36th Conference of the Middle European Cooperation in Statistical Physics (MECO 36)

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Weak-universal critical behavior in the exactly solved mixed-spin Ising model with the triplet interaction

on a centered square lattice

Jozef Streˇ cka

Department of Theoretical Physics and Astrophysics, Faculty of Science, P. J. ˇSaf´arik University, Park Angelinum 9, 040 01 Koˇsice, Slovak Republic

36th Conference of the Middle European Cooperation in Statistical Physics (MECO 36)

April 5th-7th, 2011, L’viv, Ukraine

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Outlook

1

Introduction

2

Ising model with triplet interaction Graph-theoretical approach Algebraic approach

3

The most interesting results Model A

Model B Model C Model D

4

Conclusions

J. Streˇcka (P.J. ˇSaf´arik University) MECO 36, L’viv 2 / 22

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Introduction

One of the most important concepts established in the theory of phase transitions and critical phenomena is universality hypothesis, which states that the behaviour in a close vicinity of critical points does not depend on specific details of the Hamiltonian but only upon

spatial dimensionalityof the model under investigation total number of components of theorder parameter

The universality hypothesis and its direct consequences:

critical behaviour of very different models may be characterized by the same set of critical exponents and one says that the models with the identical set of critical exponents belong to the same universality class

However, there exists few exactly solved models with critical exponents

that do depend on the interaction (Hamiltonian) parameters and hence,

they contradict the usual universality hypothesis

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Ising model with triplet interaction

Spin-1/2 Ising model with three-spin (triplet) interaction

The Hamiltonian:

H = − J X

ijk∈∆

S

i

S

j

S

k

S

i

= ± 1/2 – two-valued Ising spin variable J – three-spin (triplet) interaction X

ijk∈∆

– summation over triangular faces

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Spin-1/2 Ising model with three-spin (triplet) interaction

The Hamiltonian:

H = − J X

ijk∈∆

S

i

S

j

S

k

S

i

= ± 1/2 – two-valued Ising spin variable J – three-spin (triplet) interaction X

ijk∈∆

– summation over triangular faces

Exactly solved spin-1/2 Ising models with the triplet interaction:

centered square lattice,α= 1/2

A. Hintermann, D. Merlini,Phys. Lett. 41A, 208 (1972) triangular lattice,α= 2/3

R. J. Baxter, F. Y. Wu,PRL31, 1294 (1973);Aust. J. Phys. 27, 357 (1974) decorated triangular lattice,α≈0 (logarithmic singularity)

D. W. Wood,J. Phys. C: Solid St. Phys.6, L135 (1973) kagom´e lattice, no phase transition at all

J. H. Barry, F. Y. Wu,Int. J. Mod. Phys. 3, 1247 (1989)

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Mixed spin-(1/2,S) Ising model on a centered square lattice

V. Urumov,Ordering in Two Dimensions, Elsevier, New York, 1980, 361-364.

J1

J2

J3

J4

si , j=±1/2 S =i , j -S, -S+1,...,S

The total Hamiltonian:

H=−J1

▽

X

i,j

Si,jσi,jσi+1,j−J2

⊳

X

i,j

Si,jσi+1,jσi+1,j+1

−J3

△

X

i,j

Si,jσi,j+1σi+1,j+1−J4

⊲

X

i,j

Si,jσi,jσi,j+1

−DX

i,j

S_i,j²

Ji– triplet interactions for 4 triangular faces (i= 1−4) D– single-ion anisotropy acting on spin-S sites

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Mixed spin-(1/2,S) Ising model on a centered square lattice

V. Urumov,Ordering in Two Dimensions, Elsevier, New York, 1980, 361-364.

J1

J2

J3

J4

si , j=±1/2 S =i , j -S, -S+1,...,S

H=−J1

▽

X

i,j

⊳

X

i,j

Si,jσi+1,jσi+1,j+1

−J3

△

X

i,j

Si,jσi,j+1σi+1,j+1−J4

⊲

X

i,j

Si,jσi,jσi,j+1

−DX

i,j

S_i,j²

Ji– triplet interactions for 4 triangular faces (i= 1−4) D– single-ion anisotropy acting on spin-S sites

s

_{i , j}

S

_{i , j}

s

_i₊₁_{, j}

J

1

J

2

J

3

J

4

s

i , j+1

s

i+1 +1, j Elementary square face

Alternative definition of the total Hamiltonian:

H=

X

i,j

Hi,j,

where the summation extends over all square faces:

Hi,j=−J1Si,jσi,jσi+1,j−J2Si,jσi+1,jσi+1,j+1

−J3Si,jσi,j+1σi+1,j+1−J4Si,jσi,jσi,j+1−DS_i,j²

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Ising model with triplet interaction Graph-theoretical approach

I. Exact mapping equivalence with the zero-field eight-vertex model

H=

X

i,j

Hi,j; Hi,j= − J1Si,jσi,jσi+1,j−J2Si,jσi+1,jσi+1,j+1−J3Si,jσi,j+1σi+1,j+1

− J4Si,jσi,jσi,j+1−DS²_i,j The partition function:

Z = X

{σi,j}

Y

i,j S

X

S_i,j=−S

exp(−βHi,j) = X

{σi,j}

Y

i,j

ω(σi,j, σi+1,j, σi+1,j+1, σi,j+1)

ω(a, b, c, d) =

S

X

n=−S

exp(βDn²) cosh [βn(J1ab+J2bc+J3cd+J4da)],

whereβ= 1/(k_BT),k_Bis Boltzmann’s constant andT is the absolute temperature.

