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Order-disorder in hexagonal lattices
P h y s i c a X V I , no 5
Mei 1950
ORDER-DISORDER IN HEXAGONAL LATTICES by R. M. F. H O U T A P P E L Instituut voor theoretisehe natuurkunde, Rijksuniversiteit, Leiden, Nederland
Synopsis S o m e relations for t h e p a r t i t i o n f u n c t i o n s of t w o - d i m e n s i o n a l infinite t r i a n g u l a r a n d h o n e y c o m b lattices w i t h I s i n g i n t e r a c t i o n b e t w e e n n e i g h b o u r s w i t h o u t m a g n e t i c field are g i v e n in t h e general case of d i f f e r e n t i n t e r a c t i o n s in t h e t h r e e directions. E x a c t e v a l u a t i o n of these p a r t i t i o n f u n c t i o n s is o b t a i n e d w i t h t h e aid of t h e m e t h o d of 1 3 r u r i a Kaufm a n . T h e t h e o r y of this m e t h o d is simplified b y a v o i d i n g t h e use of a b s t r a c t g r o u p - p r o p e r t i e s . S o m e t h e r m o d y n a m i c p r o p e r t i e s of t h e isotropic h e x a g o n a l lattices are given. These are c o m p a r e d w i t h t h e c o r r e s p o n d i n g p r o p e r t i e s of t h e q u a d r a t i c lattice. All h o n e y c o m b and n e a r l y all t r i a n g u l a r lattices h a v e a t r a n s i t i o n t e m p e r ature. A t this t e m p e r a t u r e t h e e n e r g y is c o n t i n u o u s and t h e specific h e a t b e c o m e s infinite as - - In [ T - - T c J, as in t h e r e c t a n g u l a r case, s o l v e d b y Onsager and Kaufman. T h e r e are e x c e p t i o n a l t r i a n g u l a r l a t t i c e s w i t h o u t t r a n s i t i o n t e m p e r a t u r e : for e x a m p l e t h o s e w i t h e q u a l n e g a t i v e interactions.
1. Introduction. K r a m e r s and W a n n i e r l ) showed, that the computation of the partition function of a crystal-lattice with I s i n g interaction can be reduced to an eigenvalue problem. This problem, in case of the two-dimensional rectangular lattice with interaction between neighbours only, has been solved first by 0 ns a g e r 2) and afterwards in a shorter way by K a u f m a n 3). In this article 4) we will consider the partition functions of the twodimensional triangular and honeycomb lattices with I s i n g interaction between nearest neighbours only. We will consider the general case of different interaction energies, J1, J2 and Ja, of various magnitudes and signs in the three directions. If we indicate the state of each atom i by its spinvariable #,~ ---- 4- 1, the energy of a microscopic state of both lattices will be E =
-
-
fl
X(ik)l ~i/Lgk
-
-
J2 z(it)2/~#*t f3 X(i.03/zi#*m -
-
(l)
(ik), indicating that the sum is to be taken over pairs of neighbours with mutual interaction energy Js. 425 -Physica XVI
27*
426
R.M.F. HOUTAPPEL
In sections 2 and 3 we set up the eigenvalue problems related to the triangular and honeycomb lattices. In sections 4 and 5 we derive two different relations between the partition functions of the two types of lattices. In section 6 we use these relations to derive properties of invariance with respect to typical substitutions, both for' the partition functions of the triangular and of the honeycomb lattices. These properties are used in section 7 to construct series expansions valid both at high and at low temperatures. The main subject of this article is the exact evaluation of the partition functions. This is carried out in sections 8 to 11 independently of the considerations in sections 5, 6 and 7. Although we follow essentially the method of B r u r i a K a u f m a n 3), we present it in a somewhat different fashion, simplifying the theory by neither referring to, nor using, abstract group-properties. In section 12 we examen the transition temperature. In the last section, 13, some thermodynamic properties of isotropic triangular and honeycomb lattices are discussed and compared with the corresponding properties of a quadratic I s i n g lattice.
2. The eigenvalue problem related to the triangular lattice. We
a
J
2
J
l
J i
2
3
¢
2n
Fig. 1. Triangular lattice. How rows (thick zigzag lines) and columns are counted.
I
denote each atom in the strip of triangular lattice in fig. 1 by two indices, the first referring to the row and the second to the column in which it is situated. We transform this strip to a cylinder so that each atom k, I will be identical with k, l + 2n and then this cylinder to a torus so that each atom k, l will be identical with m + k, I. The partition function of this torus-shaped two-dimensional triangular lattice of 2nm atoms with interaction energies J1, J2 and J3, as indicated in fig. 1,
ORDER-DISORDER
IN HEXAGONAL
427
LATTICES
will be : Z~,2 ...... =
~ /tk, 1 =
[1I,-"=1 e x p [E~'=I(Kt#i,2i_IF*;,2i + K2/ti,2ifq,2i+t + ± 1
Jr- K 3 # i , 2 j - - l / l i + l , 2 ] - - I
~'- K 2 # i + 1 , 2 1 - 1 # i , 2 ]
~f- K 1 / t i , 2 j / l i + l , 2 i + l
with W e see t h a t : ) Z ~. )2 ..... ~
K, = J,/kT,
E l k l _-± i
i ~ 1, 2, 3.
V~,2n(/ti,l,/tl,2)
",#I,2n;/t2,1
,
"V~,2n(]~2,1, /*2,2 . . . .
•.
Jr-
Ka#~,2it*i+ 1,2j)]j
÷
)
~t2,2, ) . . )
/t2,2n ; //3,1, ,tt3,2 . . . .
....
"V:~,2,,(/t,.,l,/t,,,,2
(2)
/~,,,,2,, ; / q , l , / q , 2
#3,2n)"
""
.... /q,2,,) 22n m
m
:
/t2,2n)"
T r a c e V~,2, , = Ei= 1 ~'i'
(3)
In (3) V~.2, , *) is a 22"-dimensional m a t r i x , whose e l e m e n t in the row, d e n o t e d b y the set/q,/~2 . . . . . /t2, , ( / t , = 4 - 1 ), a n d in the c o l u m n , t t t t d e n o t e d b y the set F,, F2 . . . . . /~2,, (/t, = 4- 1), has t h e v a l u e : V'z,2,,(#l, /t2,
F2,,; /'.l,' F2,,
• .,
.
., 1~2,,) -- e x p [Zi= " I (Kltt2i-|/t2i '
t
!
i
-~t
+ K2/t2i/t2i + t-k KaF2i_~,.2i_~ + K2#2i_~/z2FF K~/t2fft2i + ~-+-Ka/t2i/~2i) l (4)
(/t2,,+
1 =
'
/q,/t2,,+1
:
#
,) •
/t 1, t 2. . . . are the 22" e i g e n v a l u e s of V~a,,. B e c a u s e all e l e m e n t s of V~.,2,, are positive, its largest e i g e n v a l u e 2.~,2, is not d e g e n e r a t e s). T h e r e f o r e the p a r t i t i o n f u n c t i o n per a t o m of this lattice for m -+ oo will b e '
Lim,,,_+oo 7~1, 2' 2nn,mm
:
]I/2n
"'~,2~,,~
One recognizes t h a t (with the n o t a t i o n ~(v) = 1, if v = 0, a n d a(v) :
O, i f v :# 0 ) :
v Z,2n (/ t l , -
-
"',#2n,/
.,;,
E
" [exp [~,i=l(Kllt2i_l/t2i-+-K2tt2s/t2i+l)3
It III I I" & . t*i'tti'Hi =_1
• ",/
=
n
• exp
[E]= t
IV
,t
.1
K3/t2i_l/t2i_lj
• exp rv,, L ~ i = ' ( K 2 / t 2 i _"l / t 2
n
•
• n_]= 2" ,a(iti--t,~"
I I"
Hi= t (~(/+2i
--/t2j
• e x p let'= t Ka#2"j/t'2i]. 1I}'= t 6(/t2i_ , ~tt /tl ' /t2n+I ,t •
=
)"
"i ..{_ K l / t 2 i / "1 2 i " + l) ] . v[2,,..i= i
tt
tt (,tt2n+l
ut
:
ut
(~(/ti" - - / t i " ) "
t --/'~q-t)j
tiE,"
/tl ; / 2n+l
).
{V
:
~
)"
• ) Throughoutthisarticleweshallusethiekletterstoindieate2 -dimensionalmatriees a n d G o t h i c l e t t e r s of o r d i n a r y size to i n d i c a t e t h e 4 * , - d i m e n s i o n a l r o t a t i o n s i n t r o d u c e d in s e c t i o n 8. T h e s u b s c r i p t s ~., ,9, 9t refer to l a t t i c e t y p e s .
428
R. M. F. H O U T A P P E L
Or V%,2. = V 1V~V3V'4, with: l]2n
"
V, = e x p [Ei= I (Ki/~2i_ltu2j --~ K2lu2i~u2i+I)]'~H= 1 ~ ( ~ i - / z i ) ' t
)!
_ _
V'2 = exp [E;'= l K3#2i_1~2i_1].Hi= l 6(/~2i .
V3 = e x p [Zi=l (K2~2i-1~2i + Klfl2iU2i+l)]
'
r
P2i), . ]-]2n
'
"'/'=1 ( ~ ( / ' t i - Fti),
" 1 K 3/~2i/~2/] ' " YIi=1 " ' V~' = exp [Zi= ~(#2i-1 - - ~u2j-l)-
It is convenient to interpret these matrices as linear operators, acting on the 22"-dimensional vectors ~P(/~I,/~2. . . . /~2,.) according to : ~'(~,
....
~2,,) = v~(~,
....
t,~,,) = t
Hi=:kl
•.,
t
t
..
, •.,
So we see, that V 1 and V 3, which are diagonal matrices, can be written in the form : V 1 = exp [K l(sts 2 + sas 4 + .. + $2n__1S2, ) + +
V3 = exp [ K 2 ( s I s 2 -]- s 3 s 4 +
K2(s2s3 -~
s4s s +
. . -~ $ 2 . _ 1 s 2 . )
..
+
s2,,sl)l,
-}-
-~- K I ( S 2 s 3 -~- $4s 5 -{- . . Jr- S2nSl)],
where the operators s~ are defined by s, ~(F,,
....
~, ....
~2.) =
t,, ~(~1
....
~,, ....
t,2,,)
(s)
V~ and V'4 are direct products of two-dimensional matrices, 8(/z,--/~) and exp(K 3 #q#,;), which can be considered as operators, each acting on one of the parameters #~ of the 22"-dimensional vector %V(/Zl, /~2 .... /~2,,)" b(~*i- - / ~ ' ) is the two-dimensional unit-matrix and ! exp ( K 3 # # q ) is the matrix exp K~ exp ( - -
exp(-- K3)) K3) exp K 3
(6)
It is convenient to put these matrices also in the form exp (matrix) This is possible by writing exp K~ exp(--K~) ~ ( e x p K~ exp(--K3) ~ T_ 1 exp(--K~) expKa / = T . T -1. \exp(--K~) e x p K 3 / . T .
429
O R D E R - D I S O R D E R IN H E X A G O N A L LATTICES
where T is the matrix which transforms (6) into diagonal form : 0 ) =expk[lna'0 0 ) a2 In a 2
--1 /expKa exp(--Ka)~.T=(;' I - • ~exp (--Ka) exp K a / Then : expK exp (--Ka, ~ exp (--K3) exp K 3 /
(1;a, -----
T" exp
0 ).T_, In a 1
=
. ~ /In a I 0 ) . T _ l ] =exp[l.~ o lna 1 In this w a y we find: , ( e x p Ka e x p ( - - Ka)) = exp (Ka/~¢~i ) = \ e x p (--Ka) exp K a
= exp
ln
with:
(2 Sh 2Ka)~ K* (2 Sh 2Ka) ~ exp (K~'ei) (7) K* In (2 Sh 2K3)~/= K* = ½ In Coth K~,
(K*)* = K e
(8)
K~' satisfies the relations, given by K r a m e r s and W a n n i e r 1). Sh 2K i Sh 2K* ---- Ch 2K i T a n h 2K* = Tanh 2K~ Ch 2K* = 1 (9) e i (in 0 n s a g e r's 2) notation Ci) is defined by: ei ~o(/~ l . . . .
#i ....
# 2 . ) -= YJ(#l . . . .
--#i ....
/z2,,).
(10)
For reference, we tabulate here the algebraic properties of s~ and follow from (5) and (10), and have been given b y 0 nsager2):
ci, which
S~= ~ = SiS k - ~ SkSi,
sick=
cks i
1, SiCi=--CiSi; CiCk =
CkCi;
(11)
(i :/:k).
Using (7) and the c o m m u t a t i v i t y of the ei's we can write V£ = (2 Sh 2Ka) '''2 exp [K*(e t + e 3 + .. + e2,,_,)], V~ = (2 Sh
2Ka)'':2exp
[K*(c 2 + c 4 + .. + c2,,)].
Thus we find that the partition function per atom X~. of a two w a y infinite triangular lattice is equal to ~.2,,, ~tz ----- Lim,,~oo ~1/2,,
(12)
430
R. M. F. HOUTAPPEL
2z,2,, being the largest eigenvalue of the 22"-dimensional m a t r i x Vz,2n, defined by: V~,2,, = (2 Sh 2K3)" V, V2VsV4, (13) V 1= e x p [K 1(SIS2"~-$3S 4 "~-.. "~$2,,__IS2n ) - ~ - K 2 ( s 2 s 3 + s 4 s s - ~ . . V2 :
"~-S2,,SI)~,
exp [K~(c t + c 3 + .. + c2,,_t) J,
V 3 :exp[K2(sls2+s3s4+..
2f-s2,t_lS2n)-+-gl(S2S3-~-S4Ss-~-..-~-s2nsl)
V4 ---- e x p ~K~(c 2 + c 4 + . . .
J,
+ C2n)3.
3. The eigenvalue problem related to the honeycomb lattice. As in the case of the triangular lattice, we denote each a t o m in the strip of honeycomb lattice in fig. 2 by two indices, the first referring to the row and the second to the column in which 5 it is situated. This strip we also 4 transform to a cylinder so t h a t 3 each a t o m k, l will be identical 2 with k, 1 + 2n, and t h e n this cylinder to a torus so t h a t each 1 2 3 4 2rl i atom k, l will be identical with Fig. 2. Honeycomb lattice. How rows k + 2m, l. The partition functand columns (thick zigzag lines) are ion of this torus-shaped twocounted. dimensional h o n e y c o m b lattice of 4nm atoms with interaction energies J l , J2 and f3, as indicated in fig. 2, will be:
Z~o,2,,,2m =
• [HILl exp [Y'~'=1 (Kl#2i--l,2j--l#2i,2i--I -~Jtk,l= 4-1
+ K2#2i--1,2i#2i,2i + K3#2i,2i#2i.2i+! -}- K2#2i,2i_1#2i+1,2i_ 1 + '~ K1#2i,2i#2i+1,2i -~- K3#2i + l,2f--l#2i + l.2i) ] ] . We can see: Z¢~,2,,2,,, = Trace lr~',2,,,
V~,2,, = V , . V b.
(14)
Va and Vb are two 22"-dimensional matrices, whose elements in the row, denoted b y the set #1,/*2 . . . . . #2,, (#k = + 1), and in t h e
O R D E R - D I S O R D E R IN H E X A G O N A L LATTICES t
!
t
column, denoted by the set #1,/~2 . . . . . #2, (#'k -~ i ively the values : Va(/Ll . . . .
~22n; ~ l . . . .
--
--
exp
t
l
!
[El'= t (K1/12i_1/12i_ l "~- K2/12i/~2i -~- K3//2//~2i+t)~ ,
t /t2n; /*1 . . . .
Vb(/~l . . . .
1), have respect-
/~2,,) = l
:= exp
r~,,
L~i=I
431
! /t2,,) =
' ' ' (K2#t2i--1/t2i--I "~- Kl/t2i#*2i' -~- K3122i_#,2i)1.
]
(15)
Reasoning in the same way as we did for the triangular lattice, we can conclude finally, t h a t the partition function per a t o m ~9 of the two w a y infinite h o n e y c o m b lattice is equal to
(16)
~s~ ~- Lim,,_+oo "'$O,2n'~liO'
~-~,2,, being the largest eigenvalue of the 22"-dimensional m a t r i x V~,2,,, defined by: V~,2, = (2 Sh 2K~)" (2 Sh 2K2)" WtV¢2W3V¢4,
(17)
W t = exp [K~ (c I + c 3 + .. + c2,;_l) + K* (c 2 @ c 4 + .. + e2n)] , W2 =
exp [K 3 (S2S 3 + $4S5 +
W3=exp[K~(c
l+c 3+
.. +
S2nSl)I,
.. + c 2 , _ t ) + K ~ ( c
2 + c 4 + .. +e2,,) 1,
W 4 = exp [ K 3 (SIS 2 -~- SSS4 "-{- .. "-~ S2n__lS2n)l.
4. Dual relation between 2~ and 2~. We shall now derive a relation between 2~ and ~ , using the m e t h o d b y which 0 n s a g e r e) showed the dual relation between the values of the partition function of a quadratic lattice for high and low temperatures. All eigenvectors of V~.2, and V~,2,, belonging to non-degenerate eigenvalues are either even, or odd,
~v(/~l, #e2. . . . #2,) = - - ~v(-- #l, - - / ' 2 . . . . - - #2,,),
because all operators s~s~+t and e~ transform an even vector into an even one and odd vector into an odd one. Substituting in the expression for V~.2,,, (17), sisi+ 1 for c~ and ci+l for sisi+l (s2,,+ t - - s t ; e2,+1 = et) we get, in view of (13)"
Vs~,2,,(Kt,K2,K3)-+ (2 Sh 2Kt)"(2 Sh 2K)" (2 <~ °m*'-"V3j~.2,,,IK*t, K2,* K~) = = (2 Sh 2Kt Sh 2K2 Sb 2K3)" V~.,2,,(K*, K*, K~). (18)
432
R. M.F. HOUTAPPEL
If not only K 1, K 2 and K 3 but also K*, K* and K* are real, all elements of both V~,2,(K ,, K 2, Ka) and V~,2,,(K*, K*2, K~) are positive and the largest eigenvalues of both will belong to even vectors 5). Since, as 0 n s a g e r showed 6), the eigenvalues belonging to even vectors are not changed b y the above substitution, it follows t h a t then: ,ts~,2,,(K`, K 2, Ka) = (2 Sh 2K, Sh 2K 2 Sh 2Ka)" ,t~,2,,(K*, K*, K*). (19) However, if some Ki or K* are negative, the corresponding K* or Ki respectively, will be complex. This case therefore needs f u r t h e r discussion. All elements of the 22"-dimensional matrices V~.2,(K 1, K 2, Ks) and Vg,2,(K1, K 2, K3) are analytical functions of K1, K 2 and K a. Therefore ~l~.2,(K1, K 2, K3) and ~.9.2,(K1, K 2, K3), which are nondegenerate eigenvalues, defined for all real values of K 1, K 2 and K 3, are also analytical for these values of the parameters, and can be continuated analytically to complex values of one or more of them. Hence the above relation, (l 9), will be valid for all values of K 1, K 2 and K a if we obtain ;t~,2,, and ~,~,2,,, for complex values of their arguments, b y appropriate analytical continuation from real values of these arguments. Using (12), (16) and (19) we derive : ~.~(K 1, K 2, K3) = (2 Sh 2K 1 Sh 2 K 2 Sh 2Ka) ½t ~ ( K * , K 2*, K a )*.
