Non-universal critical behaviour of a mixed-spin Ising model on the extended Kagome lattice
The mixed spin-1/2 and spin-3/2 Ising model on the extended Kagome lattice is solved by establishing a mapping correspondence with the eight-vertex model. When the parameter of uniaxial single-ion anisotropy tends to infinity, the model system becomes exactly solvable as the staggered eight-vertex...
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irk-123456789-1213122017-06-15T03:05:16Z Non-universal critical behaviour of a mixed-spin Ising model on the extended Kagome lattice Strecka, J. Canova, L. The mixed spin-1/2 and spin-3/2 Ising model on the extended Kagome lattice is solved by establishing a mapping correspondence with the eight-vertex model. When the parameter of uniaxial single-ion anisotropy tends to infinity, the model system becomes exactly solvable as the staggered eight-vertex model satisfying the free-fermion condition. The critical points within this manifold can be characterized by critical exponents from the standard Ising universality class. The critical points within another subspace of interaction parameters, which corresponds to a coexistence surface between two ordered phases, can be approximated by corresponding results of the uniform eight-vertex model satisfying the zero-field condition. This coexistence surface is bounded by a line of bicritical points that have non-universal continuously varying critical indices 2006 Article Non-universal critical behaviour of a mixed-spin Ising model on the extended Kagome lattice / J. Strecka, L. Canova // Condensed Matter Physics. — 2006. — Т. 9, № 1(45). — С. 179–186. — Бібліогр.: 15 назв. — англ. 1607-324X PACS: 75.10.Hk, 05.50.+q, 75.40.Cx DOI:10.5488/CMP.9.1.179 http://dspace.nbuv.gov.ua/handle/123456789/121312 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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The mixed spin-1/2 and spin-3/2 Ising model on the extended Kagome lattice is solved by establishing a mapping correspondence with the eight-vertex model. When the parameter of uniaxial single-ion anisotropy
tends to infinity, the model system becomes exactly solvable as the staggered eight-vertex model satisfying
the free-fermion condition. The critical points within this manifold can be characterized by critical exponents
from the standard Ising universality class. The critical points within another subspace of interaction parameters,
which corresponds to a coexistence surface between two ordered phases, can be approximated by
corresponding results of the uniform eight-vertex model satisfying the zero-field condition. This coexistence
surface is bounded by a line of bicritical points that have non-universal continuously varying critical indices |
| format |
Article |
| author |
Strecka, J. Canova, L. |
| spellingShingle |
Strecka, J. Canova, L. Non-universal critical behaviour of a mixed-spin Ising model on the extended Kagome lattice Condensed Matter Physics |
| author_facet |
Strecka, J. Canova, L. |
| author_sort |
Strecka, J. |
| title |
Non-universal critical behaviour of a mixed-spin Ising model on the extended Kagome lattice |
| title_short |
Non-universal critical behaviour of a mixed-spin Ising model on the extended Kagome lattice |
| title_full |
Non-universal critical behaviour of a mixed-spin Ising model on the extended Kagome lattice |
| title_fullStr |
Non-universal critical behaviour of a mixed-spin Ising model on the extended Kagome lattice |
| title_full_unstemmed |
Non-universal critical behaviour of a mixed-spin Ising model on the extended Kagome lattice |
| title_sort |
non-universal critical behaviour of a mixed-spin ising model on the extended kagome lattice |
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Інститут фізики конденсованих систем НАН України |
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2006 |
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http://dspace.nbuv.gov.ua/handle/123456789/121312 |
| citation_txt |
Non-universal critical behaviour of a mixed-spin Ising model on the extended Kagome lattice / J. Strecka, L. Canova // Condensed Matter Physics. — 2006. — Т. 9, № 1(45). — С. 179–186. — Бібліогр.: 15 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT streckaj nonuniversalcriticalbehaviourofamixedspinisingmodelontheextendedkagomelattice AT canoval nonuniversalcriticalbehaviourofamixedspinisingmodelontheextendedkagomelattice |
| first_indexed |
2025-07-08T19:38:03Z |
| last_indexed |
2025-07-08T19:38:03Z |
| _version_ |
1837108811666554880 |
| fulltext |
Condensed Matter Physics 2006, Vol. 9, No 1(45), pp. 179–186
Non-universal critical behaviour
of a mixed-spin Ising model on
the extended Kagom é lattice ∗
J.Strečka, L.Čanová
Department of Theoretical Physics and Astrophysics,
Faculty of Science, P. J. Šafárik University,
Park Angelinum 9, 040 01 Košice, Slovak Republic
Received July 18, 2005, in final form November 16, 2005
The mixed spin-1/2 and spin-3/2 Ising model on the extended Kagomé lattice is solved by establishing a
mapping correspondence with the eight-vertex model. When the parameter of uniaxial single-ion anisotropy
tends to infinity, the model system becomes exactly solvable as the staggered eight-vertex model satisfying
the free-fermion condition. The critical points within this manifold can be characterized by critical exponents
from the standard Ising universality class. The critical points within another subspace of interaction param-
eters, which corresponds to a coexistence surface between two ordered phases, can be approximated by
corresponding results of the uniform eight-vertex model satisfying the zero-field condition. This coexistence
surface is bounded by a line of bicritical points that have non-universal continuously varying critical indices.
