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8440 A. J. MILLIS 53

y
z
choices @(n ,n )5(0,1), (1,2), (0,2)#, each picks out a pre- I ,I are obtained by permuting both row and column in the
1
2
ferred axis a5x,y,z. We have @g 512(cosk 1cosk y obvious way.
k
x
1cosk )# The term of order 1/a is a rather complicated three-site
z
5v 56acos6d14cosS p interaction; however, the important physics of this term is the
coupling it induces between sites on the same sublattice. To
v ~0,1! x 24dD g determine this coupling it is convenient to restrict attention
k k 3 k
to configurations ~favored by the order 1 term! in which ad-
1 jacent sites are in different Potts states, i.e., to terms in the
24F cosk 2 ~cosk 1cosk !G . ~C12! order 1/a term in Eq. ~D2! in which bÞg. If site i is taken
x
y
z
2
to be in state a5x, then the only nonconstant terms are
0,2
1,2
z
y
Similarly v 5v and v 5v . Note that because of the when a52bÞx ˆ .If a56y ˆ , then the energy is 24/9a if
relation of the angles c to the physical lattice distortions, a both sites are in the ‘‘y’’ state, 21/9a if both are in the ‘‘z’’
nearly uniform variation of c corresponds to a nearly stag- state, and 1/9a if the two are in different states; i.e., we may
gered variation of the physical lattice distortions. For physi- write
cally relevant values of a the gap is relatively large and the
dispersion small.
1 a a
W
W
J Q
E ~2! 52 ( Q ~Q i1b b W i1b 8 , ~D4!
i
APPENDIX D: DERIVATION OF THE POTTS MODEL 9a ib8
This appendix gives the details of the derivation of Eq.
~17! from ~Eq. 7!. It is assumed A is so large that only angles with b56x,y,z and
near those minimizing the anharmonicity energy Acos6u are
i
allowed. Thus write
a J 5S 0 0 0 D
u 5f 1d , ~D1! x b 0 4 22 , ~D5!
i
i
i
a
z
y
x
with f one of f 55p/6, f 5p/6, and f 5p/2, and d a 0 22 1
i i
small deviation. Note that the Jahn-Teller distortion corre- etc.
1
a
sponding to f is u 2 2 (u 1u ). Substituting Eq. ~D1! into The principal effect of E (2) is to lift the degeneracies of
a
c
b
Eq. ~7! and expanding gives
the antiferromagnetic three-state Potts model; this effect may
be mimicked by a simple second-neighbor ferromagnetic in-
E Na 1 a a teraction with magnitude J8 fixed, e.g., by the requirement
52 1 ( cos@2f 12c #cos@2f 12c #
3k 3 3 ia i a i1a a that it reproduce Eq. ~C9!.
8
a
¯ 2
16a( d 2 a( sin@2f 12c #sin@2f a
i
a
i
i
i 27 iab
APPENDIX E: MEAN-FIELD T s WITH HOLES
12c #cos@2f b 12c #cos@2f g 12c #. ~D2!
b i1a c i1b b
In the presence of a concentration x of holes, one expects
T (x)5T (12ax). In this Appendix a is derived using a
c0
s
Here N is the number of sites in the crystal, a5A/k, and
¯ d 5d 2d min , with ]E/]d min 50. In the large-A limit the co- mean-field theory. In leading order in x one need only con-
i
i
i
sider configurations in which one of the six neighbors of the
i
¯
¯
efficient of the d term is large, and so fluctuations in d may
distinguished site, say, the one in the b direction, has a hole.
be neglected. From Eq. ~21! one has
The energy may be more conveniently written in a dis-
crete notation. Denote the state on site i by the continuous
variable d and define a discrete quantity Q which indicates E ~u!52kcos~2u12c !@ccos2c 2ssin2c 1b#
i
i
to which of f ,f ,f the angle u is nearest. Choose Q to b b b b
i
x
y
z
i
be a three-component vector witha1inone place and 0 in 14 ( cosS 2u12c 1 DF ccosS 2c 1 D
2pp
2pp
the other two. Q 5(1,0,0) means u i is close to f , p561 b 3 b 3
i
x
Q 5(0,1,0) means u is close to f and Q 5(0,0,1) means
i
i
i
y
u is close to f . The interaction term of order is then a 2pp
z
i
a
333 matrix I , which depends on the direction a ˆ of the 2ssinS 2c 1 DG . ~E1!
b
3
bond connecting the two sites. One finds
3S 1 21/2 21/2 D c and s, rearranging, and discarding terms proportional to
Substituting Eq. ~E1! into Eqs. ~19!, ~20!, expanding in
1
x
I 5 21/2 1/4 1/4 . ~D3! cos2usin2c or sin2ucos2c , which will not contribute to
b
b
21/2 1/4 1/4 averages of interest, gives