(20)
In the case where one or more of the parameters K i and K~' are complex, ~ and t0 denote the appropriate analytical continuations to these complex values. 5. Star-triangle relation between ~,0 and ~ . Following a m e t h o d of W a n n i e r 7) for K 1 = K 2 = K 3, we shall derive for the general case K t =/= K 2 :# K 3 a second relation between ~ and Jl~. In this m e t h o d the #'s of half of the atoms are eliminated from the partition function of a h o n e y c o m b lattice and new interactions between their neighbours are introduced. The partition function Z~,2n,2 m of the same torus-shaped honeycomb lattice of 4 n m atoms, as we constructed in section 3, is a sum over the possible values, + 1 and - - l , of the spin variables of a product of 2 n m factors of the form exp (K1#IF 0 + K#%t, 0 + K3#aF0).
ORDER-DISORDER IN HEXAGONAL LATTICES
I I
-. ".
1
433
Here each/~0 represents the spin variable of an a t o m denoted by a circle in fig. 3 and appears in one of these factors only. B y carrying out the s u m m a t i o n over the values + 1 a n d - - 1 of these /~0's first, we see t h a t Z9,2,,,2m is the sum over the possible values of the other spin variables, of a product of 2nm factors of the form: E/~o=±1 exp (Kg~#,o+ KzuzUo+ K3/t3F0).
Fig.3. E x p l a i n i n g W a n n i e r ' s star-triangle transformation.
This form can take only four values for ~*1, ~=2,#L3 = 4- I and we can easily verify t h a t
Xa0=el exp (Kg~lp o + Kzuzu 0 + K3/~3/~o) = r
t
p
--= S(K t, K 2, Ka) exp (Ktpzu 3 + K2~,3/~ + K3/qZ2) (21) with
S(K1, K2, K3) = = 2~¢/Ch(K1+ K2+ K3)Ch(--KI + K2+ K3)Ch(KI--K2+ K3)Ch(Kt + K2--K~)
(22)
and exp (2K'2 + 2K'3) = Ch(K, + K 2 + K3)/Ch ( - - K , + K 2 + K3), cycl. So we see, t h a t b y the above relation, (21), it is possible to eliminate the F0's if we introduce at the same time the interactions J'l = kTK'I, t t t J2 : kTK~ and J3 = kTK3, between the atoms 2 and 3, 3 and I and 1 and 2 respectively. These new interactions are indicated by dashed lines in fig. 3. We find:
Z~,2,,,2,,, -' - - {S(K1, K2, K3)} 2.... Z ~,2 ...... ( K '1, K 2,
K~)
Z~,2,,,2,,, being the partition function of the torus-shaped triangular lattice of 2nm atoms, which we constructed in section 2. So we can conclude t h a t : !
t
a~(K l, K 2, K3) = S(K,, K 2, K 3) ~r(K,, K 2, K~). Physica XVI
(24) 28
(23)
434
R. M.F. HOUTAPPEL
6. Conclusions drawn/rom the dual and star-triangle relations with respect to the/orm o/ the /unctions ~z(K l, K 2, K3) and A~(K t, 1£2, K3). B y eliminating ~ and ~z respectively, from the relations (20) and (24), we derive: ~(K,) :
Fz(Ki) ) .4~ ( K ~ o) ,
(25)
~t~(K,) • -~
C~(K,) ~s~(K, 4
(26)
),
with
F~(Ki) = 16(Ch K~ Ch K 2 Ch K a + Sh K l Sh K 2 Sh Ka). •( S h K 1 C h K 2 C h K a + C h K
1 S h K 2ShK3).
•( C h K 1 S h K 2 C h K a + S h K
1ChK2ShKa).
(27)
•(Ch K~ Ch K 2 Sh K a + Sh Kt Sh K 2 Ch Ka), Go(K,) -(Sh 2K1) 2 (Sh 2K2) 2 (Sh 2K3) 2
4Ch(Kl + K2+ K3)Ch(--K1 + K2+ K3)Ch(Kl--K2+ K3)Ch(Kt + K2__Ka ) , (28)
e2Kx+2Ke+2K3 --~ eeKx -J- e21(°-+ e2Kz
/
exp (2K~ + 2K~) = ee,~1+2~2+eK3 + e2~1__ e2~2__ e2~3 , cycl.,
(29)
(K~)° = K , and Coth K~ Coth K~ = Ch(K1 + K 2 + K3)/Ch(--K ~--F K2 + K3), cycl., } (K~) ~ = K , .
(30)
The formulae (29) and (30) connect sets of high values of K 1, K 2 and K 3 with sets of low values of these parameters. However (25) and (26) cannot, in general, be considered as relations between the values of the partition function of one triangular or honeycomb lattice at high and low temperatures. This is because the ratios of the K{s are changed b y the substitutions (29) and (30), unless K 1 ---- K 2 = K3, for which case the special forms of (25) and (29) were given b y W a n n i e r 7). Because of (29) and (30) the relations (25) and (26) can be brought into a symmetrical form:
/~ (K,) : ~8z (K,)/F~ (K,) = ~z(K,)/F~(K,) s o o =/~rfK,), o go (K,) = ~
(K,)/Gv(K,)
= ~ (K~)/G.~ (K~) =
g.9(,). K~
(31)
(32)
ORDER-DISORDER
IN HEXAGONAL
LATTICES
435
W i t h the aid of (20) we can derive from these symmetrical relations the two other symmetrical ones: /~(K,) =/~.(K*) = ~ ( K , ) / F ~ ( K , ) = ~ ~6(K,• )/F#(K,¢ ) = / , ~ ( K ,¢ ) (33) and gz(K,) = g#(K*) = ~ ( K , ) / G ~ . ( K 3 _. ~4( K i )o/ G ~ ( K , o) : g~r(Ki), o (34) with
Fs~(Ki) ---- (Sh 2K 1 Sh 2K 2 Sh 2K3)4/G~(K~)
(35)
and
G~(Ki) :
(36)
~ (Sh 2K I Sh 2K 2 Sh 2K3)2/Fz(K~).
F r o m the invariance of / ~ ( K l, K 2, K3) and g~(K 1, K2, K3) with respect to the substitution (29) and their s y m m e t r y in K 1, K 2 and K 3 we can prove, t h a t for positive values of K I + K 2, K 2 + K 3 and K3 + K1, I~(K,, K 2, K3) ---- 9(y,, Y2, 3'3) (37) and
g~(K1, K2, K3) :
Z(Yl, Y2, Ya),
(38)
where 9(Yl, Y2, Y3) and 7.(Yl, Y2, Y3) are one-valued functions of the variables Yl ~--i1 Sh 2K1/(Ch 2K~Ch 2K 2 Ch 2 K 3 + Sh 2K1 S h 2 K 2 Sh 2K3) ,cycl. (39) These y~ vanish both at very high and v e r y low temperatures and are also invariant with respect to substitution (29). The possibility of using such y's is suggested by the appearance of the parameters _ 1 Sh 2
2KI/(Ch 2K 1 Ch 2K2) and i1 Sh 2K2/(Ch 2K 1 Ch 2K2)
in the partition function of a rectangular lattice (see formulae 109a and 109b in the article of 0 n s a g e r 2)). The proof is as follows. Denoting b y ~i(x 1, x 2, x3) one-valued functions of the parameters x,, x 2 and x 3, we recognize easily, t h a t for positive values of K, + K 2, K 2 -Jr- K 3 and K3 + K1: / z ( K , , K 2, K3) = ~, (p, q, r) with
p = exp (-- 2K 2 - - 2K3),
q -- exp (-- 2K 3 - - 2K~), r :
exp (-- 2K~ - - 2K2).
(40)
F r o m (29~, (31) and (40) we conclude ~I(P, q, r) = (l(p °, qo, ro)
(41)
436
with
R.M.F. HOUTAPPEL
p°=
l+p--q--r l+p+q+r
,
qo~
1--p+q--r l+p+q+r 1--p--q+r l+p+q+r'
(42)
and we see t h a t
Yl ----- ( P -
qr)/(1 + p2 + q2 + r 2) --_ = (#o - - qo ro)/(l +/002 + qo2 + to2), cycl. (43)
The substitution (42) can be brought, by the transformation,
u=
( r + 1)~(p--q),
v=
(1 + p + q - - r ) / ( p - - q ) ,
w -~ (p + q + r - - 1 ) / ( p - - q ) ; q = (v + w - - 2 ) / ( 2 u +
p = (v + w + 2)/(2u + v - - w ) ,
v - - w ) , r = ( 2 u - - v + w)/(2u + v - - w ) , (44)
with the corresponding expressions of u °, v ° and w ° in po, qO and r °, into the form u °=u,
Writing
v°--v,
w °---w.
(45)
/~(K 1, K 2, K 3) = ~j(p, q, r) = ~2(u, v, w),
we derive from (41), (44) and (45) ~2(u, v, w) -~ ~2(u, v , - - w ) . Consequently: [~(K~, K2, K3) = ~2(u, v, w) = ~3(u, v, w2). F r o m [¢(K1, K2, Ka) = ]~.(K2, K1, K3) , (40) and (44) it follows t h a t ~3(u, v, w 2) = ~ 3 ( - - u, - - v, w2).
Therefore" [~.(Kl, K2, Ka) = ~3(u, v, w) = ~4(u/v, v 2, w2). If K x + / { 2 , K2 -J- K3 and K 3 + K 1 are positive, we see by (40) and (44), t h a t u/v will be positive too, and we can write
/~ (K1, K2, K3) ---- ~4(u/v, v 2, w21 --_ ~5(u~-/v2, v 2, w2). F r o m (43) and (44) we find that" 2yt -
(v 2 - - w 2 +
4u)/(v 2 +
w2 +
2u 2 +
2),
2y2 = (V2 - w 2 ~ 4 u ) / ( v 2 + w 2-{- 2u2 + 2), 2y 3 = (2U2 - - v 2 - w
2 + 2 ) / ( v 2 + w 2 + 2u ~-+ 2).
We can conclude t h a t u 2, v 2 and w2 are one-valued functions of y~, Y2 and Y3 if u is a one-valued function of yt, Y2 and Y3. Now, by eliminating v2 and w 2 we obtain (Yl--Y2) (u2 + 1) - - (1 + 2ya) u -~ 0,
ORDER-DISORDER IN HEXAGONAL LATTICES
437
which is a reciprocal e q u a t i o n of second degree for u and b y which u is defined in unique w a y for K 1 + K 2 > 0, K 2 + K 3 > 0 and K 3 + K~ > 0, in view of the fact t h a t then, according to (40) and (44), I z~l > 1, Finally we conclude, t h a t for positive K 1 + K 2, K 2 + K 3 and K , + K 1, [~(K t, K 2, K3) is a one-valued function of Yl, Y2 and y,, q.e.d.. The same proof applies to g~(K 1, K 2, K3). If K;, K 2 and K 3 are positive, K~, K * and K * will be positive too and vice versa. Hence we conclude from (9), (33), (34), (37), (38) and (39) t h a t in this case
/~(K t, K2,
K3) =
(46)
9 ( z l , z 2, z3),
g#(K 1, K2, K3) ~-~ Z(zl, z2, z3) ,
(47)
with z I -----½(Sh 2 K 2 Sh 2K3)/(Ch 2 K t Ch 2 K 2 Ch 2 K 3 + 1), cycl. (48) F u r t h e r we derive from (37) and (38) and from the fact t h a t ~ and F~G~ are positive for positive values of K 1 + K 2, K 2 -~- K 3 and K 3 + K 1, t h a t then:
h~.(Kl, K2, K3) = 1 2
= ~~¢ (K1, K v K3)/(Ch 2 K 1Ch 2 K 2 Ch 2 K 3 + Sh 2 K 1Sh 2K 2 Sh 2K3) =
= ( / ~ g ~ y o22p2 y 3 2/ 2 ) B
1,6
= ~(y,, y2, y3),
~(Yt, Y2, Y3) being a one-valued function of Yl, 3'2 and Y3 also. In addition, we derive with the aid of (20) for positive K 1, K 2 and K 3"
h.~(Kl, K2, K3) = 4 -- ~1 ,t.~(K 1, K 2, K3)/(Ch 2 K 1 Ch 2 K 2 Ch 2 K 3 + 1) = $(z 1, z2, z3).
Resuming the main results of this section, we state t h a t for positive values of K 1 + K 2, K 2 + K 3 and K 3 + K1:
h% = ¼ ~2/ ( C h 2Kt Ch 2 K 2 Ch 2K~ + Sh 2K~ Sh 2K2 Sh 2K~) == ~(Y~, Y2, Y~), and that for positive values of K l, K2 and K~ : •t~. =
~6/~., .~,_~ = ~0(z~, z~, z3),
g.~ = ~8I G ~ = X(z~, z~, Z3 ) , h.9 = ~2~/(Ch 2K~ Ch 2 K 2 Ch 2Ka + I) = ~(z~, z2, z3),
(49)
438
R. M. Y. HOUTAPPEL
where 50, Z and ~ are one-valued functions of their three parameters, and where F¢, G.~, F&, G~, Yi and z~ are defined by (27), (28), (35), (36), (39) and (48). Yi and zi vanish both at very high and very low temperatures. The first three functions,/~, g.~ and h E, are, for all real values of K 1, K 2 and K a, invariant with respect to substitution (29) and the last three functions,/.~, g.~, and h.w with respect to substitution (30). Later, in section 11, it will be shown t h a t the third and sixth of the relations (49) will be valid with a one-valued function ~ for 'all real values of K 1, K 2 and K 3 and the exact form of t h a t function will be given as a double integral. 7. The development o/ 9, Z and ~ in power series. We can use the relations (49) to construct expressions of 2~ and ,tW which will be valid both at very high and very low temperatures, in the form of a product or a quotient of an e l e m e n t a r y function and a power series in Yi respectively z;. To f i n d t h e s e power series we can develop the partition function 2z for positive interaction energies J l , J2 and J3 in an elementary way, as K r a m e r s and W a n n i e r S ) did for the quadratic lattice, both in a power series of K~, K 2 and K 3 valid at high temperature, and in a power series of exp(--K1), exp(--K2) and e x p ( - - K a ) , valid at low temperature. Only one of these expansions is needed to construct the power series for 50, Z and ~, but the other can be used as a check. We find (denoting by Ep the sum over the six permutations of the indices 1, 2, 3): 1 Zp (2 + 3 K~ + 4 K I K 2 K a + 4KI 1 4 -}- 721 K 12K 22 + 14 K~K2Ka + ;tz = ~, I I O~ K~K 4 22 17~ Y'¢"iTc'2Y¢"2+ . . . ) (50)
for high and ~t~.= eKI+K2+K3 • gXt,(l 1 + L i L 2 L a + 3 L 21L2La-2 2 2 + 3L~L~La+ L 21L2La -~ 9 T3T2T2 9 T3T3T2 (Li -~ exp ( - - 4Ki) ) for low temperatures. And we derive : 16/50(y,, Y2, Ya) = ~ Ep(Y,V2Ya + 3 YlY2 2 2 + 12 Y~YzYa + 24 Y~Y242+ . + 2 8 .YtY~3 2 . 2 .2 + 42 Y~Y2Ya + 168 YlYfiYa 6 2 + 120 4 4 + 3 3 + 84 YxY2 YlY2 + 762 y4y~y~ + 144 y7yz,% + 2592 YtY~Ya ~ a + 1560 YtY~Ya3 a 3 . . ) , (52)
ORDER-DISORDER
1
. 2.2.2
16 %(Y,- , Y2, Ya) Y , Y # 3 :
IN HEXAGONAL
1
439
LATTICES
22
~ Ep(Y,YzYa + 3 Y,Y2-- 6 Y~YzY3- - 12y4y22 222 1 4 y l y ~ y 3 3 y l y 2..5y a 48y~y~y 3 6 Y 6l 2Y 2 - - 2 4 y l4y 2 4 --4 2 2 - - 6 YlY2Ya 7 - - 147 YlY~Ya - - 3 9 6 [email protected] - 300 YlYp.W3 . a. a. a . . ), (53) - -
- -
.
~(Y~, Y2, Y3) = ~I Y>(1 - - 3 y~ - - 4 Y l Y z % 3 y2 60yaly2y a 6 y ~ 1 3 8 4 2 -YlY2 --- -
- -
2,2 - 15 YlY2 2 2 2 ..). 125 YiY~Ya
(54)
I n t r o d u c t i o n o/ r o t a t i o n s i n 4 n - d i m e n s i o n a l space. We shall now show how our p r o b l e m , as in the r e c t a n g u l a r case, can be reduced to the t r e a t m e n t of 4n-dimensional r o t a t i o n s only. I t will be unders t o o d t h a t a 4 n - d i m e n s i o n a l r o t a t i o n is a linear t r a n s f o r m a t i o n of 4n coordinates, which leaves the s u m of their quares i n v a r i a n t . We label it p r o p e r or i m p r o p e r according to w h e t h e r the d e t e r m i n a n t of its m a t r i x is e q u a l to + 1 or - - 1. F r o m the t r e a t m e n t of the r e c t a n g u l a r lattice b y K a u f m a n 3) it can be u n d e r s t o o d t h a t the i n t r o d u c t i o n of 4n o p e r a t o r s / 1 , / 2 , . • F4,, 8 .
I*2r--I
=
I~2r
---
ClC2
" " Or--1 Sr,
(55)
i C l C 2 • . Cr__ 1 CrSr,
will g r e a t l y simplify the calculation of, s a y ;t~. T h e o p e r a t o r s / ' k are, as was shown b y K a u f m a n, related to r o t a t i o n s in 4n-dimensional space in the following way. F r o m the algebraic p r o p e r t i e s of si a n d c i, (11), it follows r~r, + r,r~ :
(56)
2 ~,.