Key words: Ising model, eight-vertex model, bicritical points, non-universality
PACS: 75.10.Hk, 05.50.+q, 75.40.Cx
1. Introduction
Investigation of phase transitions and critical phenomena belongs to the most intensively studied
topics in the equilibrium statistical physics. A considerable progress in the understanding of order-
disorder phenomena has been achieved by solving planar Ising models that represent valuable
exceptions of exactly solvable lattice-statistical models with a non-trivial critical behaviour [1].
Although phase transitions of planar Ising models have already been understood in many respects,
there are still a lot of obscurities connected with a criticality of more complicated spin systems
exhibiting reentrant transitions, non-universal critical behaviour, tricritical phenomena, etc. It
is worth mentioning, however, that several complicated Ising models can be exactly treated by
transforming them to the solvable vertex models. A spin-1/2 Ising model on the union jack (centered
square) lattice, which represents a first exactly solvable system exhibiting reentrant transitions [2],
can be for instance reformulated as a free-fermion eight-vertex model [3]. It should be also pointed
out that an equivalence with the vertex models has already provided a precise confirmation of the
reentrant phenomenon in the anisotropic spin-1/2 Ising models on extended Kagomé lattice [4]
and centered honeycomb lattice [5] as well.
Despite the significant amount of effort, there are only a few exactly solvable Ising models con-
sisting of mixed spins of different magnitudes, which are usually called as mixed-spin Ising models.
A strong scientific interest focused on the mixed-spin systems arises partly due to a much richer
critical behaviour they display compared with their single-spin counterparts and partly due to the
fact that they represent the most simple models of ferrimagnets having a wide potential applica-
bility in practice. Using the extended versions of decoration-iteration and star-triangle mapping
transformations, Fisher [6] has derived exact solutions of the mixed spin-1/2 and spin-S (S > 1)
∗This work was financially supported under the grants VEGA 1/2009/05 and APVT 20–005204.
c© J.Strečka, L.Čanová 179
J.Strečka, L.Čanová
Ising models on the honeycomb, diced and decorated honeycomb lattices. Notice that these map-
ping transformations were later on further generalized in order to also account for the single-ion
anisotropy effect. The effect of uniaxial and biaxial single-ion anisotropies has been precisely inves-
tigated on the mixed-spin honeycomb lattice [7] as well as on some decorated planar lattices [8].
With the exception of several mixed-spin models formulated on the Bethe (Cayley tree) lattices,
which can be accurately treated within a discrete non-linear map [9] or exact recursion equations
[10], these are the only mixed-spin planar Ising models with generally known exact solutions, yet.
One of the most outstanding findings emerging in the phase transition theory is a non-universal
critical behaviour of some planar Ising models, which is in obvious contradiction with the idea of
universality hypothesis [11]. The mixed spin-1/2 and spin-S Ising model on the union jack lattice
[12] represents very interesting system from this viewpoint as it exhibits a remarkable line of bicrit-
ical points that have continuously varying critical indices obeying the weak universality hypothesis
[13]. In the present article, we shall investigate a topologically similar mixed spin-1/2 and spin-
3/2 Ising model on the extended Kagomé lattice by establishing a mapping correspondence with
the staggered and uniform eight-vertex models, respectively. In a certain subspace of interaction
parameters, the model under investigation becomes exactly solvable as the staggered eight-vertex
model satisfying the free-fermion condition [14]. Even if a non-validity of the free-fermion condi-
tion in the rest of parameter space is simply ignored, one still obtains a rather reliable estimate
of the criticality within free-fermion approximation [15]. Finally, the critical points within another
subspace of interaction parameters can be approximated from the relevant solution of the uniform
eight-vertex model satisfying the zero-field condition.
