e AM - s , and t h e n u m b e r s и A are the c o o r d i n a t e s o f t h e d o u b l y o c c u p i e d s i tes in t h e chain c o n t a i n i n g o n l y the d o u b l y and u n o c c u p i e d sites. T h e a m p l i t u d e of the c o n f i g u r a t i o n s in w h i c h t h e n u m b e r o f d o u b l y o c c u p i e d sites is m o r e or less t h a n L v a n i s h e s at l e a s t l i k e */ц as

U - * oo

T o u n d e r s t a n d (4.4) let us c o n s i d e r a c o n f i g u r a t i o n in

w h i c h the f i r s t N - H - 1 L s ites in the c h a i n a r e s i n g l y o c c u p i e d , a n d _ t h e r e m a i n i n g H + I L s i t e s a r e the e m p t y or d o u b l y o c c u p i e d ones. In t h i s c o n f i g u r a t i o n t h e e l e c t r o n s c a n n o t m o v e (except

the last one) as e i t h e r the P a u l i p r i n c i p l e or t h e l a rge on­

site r e p u l s i o n p r e v e n t s it. A l t h o u g h in this c o n f i g u r a t i o n t h e r e is n o d i r e c t i n t e r a c t i o n b e t w e e n the e l e c t r o n s , t h r o u g h an i n t e r m e d i a t e s t a t e w i t h e n e r g y U n e i g h b o u r i n g e l e c t r o n s c a n see e a c h o t h e r ' s spins, and e l e c t r o n s w i t h d i f f e r e n t s p ins can c h a n g e p o s i t i o n , i.e. t h e spins c a n m o v e in t h e same w a y as spins m o v e in a H e i s e n b e r g chain, t h e d i s t r i b u t i o n of the s p ins w i l l c o r r e s p o n d t o the e i g e n s t a t e s o f t h e H e i s e n b e r g H a m i l t o n i a n . T h e s i t u a t i o n w i t h the e m p t y and d o u b l y o c c u p i e d s i t e s is similar:

N e i g h b o u r i n g s i t e s can o b s e r v e e a c h o t h e r s o c c u p a n c y t h r o u g h an i n t e r m e d i a t e s t a t e of r e l a t i v e e n e r g y - U ; and a l s o the same

i n t e r m e d i a t e s t a t e m a k e s p o s s i b l e for an e m p t y and d o u b l y o c c u p i e d site to c hange p o s i t i o n . Thu s t h e d i s t r i b u t i o n of t h e e m p t y and d o u b l y o c c u p i e d sites w i l l be t h e s a m e as t h e d i s t r i b u t i o n of u p and do w n s p i n s in a H e i s e n b e r g c h a i n . (See a l s o C h a p t e r 3.) It is c l e a r th a t n e i t h e r t h e spin d i s t r i b u t i o n n o r the r e l a t i v e d i s t r i b u t i o n of t h e e m p t y a n d d o u b l y o c c u p i e d s i t e s d o e s c h a n g e if the c h a i n of s i n g l y o c c u p i e d s i t e s is " d i l u t e d " by e m p t y and d o u b l y o c c u p i e d s i t e s m a k i n g p o s s i b l e a l s o d i r e c t p r o p a g a t i o n

for the e l e c t r o n s .

Now, h aving t h e U ■* oo f o r m of t h e e i g e n f u n c t i o n at hand, w e can see, th a t c o m p l e x к p a i r s in a s o l u t i o n of the L i e b s - W u e q u a t i o n s c o r r e s p o n d to d o u b l y o c c u p i e d and e m p t y s i t e s if U is large. Th u s in t h i s l i m i t the q u a s i p a r t i c l e s c o r r e s p o n d i n g to t h e s e e x c i t e d s t a t e s s h o u l d be i d e n t i f i e d w i t h t h e s e o b j e c t s . W e c a n see also, t h a t if w e t r e a t t h e m as p a r t i c l e and h o l e like

ones, t h e n the f i l l e d b a n d c o r r e s p o n d s t o all s i t e s b e i n g o c c u p i e d b y o n e e l e c t r o n , the h o l e s c o r r e s p o n d to t h e e m p t y sites and the p a r t i c l e s to the "sec o n d " e l e c t r o n s a t the d o u b l y o c c u p i e d sites. T h i s is in a c c o r d a n c e w i t h the i n t u i t i v n o t i o n of e x c i t i n g a n u m b e r of c a r r i e r s a c r o s s a M o t t - H u b b a r d gap.

It can be seen a l s o tha t t h e a l t e r n a t i v e i n t e r p r e t a t i o n is e q u i v a l e n t l y good: in thi s c a s e the c a r r i e r s a r e the not s i n g l y o c c u p i e d sites, and the i s o s p i n t e l l s us w e t h e r a c a r r i e r is an e m p t y or a d o u b l y o c c u p i e d site.