F r o m this we derive /'1.
if i # k , l ,
exp (~0 rkr,), r/. exp (--{o r~r,)= c o s O F k - - s i n O F l
if ] = k ,
(57)
sin 0 Fk + cos 0 F~ if ] = 1. I t follows t h a t e x p ( ½ 0 P ~ P l ) ~/Y:" /=1 x / I / ) e x p ( - - + + r : , ) xi, :
where
= ~ ' " x~' < , ~/=1
X~4n
z-.k=1 ~k/Xk •
(58) (59)
This r e p r e s e n t s a p r o p e r r o t a t i o n © t h r o u g h the angle 0, in which only x k a n d x I are involved. Expressing, f r o m the definition (55), the old o p e r a t o r s in t e r m s of the new ones, ek = -
i r~_,r2k,
s, = (-
i) ~-' r , & .. r ~ _ , , (60)
sks~+ I =
- - i F2~ P2k +1,
S2,,Sl = + i U r4,, Fl,
440
R. M.F. HOUTAPPEL
where U is u
=
=
r4,,,
we find t h a t I71, V2, Va and V4 in (13) are products of factors of the form e x p ( - - i KiF~11~+t) and exp(iK iU11.,110. All factors, except those of the latter form, induce b y (58/ rotations on (x t, x2, x 3. . . . . x4. ). Because of the factors of the latter form, the product V I V 2 V 3 V 4 itself does not induce b y
(V, V~V3V4) (Z~, xirj) (V, V~ V3 V4)-' a rotation on (x 1, x 2. . . . . x4. ). To bring these annoying factors in line with the others it is necessary to split up V~,2,, into a part, acting on even vectors only, and a part, acting on odd vectors only, b y V~,~,, = ~(1 + U)V~,~,, + }(1 - - U)V~,~,,.
(62)
It follows from (10) and (61) t h a t }(1 + U) acts as a u n i t - m a t r i x on the 22"-kdimensional space of even vectors and reduces to zero all odd vectors and t h a t ½(1 - - U) acts as a u n i t - m a t r i x on the 22'`-'dimensional space of odd vectors reducing to zero all even vectors. Therefore ½(1 + U)V~,2, , has the same eigenvalues belonging to even eigenvectors as V~,2,,, while its eigenvalues belonging to odd eigenvectors are all zero. On the other h a n d ½(1 - - U) Vz,2. has the same eigenvalues belonging to odd eigenvectors as Vz,2,,, while its eigen values belonging to even eigenvectors are all zero. Since (1 + U) U =
1+ U
and
(1--U)
U=--(1--U)
and since both 1 + U and 1 - - U c o m m u t e with all products PkPl, we see t h a t (1 + U) exp (iKiU114, , Yl) = (1 + U) exp ( i K i P4,,11,) and
(63)
(1 - - U) exp (iKiU114, , f11) = (1 --- U) exp (-- i K i 114,,110 (64)
and t h a t we therefore can omit U in the factors of V~ and V3 in ½(1 + U)V~,2. and replace it by - - 1 in the factors V, and V3 in -
-
tr)vz,2,,.
So we find: Vz,2. - - ½ ( I + U ) (2 Sh 2K3)" V o + ½ ( 1 with
V~ = V ~ V 2 V ~ V 4 ,
U) (2 Sh 2K3)" Vo
Vo = It'~V2V~V4;
(65) (66)
441
O R D E R - D I S O R D E R IN HEXAGONAL LATTICES
V~ = e x p •exp V2 = exp W = exp •exp
(--iK,PJ'a).ex p ( - - i K , P 6 f ' 7 ) .. .. exp (--iK,f',,,_21"4,,_l) ( - - iK2P4Ps), e x p ( - - iK21"sP9).. .. exp ( ± iK2I',,Pt) , (--iK;I']I'a).ex p (-- iK*I'5I'6).. ." exp (--I,K3/~4,v__3P4n_2), ( - - iKJ'=r3), e x p ( - - iK=r6rT).. ." exp ( - - i K 2 r,,,_~r,._,) ( - - iK,I'4I's) . e x p (-- i K , r s F g ) .. ." exp (4- iKll",,I'l), •
,
V4 = exp (--iK~l'aI'4).ex p ( - - i K ~ / ' 7 / ' 8 ) . . . . exp (-- iK~/'4,_,['4, ). B y (65) and (66) we have reached the aim of the present section.
9. Explicit representation o/ Vo+ and V o as rotations. F r o m (58), (59) and (66) we find t h a t V#, V2, Va~, V4 and their products Vo+ and V~ induce b y V(Z2"__,xkFk)V - t ( V = V~, V 2, V~, V 4, Vo~) on (Xl, X2 . . . . X4n ) the rotations ~ + , ~]-, ~ 2 ' " ~ + ' ~:)3-, ~)4' ~)+ : J~)4~+~)2 ~)+ and £)0 = ~4t~3£)2t~] -. For their matrix-representations we find:
• C1 iSI --iS l C 1
~=
c;
is; I o
C2 iS2
o I
, ~2=
C;
- - i S 2 C2.
,
'i~;
- - ~ S 3 C3 .
i o/
=CiS 2
o 1
(
1 o
t CI
~=
C2 iS2 --iS 2 C2
C] --iS1
iS]
Cl. C]
TiS[
C3 zS3
-is; c;
1 o
o I.
"C; --iS a
is; I ' C3
with the abbreviations: Ci - - C h 2 K i,
Si = S h 2 K
i,
C*=Ch2K*,
S*=Sh2K*.
Carrying out the multiplications O + = ~ 4 " ~ + ' ~ 2 " ~ + ~ o ---- 94" 9 7 " 02" ~]- we find: a c
b a
and
(68)
b •
b
(67)
•
b
c
a
442
R. M. F. H O U T A P P E L
a, b and c being the 4-dimensional matrices' +CIC2C; + S l S 2 a =
b=
--ClSlS;
-c s c;s;
,-ic,s
+iC2S2S~ '
, --CIS2C;S;--SIC2N~ , --iNIS2C;S;--iC1C2G;
c;'-is,c2c;,
,
0
,
0
0
,
0
,
0
\+~CIS2C3 +~SIC2C 3 , 0 0
,
,
+SxS C;:+c,c c;
0
--Cl S2S3--SIC2C3S3,
I
,
+$C*1S;
;
+ic,c
s;+is,s
cb;]
+C1C2C1+SIS2C 1' /
o ) 0
,
- - , ClSlS3
,
+SIS 3
--CISIC;S;
,
- - i SIC3S3
O
,
0
, --i S1C2--i CIS2C; \
0
,
0
--C2S2S~
0
+$ 82C3S3
0
0
0
0
,
0
0
0
•
,
2
*
)
*
+S;S;'
10. Diagonalization o/ ½(1 + U)V~,2, ,. To compute the partition function per atom t~ of a two way infinite triangular lattice we need only calculate the largest eigenvalue of V~,2, ,. Since this eigenvalue is positive and related to an even eigenvector, it is the largest eigenvalue of 2(1 + U)Vz,2, too. Therefore we shall diagonalize only ½(1 + U)V~,2, ,, although the same procedure is also applicable
to ½(1 - Each proper 4n-dimensional rotation which can be diagonalized, can be transformed by a suitable rotation of the coordinate system to the standard form 9),
- - i Sh 71
Ch 71
Ch 72
i Sh 72
--iSh72
Ch72
(7o)
.. Ch 72n --iSh72.
i Sh 72. Ch72. /
which represents a product of 2n comnmtative rotations through the angles #~ -- - - i y i . By applying a reflection, which is an improper rotation, relatively to the coordinate x2~, we can make ~gk and 7h change sign. £)+ can be diagonalized as will be shown in next section. Thus there exists a proper or improper rotation, ~ , such that ~3o+ = ~ . £ 3 + .~R-'
(71)
has the form (70), in which the real parts of 71, 72. . . . , 72. are all
443
ORDER-DISORDER IN HEXAGONAL LATTICES
non-negative. The latter will be used later on. The eigenvalues ~i, 1/~/i of 9 +, and therefore also of ~D~-, are given b y ~i = exp 7i.
(72)
In next section we shall indicate how t h e y can be calculated. We shall now first show how the diagonal form of ½(1 + U)Vza,, is related to the Yi. It can be shown t h a t there exists an operator M(~R), which induces the just mentioned rotation ~R, while (M(~R)) -1 induces ~R-1, and therefore has the p r o p e r t y t h a t V + ---- M(,gt). V + . (M(9t)) - I
(73)
induces the rotation t0 +. This follows from the fact t h a t each proper rotation can be obtained b y an appropriate succession of twodimensional proper rotations, each involving only two coordinates, while for an improper rotation just one additional reflection, relative to, say x I, is needed 1°/. Actually, a rotation through the angle O, involving x k and x~ only, is caused b y transformation of _,=E 4'' i x~Fi w i t h exp (½0/'k~) and the reflection relative to x 1 b y transformation with/~2/'3 " ' " 1~4,," B u t the identical rotation ~00+ is induced also b y V0+ = exp (½ 0, P~/'2) • exp (½ 02/'aP4/• . . . e x p (½ 0 2 , / ' . _ ~ r . , ) = -- exp [½ (y,e, + Y2e2 + .. + y2,,e2,,)], (74) i.e.
Vo+ ,-,=(E4"1 x~ /';) (V+o)-1 :
for all sets of x t, x 2. . . . .
IT+ v-"~=1~'~4"x~ F;) (V~-)-'
x4,, and thus
( V + ) -1 V + --i=~'~4"1 Xi Ill __. ,{~4"--i=1 Xi 1"~i) (VO+)-1 v+"
Now, it can be shown from (5), (10) and (60), t h a t each 22"-dimens ional m a t r i x can be written as a sum of products of I"1,/'2 . . . . /'4.. Consequently (V+) - 1 . V0+ c o m m u t e s with all 22"-dimensional matrices and is therefore a multiple of the 22"-dimensional unitmatrix, so t h a t : vo+ = Q vo*, (75) where Q is a scalar. We conclude t h a t q22,, = I
(76)
from the fact t h a t b o t h Vo+ and Vo+ are unimodular, because all
444
R. M . F . HOUTAPPEL
their factors are so. The latter can be recognized on writing the factors exp ( - - i K i F 2 ~ _ l F 2 k ) ,
exp (--iKiF2~F2h+l)
and exp ( + iKiT'4.['t)
in the forms exp ( K i ch), exp ( K i
SkSk+l)
and exp ( - - K i c l c 2 .. c2,,s2,sl),
diagonalizing afterwards the first and the last form respectively, by transformation with 2-4(sk + ck) and 2-"(s I + i c l s l ) (s2, + ie2,,s2, ) r12,-1 xxt~ 2 (% + c,) into exp(Kish) and exp(Kisls2..s2, ). Indeed, the diagonal matrices exp ( K i sk), exp (Ki s,sk=l) and exp ( K i sis 2 .. *2,) are unimodular, because half of their diagonal elements are the reciprocal of the others. Thus, we see from (65), (73), (74) and (75) that
M(~). ~(I + U)V~,2,,. (M(~))-' ---2. 7ici) • (77) -- M(~R).½(1 + U)" (M(~)) -1. (2Sh 2K3)"0 exp(~1 Y'i=t - -
As to the transformation of 1 -~ U we remark that 1 + U will be invariant with respect to transformation by each factor exp(½ o r h r ~ ) of M(~), because exp (½ ~9r~F~) . r i P 2 . .
[4,.exp
(-- ½t~ P~F~) = F i r 2 . .
F4,,
(78)
and that 1 + U is changed into 1 - - U by transformation with a factor F 2 F 3 .. r 4 , , occurring in M(~R) if ~R is an improper rotation, because (r~r3 . . r,,) . rtr~
. . v~,,. ( r ~ r ~ . . r ~ , ) - ' = -
r,r~
. . r~,. (79)
Consequently we conclude from (61), (77), (78) and (79) that M(~R).½(I + U ) V ~ , 2 , .
(M(~))-'
=
---- ½ (1 + U) (2 Sh 2K3)"@ exp(½ "~i=*~2"7ici ) = :
½ (l ~:_
2- 17ici) (80) • . C2n ) (2 Sh 2K3)"0 exp(½ Zi=
ClC 2
with + or - - according to whether 9%is a proper rotation or not. By an additional transformation with T = T - l = 2-" 2. l(s; + c;) each ei will be changed into the corresponding s~ so that T.M(~R). ½(1 + U)V~,2,. ( M ( ~ ) ) - ' . T - t --- ½ (l 4-
s,s 2
.
.
=
s2.) (2 Sh '2K2)"~ exp (½
2. r;%),
(81)
O R D E R - D I S O R D E R IN H E X A G O N A L LATTICES
445
the last form being a diagonal matrix, in which the + or ~ sign m u s t be used according to w h e t h e r ~t is a proper rotation or not, and in which 0, with modulus 1, must be chosen so, that the largest eigen-value is real and positive.
1 I. Computation o / ~ . and o~ ~s~. Since ~,2,, is real and positive, it must be equal to the modulus of the largest eigenvalue of ½(l + s t s 2 .. s2,,) (2 Sh 2K3)" exp (} Ej=12, Yi si) or of
{(l - -
SIS 2 ..
$ 2 . ) (2
Sh 2K3)" exp (} X~"__l yis/),
depending on whether ~R is a proper rotation or not. Since no yi has a negative real part, it is easily seen that this largest eigenvalue will be (2 Sh 2K3)" exp (} E2"=1 Yi) (82) in the first and
(2 Sh 2K3)" exo- (} ~2,, ~i= 1 7i
--
Y ,)
(83)
in the second case, y' being the Yi with smallest real part. In the calculation Of t~ Lim,, ->oo~1,2,, b o t h formulae lead to the same limit and we can write In 2z = Re[Lim,,~oo (1/4n) Y~i=l 2n Yi + } In (2 Sh 2K3) ]. (84) Here we need not consider further the form of ~R or whether it is a proper rotation or not. We have only used the fact that such a rotation, with p r o p e r t y (71), and consequently the matrix M ( ~ , exists. Using the identity Yi = (1/2z~) f02= In (2 Ch Yi - - 2 cos co) dco,
(85)
valid for Re[yj] > 0, which is the case here, we conclude In ~Lz :
Re[} In (2 Sh 2K3) +
+ Lim._+oo (1/8~n) ~2,, "~'d/=1Jf2= 0 In {(~i + 1/~li) - - 2 cos w} do~], (86) with ~1i, defined b y (72), involving half of the eigenvalues of ~ - . We shall show now how the eigenvalues, ~?i, 1/~i, of £)0+ can be calculated. If, in the formula (68) for £)0+, a, b and c were scalars, ~3o+ being an n-dimensional matrix, its n eigenvectors vr would have the form (vr)k = ek(2,-l) (k = 1, 2, . . . , n), with
e = exp (iz~/n),
(87)
446
R . M . F . HOUTAPPEL
and the corresponding eigenvalues would b e : ~/r =
a +
e 2'-1 b +
e - ( 2 ' - 1 ) c.
B y a slight generalization we find that for a, b and c being 4-dimensional matrices, 9 + can be reduced to a form in which n 4-dimensional matrices, d, =
a +
b e 2'-1 +
c ~-(2,-1)
(r = 1, 2 . . . . .
n),
(88)
are placed along the diagonal. Using (69) we find that the eigenvalues of d, are the roots of the fourth degree equation :
~14__ (A, + iA ',) ~13 + B, ~2 __ ( A , -
iA ',) ~; + 1 = 0,
(89)
with
A, ---- 4C~C2C~ -}- 2S,$2 (1 + C .2) + (S,'2+ $2)2 Sa,2cos
[(2r--ll=/n],
A; = (S2 -- S2) S.2 sin [ ( 2 r - - 1)~/n], B, = 2C .2 + 2(C2C~ +
2 2 (a s,s2)
+ 8 s,s
c,c
+
(90) +
c* - - 4 s,s
s
cos [(2r - -
(Abbreviations: see (67)). The fact, that equation (89) has, in general, four different roots, shows that d, and therefore 9 + can be diagonalized as was stated in section 10. Fortunately, the computation of 2~ does not involve the explicit solution of (89). To see this, we consider the n reciprocal eighth degree equations ~;s-- 2Aj;~ + (A,2 + A~2 + 2B,)~;6-- 2A,(1 + B,)~;s + + (2 + 2A,2-- 2A; 2 + B2,)~4--2Ar(1 + Br)~'l 3 Jr+ (A~ + A; 2 + 2B,)~2-- 2A,~? + 1 =
O,
(91)
which we obtain b y multiplying each fourth degree equation (89), for which r = rl, with that for which r = r 2 = n + 1 - - r I. We see then that the 8n roots, ~r,l' 1/r/r,l, Tit,2, 1~'fir,2, T]r,3, l /'r]r,3, TIr,4, l /~r,4,
(r :
1, 2 . . . .
n),
of these n eighth degree equations involve twice all the eigenvalues of 9 + and that In 2z = Re[½ In (2 Sh 2Ka) + + Lim,,___>oo (l/16~n) ~,=ls0s~" f2,,s'4~i=l In {(~,.i+ liar,i)--2cosco} dw]. (92)
ORDER-DISORDER IN HEXAGONAL LATTICES
447
W e now replace the sum u n d e r the integral sign b y a single logar i t h m ; use the relations between the s y m m e t r i c a l functions of an eighth degree e q u a t i o n and its coefficients; continue to the limit n - + oo b y changing the s u m m a t i o n over r into an integral o v e r co' = (2r - - 1)rr/n and bring the first t e r m u n d e r the double integral also. Using the relations (9) we get then a double integral of a logar i t h m of a f o r m / , which can be factorized into / ----/t[2 so t h a t 1
In (,1~./2) -- 327r2 [f~'~f2o~ In [1 d~, d a / + f~"~'~ In/2 &o do'i, (93) with
1,,I2= (c,c2c3+ s,s2s3--$3 cos o,)~--½{s~+ s~+ 2s,ss(cos o~+ cos o~')+ + (S~ + S~) cos oJ cos co' + (S~ - - $22) sin co sin co'},
(94)
the + sign referring to /, and the - - sign to Is. The two double integrals in (93) are equal. After s u b s t i t u t i n g co = o I + c% and oJ' = co1 --cos, we see easily t h a t 1, can be factorized again, and we obtain finally: in (,1~/2) =
~fo2"~fo2'~In {Ch2K1Ch2KsCh2K 3 +
Sh2K1Sh2KsSh2K 3 - -
- - Sh2K, cos co1- - S h 2 K 2 cos c % - - S h 2 K 3 cos (~o, + c%)}doJ, dw 2. (95) F r o m (95) and the dual relation, (20), we derive for the partition per atom, ,1,9, of a two w a y infinite h o n e y c o m b lattice: In (,l.9/2) = 16-~2fo2'~f~'~In {{Ch2K,Ch2KsCh2K3+ 1 - - S h 2 K s S h 2 K 3 cos w , - - - Sh2K3Sh2K , cos co2 - S h 2 K , S h 2 K s cos (oJ, + c%)} dco, do~s. (96) F r o m formulae (95) and (96) the s y m m e t r y of ,1~ and ,1~ with respect to K,, K s and K 3 is easily verified. In accordance with the third and sixth of the relations (49) and our r e m a r k at the end of section 6, we conclude from (95) and (96), t h a t for all real values of K,, K 2 and K 3 ¼ ,1~:/(Ch2K, Ch2K s Ch2K 3 + Sh2K, S h 2 K s Sh2K3) =
= E (Y,, Ys, Y3), (97) ,1~/(Ch2K 1 Ch2K s Ch2K 3 + 1) ---- ~(z 1, z s, z3),
(98)
448
R. M.F. HOUTAPPEL
where Yi and ::i are defined b y (39) and (48). So we find for ~ the expression : In ~(x t,
I
x 2, xa) 2~r
=
2"n-
-- 4~2fo ~o ln{1--2XlC°S°~l--2x2c°sc°2--2x3c°s(°)l+°@}d°)Idr%" (99) This double integral is convergent for all leal values of K~, K 2 and K a, if we read y~ or z~ for x v We recognize that the power series, (54), which can be deduced from this expression, is also convergent for all real values of Ki, K 2 and K 3, and this remains true if we replace in it Yi b y z~. O n s a g e r's result for the partition function per atom, ~ , of an infinite rectangular lattice, In ( ~ / 2 ) = 8@f02'~f02'~In (Ch 2K~ Ch 2K 2 - Sh 2K, cos co~---
Sh 2 K 2 cos w2) &o 1do)2,
(100)
can be obtained both from the expression (95), by using
Jl~i(KI,K2) = 2t~ (K1, K2, 0), and from the expression (96), b y using 2~(K1, K2)
=
Limjs___~o o
(~,?(K1, K2, Ka)) 2 exp (--K3).
