378
The European Physical Journal B
for all values of a, i.e. the model is exactly solvable.
1
0.1
In order to determine the phase transition point(s) of
the model we investigate directly the general partition
function of a free-fermionic 8-vertex model given in [13]
0.01
Z
Z
2π
2π
1
8π2
ln Z =
ln [2p + 2q1 cos θ + 2q2 cos φ
0.001
0.0001
1e-05
1e-06
1e-07
p
↓↑
0
0
(5)
+2q3 cos(θ − φ) + 2q4 cos(θ + φ)] dθ dφ,
where
1
p = (ω12 + ω22 + ω32 + ω42)
2
q1 = ω1ω3 − ω2ω4
q2 = ω1ω4 − ω2ω3
q3 = ω3ω4 − ω7ω8
q4 = ω3ω4 − ω5ω6.
(6)
0
1
2
3
4
5
6
7
8
9
10
a2
In case of our special set of vertex weights we have q1 = q2
and q3 = q4. This simplifies the partition function to
Fig. 2. Probability p for finding an antiparallel arrow pair on
↓↑
a bond within a plaquette (solid) and between two plaquettes
(dashed).
Z
Z
2π
2π
1
8π2
ln Z =
ln[2p + 2q1(cos θ + cos φ)
0
0
+ 4q3 cos θ cos φ]dθ dφ.
(7)
In order to calculate properties of the vertex state
model we investigate the inner product hΨ0|Ψ0i. As ex- As explained in [13] the θ-integration can be performed
plained in [3] this inner product is given by the partition by rewriting (7) in the following form
Z
Z
function of a classical vertex model with two arrow vari-
ables between each pair of adjacent sites. Motivated by the
results on the hexagonal lattice we have measured numer-
2π
2π
1
ln Z =
ln [2A + 2B cos θ] dθ dφ
(8)
(9)
8π2
0
0
Z
ically the probability p for finding an antiparallel arrow
↓↑
2π
h
i
p
1
4π
pair on a bond. As shown in Figure 2, p decays exponen-
↓↑
=
ln A + Q(φ) dφ,
tially as a function of a2, but for bonds within a plaquette
the decay is slower than between two plaquettes. Numer-
ically we have also found a second order phase transition
0
where
A = p + q1 cos φ, B = q1 + 2q3 cos φ, Q(φ) = A2 − B2.
at a2 ≈ 7.0. In this regime p ≈ 0.002, hence we neglect
↓↑
c
(10)
all vertices with antiparallel arrow pairs in the following
consideration, i.e. we reduce the model to a vertex model
with only eight different classical vertices at each site.
The next step is to sum out the four interior bonds
on each plaquette, which yields a 16-vertex model on the
square lattice. The vertex weights are invariant under a
simultaneous flip of all four arrows, i.e. the model is field-
Now we can follow the argument given in [13]. By explic-
itly calculating the function Q(φ) in terms of the model
parameter a we observe that it is not a complete square,
so (9) is analytic unless
Q(φ) = 0.
(11)
free. Thus we can transform this 16-vertex model to an The only real non-negative solution of this equation is
q
8-vertex model by attaching the orthogonal 2 × 2-matrix
√
√
φc = 0 and a2c = 1 + 2 + 2(5 + 4 2) ≈ 7.03.
ꢀ
ꢁ
1
2
1
1
(12)
√
u =
(2)
1 −1
This is consistent with our numerical result.
The phase transition is of the same type as in [3]. It
corresponds to two simultaneous Ising transitions from a
disordered phase (a2 < ac2) with exponentially decaying
correlation functions to a N´eel ordered phase (a2 > ac2)
with alternating long-range correlations.
In summary we have applied the vertex state model
approach to a spin-3/2 antiferromagnet on a decorated
square lattice. The local interaction and the vertices used
for the vertex state model are the same as those used in [3]
on the hexagonal lattice. As a function of the anisotropy
parameter the resulting global ground state exhibits a sec-
ond order transition from a disordered phase to a N´eel
ordered phase. The phase transition corresponds to two
simultaneous Ising transitions.
to each bond. If the usual notation for the 8-vertex model
is used, the resulting vertex weights are
ꢂ
ꢂ
ꢂ
ꢃ
1
2
1
2
1
ω1
ω2
=
=
41 + 52 a2 + 30 a4 + 4 a6 + a8
ꢃ
4
−1 + a2
ꢃ ꢂ
ꢃ
3
ω3 = ω4 = −2 −1 + a2
3 + a2
(3)
ꢂ
ꢃ ꢂ
ꢃ
ꢃ
2
1
ω5 = ω6 =
−1 + a2
5 + 2 a2 + a4
5 + 2 a2 + a4
2
ꢂ
ꢃ ꢂ
2
1
ω7 = ω8 = −2 −1 + a2
.
As in the case of the corresponding model on the hexago-
nal lattice, these weights fulfil the free-fermion condition
ω1ω2 + ω3ω4 = ω5ω6 + ω7ω8
(4)
Tanaka