The outline of this paper is as follows. In section 2, a detailed formulation of the model is
presented and subsequently, the mapping correspondence that ensures an equivalence with the
eight-vertex models will be derived. The most interesting numerical results for a critical behaviour
will be presented and particularly discussed in section 3. Finally, some concluding remarks are
drawn in section 4.
2. Formulation
Let us begin with considering the mixed spin-1/2 and spin-3/2 Ising model on the extended
Kagomé lattice L schematically illustrated in figure 1. The mixed-spin Kagomé lattice consists of
the spin-1/2 (empty) and spin-3/2 (filled circles) atoms placed on the six- and four-coordinated
sites, respectively. The total Hamiltonian defined upon the underlying lattice L reads:
Hmix = −J
2N
∑
(i,j)⊂J
Siσj − J ′
2N
∑
(k,l)⊂K
σkσl − D
N/2
∑
i=1
S2
i , (1)
where σj = ±1/2 and Si = ±1/2,±3/2 are Ising spin variables, J denotes the exchange interaction
between nearest-neighbouring spin-1/2 and spin-3/2 pairs and J ′ labels the interaction between
the spin-1/2 pairs that are next-nearest-neighbours on the extended Kagomé lattice L. Finally, the
parameter D measures a strength of the uniaxial single-ion anisotropy acting on the spin-3/2 sites
and N denotes the total number of the spin-1/2 sites.
In order to proceed further with calculation, the central spin-3/2 atoms should be firstly dec-
imated from all faces of extended Kagomé lattice L. After the decimation, i.e. after performing a
summation over spin degrees of freedom of the spin-3/2 sites (filled circles), the partition function
of the mixed-spin system can be rewritten as:
Zmix =
∑
{σ}
N/2
∏
m=1
ωA
m(σi, σj , σk, σl)
N/2
∏
n=1
ωB
n (σi, σj , σk, σl). (2)
Above, the summation is performed over all possible spin configurations available at the spin-1/2
sites and the first (second) product is over N/2 faces having four spin-1/2 sites σi, σj , σk, σl placed
in the corners of square plaquettes with (without) a central spin-3/2 site in the middle of these
180
Non-universal critical behaviour of a mixed-spin Ising model
Sn
si
sj
sk
sl
J
J’
Figure 1. Diagrammatic representation of the extended Kagomé lattice composed of the mixed
spin-1/2 (empty circles) and spin-3/2 (filled circles) sites, respectively. The solid (broken) lines
depict the nearest-neighbour (next-nearest-neighbour) interactions.
plaquettes (see figure 1). The Boltzmann factors ωA(a, b, c, d) and ωB(a, b, c, d) assigned to two
different kinds of alternating faces, which constitute the checkerboard lattice, can be defined as:
ωA(a, b, c, d) = 2 exp[K ′(ab + bc + cd + da)/2 + ∆/4]
×
{
exp(2∆) cosh[3K(a + b + c + d)/2] + cosh[K(a + b + c + d)/2]
}
,
ωB(a, b, c, d) = exp[K ′(ab + bc + cd + da)/2], (3)
where K = J/(kBT ), K ′ = J ′/(kBT ), ∆ = D/(kBT ), kB is Boltzmann’s constant, and T stands
for the absolute temperature.
+ +
+ +
w
1
+
_
_
+
w
2
+
_
+
_
w
3
_
w
4
+ +
_
+
_
+ +
w
5
_
+
w
6
+ +
_
+
w
7
+ + +
_
+ +
w
8
Figure 2. The eight possible line arrangements at a vertex of the dual lattice.
At this stage, the model under investigation can be rather straightforwardly mapped onto the
staggered eight-vertex model defined on a dual checkerboard lattice LD, since Boltzmann factors
ωA(a, b, c, d) and ωB(a, b, c, d) are being invariant under the reversal of all four spin variables.
Actually, there are maximally eight different spin arrangements giving different Boltzmann weights
ωA(a, b, c, d) and ωB(a, b, c, d) for each kind of face. Diagrammatic representation of eight possible
spin arrangements and their corresponding line coverings of the eight-vertex model is shown in
figure 2. If, and only if, the adjacent spins are aligned opposite to each other, then solid lines are
drawn on the edges of the dual lattice LD, otherwise they are drawn as broken lines. It can be easily
understood that eight possible line coverings around each vertex of the dual checkerboard lattice
always correspond to two spin configurations, one is being obtained from the other by reversing
all spins. Since there is an even number of solid (broken) lines incident to each vertex of the dual
lattice LD, the model becomes equivalent to the staggered eight-vertex model.
The Boltzmann weights ωA(a, b, c, d) and ωB(a, b, c, d), which correspond to eight possible line
coverings emerging at vertices of the dual checkerboard lattice, can directly be calculated from
equation (3):
ωA
1 = 2 exp(K ′/2 + ∆/4)[exp(2∆) cosh(3K) + cosh(K)],
ωA
2 = 2 exp(−K ′/2 + ∆/4)[exp(2∆) + 1],
ωA
3 = ωA
4 = 2 exp(∆/4)[exp(2∆) + 1],
ωA
5 = ωA
6 = ωA
7 = ωA
8 = 2 exp(∆/4)[exp(2∆) cosh(3K/2) + cosh(K/2)]; (4)
181
J.Strečka, L.Čanová
ωB
1 = exp(K ′/2), ωB
2 = exp(−K ′/2),
ωB
3 = ωB
4 = ωB
5 = ωB
6 = ωB
7 = ωB
8 = 1. (5)
Unfortunately, there does not exist a general exact solution to the staggered eight-vertex model
with arbitrary Boltzmann weights ωA
i and ωB
j (i, j = 1 − 8). However, if the weights (4) and (5)
satisfy the so-called free-fermion condition:
Ω1Ω2 + Ω3Ω4 = Ω5Ω6 + Ω7Ω8 , (6)
the staggered eight-vertex model then becomes exactly solvable as the free-fermion model solved
several years ago by Hsue, Lin and Wu [14]. The expressions which enter into the free-fermion
condition (6) can be defined through:
Ω1 = ωA
1 ωB
1 + ωA
2 ωB
2 , Ω2 = ωA
3 ωB
3 + ωA
4 ωB
4 ,
Ω3 = ωA
5 ωB
6 + ωB
5 ωA
6 , Ω4 = ωA
7 ωB
8 + ωB
7 ωA
8 ,
Ω5Ω6 = ωA
1 ωB
1 ωA
3 ωB
3 + ωA
2 ωB
2 ωA
4 ωB
4 + ωA
5 ωB
6 ωA
7 ωB
8 + ωB
5 ωA
6 ωB
7 ωA
8 ,
Ω7Ω8 = ωA
1 ωB
1 ωA
4 ωB
4 + ωA
2 ωB
2 ωA
3 ωB
3 + ωA
5 ωB
6 ωB
7 ωA
8 + ωB
5 ωA
6 ωA
7 ωB
8 . (7)
It can be readily proved that the free-fermion condition (6) holds in our case just as D → ±∞,
or T → ∞. The restriction to infinitely strong single-ion anisotropy consequently leads to the
familiar phase transitions from the standard Ising universality class, because in this case our model
effectively reduces to a simple spin-1/2 Ising model on the extended Kagomé lattice. Within the
manifold given by the constraint (6), the free-fermion model becomes critical as long as:
Ω1 + Ω2 + Ω3 + Ω4 = 2max{Ω1, Ω2, Ω3, Ω4}. (8)
It is noteworthy, however, that the critical condition (8) yields a rather reliable estimate of the
criticality within the so-called free-fermion approximation [15] even if a non-validity of the free-
fermion condition (6) is simply ignored.
Now, we shall establish an approximate mapping between the staggered and uniform eight-
vertex models, since the second branch of exact solution is available just for the latter model
under the zero-field condition [1]. For this purpose, let us define average Boltzmann weights of
the staggered eight-vertex model, which would approximately transform the staggered eight-vertex
model into the uniform one:
ω̃i = ωA
i ωB
i , (i = 1 − 8). (9)
Note that the uniform eight-vertex model satisfies the zero-field condition just when its Boltzmann
weights are pairwise and symmetrically equal to each other:
ω̃1 = ω̃2, ω̃3 = ω̃4, ω̃5 = ω̃6, ω̃7 = ω̃8 . (10)
As we already have ω̃3 = ω̃4, ω̃5 = ω̃6, and ω̃7 = ω̃8, the zero-field case is consequently reached by
imposing the condition ω̃1 = ω̃2 only, or equivalently:
exp(2∆) =
exp(−2K ′) − cosh(K)
cosh(3K) − exp(−2K ′)
. (11)
According to Baxter’s exact solution [1], the zero-field eight-vertex model becomes critical on the
manifold (10) if:
ω̃1 + ω̃3 + ω̃5 + ω̃7 = 2max{ω̃1, ω̃3, ω̃5, ω̃7}. (12)
It is easy to check that in our case ω̃1 represents the largest Boltzmann weight. Thus, the condition
determining the criticality can also be written in this equivalent form:
exp(K ′
c)[exp(2∆c) cosh(3Kc) + cosh(Kc)] =
= 1 + exp(2∆c) + 2 exp(2∆c) cosh(3Kc/2) + 2 cosh(Kc/2), (13)
182
Non-universal critical behaviour of a mixed-spin Ising model
where Kc = J/(kBTc), K ′
c = J ′/(kBTc), ∆c = D/(kBTc), and Tc denotes the critical temperature.
It should be stressed, nevertheless, that the critical exponents (with the exception of δ and η)
describing a phase transition of the zero-field eight-vertex model depend on the function µ =
2 arctan(ω̃5ω̃7/ω̃1ω̃3)
1/2, in fact:
α = α′ = 2 −
π
µ
, β =
π
16µ
, ν = ν′ =
π
2µ
, γ =
7π
8µ
, δ = 15, η =
1
4
. (14)
Finally, let us explicitly evaluate the critical exponent β that determines the disappearance of the
spontaneous order as the critical temperature is approached from below:
β−1 =
32
π
arctan
{
exp(2∆c) cosh(3Kc/2) + cosh(Kc/2)
[exp(2∆c) + 1]3/4[exp(2∆c) cosh(3Kc) + cosh(Kc)]1/4
}
. (15)
3. Results and discussion
Now, let us turn our attention to a discussion of the most interesting results obtained for the
ground-state and finite-temperature phase diagrams. Solid lines displayed in figure 3 represent
ground-state phase boundaries separating four distinct long-range ordered phases that emerge
in the ground state when J > 0. Spin order drawn in broken rectangles shows a typical spin
configuration within basic unit cell of each phase. As could be expected, a sufficiently strong
antiferromagnetic next-nearest-neighbour interaction J ′ alters the structure of the ground state
owing to a competing effect with the nearest-neighbour interaction J . Due to a competition between
the interactions, the central spins are free to flip within the phases III and IV and thus, these phases
exhibit a remarkable coexistence of the order and the disorder. At last, it is worthwhile mentioning
that a broken line connecting both triple points depicts a projection of the approximate critical line
(13) into the J ′−D plane. As this projection crosses zero-temperature plane along the ground-state
transition line D/J = −3/2− J ′/J between the phases I and IV, it is quite reasonable to suspect
that this line represents a location of phase transitions between these phases.
Figure 3. Ground-state phase diagram in the J ′
− D plane when J > 0. Broken rectangles
schematically illustrate a typical spin configuration within each phase. Broken line connecting
both triple points shows a projection of the critical line (13) into the J ′
− D plane.
Let us more deeply investigate this line of critical points. The critical temperatures calculated
from the uniform zero-field eight-vertex model must simultaneously obey both the zero-field condi-
tion (11) and the critical condition (13). It is easy to check that the former condition necessitates
−1.5 < J ′/J < −0.5 and −1.0 < D/J < 0.0. Figure 4(a) displays a projection of this critical line
into the J ′ − Tc plane (the dependence scaled to the left axis) and respectively, a projection into
the J ′ − D plane which is scaled to the right axis. Along this critical line, the critical exponents
are expected to vary with interaction parameters as they have to follow the equations (14). For
illustration, figure 4(b) shows how the critical index β changes along the critical line. Apparently,
183
J.Strečka, L.Čanová
Figure 4. (a) The curve scaled to the left axis shows how the critical temperature changes with a
strength of the next-nearest-neighbour coupling J ′/J , the curve scaled to the right axis depicts
a variation of the single-ion anisotropy parameter along this line; (b) The same as for figure 4(a),
but the critical exponent β is now scaled to the left axis. In both figures broken lines are guides
for eyes only.
the exponent β approaches its smallest possible value 1/16 by reaching both triple points with
zero critical temperature. However, it is also quite interesting to ascertain that its greatest value
is below the value 1/8 that predicts the universality hypothesis for planar Ising systems [1].
Figure 5. A plot critical temperature versus the ratio J ′/J for several values of the single-ion
anisotropy term D/J . For details see the text.