The reason for the former is trivial; the latter can be understood bv observing that, if the interaction Ja is very large, the spin variables of each pair of atoms between which the interaction fa takes place, will always have equal values, i.e. such a pair behaves as one atom with constant inner energy - - f a . For the computation of the partition functions of finite hexagonal lattices it is necessary to know all eigenvalues of V~,2,,,,,, and V~o,2,,2,,. The eigenvalues belonging to even eigenvectors can be found from the explicit solution of the fourth degree equations (89), if we know whether ill in formula (71) is a proper rotation or not. Those belonging to odd eigenvectors can be found in an analogous w a y starting from V o (see (66)). The exact forms of the rotation ~R related to V +, and of the corresponding rotation related to Vo, are needed moreover if we wish to calculate the average correlation between the spin variables of two non-neighbouring sites in the lattice according to the method of K a u f m a n and O n s a g e r 11). However, we shall not enter here into these topics.
O R D E R - D I S O R D E R IN H E X A G O N A L LATTICES
449
12. Transition temperature. T h e p a r t i t i o n functions Jlz and ~ d e p e n d on the t e m p e r a t u r e from the relations K 1= J l / k T , K 2 = J J k T and K 3 = J 3 / k T only. T h e interaction energies, J1, J2 and J3, are c o n s t a n t s belonging to the lattice. In the expressions (95) and (96) for Jl~ and ;~ the a r g u m e n t of the l o g a r i t h m u n d e r the double integral sign, Ch 2 K 1 Ch 2K2 Ch 2 K 3 + Sh 2 K 1 Sh 2 K 2 Sh 2 K 3 - - - Sh 2 K 1 cos coI - - Sh 2 K 2 cos a~2--Sh 2 K 3 cos (~oI + o~2). (101) and ½ {Ch 2 K 1 Ch 2 K 2 Ch 2K~ + 1 - - Sh 2 K 2 Sh 2 K 3 cos ~oI - - - Sh 2 K 3 Sh 2 K 1 cos ~o2 - - Sh 2 K 1 Sh 2 K 2 cos (oJ1 + c%)} (102) respectively, c a n n o t be negative for real K 1, K 2 and K 3. ~ , t a and all their higher derivatives with respect to the variables K i are continuous, e x c e p t for those sets of values of K 1, K 2 and K 3 which, for a p p r o p r i a t e values of a~l and oJ2, cause (101) or (102) respectively, to vanish. F o r t h e s e sets the second derivatives will diverge. So we find t h a t at a t e m p e r a t u r e , To, at which K 1 = Jl/kTc, K2 ~ J2/kTc and K 3 = J3/kT~ form such an exceptional set of values, the specific heat per atom, C
:
k
El.i= 1,2,3K i K i ~ In 2/aKi~K i
(4 = 2z or 1s~),
becomes infinite as - - In [ T ~ Tc I (as in the case of the r e c t a n g u l a r lattice). F o r given f l , .72 and Ja not more t h a n one such Tc can appear. On the analogy of the rectangular lattice we assume t h a t this T~ is a transition t e m p e r a t u r e between two phases. At T < T, the interactions between the spins of neighbouring atoms would m a i n t a i n a long-distance order, while at T > T~ only a short-range order would obtain. While e v e r y h o n e y c o m b lattice has a transition t e m p e r a t u r e , it t u r n s out t h a t t h e r e are exceptional t r i a n g u l a r lattices for which no transition t e m p e r a t u r e exists. These are the lattices for which one or all three interactions f~. are negative, while all three interactions, or at least the two weakest, are equal in strength. As to the calculation of Tc in all o t h e r cases, we notice t h a t ~ is i n v a r i a n t with respect to the simultaneous reversal of the Signs of two interaction energies J i and t h a t we can reduce all t r i a n g u l a r lattices, b y Physica XVI
29
450
R.M.F. HOUTAPPEL
such an operation, to one in which the two strongest interactions are positive, i.e. Jl + J2 > o, J2 + J3 > 0 and J3 + f l > 0. In the latter case Tc will be found from Sh (2]l/kTc) Sh (2]2/kTc) + Sh (2J2/kT~) Sh (2]~/kT~) + + Sh (2J3/kTc) Sh (2]i/kT~) ---- 1. (103) In fig. 4a we have illustrated in a triangular diagram the dependence of kTJ(Jl + J2 + J3) on the ratios of Jl, J2 and J3 if the two strongest of these interactions are positive. In this case -(fl +J2+J3) is the energy per atom in the completely ordered state at T = 0. We see that T, vanishes on the external bisectors of the basic triangle, representing thus the cases in which no transition temperature appears. The special case J r = J2---J3 < 0 is represented, after reduction as mentioned above, by the intersections of the external bisectors. A
o
A o
C Fig. 4a. Triangular lattice. Curves of constant kTd(Jl + J2 + J3) for ]x + J2 > o, J2 + J3 > o and J3 + JI > 0, calculated from formula (103).
Fig. 4b. Honeycomb lattice. Curves of constant kTJ½(J l + J2 + J3) for J l > 0, J2 > 0 and J3 > 0, calculated from formula (104).
Jl, J2 and 9r3 are plotted as triangle coordinates with A B C as basic triangle and centre D.
The appearance of a transition temperature for IJll ~ If21 ~: IJ3[ can be understood in the following way. The two strongest interactions will, at low temperatures, build up a long-distance order, analogous to t h a t of a rectangular lattice, while the third, if it
ORDER-DISORDER IN HEXAGONAL LATTICES
451
counteracts, will only lower the transition temperature. One can also understand why there are cases, as previously mentioned, in which no transition temperature exists. This is because in these cases the tendency of the strongest interaction to build up a longdistance order, together with one of the two weakest, will be counterbalanced by the tendency to build up another long-distance order with the second of them. Unlike the triangular lattice, every honeycomb lattice with non zero values of Jl, J2 and J3 has a transition temperature, To. This T~ can be found from the relation Sh (2]J1]/kT~) + Sh (2lJ2l/kTc) + Sh (2]J31/kT,) = = Sh(2lJ~l/kT~) Sh(2lJ2]/kTc) Sh(2lJ3i/kT,). (104)
B y inspection of the honeycomb lattice one can see that its three interactions cannot counteract each other in building up a longdistance order. In Fig. 4b we have illustrated, also in a triangular diagram, the dependence of kT,/~(IJll + IJ2] + IJ31) on the ratios of [Jll, ]J2] and ]J31 for a honeycomb lattice. - - ½(]Jx] + [J2t + ]J3{) is the energy per atom of the completely ordered state at T == 0. 13. Thermodynamic properties o] large isotropic triangular and honeycomb lattices. The partition functions ,~ and 2~ yield the free energy, F, the energy, E, the specific heat, C, and the entropy, S, per atom of large triangular and honeycomb lattices b y the formulae F = E - - T S = - - k T l n 2, E = F - - T d F / d T = - - X i= 1,2,aJi 0 In 2~OKi,
(105)
C ---- k Xi,i= 1,2,3K i K i 02 In 2/OKiOKi, (4 = 7~z or 2~). W e shall n o w consider more closely the isotropic triangular and
honeycomb lattices, i.e. those with equal interaction J in the three different directions J = J1 = J2 -- J3. An isotropic honeycomb lattice has always a transition temperature, To, which depends only on the strength [J[ of the interaction b y Ch (2J/kTc) ---- 2.
(106)
An isotropic triangular lattice, on the other hand, has only a transi-
452
R. M. F. HOUTAPPEL
tion temperature if J is positive. This transition temperature, To, is then given by exp (4J/kTc) = 3, (107) as W a n n i e r 7) found already. For the energy and the specific heat of isotropic triangular and honeycomb lattices with positive or negative interaction energy, we find with the aid of (95), (96/and (105) formulae similar to those Onsager found for the quadratic lattice (see O n s a g e r 2 ) formulae (116) and (117)): E = - - J [Coth 2K + at~(fl)], (K = J/kT) (108)
C=kK2[-2/(Sh2K)2+ - ~
a3)
a
d3
2fl-~ *'(fl)+ 2fl(1-----fl)dK %(fl)3" (109)
Here we denote by et(fl) and e2(fl) the complete elliptic integrals of the first and second kind,
ea(fl) = fo2" 11 x/1--fl(sin 9)2.d%
eu(fi)= fo2" x / l - - f l (sin 9)2.d9.
Table I shows the form of the functions a(K) and fl(K) for the isotropic triangular and honeycomb lattices, and, for comparison, also for the quadratic lattice. We notice the appearance of the functions Fz(K ) = F~z(K, K, K) and Go(K ) = Go(K, K, K) from formulae (25) to (28), which are related to the invariance properties discussed in section 6. TABLE I Lattice type
Temperature (x = exp (2K)) T< T c x2--4x + I >0)
Honeycomb T>T c X 2 - - 4 x + I < O)
Quadratic
T<
Triangular J>0
T>T
Triangular J
--
3 <
~[X 2 -
l[ ( X - - 1)4
O)
1 6 ( x 5 - x4 + x3) 4
(X4-- l) ( x Z - - 4 X q- 1) G~(K)
2 ( x 4 - - 6 x 2 + I)
16(X2-- 1)2 X2
n ( x 4 - - I)
(X2 + l) 4
:~'~/(x2-- 1)~ (x2 + 3) x(x2 - - 3 ) ( x 2 + 1)
c
1
G~(K~ = (x2--l)2(x--l)
4~(x - - t)2~/x~ - - x4 + .T
4X2(X2--3) (X2 + 1)
Tc
( x 2 - 3 > 0)
(x2
(X4-- 1) ( X 2 - - 4 X + l)
~z(x2-- 1) 4xa (3 - - x2)
u(l - - x ) 2 V ' ( l
+ x~)(3--x2)
16x2
1
F~(K)
( x 2 - - Ip (x2 + 3) F~,(K)
(I --x2)3 (3 + x2) (I + x2)3 (3 - - x 2 )
ORDER-DISORDER
IN HEXAGONAL
453
LATTICES
For the first three types of lattices in table I,/3 equals 1 at the transition temperature, and e1(/5) becomes, for this value of /5, infinite as ½In {16/( 1 7- fl)}. However the factor a, which becomes zero at T~, causes the term ae1(/5 ) in the formula for the energy, (108~, to vanish. The term (da/dK)el(/5) in formula (109) causes the specific heat to become infinite as - - In I T - - T~ [. For the triangular lattice with negative interaction,/5 becomes 1 and e1(/5) infinite at T = 0. Nervertheless the factors with which it appears multiplied in the formulae (108) and (109), reduce the respective terms to zero. In table II we have given several numerical data of thermodynamic properties of the different isotropic lattices at extremely low temperatures and at the transition temperature. T A B L E II Lattice type
--
ET=
0
O)
C'(T~
kTc
E ( T c)
F(Tc)
Cc
½clJI Honeycomb
I
1
1.012
0.7698
1.0251
1.303
0.4781
Quadratic
1
1
1.135
0.7071
0.9297
1.359
0.4945
1
1
1.214
0.6667
0.8796
1.391
0.4991
Triangular
j>0 Triangular
j
2K -- _ _
3~V/3
e--8K
c = c o r r e l a t i o n n u m b e r ; i.e. n u m b e r of a t o m s w i t h w h i c h e a c h a t o m i n t e r a c t s . C' =
4cakK2 e x p ( - - 2clKI) ;
Cc = L i m r - ~ r c
k In iT - - Tel" "
All data for the hexagonal lattices in table II can be found from the formulae (95), (96) and (105) b y elementary calculation, except the free energy and the entropy at To, the computation of which involves the numerical calculation of the integral in ~(9, 9, 9) = -- 412fo2'~/o2'~ In [1 - - ~ {cos co1 + cos co2 + cos (col + co2)}]dcol dco2 = - - 0. 17642. From table II we see that the first three kinds of lattices herein are very similar. For example --E(Tc)/(½clJI), which can be considered as a measure of the mean short range order at To, appears to decrease uniformly, but slowly, with increasing correlation number c. The behaviour of the triangular lattice with f < 0 is quite different. It has no transition temperature and its specific heat vanishes
454
R. M.F. HOUTAPPEL
b o t h at v e r y high and at v e r y low t e m p e r a t u r e s and has a m a x i m u m in between, as in the case of the linear I s i n g chain 1.,). It is well known t h a t a lattice with negative I s i n g interaction has the same p a r t i t i o n function as a g r a n d ensemble of a b i n a r y solid m i x t u r e of the same lattice t y p e with equal concentrations of the components. If ei,h is the interaction e n e r g y in the solid m i x t u r e for each pair of neighbouring atoms, one of kind i and the o t h e r of kind k(i, k = 1,2), the corresponding interaction J in the I s i n g lattice will be J = ½ [ei,2 - - ½ (el,l + e2,2)1. If such a solid m i x t u r e crystallizes in a triangular lattice and J is negative, it has, of course, a t e n d e n c y to order its atoms, i.e. to form a superqattice. It is easily seen t h a t the m a x i m u m of order is o b t a i n e d if, for example, the atoms of one kind o c c u p y all positions on the odd columns and the a t o m s of the o t h e r kind the positions on the even columns in fig. 1. Such a configuration has the e n e r g y per a t o m J , which is, indeed, equal to the e n e r g y at zero t e m p e r a t u r e (compare table II). Yet we conclude t h a t this t e n d e n c y to order does not lead to a transition t e m p e r a t u r e and t h a t at no t e m p e r a t u r e a long distance order will be established. 14. Acknowledgments. I wish to express m y t h a n k s to Prof. Dr. H. A. K r a m e r s and to D r . J . K o r r i n g a for t h e i r s t i m u l a ting interest and their c o n s t a n t advice, to Mr. T. A. S p r i n g e r and Mr. J. H. v a n der M e r w e for their kind help with some of the calculations and to Mrs. B. D a n i e I s- H u n t for her assistance with the translation into English. I am v e r y grateful to the L o r e n t z F o u n d a t i o n , which enabled me, b y a grant, to continue m y s t u d y of the problem discussed in this article.
ORDER-DISORDER
IN HEXAGONAL
LATTICES
455
REFERENCES 1) 2) 3) 4)
5) 6) 7) 8) 9)
10) 11) 12)
H.A. Kramers a n d G. H . W a n n i e r , Phys. Rev. 60(1941) 252-262. L. O n s a g e r , P h y s . R e v . 6 5 (1944) 117-149. B. K a u f m a n , P h y s . R e v . 7 6 (1949) 1 2 3 2 - 1 2 4 3 . T h e m a i n r e s u l t s c o n t a i n e d in t h i s a r t i c l e h a v e b e e n a n n o u n c e d in m y l e t t e r to t h e e d i t o r , R. M. F. H o u t a p p e 1, P h y s i c a , D e n H a a g 1 6 (1950) 3 9 1 - - 3 9 2 . H e r e i n is also m e n t i o n e d u n p u b l i s h e d w o r k o n t h e s a m e p r o b l e m b y T e m p e r I e y a n d b y H u s i m i a n d S y o z i . As y e t n o f u r t h e r i n f o r m a t i o n a b o u t t h e i r m e t h o d s a n d r e s u l t s h a s c o m e to m y a t t e n t i o n . S.B. Frobenius, P r e u s s . A k a d . Wiss. (1909) 5 1 4 - 5 1 8 . See r e f e r e n c e 2) p. 123. G.H. Wannier, Rev. mod. Phys. 17(1945) 50-60. H.A. Kramers a n d G . H. W a n n i e r , Phys. Rev. 60(1941) 263-276. F o r r e a l r o t a t i o n s see for e x a m p l e : B. L. v. d. W a e r d e n , E r g e b n . M a t h . u. G r e n s g e b . , B e r l i n (1935) B n d 4 1 6 5 - 1 6 9 . T h e t h e o r e m c a n also be p r o v e d for c o m p l e x r o t a t i o n s w h i c h c a n be d i a g o n a l i z e d . L. K r o n e c k e r., S. B. p r e u s z . A k a d . W i s s . (1890) 1 0 6 3 - 1 0 8 0 . B. K a u f m a n andL. Onsager, Phys. Rev. 76(1949) 1244-1252. E. I s i n g, Z. P h y s . 21 (1925) 2 5 3 - 2 5 8 .