W e s h o u l d e m p h a s i z e , tha t w h a t h a s b e e n said a p p l i e s o n l y if U is large, and n o d i s c u s s i o n of c o m p a r a b l e s i m p l i c i t y c a n be g i v e n as o n e m o v e s aw a y f r o m t h e U -*o o limit, s i n c e it is o b v i o u s t h a t for U~ / t h e r e a r e m a n y d o u b l y o c c u p i e d sites e v e n in the g r o u n d state w h e r e t h e r e is n o c o m p l e x w a v e n u m b e r in t h e

к -set. F o r f inite U w e h a v e to be s a t i s f i e d w i t h the

e x p r e s s i o n s for the e n e r g y and m o m e n t u m o f t h ese s t a t e s w i t h o u t p u t t i n g b e h i n d t h e m a t r a n s p a r e n t p i c t u r e .

5. S U M M A R Y

In the p r e s e n t w o r k w e h a v e i n v e s t i g a t e d t h o s e e i g e n s t a t e s o f the 1-d H u b b a r d m o d e l , for w h i c h in t h e w a v e n u m b e r set t h e r e are s e v e r a l p a i r s o f c o m p l e x к -s. O u r r e s u l t s a r e the f o l l o w i n g

1. F o r s u c h s t a t e s s o l v i n g the L i e b - W u e q u a t i o n s is

e q u i v a l e n t to s o l v i n g the s y s t e m of Eqs. (2.5), (2.13) and (2.19) p r o v i d e d (2.4) is s a t i s f i e d b y the f o u n d s o l u t i o n . T h i s s y s t e m is s i m i l a r in s t r u c t u r e to t h e L i e b - W u e q u a t i o n s , t h e d i f f e r e n c e is that t h ere are n o c o m p l e x w a v e n u m b e r s in it.

2. T h i s s y s t e m can be r e d u c e d t o a s i m p l e r o n e (Eqs. (2.19) (2.25)) for t hose s t a t e s in w h i c h the s p i n d e g r e e s o f f r e e d o m a r e not excited. F o r t h ese s t a t e s the e n e r g y and m o m e n t u m c a n be g i v e n as the e n e r g y and m o m e n t u m of q u a s i p a r t i c l e s . T h e s e q u a s i p a r t i c l e s can be r e g a r d e d as p a r t i c l e - h o l e like o n e s b u t t h e y c a n be t r e a t e d as p a i r s of i d e n t i c a l p a r t i c l e s w i t h " i s o s p i n "

e q u i v a l e n t l y well. T h e f o r m of t h e e n e r g y and m o m e n t u m o f the q u a s y p a r t i c l e s is g i v e n by (1.10) and (1.13).

3. A " c o m p l e m e n t a r i t y " b e t w e e n s o l u t i o n s of t h e s y s t e m

(2.5), (2.13) and (2.19) c o r r e s p o n d i n g t o low and h i g h l y é x c i t e d s t a t e s c a n be e s t a b l i s h e d , w h i c h c a n b e u s e d to d e s c r i b e d i f f e ­ r e n t s t a t e s w i t h o n e s o l u t i o n o f the s y s t e m . In the c o m p l e m e n t a r y s t a t e s t h e p a r a m e t e r s c o n n e c t e d w i t h t h e c h a r g e and sp i n d e g r e e s o f f r e e d o m c h a n g e role.

In the p r e s e n t s t u d y we c o n c e n t r a t e d on the " c h a r g e e x c i t a ­ tions". T o iso l a t e c h a r g e r e a r r a n g e m e n t effe c t s ,

w e e x a m i n e d such s t a t e s in w h i c h the s p i n p a r t w a s in its g r o u n d state. W e pla n to e x t e n d o u r s t u d y to t h o s e s t a t e s in w h i c h a l s o the s p i n d e g r e e s of f r e e d o m are e x c i t e d . P r e l i m i n a r y r e s u l t s show, that, as it is e x p e c t e d , t h e p r e s e n c e o f spin

e x c i t a t i o n s d o e s not a f f e c t d r a s t i c a l l y t h e r e s u l t s c o n c e r n i n g the s t a t e s s t u d i e d so far, j u s t in a d d i t i o n a n e w t y p e of

" e l e m e n t a r y e x c i t a t i o n s " m u s t be i n t r o d u c e d .

I a m v e r y g r a t e f u l to Drs. A. S ü t ő and P. F a z e k a s for the m a n y v a l u a b l e d i s c u s s i o n s .

R E F E R E N C E S

E . M. Lie b F.Y. Wu Phys. Rev. Lett. 20 1445 (1968) F. W o y n a r o v i c h K F K I - 1 9 3 0 - 6 1 .

F o o t n o t e (p. 23.)

* I

T h e s tate w i t h И e x t r a e l e c t r o n s can be c o n s t r u c t e d this w a y o n l y if /V is even. If N is odd, the t r a n s f o r ­ m a t i o n b e t w e e n p a r t i c l e s and h o l e s c h a n g e s t h e p e r i o d i c b o u n d a r y c o n d i t i o n into a n t i p e r i o d i c or c h a n g e s the si g n of t h e k i n e t i c energy.

Kiadja a Központi Fizikai Kutató Intézet Felelős kiadó: Krén Emil

Szakmai lektor: Fazekas Patrik Nyelvi lektor: Bergou János

Példányszám: 465 Törzsszám: 80-517 Készült a KFKI sokszorosító üzemében Budapest, 1980. szeptember hó