Before concluding, few remarks should be addressed to a global finite-temperature phase dia-
gram plotted in figure 5, which displays the critical temperature as a function of the ratio J ′/J for
several values of the single-ion anisotropy D/J . Critical boundaries depicted as solid lines repre-
sent exact critical points obtained from the free-fermion solution (8) of the staggered eight-vertex
model obtained under the constraint (6), which is fulfilled in the limiting cases D/J → ±∞.
Dotted critical lines show the estimated critical temperatures calculated from the free-fermion
approximation simply ignoring the non-validity of the free-fermion condition (6) for any finite
value of D/J . Approximative solution related to the critical points (13) of the uniform zero-field
eight-vertex model on the variety (11) is displayed as a rounded broken line. It is quite obvious
from the ground-state phase diagram (figure 3) that a right (left) wing of the displayed critical
boundaries corresponds to the phase I (III) if D/J > 0.0, while it corresponds to the phase II
(IV) if D/J < −1.0. Actually, both the exact and the approximate critical points resulting from
the free-fermion solution correctly reproduce the ground-state boundaries between these phases.
When the single-ion anisotropy parameter is selected within the range −1.0 < D/J < 0.0 (see
for instance the curve for D/J = −0.5), however, the critical line obtained from the free-fermion
approximation meets at a bicritical (circled) point with the critical line of the equivalent uniform
zero-field eight-vertex model as it has been already reasoned by Lipowski and Horiguchi [12] who
have solved similar spin system on the union-jack lattice. In such a case, the right and left part
(with respect to the bicritical point) of this critical line separate the phases I and IV, respectively,
184
Non-universal critical behaviour of a mixed-spin Ising model
and a line of first-order phase transitions is expected to terminate at this special multicritical point.
There are strong indications supporting this concept [12], actually, the almost straight broken line
depicting the zero-field condition (11) should always show the coexistence of these two phases as
it starts from the point that determines their coexistence in the ground state. With regard to the
aforementioned arguments one may conclude that the coexistence surface between the phases I
and IV lies inside the area, which is bounded by the line of bicritical points (rounded broken line)
having the non-universal interaction-dependent critical exponents.
Finally, we should remark a feasible appearance of reentrant transitions which can be observed
in the critical lines near the coexistence points D/J = 0.0 and −1.0. It is quite apparent that
the observed reentrance can be explained in terms of the coexistence of a partial order and partial
disorder emerging in the both high-temperature reentrant phases III and IV. As a matter of fact, the
partial disorder of the spin-3/2 atoms can compensate a loss of entropy that occurs in these phases
due to a thermally induced partial ordering of the spin-1/2 atoms what is in a good accordance
with the necessary condition conjectured for the appearance of reentrant phase transitions [4,5].
4. Concluding remarks
The work reported in the present article provides a relatively precise information on the crit-
ical behaviour of the mixed spin-1/2 and spin-3/2 Ising model on the extended Kagomé lattice
by establishing a mapping correspondence with the staggered and uniform eight-vertex models,
respectively. The main focus of the present work has been aimed at the examination of the criti-
cality depending on the single-ion anisotropy strength as well as on the strength of the competing
next-nearest-neighbour interaction. The location of the critical boundaries has accurately been
determined from the free-fermion solution of the staggered eight-vertex model and the zero-field
solution of the uniform eight-vertex model, respectively, whereas the validity of both mappings
is restricted to certain subspaces of interaction parameters only. In the rest of parameter space,
the free-fermion approximation has been used to estimate the critical boundaries as this method
should provide a meaningful approximation giving a rather reliable estimate to the true transition
temperatures.
The remarkable critical line consisting of bicritical points, which bounds the coexistence surface
between two long-range ordered phases is of theoretical interest in this model. The bicritical points
can be characterized by non-universal interaction-dependent critical exponents that satisfy the
weak universality hypothesis. Moreover, the same arguments as those suggested by Lipowski and
Horiguchi [12] have enabled us to identify the zero-field condition (11) with the location of the
first-order transition lines separating these two ordered phases.
It should be remarked that the spin system considered also shows reentrant phase transitions
on account of the competition between the nearest- and next-nearest-neighbour interactions. Our
results are in agreement with the conjecture [4] stating that the reentrance appears as a consequence
of the coexistence of a partial order and disorder, namely, the partial disorder induced among spin-
3/2 atoms can compensate for the loss of entropy, which occurs on behalf of the partial ordering
of the spin-1/2 atoms in the both high-temperature partially ordered phases.
185
J.Strečka, L.Čanová
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186
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