Mei 1950
ORDER-DISORDER IN HEXAGONAL LATTICES by R. M. F. H O U T A P P E L Instituut voor theoretisehe natuurkunde, Rijksuniversiteit, Leiden, Nederland
Synopsis S o m e relations for t h e p a r t i t i o n f u n c t i o n s of t w o - d i m e n s i o n a l infinite t r i a n g u l a r a n d h o n e y c o m b lattices w i t h I s i n g i n t e r a c t i o n b e t w e e n n e i g h b o u r s w i t h o u t m a g n e t i c field are g i v e n in t h e general case of d i f f e r e n t i n t e r a c t i o n s in t h e t h r e e directions. E x a c t e v a l u a t i o n of these p a r t i t i o n f u n c t i o n s is o b t a i n e d w i t h t h e aid of t h e m e t h o d of 1 3 r u r i a Kaufm a n . T h e t h e o r y of this m e t h o d is simplified b y a v o i d i n g t h e use of a b s t r a c t g r o u p - p r o p e r t i e s . S o m e t h e r m o d y n a m i c p r o p e r t i e s of t h e isotropic h e x a g o n a l lattices are given. These are c o m p a r e d w i t h t h e c o r r e s p o n d i n g p r o p e r t i e s of t h e q u a d r a t i c lattice. All h o n e y c o m b and n e a r l y all t r i a n g u l a r lattices h a v e a t r a n s i t i o n t e m p e r ature. A t this t e m p e r a t u r e t h e e n e r g y is c o n t i n u o u s and t h e specific h e a t b e c o m e s infinite as - - In [ T - - T c J, as in t h e r e c t a n g u l a r case, s o l v e d b y Onsager and Kaufman. T h e r e are e x c e p t i o n a l t r i a n g u l a r l a t t i c e s w i t h o u t t r a n s i t i o n t e m p e r a t u r e : for e x a m p l e t h o s e w i t h e q u a l n e g a t i v e interactions.
1. Introduction. K r a m e r s and W a n n i e r l ) showed, that the computation of the partition function of a crystal-lattice with I s i n g interaction can be reduced to an eigenvalue problem. This problem, in case of the two-dimensional rectangular lattice with interaction between neighbours only, has been solved first by 0 ns a g e r 2) and afterwards in a shorter way by K a u f m a n 3). In this article 4) we will consider the partition functions of the twodimensional triangular and honeycomb lattices with I s i n g interaction between nearest neighbours only. We will consider the general case of different interaction energies, J1, J2 and Ja, of various magnitudes and signs in the three directions. If we indicate the state of each atom i by its spinvariable #,~ ---- 4- 1, the energy of a microscopic state of both lattices will be E =
-
-
fl
X(ik)l ~i/Lgk
-
-
J2 z(it)2/~#*t f3 X(i.03/zi#*m -
-
(l)
(ik), indicating that the sum is to be taken over pairs of neighbours with mutual interaction energy Js. 425 -Physica XVI
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426
R.M.F. HOUTAPPEL
In sections 2 and 3 we set up the eigenvalue problems related to the triangular and honeycomb lattices. In sections 4 and 5 we derive two different relations between the partition functions of the two types of lattices. In section 6 we use these relations to derive properties of invariance with respect to typical substitutions, both for' the partition functions of the triangular and of the honeycomb lattices. These properties are used in section 7 to construct series expansions valid both at high and at low temperatures. The main subject of this article is the exact evaluation of the partition functions. This is carried out in sections 8 to 11 independently of the considerations in sections 5, 6 and 7. Although we follow essentially the method of B r u r i a K a u f m a n 3), we present it in a somewhat different fashion, simplifying the theory by neither referring to, nor using, abstract group-properties. In section 12 we examen the transition temperature. In the last section, 13, some thermodynamic properties of isotropic triangular and honeycomb lattices are discussed and compared with the corresponding properties of a quadratic I s i n g lattice.
2. The eigenvalue problem related to the triangular lattice. We
a
J
2
J
l
J i
2
3
¢
2n
Fig. 1. Triangular lattice. How rows (thick zigzag lines) and columns are counted.
I
denote each atom in the strip of triangular lattice in fig. 1 by two indices, the first referring to the row and the second to the column in which it is situated. We transform this strip to a cylinder so that each atom k, I will be identical with k, l + 2n and then this cylinder to a torus so that each atom k, l will be identical with m + k, I. The partition function of this torus-shaped two-dimensional triangular lattice of 2nm atoms with interaction energies J1, J2 and J3, as indicated in fig. 1,
ORDER-DISORDER
IN HEXAGONAL
427
LATTICES
will be : Z~,2 ...... =
~ /tk, 1 =
[1I,-"=1 e x p [E~'=I(Kt#i,2i_IF*;,2i + K2/ti,2ifq,2i+t + ± 1
Jr- K 3 # i , 2 j - - l / l i + l , 2 ] - - I
~'- K 2 # i + 1 , 2 1 - 1 # i , 2 ]
~f- K 1 / t i , 2 j / l i + l , 2 i + l
with W e see t h a t : ) Z ~. )2 ..... ~
K, = J,/kT,
E l k l _-± i
i ~ 1, 2, 3.
V~,2n(/ti,l,/tl,2)
",#I,2n;/t2,1
,
"V~,2n(]~2,1, /*2,2 . . . .
•.
Jr-
Ka#~,2it*i+ 1,2j)]j
÷
)
~t2,2, ) . . )
/t2,2n ; //3,1, ,tt3,2 . . . .
....
"V:~,2,,(/t,.,l,/t,,,,2
(2)
/~,,,,2,, ; / q , l , / q , 2
#3,2n)"
""
.... /q,2,,) 22n m
m
:
/t2,2n)"
T r a c e V~,2, , = Ei= 1 ~'i'
(3)
In (3) V~.2, , *) is a 22"-dimensional m a t r i x , whose e l e m e n t in the row, d e n o t e d b y the set/q,/~2 . . . . . /t2, , ( / t , = 4 - 1 ), a n d in the c o l u m n , t t t t d e n o t e d b y the set F,, F2 . . . . . /~2,, (/t, = 4- 1), has t h e v a l u e : V'z,2,,(#l, /t2,
F2,,; /'.l,' F2,,
• .,
.
., 1~2,,) -- e x p [Zi= " I (Kltt2i-|/t2i '
t
!
i
-~t
+ K2/t2i/t2i + t-k KaF2i_~,.2i_~ + K2#2i_~/z2FF K~/t2fft2i + ~-+-Ka/t2i/~2i) l (4)
(/t2,,+
1 =
'
/q,/t2,,+1
:
#
,) •
/t 1, t 2. . . . are the 22" e i g e n v a l u e s of V~a,,. B e c a u s e all e l e m e n t s of V~.,2,, are positive, its largest e i g e n v a l u e 2.~,2, is not d e g e n e r a t e s). T h e r e f o r e the p a r t i t i o n f u n c t i o n per a t o m of this lattice for m -+ oo will b e '
Lim,,,_+oo 7~1, 2' 2nn,mm
:
]I/2n
"'~,2~,,~
One recognizes t h a t (with the n o t a t i o n ~(v) = 1, if v = 0, a n d a(v) :
O, i f v :# 0 ) :
v Z,2n (/ t l , -
-
"',#2n,/
.,;,
E
" [exp [~,i=l(Kllt2i_l/t2i-+-K2tt2s/t2i+l)3
It III I I" & . t*i'tti'Hi =_1
• ",/
=
n
• exp
[E]= t
IV
,t
.1
K3/t2i_l/t2i_lj
• exp rv,, L ~ i = ' ( K 2 / t 2 i _"l / t 2
n
•
• n_]= 2" ,a(iti--t,~"
I I"
Hi= t (~(/+2i
--/t2j
• e x p let'= t Ka#2"j/t'2i]. 1I}'= t 6(/t2i_ , ~tt /tl ' /t2n+I ,t •
=
)"
"i ..{_ K l / t 2 i / "1 2 i " + l) ] . v[2,,..i= i
tt
tt (,tt2n+l
ut
:
ut
(~(/ti" - - / t i " ) "
t --/'~q-t)j
tiE,"
/tl ; / 2n+l
).
{V
:
~
)"
• ) Throughoutthisarticleweshallusethiekletterstoindieate2 -dimensionalmatriees a n d G o t h i c l e t t e r s of o r d i n a r y size to i n d i c a t e t h e 4 * , - d i m e n s i o n a l r o t a t i o n s i n t r o d u c e d in s e c t i o n 8. T h e s u b s c r i p t s ~., ,9, 9t refer to l a t t i c e t y p e s .
428
R. M. F. H O U T A P P E L
Or V%,2. = V 1V~V3V'4, with: l]2n
"
V, = e x p [Ei= I (Ki/~2i_ltu2j --~ K2lu2i~u2i+I)]'~H= 1 ~ ( ~ i - / z i ) ' t
)!
_ _
V'2 = exp [E;'= l K3#2i_1~2i_1].Hi= l 6(/~2i .
V3 = e x p [Zi=l (K2~2i-1~2i + Klfl2iU2i+l)]
'
r
P2i), . ]-]2n
'
"'/'=1 ( ~ ( / ' t i - Fti),
" 1 K 3/~2i/~2/] ' " YIi=1 " ' V~' = exp [Zi= ~(#2i-1 - - ~u2j-l)-
It is convenient to interpret these matrices as linear operators, acting on the 22"-dimensional vectors ~P(/~I,/~2. . . . /~2,.) according to : ~'(~,
....
~2,,) = v~(~,
....
t,~,,) = t
Hi=:kl
•.,
t
t
..
, •.,
So we see, that V 1 and V 3, which are diagonal matrices, can be written in the form : V 1 = exp [K l(sts 2 + sas 4 + .. + $2n__1S2, ) + +
V3 = exp [ K 2 ( s I s 2 -]- s 3 s 4 +
K2(s2s3 -~
s4s s +
. . -~ $ 2 . _ 1 s 2 . )
..
+
s2,,sl)l,
-}-
-~- K I ( S 2 s 3 -~- $4s 5 -{- . . Jr- S2nSl)],
where the operators s~ are defined by s, ~(F,,
....
~, ....
~2.) =
t,, ~(~1
....
~,, ....
t,2,,)
(s)
V~ and V'4 are direct products of two-dimensional matrices, 8(/z,--/~) and exp(K 3 #q#,;), which can be considered as operators, each acting on one of the parameters #~ of the 22"-dimensional vector %V(/Zl, /~2 .... /~2,,)" b(~*i- - / ~ ' ) is the two-dimensional unit-matrix and ! exp ( K 3 # # q ) is the matrix exp K~ exp ( - -
exp(-- K3)) K3) exp K 3
(6)
It is convenient to put these matrices also in the form exp (matrix) This is possible by writing exp K~ exp(--K~) ~ ( e x p K~ exp(--K3) ~ T_ 1 exp(--K~) expKa / = T . T -1. \exp(--K~) e x p K 3 / . T .
429
O R D E R - D I S O R D E R IN H E X A G O N A L LATTICES
where T is the matrix which transforms (6) into diagonal form : 0 ) =expk[lna'0 0 ) a2 In a 2
--1 /expKa exp(--Ka)~.T=(;' I - • ~exp (--Ka) exp K a / Then : expK exp (--Ka, ~ exp (--K3) exp K 3 /
(1;a, -----
T" exp
0 ).T_, In a 1
=
. ~ /In a I 0 ) . T _ l ] =exp[l.~ o lna 1 In this w a y we find: , ( e x p Ka e x p ( - - Ka)) = exp (Ka/~¢~i ) = \ e x p (--Ka) exp K a
= exp
ln
with:
(2 Sh 2Ka)~ K* (2 Sh 2Ka) ~ exp (K~'ei) (7) K* In (2 Sh 2K3)~/= K* = ½ In Coth K~,
(K*)* = K e
(8)
K~' satisfies the relations, given by K r a m e r s and W a n n i e r 1). Sh 2K i Sh 2K* ---- Ch 2K i T a n h 2K* = Tanh 2K~ Ch 2K* = 1 (9) e i (in 0 n s a g e r's 2) notation Ci) is defined by: ei ~o(/~ l . . . .
#i ....
# 2 . ) -= YJ(#l . . . .
--#i ....
/z2,,).
(10)
For reference, we tabulate here the algebraic properties of s~ and follow from (5) and (10), and have been given b y 0 nsager2):
ci, which
S~= ~ = SiS k - ~ SkSi,
sick=
cks i
1, SiCi=--CiSi; CiCk =
CkCi;
(11)
(i :/:k).
Using (7) and the c o m m u t a t i v i t y of the ei's we can write V£ = (2 Sh 2Ka) '''2 exp [K*(e t + e 3 + .. + e2,,_,)], V~ = (2 Sh
2Ka)'':2exp
[K*(c 2 + c 4 + .. + c2,,)].
Thus we find that the partition function per atom X~. of a two w a y infinite triangular lattice is equal to ~.2,,, ~tz ----- Lim,,~oo ~1/2,,
(12)
430
R. M. F. HOUTAPPEL
2z,2,, being the largest eigenvalue of the 22"-dimensional m a t r i x Vz,2n, defined by: V~,2,, = (2 Sh 2K3)" V, V2VsV4, (13) V 1= e x p [K 1(SIS2"~-$3S 4 "~-.. "~$2,,__IS2n ) - ~ - K 2 ( s 2 s 3 + s 4 s s - ~ . . V2 :
"~-S2,,SI)~,
exp [K~(c t + c 3 + .. + c2,,_t) J,
V 3 :exp[K2(sls2+s3s4+..
2f-s2,t_lS2n)-+-gl(S2S3-~-S4Ss-~-..-~-s2nsl)
V4 ---- e x p ~K~(c 2 + c 4 + . . .
J,
+ C2n)3.
3. The eigenvalue problem related to the honeycomb lattice. As in the case of the triangular lattice, we denote each a t o m in the strip of honeycomb lattice in fig. 2 by two indices, the first referring to the row and the second to the column in which 5 it is situated. This strip we also 4 transform to a cylinder so t h a t 3 each a t o m k, l will be identical 2 with k, 1 + 2n, and t h e n this cylinder to a torus so t h a t each 1 2 3 4 2rl i atom k, l will be identical with Fig. 2. Honeycomb lattice. How rows k + 2m, l. The partition functand columns (thick zigzag lines) are ion of this torus-shaped twocounted. dimensional h o n e y c o m b lattice of 4nm atoms with interaction energies J l , J2 and f3, as indicated in fig. 2, will be:
Z~o,2,,,2m =
• [HILl exp [Y'~'=1 (Kl#2i--l,2j--l#2i,2i--I -~Jtk,l= 4-1
+ K2#2i--1,2i#2i,2i + K3#2i,2i#2i.2i+! -}- K2#2i,2i_1#2i+1,2i_ 1 + '~ K1#2i,2i#2i+1,2i -~- K3#2i + l,2f--l#2i + l.2i) ] ] . We can see: Z¢~,2,,2,,, = Trace lr~',2,,,
V~,2,, = V , . V b.
(14)
Va and Vb are two 22"-dimensional matrices, whose elements in the row, denoted b y the set #1,/*2 . . . . . #2,, (#k = + 1), and in t h e
O R D E R - D I S O R D E R IN H E X A G O N A L LATTICES t
!
t
column, denoted by the set #1,/~2 . . . . . #2, (#'k -~ i ively the values : Va(/Ll . . . .
~22n; ~ l . . . .
--
--
exp
t
l
!
[El'= t (K1/12i_1/12i_ l "~- K2/12i/~2i -~- K3//2//~2i+t)~ ,
t /t2n; /*1 . . . .
Vb(/~l . . . .
1), have respect-
/~2,,) = l
:= exp
r~,,
L~i=I
431
! /t2,,) =
' ' ' (K2#t2i--1/t2i--I "~- Kl/t2i#*2i' -~- K3122i_#,2i)1.
]
(15)
Reasoning in the same way as we did for the triangular lattice, we can conclude finally, t h a t the partition function per a t o m ~9 of the two w a y infinite h o n e y c o m b lattice is equal to
(16)
~s~ ~- Lim,,_+oo "'$O,2n'~liO'
~-~,2,, being the largest eigenvalue of the 22"-dimensional m a t r i x V~,2,,, defined by: V~,2, = (2 Sh 2K~)" (2 Sh 2K2)" WtV¢2W3V¢4,
(17)
W t = exp [K~ (c I + c 3 + .. + c2,;_l) + K* (c 2 @ c 4 + .. + e2n)] , W2 =
exp [K 3 (S2S 3 + $4S5 +
W3=exp[K~(c
l+c 3+
.. +
S2nSl)I,
.. + c 2 , _ t ) + K ~ ( c
2 + c 4 + .. +e2,,) 1,
W 4 = exp [ K 3 (SIS 2 -~- SSS4 "-{- .. "-~ S2n__lS2n)l.
4. Dual relation between 2~ and 2~. We shall now derive a relation between 2~ and ~ , using the m e t h o d b y which 0 n s a g e r e) showed the dual relation between the values of the partition function of a quadratic lattice for high and low temperatures. All eigenvectors of V~.2, and V~,2,, belonging to non-degenerate eigenvalues are either even, or odd,
~v(/~l, #e2. . . . #2,) = - - ~v(-- #l, - - / ' 2 . . . . - - #2,,),
because all operators s~s~+t and e~ transform an even vector into an even one and odd vector into an odd one. Substituting in the expression for V~.2,,, (17), sisi+ 1 for c~ and ci+l for sisi+l (s2,,+ t - - s t ; e2,+1 = et) we get, in view of (13)"
Vs~,2,,(Kt,K2,K3)-+ (2 Sh 2Kt)"(2 Sh 2K)" (2 <~ °m*'-"V3j~.2,,,IK*t, K2,* K~) = = (2 Sh 2Kt Sh 2K2 Sb 2K3)" V~.,2,,(K*, K*, K~). (18)
432
R. M.F. HOUTAPPEL
If not only K 1, K 2 and K 3 but also K*, K* and K* are real, all elements of both V~,2,(K ,, K 2, Ka) and V~,2,,(K*, K*2, K~) are positive and the largest eigenvalues of both will belong to even vectors 5). Since, as 0 n s a g e r showed 6), the eigenvalues belonging to even vectors are not changed b y the above substitution, it follows t h a t then: ,ts~,2,,(K`, K 2, Ka) = (2 Sh 2K, Sh 2K 2 Sh 2Ka)" ,t~,2,,(K*, K*, K*). (19) However, if some Ki or K* are negative, the corresponding K* or Ki respectively, will be complex. This case therefore needs f u r t h e r discussion. All elements of the 22"-dimensional matrices V~.2,(K 1, K 2, Ks) and Vg,2,(K1, K 2, K3) are analytical functions of K1, K 2 and K a. Therefore ~l~.2,(K1, K 2, K3) and ~.9.2,(K1, K 2, K3), which are nondegenerate eigenvalues, defined for all real values of K 1, K 2 and K 3, are also analytical for these values of the parameters, and can be continuated analytically to complex values of one or more of them. Hence the above relation, (l 9), will be valid for all values of K 1, K 2 and K a if we obtain ;t~,2,, and ~,~,2,,, for complex values of their arguments, b y appropriate analytical continuation from real values of these arguments. Using (12), (16) and (19) we derive : ~.~(K 1, K 2, K3) = (2 Sh 2K 1 Sh 2 K 2 Sh 2Ka) ½t ~ ( K * , K 2*, K a )*.