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I. Exact mapping equivalence with the zero-field eight-vertex model

H=

X

i,j

Hi,j; Hi,j= − J1Si,jσi,jσi+1,j−J2Si,jσi+1,jσi+1,j+1−J3Si,jσi,j+1σi+1,j+1

− J4Si,jσi,jσi,j+1−DS²_i,j The partition function:

Z = X

{σi,j}

Y

i,j S

X

S_i,j=−S

exp(−βHi,j) = X

{σi,j}

Y

i,j

ω(σi,j, σi+1,j, σi+1,j+1, σi,j+1)

ω(a, b, c, d) =

S

X

n=−S

exp(βDn²) cosh [βn(J1ab+J2bc+J3cd+J4da)],

whereβ= 1/(k_BT),k_Bis Boltzmann’s constant andT is the absolute temperature.

The invarianceω(a, b, c, d) =ω(−a,−b,−c,−d)implies that one can establish two-to-one mapping correspondence between Ising spin configurations and line coverings of the equivalent eight-vertex model on a dual square lattice (the sign±marksσ=±1/2).

+ + w

1

_ + w

2

+ _ w

3

_ w

4

_ + +

w

5

_ + w

6

+ _ w

7

+ + + _ + _ + + + _ + + + + _ +

+ +

w

8

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I. Exact mapping equivalence with the zero-field eight-vertex model

Two-to-one mapping correspondence between Ising spin configurations and line coverings of the equivalent eight-vertex model on a dual square lattice (the sign±marksσ=±1/2).

+ + w

₁

_ + w

₂

+ _ w

₃

_ w

₄

_ + +

w

₅

_ + w

6

+ _ w

7

+ + + _ + _ + + + _ + + + + _ +

+ + w

₈

ω(a, b, c, d) =

S

X

n=−S

exp(βDn²) cosh [βn(J1ab+J2bc+J3cd+J4da)]

The effective Boltzmann’s weights of the eight-vertex model read:

ω1 =ω2=

S

X

n=−S

exp(βDn²) cosh βn

4 (J1+J2+J3+J4)

,

ω3 =ω4=

S

X

n=−S

4 (J1−J2+J3−J4)

,

ω5 =ω6=

S

X

n=−S

4 (J1−J2−J3+J4)

,

ω7 =ω8=

S

X

n=−S

4 (J1+J2−J3−J4)

.

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I. Exact mapping equivalence with the zero-field eight-vertex model

As a result, there holds an exact mapping relationship between the partition functions:

Z(β, J1, J2, J3, J4, D) = 2Z8−vertex(ω1, ω3, ω5, ω7).

The effective Boltzmann’s weights of the eight-vertex model obey the zero-field condition:

ω1 =ω2=

S

X

n=−S

4 (J1+J2+J3+J4)

,

ω3 =ω4=

S

X

n=−S

4 (J1−J2+J3−J4)

,

ω5 =ω6=

S

X

n=−S

4 (J1−J2−J3+J4)

,

ω7 =ω8=

S

X

n=−S

4 (J1+J2−J3−J4)

.

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I. Exact mapping equivalence with the zero-field eight-vertex model

As a result, there holds an exact mapping relationship between the partition functions:

Z(β, J1, J2, J3, J4, D) = 2Z8−vertex(ω1, ω3, ω5, ω7).

The effective Boltzmann’s weights of the eight-vertex model obey the zero-field condition:

ω1 =ω2=

S

X

n=−S

4 (J1+J2+J3+J4)

,

ω3 =ω4=

S

X

n=−S

4 (J1−J2+J3−J4)

,

ω5 =ω6=

S

X

n=−S

4 (J1−J2−J3+J4)

,

ω7 =ω8=

S

X

n=−S

4 (J1+J2−J3−J4)

.

Baxter’s [1] critical condition: ω1+ω3+ω5+ω7= 2max{ω1, ω3, ω5, ω7} Critical exponents satisfy Suzuki’s weak-universal hypothesis [1,2]:

α=α^′= 2−π/µ, β=π/16µ, ν=ν^′=π/2µ, γ=γ^′= 7π/8µ, δ= 15, η= 1/4, where tan(µ/2) = (ω1ω3/ω5ω7)^1/2 on assumption that ω1= max{ω1, ω3, ω5, ω7}

.

[1] R. J. Baxter,Phys. Rev. Lett.26, 832 (1971);Ann. Phys.70, 193 (1972).

[2] M. