Volume 26A. number 11 PHYSICS L E T T E R S 22 April 1968

C O R R E L A T I O N F U N C T I O N S F O R T W O - D I M E N S I O N A L F E R R O E L E C T R I C S

B. SUTHERLAND Institute for Theoretical Physics, State University of New York, Stony Brook, N. Y., USA

Received 23 March 1968

The correlation functions of the general two-dimensional ferroelectric problem are calculated exactly for the case A = 0. This result includes the modified model of Wu at all temperatures and applied electric fields.

F r o m IAeb's exact solution of the two-d imen- sional ice problem [1], have come a s e r i e s of genera l iza t ions [2-6]; the most genera l of these is that of Sutherland, Yang and Yang [5,6]. Also, F . Y . W u has invest igated a modified model [7,8]. We cal l al l of these models two-d imens iona l f e r - roe lec t r i c s . It is the purpose of this le t te r to in - vest igate the cor re la t ion functions of such models.

In the most genera l two-d imens iona l f e r r o - e lec t r ic problem, there a r e two independent co r - re la t ion functions to be considered: the c o r r e l a - tion between two ver t i ca l spins , and the c o r r e l a - tion between a ve r t i ca l and horizontal spin. We shall here consider only the f i r s t type of c o r r e - lation. If the la t t ice s i tes a re labeled by the in - dex r = (re,n), then the cor re la t ion function G(r) is defined as the the rmal average of the product of two spins separa ted by a d is tance r . In t e r m s of the eigenvalues Aa (y) of the t r ans fe r ma t r ix A(y), when N ~ ~o,

\;to / (1)

~o(Y) is the maximum eigenvalue. This type of express ion for G(I") is well known. Its evaluation in full genera l i ty is ex t remely difficult, the p r i n - cipal problem being the evaluation of the mat r ix e lements . However, when the pa r ame te r A vanishes the e igenvectors a r e especial ly s imple , and the exact evaluation of eq. (1) is possible . It is the purpose of this le t te r to p resen t such an evaluation. Our notation is that of r e f e r ences 3, 5 and 6.

For A = 0, the eigenfunction is of d e t e r m i n e n - tal form. We choose a set of M unequal number s k l , k 2 , . . . , kM, where k i = 2y ( integer) /N. Then the unnormal ized eigenfunctions a re ~ ( x 1 . . . . , x M ) = d e t [ e x p ( i k j x D ] , 1 ~< x 1 < . . . <

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< XM ~ N. The total momentum of the s tate is K = ~ kj. We wish to evaluate s ca l a r products and ma t r ix e lements . To do this, we enlarge the region of summat ion , so that all va r i ab les extend f rom 1 to N. The norTnalization constant is then easi ly found to be N ]M. The mat r ix e lements in eq. (1) a re re la ted by (0]o2m la) -- exp (imk) × x (01aoZ]a). They may be evaluated, and one finds that the only s ta tes a which contr ibute non- vanish ing mat r ix e lements a re the ground state, and one for which one k o of the ground state is moved to k. The total momentum of this excited s tate is then K(a) = k - k o. We find the mat r ix e lements to be

(0)O~ola) = y , a : 0 , (2)

= - 2 I N , a * 0 .

Inse r t ing eq. (2) into eq. (1) gives the resu l t

1 ?~(3+y) [7 +Hexp (ik) 7 n G(r) = y2 + ~-2 ~ ( 1 - y ) LH*/exp ( ik) - l J exp (imk)dk x

(3) ' 1 ~ ( - Y ) [ r /+H exp (ik) ~-n

- ( l -y) LH~lexp(ik)-lJ exp(- imk) dk .

The f i r s t t e r m , y2, is the l imi t ing value of G(r) as /- ---, 0o.

When n = 0

I 2 sin [~rm(1-y)] 12 G(m) =2 2 - ~-~ . (4)

In general , for r large,

4 1 cos (0+~) c o s ( 0 - qb) (5) c(r) y2 +

where p2 is a quadrat ic form in m and n. As an example, the Rys F model [2] in zero

Volume 26A, number 11 P H Y S I C S L E T T E R S 22 April 1968

field, w h e r e H = ~ = 1 a n d y = 0, yields an asymptot ic behavior for the cor re la t ion function of

2 I n2 m27 cCr) r-~'J~ ~ cos ~(n - m) + ~ j . (6)

If we define two sublattices according to whether m + n is even or odd, and write r in polar coor-

dinates (r,~), then

Geven(r) ~ (2sin~V/~rr) 2 , (7a)

God d (r) ~ - (2cosff /nr ) 2 , (Tb)

The modified f e r roe l ec t r i c model of Wu [7,8] is obtained as a l imi t ing case of the genera l f e r - roe l ec t r i c when A = 0 [6]. Thus eq. (3) gives the co r re l a t ion function G(r) of Wu ' s model for all t e m p e r a t u r e s and all applied f ields, including the c r i t i ca l points. It is thus in t e res t ing to r e - fo rmula te (3), and exhibit the cor respondence . Wu cons ide r s the two-d imens iona l Sla ter KDP model in external f ield and modifies it by ex- cluding one of the two configurat ions with lower energy. In t e r m s of Wu ' s va r i ab le s K -- exp ~¢) , X - exp (2/Sex) ,

G(r) ~ 1 - r 2--~- s in (C Pc Tq~-Z~-Tc)I 2 r - ~ L~Pc J , (8) T ~ T e

where C is a constant , and

P ~ ' ~ c Pc = m + n X/(K + X) . (9)

We see that G(r) dec rease roughly as 1/r 2 for l a rge r , not exponential ly as in the Is ing model. This behavior should not be pecul iar to the point A = 0, for exponential dec rease is a r - gued to be the r e su l t of a gap in the spec t rum of In ~ (y) /N below the max imum eigenvalue. This gap does not exist for any value of A.

I would l ike to thank B. McCoy for helpful d i scuss ions .

References 1. E.H. Lieb, Phys. Rev. Letters 18 (1967) 692. 2. E.H. Lieb, Phys. Rev. Letters 18 (1967) 1046. 3. B.Sutherland, Phys. Rev. Letters 19 (1967) 103. 4. E.H. Lieb, Phys. Rev. Letters 19 (1967) 108. 5. C.P.Yang, Phys. Rev. Letters 19 (1967) 586. 6. B. Sutherland, C.N. Yang and C. P. Yang, Phys.

Rev. Letters 19 (1967) 588. 7. F.Y.Wu, Phys. Rev. Letters 18 (1967) 605. 8. F.Y.Wu, preprint and report at Yeshiva, 1967.

S P I N - W A V E S I D E L I N E IN T H E L U M I N E S C E N C E S P E C T R A O F E u 3+ I O N IN K M n F 3

S. SHIONOYA and M. HIRANO The Institute for Solid State Physics, The University of Tokyo,

Roppongi, Minato-ku, Tokyo, Japan

Received 20 March 1968

Spin-wave sidelines were observed in the luminescence spectra of Eu 3+ ion doped in antiferromagnetic KMnF 3 crystals.

Recent ly some repor t s have been published about the observat ion of sp in-wave sidebands in absorpt ion and emiss ion spec t ra [1-6] of an t i f e r - romaguet ic ma te r i a l s such as MnF 2 and KMnF 3. We wish to repor t here the observat ion of sp in - wave s ide l ines appear ing in the luminescence spec t ra of Eu 3+ ion doped in an t i f e r romague t i c KMnF 3 c rys ta l s . To our knowledge, this is the

f i r s t observa t ion of sp in-wave sideband in the op- t ical spec t rum of an impur i ty embedded in a mag- net ica l ly o rde red ma te r i a l .

T r i v a l e n t r a r e ear th ions usual ly emit , when in t roduced in sol ids , br ight luminescences due to f - f t r ans i t i ons . The spec t ra of these l u m i n e s - cences a r e composed of sharp l ines . This makes it easy to detect the in te rac t ion of r a r e ear th ions

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