(20)
In the case where one or more of the parameters K i and K~' are complex, ~ and t0 denote the appropriate analytical continuations to these complex values. 5. Star-triangle relation between ~,0 and ~ . Following a m e t h o d of W a n n i e r 7) for K 1 = K 2 = K 3, we shall derive for the general case K t =/= K 2 :# K 3 a second relation between ~ and Jl~. In this m e t h o d the #'s of half of the atoms are eliminated from the partition function of a h o n e y c o m b lattice and new interactions between their neighbours are introduced. The partition function Z~,2n,2 m of the same torus-shaped honeycomb lattice of 4 n m atoms, as we constructed in section 3, is a sum over the possible values, + 1 and - - l , of the spin variables of a product of 2 n m factors of the form exp (K1#IF 0 + K#%t, 0 + K3#aF0).
ORDER-DISORDER IN HEXAGONAL LATTICES
I I
-. ".
1
433
Here each/~0 represents the spin variable of an a t o m denoted by a circle in fig. 3 and appears in one of these factors only. B y carrying out the s u m m a t i o n over the values + 1 a n d - - 1 of these /~0's first, we see t h a t Z9,2,,,2m is the sum over the possible values of the other spin variables, of a product of 2nm factors of the form: E/~o=±1 exp (Kg~#,o+ KzuzUo+ K3/t3F0).
Fig.3. E x p l a i n i n g W a n n i e r ' s star-triangle transformation.
This form can take only four values for ~*1, ~=2,#L3 = 4- I and we can easily verify t h a t
Xa0=el exp (Kg~lp o + Kzuzu 0 + K3/~3/~o) = r
t
p
--= S(K t, K 2, Ka) exp (Ktpzu 3 + K2~,3/~ + K3/qZ2) (21) with
S(K1, K2, K3) = = 2~¢/Ch(K1+ K2+ K3)Ch(--KI + K2+ K3)Ch(KI--K2+ K3)Ch(Kt + K2--K~)
(22)
and exp (2K'2 + 2K'3) = Ch(K, + K 2 + K3)/Ch ( - - K , + K 2 + K3), cycl. So we see, t h a t b y the above relation, (21), it is possible to eliminate the F0's if we introduce at the same time the interactions J'l = kTK'I, t t t J2 : kTK~ and J3 = kTK3, between the atoms 2 and 3, 3 and I and 1 and 2 respectively. These new interactions are indicated by dashed lines in fig. 3. We find:
Z~,2,,,2,,, -' - - {S(K1, K2, K3)} 2.... Z ~,2 ...... ( K '1, K 2,
K~)
Z~,2,,,2,,, being the partition function of the torus-shaped triangular lattice of 2nm atoms, which we constructed in section 2. So we can conclude t h a t : !
t
a~(K l, K 2, K3) = S(K,, K 2, K 3) ~r(K,, K 2, K~). Physica XVI
(24) 28
(23)
434
R. M.F. HOUTAPPEL
6. Conclusions drawn/rom the dual and star-triangle relations with respect to the/orm o/ the /unctions ~z(K l, K 2, K3) and A~(K t, 1£2, K3). B y eliminating ~ and ~z respectively, from the relations (20) and (24), we derive: ~(K,) :
Fz(Ki) ) .4~ ( K ~ o) ,
(25)
~t~(K,) • -~
C~(K,) ~s~(K, 4
(26)
),
with
F~(Ki) = 16(Ch K~ Ch K 2 Ch K a + Sh K l Sh K 2 Sh Ka). •( S h K 1 C h K 2 C h K a + C h K
1 S h K 2ShK3).
•( C h K 1 S h K 2 C h K a + S h K
1ChK2ShKa).
(27)
•(Ch K~ Ch K 2 Sh K a + Sh Kt Sh K 2 Ch Ka), Go(K,) -(Sh 2K1) 2 (Sh 2K2) 2 (Sh 2K3) 2
4Ch(Kl + K2+ K3)Ch(--K1 + K2+ K3)Ch(Kl--K2+ K3)Ch(Kt + K2__Ka ) , (28)
e2Kx+2Ke+2K3 --~ eeKx -J- e21(°-+ e2Kz
/
exp (2K~ + 2K~) = ee,~1+2~2+eK3 + e2~1__ e2~2__ e2~3 , cycl.,
(29)
(K~)° = K , and Coth K~ Coth K~ = Ch(K1 + K 2 + K3)/Ch(--K ~--F K2 + K3), cycl., } (K~) ~ = K , .
(30)
The formulae (29) and (30) connect sets of high values of K 1, K 2 and K 3 with sets of low values of these parameters. However (25) and (26) cannot, in general, be considered as relations between the values of the partition function of one triangular or honeycomb lattice at high and low temperatures. This is because the ratios of the K{s are changed b y the substitutions (29) and (30), unless K 1 ---- K 2 = K3, for which case the special forms of (25) and (29) were given b y W a n n i e r 7). Because of (29) and (30) the relations (25) and (26) can be brought into a symmetrical form:
/~ (K,) : ~8z (K,)/F~ (K,) = ~z(K,)/F~(K,) s o o =/~rfK,), o go (K,) = ~
(K,)/Gv(K,)
= ~ (K~)/G.~ (K~) =
g.9(,). K~
(31)
(32)
ORDER-DISORDER
IN HEXAGONAL
LATTICES
435
W i t h the aid of (20) we can derive from these symmetrical relations the two other symmetrical ones: /~(K,) =/~.(K*) = ~ ( K , ) / F ~ ( K , ) = ~ ~6(K,• )/F#(K,¢ ) = / , ~ ( K ,¢ ) (33) and gz(K,) = g#(K*) = ~ ( K , ) / G ~ . ( K 3 _. ~4( K i )o/ G ~ ( K , o) : g~r(Ki), o (34) with
Fs~(Ki) ---- (Sh 2K 1 Sh 2K 2 Sh 2K3)4/G~(K~)
(35)
and
G~(Ki) :
(36)
~ (Sh 2K I Sh 2K 2 Sh 2K3)2/Fz(K~).
F r o m the invariance of / ~ ( K l, K 2, K3) and g~(K 1, K2, K3) with respect to the substitution (29) and their s y m m e t r y in K 1, K 2 and K 3 we can prove, t h a t for positive values of K I + K 2, K 2 + K 3 and K3 + K1, I~(K,, K 2, K3) ---- 9(y,, Y2, 3'3) (37) and
g~(K1, K2, K3) :
Z(Yl, Y2, Ya),
(38)
where 9(Yl, Y2, Y3) and 7.(Yl, Y2, Y3) are one-valued functions of the variables Yl ~--i1 Sh 2K1/(Ch 2K~Ch 2K 2 Ch 2 K 3 + Sh 2K1 S h 2 K 2 Sh 2K3) ,cycl. (39) These y~ vanish both at very high and v e r y low temperatures and are also invariant with respect to substitution (29). The possibility of using such y's is suggested by the appearance of the parameters _ 1 Sh 2
2KI/(Ch 2K 1 Ch 2K2) and i1 Sh 2K2/(Ch 2K 1 Ch 2K2)
in the partition function of a rectangular lattice (see formulae 109a and 109b in the article of 0 n s a g e r 2)). The proof is as follows. Denoting b y ~i(x 1, x 2, x3) one-valued functions of the parameters x,, x 2 and x 3, we recognize easily, t h a t for positive values of K, + K 2, K 2 -Jr- K 3 and K3 + K1: / z ( K , , K 2, K3) = ~, (p, q, r) with
p = exp (-- 2K 2 - - 2K3),
q -- exp (-- 2K 3 - - 2K~), r :
exp (-- 2K~ - - 2K2).
(40)
F r o m (29~, (31) and (40) we conclude ~I(P, q, r) = (l(p °, qo, ro)
(41)
436
with
R.M.F. HOUTAPPEL
p°=
l+p--q--r l+p+q+r
,
qo~
1--p+q--r l+p+q+r 1--p--q+r l+p+q+r'
(42)
and we see t h a t
Yl ----- ( P -
qr)/(1 + p2 + q2 + r 2) --_ = (#o - - qo ro)/(l +/002 + qo2 + to2), cycl. (43)
The substitution (42) can be brought, by the transformation,
u=
( r + 1)~(p--q),
v=
(1 + p + q - - r ) / ( p - - q ) ,
w -~ (p + q + r - - 1 ) / ( p - - q ) ; q = (v + w - - 2 ) / ( 2 u +
p = (v + w + 2)/(2u + v - - w ) ,
v - - w ) , r = ( 2 u - - v + w)/(2u + v - - w ) , (44)
with the corresponding expressions of u °, v ° and w ° in po, qO and r °, into the form u °=u,
Writing
v°--v,
w °---w.
(45)
/~(K 1, K 2, K 3) = ~j(p, q, r) = ~2(u, v, w),
we derive from (41), (44) and (45) ~2(u, v, w) -~ ~2(u, v , - - w ) . Consequently: [~(K~, K2, K3) = ~2(u, v, w) = ~3(u, v, w2). F r o m [¢(K1, K2, Ka) = ]~.(K2, K1, K3) , (40) and (44) it follows t h a t ~3(u, v, w 2) = ~ 3 ( - - u, - - v, w2).
Therefore" [~.(Kl, K2, Ka) = ~3(u, v, w) = ~4(u/v, v 2, w2). If K x + / { 2 , K2 -J- K3 and K 3 + K 1 are positive, we see by (40) and (44), t h a t u/v will be positive too, and we can write
/~ (K1, K2, K3) ---- ~4(u/v, v 2, w21 --_ ~5(u~-/v2, v 2, w2). F r o m (43) and (44) we find that" 2yt -
(v 2 - - w 2 +
4u)/(v 2 +
w2 +
2u 2 +
2),
2y2 = (V2 - w 2 ~ 4 u ) / ( v 2 + w 2-{- 2u2 + 2), 2y 3 = (2U2 - - v 2 - w
2 + 2 ) / ( v 2 + w 2 + 2u ~-+ 2).
We can conclude t h a t u 2, v 2 and w2 are one-valued functions of y~, Y2 and Y3 if u is a one-valued function of yt, Y2 and Y3. Now, by eliminating v2 and w 2 we obtain (Yl--Y2) (u2 + 1) - - (1 + 2ya) u -~ 0,
ORDER-DISORDER IN HEXAGONAL LATTICES
437
which is a reciprocal e q u a t i o n of second degree for u and b y which u is defined in unique w a y for K 1 + K 2 > 0, K 2 + K 3 > 0 and K 3 + K~ > 0, in view of the fact t h a t then, according to (40) and (44), I z~l > 1, Finally we conclude, t h a t for positive K 1 + K 2, K 2 + K 3 and K , + K 1, [~(K t, K 2, K3) is a one-valued function of Yl, Y2 and y,, q.e.d.. The same proof applies to g~(K 1, K 2, K3). If K;, K 2 and K 3 are positive, K~, K * and K * will be positive too and vice versa. Hence we conclude from (9), (33), (34), (37), (38) and (39) t h a t in this case
/~(K t, K2,
K3) =
(46)
9 ( z l , z 2, z3),
g#(K 1, K2, K3) ~-~ Z(zl, z2, z3) ,
(47)
with z I -----½(Sh 2 K 2 Sh 2K3)/(Ch 2 K t Ch 2 K 2 Ch 2 K 3 + 1), cycl. (48) F u r t h e r we derive from (37) and (38) and from the fact t h a t ~ and F~G~ are positive for positive values of K 1 + K 2, K 2 -~- K 3 and K 3 + K 1, t h a t then:
h~.(Kl, K2, K3) = 1 2
= ~~¢ (K1, K v K3)/(Ch 2 K 1Ch 2 K 2 Ch 2 K 3 + Sh 2 K 1Sh 2K 2 Sh 2K3) =
= ( / ~ g ~ y o22p2 y 3 2/ 2 ) B
1,6
= ~(y,, y2, y3),
~(Yt, Y2, Y3) being a one-valued function of Yl, 3'2 and Y3 also. In addition, we derive with the aid of (20) for positive K 1, K 2 and K 3"
h.~(Kl, K2, K3) = 4 -- ~1 ,t.~(K 1, K 2, K3)/(Ch 2 K 1 Ch 2 K 2 Ch 2 K 3 + 1) = $(z 1, z2, z3).
Resuming the main results of this section, we state t h a t for positive values of K 1 + K 2, K 2 + K 3 and K 3 + K1:
h% = ¼ ~2/ ( C h 2Kt Ch 2 K 2 Ch 2K~ + Sh 2K~ Sh 2K2 Sh 2K~) == ~(Y~, Y2, Y~), and that for positive values of K l, K2 and K~ : •t~. =
~6/~., .~,_~ = ~0(z~, z~, z3),
g.~ = ~8I G ~ = X(z~, z~, Z3 ) , h.9 = ~2~/(Ch 2K~ Ch 2 K 2 Ch 2Ka + I) = ~(z~, z2, z3),
(49)
438
R. M. Y. HOUTAPPEL
where 50, Z and ~ are one-valued functions of their three parameters, and where F¢, G.~, F&, G~, Yi and z~ are defined by (27), (28), (35), (36), (39) and (48). Yi and zi vanish both at very high and very low temperatures. The first three functions,/~, g.~ and h E, are, for all real values of K 1, K 2 and K a, invariant with respect to substitution (29) and the last three functions,/.~, g.~, and h.w with respect to substitution (30). Later, in section 11, it will be shown t h a t the third and sixth of the relations (49) will be valid with a one-valued function ~ for 'all real values of K 1, K 2 and K 3 and the exact form of t h a t function will be given as a double integral. 7. The development o/ 9, Z and ~ in power series. We can use the relations (49) to construct expressions of 2~ and ,tW which will be valid both at very high and very low temperatures, in the form of a product or a quotient of an e l e m e n t a r y function and a power series in Yi respectively z;. To f i n d t h e s e power series we can develop the partition function 2z for positive interaction energies J l , J2 and J3 in an elementary way, as K r a m e r s and W a n n i e r S ) did for the quadratic lattice, both in a power series of K~, K 2 and K 3 valid at high temperature, and in a power series of exp(--K1), exp(--K2) and e x p ( - - K a ) , valid at low temperature. Only one of these expansions is needed to construct the power series for 50, Z and ~, but the other can be used as a check. We find (denoting by Ep the sum over the six permutations of the indices 1, 2, 3): 1 Zp (2 + 3 K~ + 4 K I K 2 K a + 4KI 1 4 -}- 721 K 12K 22 + 14 K~K2Ka + ;tz = ~, I I O~ K~K 4 22 17~ Y'¢"iTc'2Y¢"2+ . . . ) (50)
for high and ~t~.= eKI+K2+K3 • gXt,(l 1 + L i L 2 L a + 3 L 21L2La-2 2 2 + 3L~L~La+ L 21L2La -~ 9 T3T2T2 9 T3T3T2 (Li -~ exp ( - - 4Ki) ) for low temperatures. And we derive : 16/50(y,, Y2, Ya) = ~ Ep(Y,V2Ya + 3 YlY2 2 2 + 12 Y~YzYa + 24 Y~Y242+ . + 2 8 .YtY~3 2 . 2 .2 + 42 Y~Y2Ya + 168 YlYfiYa 6 2 + 120 4 4 + 3 3 + 84 YxY2 YlY2 + 762 y4y~y~ + 144 y7yz,% + 2592 YtY~Ya ~ a + 1560 YtY~Ya3 a 3 . . ) , (52)
ORDER-DISORDER
1
. 2.2.2
16 %(Y,- , Y2, Ya) Y , Y # 3 :
IN HEXAGONAL
1
439
LATTICES
22
~ Ep(Y,YzYa + 3 Y,Y2-- 6 Y~YzY3- - 12y4y22 222 1 4 y l y ~ y 3 3 y l y 2..5y a 48y~y~y 3 6 Y 6l 2Y 2 - - 2 4 y l4y 2 4 --4 2 2 - - 6 YlY2Ya 7 - - 147 YlY~Ya - - 3 9 6 [email protected] - 300 YlYp.W3 . a. a. a . . ), (53) - -
- -
.
~(Y~, Y2, Y3) = ~I Y>(1 - - 3 y~ - - 4 Y l Y z % 3 y2 60yaly2y a 6 y ~ 1 3 8 4 2 -YlY2 --- -
- -
2,2 - 15 YlY2 2 2 2 ..). 125 YiY~Ya
(54)
I n t r o d u c t i o n o/ r o t a t i o n s i n 4 n - d i m e n s i o n a l space. We shall now show how our p r o b l e m , as in the r e c t a n g u l a r case, can be reduced to the t r e a t m e n t of 4n-dimensional r o t a t i o n s only. I t will be unders t o o d t h a t a 4 n - d i m e n s i o n a l r o t a t i o n is a linear t r a n s f o r m a t i o n of 4n coordinates, which leaves the s u m of their quares i n v a r i a n t . We label it p r o p e r or i m p r o p e r according to w h e t h e r the d e t e r m i n a n t of its m a t r i x is e q u a l to + 1 or - - 1. F r o m the t r e a t m e n t of the r e c t a n g u l a r lattice b y K a u f m a n 3) it can be u n d e r s t o o d t h a t the i n t r o d u c t i o n of 4n o p e r a t o r s / 1 , / 2 , . • F4,, 8 .
I*2r--I
=
I~2r
---
ClC2
" " Or--1 Sr,
(55)
i C l C 2 • . Cr__ 1 CrSr,
will g r e a t l y simplify the calculation of, s a y ;t~. T h e o p e r a t o r s / ' k are, as was shown b y K a u f m a n, related to r o t a t i o n s in 4n-dimensional space in the following way. F r o m the algebraic p r o p e r t i e s of si a n d c i, (11), it follows r~r, + r,r~ :
(56)
2 ~,.
F r o m this we derive /'1.
if i # k , l ,
exp (~0 rkr,), r/. exp (--{o r~r,)= c o s O F k - - s i n O F l
if ] = k ,
(57)
sin 0 Fk + cos 0 F~ if ] = 1. I t follows t h a t e x p ( ½ 0 P ~ P l ) ~/Y:" /=1 x / I / ) e x p ( - - + + r : , ) xi, :
where
= ~ ' " x~' < , ~/=1
X~4n
z-.k=1 ~k/Xk •
(58) (59)
This r e p r e s e n t s a p r o p e r r o t a t i o n © t h r o u g h the angle 0, in which only x k a n d x I are involved. Expressing, f r o m the definition (55), the old o p e r a t o r s in t e r m s of the new ones, ek = -
i r~_,r2k,
s, = (-
i) ~-' r , & .. r ~ _ , , (60)
sks~+ I =
- - i F2~ P2k +1,
S2,,Sl = + i U r4,, Fl,
440
R. M.F. HOUTAPPEL
where U is u
=
=
r4,,,
we find t h a t I71, V2, Va and V4 in (13) are products of factors of the form e x p ( - - i KiF~11~+t) and exp(iK iU11.,110. All factors, except those of the latter form, induce b y (58/ rotations on (x t, x2, x 3. . . . . x4. ). Because of the factors of the latter form, the product V I V 2 V 3 V 4 itself does not induce b y
(V, V~V3V4) (Z~, xirj) (V, V~ V3 V4)-' a rotation on (x 1, x 2. . . . . x4. ). To bring these annoying factors in line with the others it is necessary to split up V~,2,, into a part, acting on even vectors only, and a part, acting on odd vectors only, b y V~,~,, = ~(1 + U)V~,~,, + }(1 - - U)V~,~,,.
(62)
It follows from (10) and (61) t h a t }(1 + U) acts as a u n i t - m a t r i x on the 22"-kdimensional space of even vectors and reduces to zero all odd vectors and t h a t ½(1 - - U) acts as a u n i t - m a t r i x on the 22'`-'dimensional space of odd vectors reducing to zero all even vectors. Therefore ½(1 + U)V~,2, , has the same eigenvalues belonging to even eigenvectors as V~,2,,, while its eigenvalues belonging to odd eigenvectors are all zero. On the other h a n d ½(1 - - U) Vz,2. has the same eigenvalues belonging to odd eigenvectors as Vz,2,,, while its eigen values belonging to even eigenvectors are all zero. Since (1 + U) U =
1+ U
and
(1--U)
U=--(1--U)
and since both 1 + U and 1 - - U c o m m u t e with all products PkPl, we see t h a t (1 + U) exp (iKiU114, , Yl) = (1 + U) exp ( i K i P4,,11,) and
(63)
(1 - - U) exp (iKiU114, , f11) = (1 --- U) exp (-- i K i 114,,110 (64)
and t h a t we therefore can omit U in the factors of V~ and V3 in ½(1 + U)V~,2. and replace it by - - 1 in the factors V, and V3 in -
-
tr)vz,2,,.
So we find: Vz,2. - - ½ ( I + U ) (2 Sh 2K3)" V o + ½ ( 1 with
V~ = V ~ V 2 V ~ V 4 ,
U) (2 Sh 2K3)" Vo
Vo = It'~V2V~V4;
(65) (66)
441
O R D E R - D I S O R D E R IN HEXAGONAL LATTICES
V~ = e x p •exp V2 = exp W = exp •exp
(--iK,PJ'a).ex p ( - - i K , P 6 f ' 7 ) .. .. exp (--iK,f',,,_21"4,,_l) ( - - iK2P4Ps), e x p ( - - iK21"sP9).. .. exp ( ± iK2I',,Pt) , (--iK;I']I'a).ex p (-- iK*I'5I'6).. ." exp (--I,K3/~4,v__3P4n_2), ( - - iKJ'=r3), e x p ( - - iK=r6rT).. ." exp ( - - i K 2 r,,,_~r,._,) ( - - iK,I'4I's) . e x p (-- i K , r s F g ) .. ." exp (4- iKll",,I'l), •
,
V4 = exp (--iK~l'aI'4).ex p ( - - i K ~ / ' 7 / ' 8 ) . . . . exp (-- iK~/'4,_,['4, ). B y (65) and (66) we have reached the aim of the present section.
9. Explicit representation o/ Vo+ and V o as rotations. F r o m (58), (59) and (66) we find t h a t V#, V2, Va~, V4 and their products Vo+ and V~ induce b y V(Z2"__,xkFk)V - t ( V = V~, V 2, V~, V 4, Vo~) on (Xl, X2 . . . . X4n ) the rotations ~ + , ~]-, ~ 2 ' " ~ + ' ~:)3-, ~)4' ~)+ : J~)4~+~)2 ~)+ and £)0 = ~4t~3£)2t~] -. For their matrix-representations we find:
• C1 iSI --iS l C 1
~=
c;
is; I o
C2 iS2
o I
, ~2=
C;
- - i S 2 C2.
,
'i~;
- - ~ S 3 C3 .
i o/
=CiS 2
o 1
(
1 o
t CI
~=
C2 iS2 --iS 2 C2
C] --iS1
iS]
Cl. C]
TiS[
C3 zS3
-is; c;
1 o
o I.
"C; --iS a
is; I ' C3
with the abbreviations: Ci - - C h 2 K i,
Si = S h 2 K
i,
C*=Ch2K*,
S*=Sh2K*.
Carrying out the multiplications O + = ~ 4 " ~ + ' ~ 2 " ~ + ~ o ---- 94" 9 7 " 02" ~]- we find: a c
b a
and
(68)
b •
b
(67)
•
b
c
a
442
R. M. F. H O U T A P P E L
a, b and c being the 4-dimensional matrices' +CIC2C; + S l S 2 a =
b=
--ClSlS;
-c s c;s;
,-ic,s
+iC2S2S~ '
, --CIS2C;S;--SIC2N~ , --iNIS2C;S;--iC1C2G;
c;'-is,c2c;,
,
0
,
0
0
,
0
,
0
\+~CIS2C3 +~SIC2C 3 , 0 0
,
,
+SxS C;:+c,c c;
0
--Cl S2S3--SIC2C3S3,
I
,
+$C*1S;
;
+ic,c
s;+is,s
cb;]
+C1C2C1+SIS2C 1' /
o ) 0
,
- - , ClSlS3
,
+SIS 3
--CISIC;S;
,
- - i SIC3S3
O
,
0
, --i S1C2--i CIS2C; \
0
,
0
--C2S2S~
0
+$ 82C3S3
0
0
0
0
,
0
0
0
•
,
2
*
)
*
+S;S;'
10. Diagonalization o/ ½(1 + U)V~,2, ,. To compute the partition function per atom t~ of a two way infinite triangular lattice we need only calculate the largest eigenvalue of V~,2, ,. Since this eigenvalue is positive and related to an even eigenvector, it is the largest eigenvalue of 2(1 + U)Vz,2, too. Therefore we shall diagonalize only ½(1 + U)V~,2, ,, although the same procedure is also applicable
to ½(1 - Each proper 4n-dimensional rotation which can be diagonalized, can be transformed by a suitable rotation of the coordinate system to the standard form 9),
- - i Sh 71
Ch 71
Ch 72
i Sh 72
--iSh72
Ch72
(7o)
.. Ch 72n --iSh72.
i Sh 72. Ch72. /
which represents a product of 2n comnmtative rotations through the angles #~ -- - - i y i . By applying a reflection, which is an improper rotation, relatively to the coordinate x2~, we can make ~gk and 7h change sign. £)+ can be diagonalized as will be shown in next section. Thus there exists a proper or improper rotation, ~ , such that ~3o+ = ~ . £ 3 + .~R-'
(71)
has the form (70), in which the real parts of 71, 72. . . . , 72. are all
443
ORDER-DISORDER IN HEXAGONAL LATTICES
non-negative. The latter will be used later on. The eigenvalues ~i, 1/~/i of 9 +, and therefore also of ~D~-, are given b y ~i = exp 7i.
(72)
In next section we shall indicate how t h e y can be calculated. We shall now first show how the diagonal form of ½(1 + U)Vza,, is related to the Yi. It can be shown t h a t there exists an operator M(~R), which induces the just mentioned rotation ~R, while (M(~R)) -1 induces ~R-1, and therefore has the p r o p e r t y t h a t V + ---- M(,gt). V + . (M(9t)) - I
(73)
induces the rotation t0 +. This follows from the fact t h a t each proper rotation can be obtained b y an appropriate succession of twodimensional proper rotations, each involving only two coordinates, while for an improper rotation just one additional reflection, relative to, say x I, is needed 1°/. Actually, a rotation through the angle O, involving x k and x~ only, is caused b y transformation of _,=E 4'' i x~Fi w i t h exp (½0/'k~) and the reflection relative to x 1 b y transformation with/~2/'3 " ' " 1~4,," B u t the identical rotation ~00+ is induced also b y V0+ = exp (½ 0, P~/'2) • exp (½ 02/'aP4/• . . . e x p (½ 0 2 , / ' . _ ~ r . , ) = -- exp [½ (y,e, + Y2e2 + .. + y2,,e2,,)], (74) i.e.
Vo+ ,-,=(E4"1 x~ /';) (V+o)-1 :
for all sets of x t, x 2. . . . .
IT+ v-"~=1~'~4"x~ F;) (V~-)-'
x4,, and thus
( V + ) -1 V + --i=~'~4"1 Xi Ill __. ,{~4"--i=1 Xi 1"~i) (VO+)-1 v+"
Now, it can be shown from (5), (10) and (60), t h a t each 22"-dimens ional m a t r i x can be written as a sum of products of I"1,/'2 . . . . /'4.. Consequently (V+) - 1 . V0+ c o m m u t e s with all 22"-dimensional matrices and is therefore a multiple of the 22"-dimensional unitmatrix, so t h a t : vo+ = Q vo*, (75) where Q is a scalar. We conclude t h a t q22,, = I
(76)
from the fact t h a t b o t h Vo+ and Vo+ are unimodular, because all
444
R. M . F . HOUTAPPEL
their factors are so. The latter can be recognized on writing the factors exp ( - - i K i F 2 ~ _ l F 2 k ) ,
exp (--iKiF2~F2h+l)
and exp ( + iKiT'4.['t)
in the forms exp ( K i ch), exp ( K i
SkSk+l)
and exp ( - - K i c l c 2 .. c2,,s2,sl),
diagonalizing afterwards the first and the last form respectively, by transformation with 2-4(sk + ck) and 2-"(s I + i c l s l ) (s2, + ie2,,s2, ) r12,-1 xxt~ 2 (% + c,) into exp(Kish) and exp(Kisls2..s2, ). Indeed, the diagonal matrices exp ( K i sk), exp (Ki s,sk=l) and exp ( K i sis 2 .. *2,) are unimodular, because half of their diagonal elements are the reciprocal of the others. Thus, we see from (65), (73), (74) and (75) that
M(~). ~(I + U)V~,2,,. (M(~))-' ---2. 7ici) • (77) -- M(~R).½(1 + U)" (M(~)) -1. (2Sh 2K3)"0 exp(~1 Y'i=t - -
As to the transformation of 1 -~ U we remark that 1 + U will be invariant with respect to transformation by each factor exp(½ o r h r ~ ) of M(~), because exp (½ ~9r~F~) . r i P 2 . .
[4,.exp
(-- ½t~ P~F~) = F i r 2 . .
F4,,
(78)
and that 1 + U is changed into 1 - - U by transformation with a factor F 2 F 3 .. r 4 , , occurring in M(~R) if ~R is an improper rotation, because (r~r3 . . r,,) . rtr~
. . v~,,. ( r ~ r ~ . . r ~ , ) - ' = -
r,r~
. . r~,. (79)
Consequently we conclude from (61), (77), (78) and (79) that M(~R).½(I + U ) V ~ , 2 , .
(M(~))-'
=
---- ½ (1 + U) (2 Sh 2K3)"@ exp(½ "~i=*~2"7ici ) = :
½ (l ~:_
2- 17ici) (80) • . C2n ) (2 Sh 2K3)"0 exp(½ Zi=
ClC 2
with + or - - according to whether 9%is a proper rotation or not. By an additional transformation with T = T - l = 2-" 2. l(s; + c;) each ei will be changed into the corresponding s~ so that T.M(~R). ½(1 + U)V~,2,. ( M ( ~ ) ) - ' . T - t --- ½ (l 4-
s,s 2
.
.
=
s2.) (2 Sh '2K2)"~ exp (½
2. r;%),
(81)
O R D E R - D I S O R D E R IN H E X A G O N A L LATTICES
445
the last form being a diagonal matrix, in which the + or ~ sign m u s t be used according to w h e t h e r ~t is a proper rotation or not, and in which 0, with modulus 1, must be chosen so, that the largest eigen-value is real and positive.
1 I. Computation o / ~ . and o~ ~s~. Since ~,2,, is real and positive, it must be equal to the modulus of the largest eigenvalue of ½(l + s t s 2 .. s2,,) (2 Sh 2K3)" exp (} Ej=12, Yi si) or of
{(l - -
SIS 2 ..
$ 2 . ) (2
Sh 2K3)" exp (} X~"__l yis/),
depending on whether ~R is a proper rotation or not. Since no yi has a negative real part, it is easily seen that this largest eigenvalue will be (2 Sh 2K3)" exp (} E2"=1 Yi) (82) in the first and
(2 Sh 2K3)" exo- (} ~2,, ~i= 1 7i
--
Y ,)
(83)
in the second case, y' being the Yi with smallest real part. In the calculation Of t~ Lim,, ->oo~1,2,, b o t h formulae lead to the same limit and we can write In 2z = Re[Lim,,~oo (1/4n) Y~i=l 2n Yi + } In (2 Sh 2K3) ]. (84) Here we need not consider further the form of ~R or whether it is a proper rotation or not. We have only used the fact that such a rotation, with p r o p e r t y (71), and consequently the matrix M ( ~ , exists. Using the identity Yi = (1/2z~) f02= In (2 Ch Yi - - 2 cos co) dco,
(85)
valid for Re[yj] > 0, which is the case here, we conclude In ~Lz :
Re[} In (2 Sh 2K3) +
+ Lim._+oo (1/8~n) ~2,, "~'d/=1Jf2= 0 In {(~i + 1/~li) - - 2 cos w} do~], (86) with ~1i, defined b y (72), involving half of the eigenvalues of ~ - . We shall show now how the eigenvalues, ~?i, 1/~i, of £)0+ can be calculated. If, in the formula (68) for £)0+, a, b and c were scalars, ~3o+ being an n-dimensional matrix, its n eigenvectors vr would have the form (vr)k = ek(2,-l) (k = 1, 2, . . . , n), with
e = exp (iz~/n),
(87)
446
R . M . F . HOUTAPPEL
and the corresponding eigenvalues would b e : ~/r =
a +
e 2'-1 b +
e - ( 2 ' - 1 ) c.
B y a slight generalization we find that for a, b and c being 4-dimensional matrices, 9 + can be reduced to a form in which n 4-dimensional matrices, d, =
a +
b e 2'-1 +
c ~-(2,-1)
(r = 1, 2 . . . . .
n),
(88)
are placed along the diagonal. Using (69) we find that the eigenvalues of d, are the roots of the fourth degree equation :
~14__ (A, + iA ',) ~13 + B, ~2 __ ( A , -
iA ',) ~; + 1 = 0,
(89)
with
A, ---- 4C~C2C~ -}- 2S,$2 (1 + C .2) + (S,'2+ $2)2 Sa,2cos
[(2r--ll=/n],
A; = (S2 -- S2) S.2 sin [ ( 2 r - - 1)~/n], B, = 2C .2 + 2(C2C~ +
2 2 (a s,s2)
+ 8 s,s
c,c
+
(90) +
c* - - 4 s,s
s
cos [(2r - -
(Abbreviations: see (67)). The fact, that equation (89) has, in general, four different roots, shows that d, and therefore 9 + can be diagonalized as was stated in section 10. Fortunately, the computation of 2~ does not involve the explicit solution of (89). To see this, we consider the n reciprocal eighth degree equations ~;s-- 2Aj;~ + (A,2 + A~2 + 2B,)~;6-- 2A,(1 + B,)~;s + + (2 + 2A,2-- 2A; 2 + B2,)~4--2Ar(1 + Br)~'l 3 Jr+ (A~ + A; 2 + 2B,)~2-- 2A,~? + 1 =
O,
(91)
which we obtain b y multiplying each fourth degree equation (89), for which r = rl, with that for which r = r 2 = n + 1 - - r I. We see then that the 8n roots, ~r,l' 1/r/r,l, Tit,2, 1~'fir,2, T]r,3, l /'r]r,3, TIr,4, l /~r,4,
(r :
1, 2 . . . .
n),
of these n eighth degree equations involve twice all the eigenvalues of 9 + and that In 2z = Re[½ In (2 Sh 2Ka) + + Lim,,___>oo (l/16~n) ~,=ls0s~" f2,,s'4~i=l In {(~,.i+ liar,i)--2cosco} dw]. (92)
ORDER-DISORDER IN HEXAGONAL LATTICES
447
W e now replace the sum u n d e r the integral sign b y a single logar i t h m ; use the relations between the s y m m e t r i c a l functions of an eighth degree e q u a t i o n and its coefficients; continue to the limit n - + oo b y changing the s u m m a t i o n over r into an integral o v e r co' = (2r - - 1)rr/n and bring the first t e r m u n d e r the double integral also. Using the relations (9) we get then a double integral of a logar i t h m of a f o r m / , which can be factorized into / ----/t[2 so t h a t 1
In (,1~./2) -- 327r2 [f~'~f2o~ In [1 d~, d a / + f~"~'~ In/2 &o do'i, (93) with
1,,I2= (c,c2c3+ s,s2s3--$3 cos o,)~--½{s~+ s~+ 2s,ss(cos o~+ cos o~')+ + (S~ + S~) cos oJ cos co' + (S~ - - $22) sin co sin co'},
(94)
the + sign referring to /, and the - - sign to Is. The two double integrals in (93) are equal. After s u b s t i t u t i n g co = o I + c% and oJ' = co1 --cos, we see easily t h a t 1, can be factorized again, and we obtain finally: in (,1~/2) =
~fo2"~fo2'~In {Ch2K1Ch2KsCh2K 3 +
Sh2K1Sh2KsSh2K 3 - -
- - Sh2K, cos co1- - S h 2 K 2 cos c % - - S h 2 K 3 cos (~o, + c%)}doJ, dw 2. (95) F r o m (95) and the dual relation, (20), we derive for the partition per atom, ,1,9, of a two w a y infinite h o n e y c o m b lattice: In (,l.9/2) = 16-~2fo2'~f~'~In {{Ch2K,Ch2KsCh2K3+ 1 - - S h 2 K s S h 2 K 3 cos w , - - - Sh2K3Sh2K , cos co2 - S h 2 K , S h 2 K s cos (oJ, + c%)} dco, do~s. (96) F r o m formulae (95) and (96) the s y m m e t r y of ,1~ and ,1~ with respect to K,, K s and K 3 is easily verified. In accordance with the third and sixth of the relations (49) and our r e m a r k at the end of section 6, we conclude from (95) and (96), t h a t for all real values of K,, K 2 and K 3 ¼ ,1~:/(Ch2K, Ch2K s Ch2K 3 + Sh2K, S h 2 K s Sh2K3) =
= E (Y,, Ys, Y3), (97) ,1~/(Ch2K 1 Ch2K s Ch2K 3 + 1) ---- ~(z 1, z s, z3),
(98)
448
R. M.F. HOUTAPPEL
where Yi and ::i are defined b y (39) and (48). So we find for ~ the expression : In ~(x t,
I
x 2, xa) 2~r
=
2"n-
-- 4~2fo ~o ln{1--2XlC°S°~l--2x2c°sc°2--2x3c°s(°)l+°@}d°)Idr%" (99) This double integral is convergent for all leal values of K~, K 2 and K a, if we read y~ or z~ for x v We recognize that the power series, (54), which can be deduced from this expression, is also convergent for all real values of Ki, K 2 and K 3, and this remains true if we replace in it Yi b y z~. O n s a g e r's result for the partition function per atom, ~ , of an infinite rectangular lattice, In ( ~ / 2 ) = 8@f02'~f02'~In (Ch 2K~ Ch 2K 2 - Sh 2K, cos co~---
Sh 2 K 2 cos w2) &o 1do)2,
(100)
can be obtained both from the expression (95), by using
Jl~i(KI,K2) = 2t~ (K1, K2, 0), and from the expression (96), b y using 2~(K1, K2)
=
Limjs___~o o
(~,?(K1, K2, Ka)) 2 exp (--K3).
The reason for the former is trivial; the latter can be understood bv observing that, if the interaction Ja is very large, the spin variables of each pair of atoms between which the interaction fa takes place, will always have equal values, i.e. such a pair behaves as one atom with constant inner energy - - f a . For the computation of the partition functions of finite hexagonal lattices it is necessary to know all eigenvalues of V~,2,,,,,, and V~o,2,,2,,. The eigenvalues belonging to even eigenvectors can be found from the explicit solution of the fourth degree equations (89), if we know whether ill in formula (71) is a proper rotation or not. Those belonging to odd eigenvectors can be found in an analogous w a y starting from V o (see (66)). The exact forms of the rotation ~R related to V +, and of the corresponding rotation related to Vo, are needed moreover if we wish to calculate the average correlation between the spin variables of two non-neighbouring sites in the lattice according to the method of K a u f m a n and O n s a g e r 11). However, we shall not enter here into these topics.
O R D E R - D I S O R D E R IN H E X A G O N A L LATTICES
449
12. Transition temperature. T h e p a r t i t i o n functions Jlz and ~ d e p e n d on the t e m p e r a t u r e from the relations K 1= J l / k T , K 2 = J J k T and K 3 = J 3 / k T only. T h e interaction energies, J1, J2 and J3, are c o n s t a n t s belonging to the lattice. In the expressions (95) and (96) for Jl~ and ;~ the a r g u m e n t of the l o g a r i t h m u n d e r the double integral sign, Ch 2 K 1 Ch 2K2 Ch 2 K 3 + Sh 2 K 1 Sh 2 K 2 Sh 2 K 3 - - - Sh 2 K 1 cos coI - - Sh 2 K 2 cos a~2--Sh 2 K 3 cos (~oI + o~2). (101) and ½ {Ch 2 K 1 Ch 2 K 2 Ch 2K~ + 1 - - Sh 2 K 2 Sh 2 K 3 cos ~oI - - - Sh 2 K 3 Sh 2 K 1 cos ~o2 - - Sh 2 K 1 Sh 2 K 2 cos (oJ1 + c%)} (102) respectively, c a n n o t be negative for real K 1, K 2 and K 3. ~ , t a and all their higher derivatives with respect to the variables K i are continuous, e x c e p t for those sets of values of K 1, K 2 and K 3 which, for a p p r o p r i a t e values of a~l and oJ2, cause (101) or (102) respectively, to vanish. F o r t h e s e sets the second derivatives will diverge. So we find t h a t at a t e m p e r a t u r e , To, at which K 1 = Jl/kTc, K2 ~ J2/kTc and K 3 = J3/kT~ form such an exceptional set of values, the specific heat per atom, C
:
k
El.i= 1,2,3K i K i ~ In 2/aKi~K i
(4 = 2z or 1s~),
becomes infinite as - - In [ T ~ Tc I (as in the case of the r e c t a n g u l a r lattice). F o r given f l , .72 and Ja not more t h a n one such Tc can appear. On the analogy of the rectangular lattice we assume t h a t this T~ is a transition t e m p e r a t u r e between two phases. At T < T, the interactions between the spins of neighbouring atoms would m a i n t a i n a long-distance order, while at T > T~ only a short-range order would obtain. While e v e r y h o n e y c o m b lattice has a transition t e m p e r a t u r e , it t u r n s out t h a t t h e r e are exceptional t r i a n g u l a r lattices for which no transition t e m p e r a t u r e exists. These are the lattices for which one or all three interactions f~. are negative, while all three interactions, or at least the two weakest, are equal in strength. As to the calculation of Tc in all o t h e r cases, we notice t h a t ~ is i n v a r i a n t with respect to the simultaneous reversal of the Signs of two interaction energies J i and t h a t we can reduce all t r i a n g u l a r lattices, b y Physica XVI
29
450
R.M.F. HOUTAPPEL
such an operation, to one in which the two strongest interactions are positive, i.e. Jl + J2 > o, J2 + J3 > 0 and J3 + f l > 0. In the latter case Tc will be found from Sh (2]l/kTc) Sh (2]2/kTc) + Sh (2J2/kT~) Sh (2]~/kT~) + + Sh (2J3/kTc) Sh (2]i/kT~) ---- 1. (103) In fig. 4a we have illustrated in a triangular diagram the dependence of kTJ(Jl + J2 + J3) on the ratios of Jl, J2 and J3 if the two strongest of these interactions are positive. In this case -(fl +J2+J3) is the energy per atom in the completely ordered state at T = 0. We see that T, vanishes on the external bisectors of the basic triangle, representing thus the cases in which no transition temperature appears. The special case J r = J2---J3 < 0 is represented, after reduction as mentioned above, by the intersections of the external bisectors. A
o
A o
C Fig. 4a. Triangular lattice. Curves of constant kTd(Jl + J2 + J3) for ]x + J2 > o, J2 + J3 > o and J3 + JI > 0, calculated from formula (103).
Fig. 4b. Honeycomb lattice. Curves of constant kTJ½(J l + J2 + J3) for J l > 0, J2 > 0 and J3 > 0, calculated from formula (104).
Jl, J2 and 9r3 are plotted as triangle coordinates with A B C as basic triangle and centre D.
The appearance of a transition temperature for IJll ~ If21 ~: IJ3[ can be understood in the following way. The two strongest interactions will, at low temperatures, build up a long-distance order, analogous to t h a t of a rectangular lattice, while the third, if it
ORDER-DISORDER IN HEXAGONAL LATTICES
451
counteracts, will only lower the transition temperature. One can also understand why there are cases, as previously mentioned, in which no transition temperature exists. This is because in these cases the tendency of the strongest interaction to build up a longdistance order, together with one of the two weakest, will be counterbalanced by the tendency to build up another long-distance order with the second of them. Unlike the triangular lattice, every honeycomb lattice with non zero values of Jl, J2 and J3 has a transition temperature, To. This T~ can be found from the relation Sh (2]J1]/kT~) + Sh (2lJ2l/kTc) + Sh (2]J31/kT,) = = Sh(2lJ~l/kT~) Sh(2lJ2]/kTc) Sh(2lJ3i/kT,). (104)
B y inspection of the honeycomb lattice one can see that its three interactions cannot counteract each other in building up a longdistance order. In Fig. 4b we have illustrated, also in a triangular diagram, the dependence of kT,/~(IJll + IJ2] + IJ31) on the ratios of [Jll, ]J2] and ]J31 for a honeycomb lattice. - - ½(]Jx] + [J2t + ]J3{) is the energy per atom of the completely ordered state at T == 0. 13. Thermodynamic properties o] large isotropic triangular and honeycomb lattices. The partition functions ,~ and 2~ yield the free energy, F, the energy, E, the specific heat, C, and the entropy, S, per atom of large triangular and honeycomb lattices b y the formulae F = E - - T S = - - k T l n 2, E = F - - T d F / d T = - - X i= 1,2,aJi 0 In 2~OKi,
(105)
C ---- k Xi,i= 1,2,3K i K i 02 In 2/OKiOKi, (4 = 7~z or 2~). W e shall n o w consider more closely the isotropic triangular and
honeycomb lattices, i.e. those with equal interaction J in the three different directions J = J1 = J2 -- J3. An isotropic honeycomb lattice has always a transition temperature, To, which depends only on the strength [J[ of the interaction b y Ch (2J/kTc) ---- 2.
(106)
An isotropic triangular lattice, on the other hand, has only a transi-
452
R. M. F. HOUTAPPEL
tion temperature if J is positive. This transition temperature, To, is then given by exp (4J/kTc) = 3, (107) as W a n n i e r 7) found already. For the energy and the specific heat of isotropic triangular and honeycomb lattices with positive or negative interaction energy, we find with the aid of (95), (96/and (105) formulae similar to those Onsager found for the quadratic lattice (see O n s a g e r 2 ) formulae (116) and (117)): E = - - J [Coth 2K + at~(fl)], (K = J/kT) (108)
C=kK2[-2/(Sh2K)2+ - ~
a3)
a
d3
2fl-~ *'(fl)+ 2fl(1-----fl)dK %(fl)3" (109)
Here we denote by et(fl) and e2(fl) the complete elliptic integrals of the first and second kind,
ea(fl) = fo2" 11 x/1--fl(sin 9)2.d%
eu(fi)= fo2" x / l - - f l (sin 9)2.d9.
Table I shows the form of the functions a(K) and fl(K) for the isotropic triangular and honeycomb lattices, and, for comparison, also for the quadratic lattice. We notice the appearance of the functions Fz(K ) = F~z(K, K, K) and Go(K ) = Go(K, K, K) from formulae (25) to (28), which are related to the invariance properties discussed in section 6. TABLE I Lattice type
Temperature (x = exp (2K)) T< T c x2--4x + I >0)
Honeycomb T>T c X 2 - - 4 x + I < O)
Quadratic
T<
Triangular J>0
T>T
Triangular J
--
3 <
~[X 2 -
l[ ( X - - 1)4
O)
1 6 ( x 5 - x4 + x3) 4
(X4-- l) ( x Z - - 4 X q- 1) G~(K)
2 ( x 4 - - 6 x 2 + I)
16(X2-- 1)2 X2
n ( x 4 - - I)
(X2 + l) 4
:~'~/(x2-- 1)~ (x2 + 3) x(x2 - - 3 ) ( x 2 + 1)
c
1
G~(K~ = (x2--l)2(x--l)
4~(x - - t)2~/x~ - - x4 + .T
4X2(X2--3) (X2 + 1)
Tc
( x 2 - 3 > 0)
(x2
(X4-- 1) ( X 2 - - 4 X + l)
~z(x2-- 1) 4xa (3 - - x2)
u(l - - x ) 2 V ' ( l
+ x~)(3--x2)
16x2
1
F~(K)
( x 2 - - Ip (x2 + 3) F~,(K)
(I --x2)3 (3 + x2) (I + x2)3 (3 - - x 2 )
ORDER-DISORDER
IN HEXAGONAL
453
LATTICES
For the first three types of lattices in table I,/3 equals 1 at the transition temperature, and e1(/5) becomes, for this value of /5, infinite as ½In {16/( 1 7- fl)}. However the factor a, which becomes zero at T~, causes the term ae1(/5 ) in the formula for the energy, (108~, to vanish. The term (da/dK)el(/5) in formula (109) causes the specific heat to become infinite as - - In I T - - T~ [. For the triangular lattice with negative interaction,/5 becomes 1 and e1(/5) infinite at T = 0. Nervertheless the factors with which it appears multiplied in the formulae (108) and (109), reduce the respective terms to zero. In table II we have given several numerical data of thermodynamic properties of the different isotropic lattices at extremely low temperatures and at the transition temperature. T A B L E II Lattice type
--
ET=
0
O)
C'(T~
kTc
E ( T c)
F(Tc)
Cc
½clJI Honeycomb
I
1
1.012
0.7698
1.0251
1.303
0.4781
Quadratic
1
1
1.135
0.7071
0.9297
1.359
0.4945
1
1
1.214
0.6667
0.8796
1.391
0.4991
Triangular
j>0 Triangular
j
2K -- _ _
3~V/3
e--8K
c = c o r r e l a t i o n n u m b e r ; i.e. n u m b e r of a t o m s w i t h w h i c h e a c h a t o m i n t e r a c t s . C' =
4cakK2 e x p ( - - 2clKI) ;
Cc = L i m r - ~ r c
k In iT - - Tel" "
All data for the hexagonal lattices in table II can be found from the formulae (95), (96) and (105) b y elementary calculation, except the free energy and the entropy at To, the computation of which involves the numerical calculation of the integral in ~(9, 9, 9) = -- 412fo2'~/o2'~ In [1 - - ~ {cos co1 + cos co2 + cos (col + co2)}]dcol dco2 = - - 0. 17642. From table II we see that the first three kinds of lattices herein are very similar. For example --E(Tc)/(½clJI), which can be considered as a measure of the mean short range order at To, appears to decrease uniformly, but slowly, with increasing correlation number c. The behaviour of the triangular lattice with f < 0 is quite different. It has no transition temperature and its specific heat vanishes
454
R. M.F. HOUTAPPEL
b o t h at v e r y high and at v e r y low t e m p e r a t u r e s and has a m a x i m u m in between, as in the case of the linear I s i n g chain 1.,). It is well known t h a t a lattice with negative I s i n g interaction has the same p a r t i t i o n function as a g r a n d ensemble of a b i n a r y solid m i x t u r e of the same lattice t y p e with equal concentrations of the components. If ei,h is the interaction e n e r g y in the solid m i x t u r e for each pair of neighbouring atoms, one of kind i and the o t h e r of kind k(i, k = 1,2), the corresponding interaction J in the I s i n g lattice will be J = ½ [ei,2 - - ½ (el,l + e2,2)1. If such a solid m i x t u r e crystallizes in a triangular lattice and J is negative, it has, of course, a t e n d e n c y to order its atoms, i.e. to form a superqattice. It is easily seen t h a t the m a x i m u m of order is o b t a i n e d if, for example, the atoms of one kind o c c u p y all positions on the odd columns and the a t o m s of the o t h e r kind the positions on the even columns in fig. 1. Such a configuration has the e n e r g y per a t o m J , which is, indeed, equal to the e n e r g y at zero t e m p e r a t u r e (compare table II). Yet we conclude t h a t this t e n d e n c y to order does not lead to a transition t e m p e r a t u r e and t h a t at no t e m p e r a t u r e a long distance order will be established. 14. Acknowledgments. I wish to express m y t h a n k s to Prof. Dr. H. A. K r a m e r s and to D r . J . K o r r i n g a for t h e i r s t i m u l a ting interest and their c o n s t a n t advice, to Mr. T. A. S p r i n g e r and Mr. J. H. v a n der M e r w e for their kind help with some of the calculations and to Mrs. B. D a n i e I s- H u n t for her assistance with the translation into English. I am v e r y grateful to the L o r e n t z F o u n d a t i o n , which enabled me, b y a grant, to continue m y s t u d y of the problem discussed in this article.
ORDER-DISORDER
IN HEXAGONAL
LATTICES
455
REFERENCES 1) 2) 3) 4)
5) 6) 7) 8) 9)
10) 11) 12)
H.A. Kramers a n d G. H . W a n n i e r , Phys. Rev. 60(1941) 252-262. L. O n s a g e r , P h y s . R e v . 6 5 (1944) 117-149. B. K a u f m a n , P h y s . R e v . 7 6 (1949) 1 2 3 2 - 1 2 4 3 . T h e m a i n r e s u l t s c o n t a i n e d in t h i s a r t i c l e h a v e b e e n a n n o u n c e d in m y l e t t e r to t h e e d i t o r , R. M. F. H o u t a p p e 1, P h y s i c a , D e n H a a g 1 6 (1950) 3 9 1 - - 3 9 2 . H e r e i n is also m e n t i o n e d u n p u b l i s h e d w o r k o n t h e s a m e p r o b l e m b y T e m p e r I e y a n d b y H u s i m i a n d S y o z i . As y e t n o f u r t h e r i n f o r m a t i o n a b o u t t h e i r m e t h o d s a n d r e s u l t s h a s c o m e to m y a t t e n t i o n . S.B. Frobenius, P r e u s s . A k a d . Wiss. (1909) 5 1 4 - 5 1 8 . See r e f e r e n c e 2) p. 123. G.H. Wannier, Rev. mod. Phys. 17(1945) 50-60. H.A. Kramers a n d G . H. W a n n i e r , Phys. Rev. 60(1941) 263-276. F o r r e a l r o t a t i o n s see for e x a m p l e : B. L. v. d. W a e r d e n , E r g e b n . M a t h . u. G r e n s g e b . , B e r l i n (1935) B n d 4 1 6 5 - 1 6 9 . T h e t h e o r e m c a n also be p r o v e d for c o m p l e x r o t a t i o n s w h i c h c a n be d i a g o n a l i z e d . L. K r o n e c k e r., S. B. p r e u s z . A k a d . W i s s . (1890) 1 0 6 3 - 1 0 8 0 . B. K a u f m a n andL. Onsager, Phys. Rev. 76(1949) 1244-1252. E. I s i n g, Z. P h y s . 21 (1925) 2 5 3 - 2 5 8 .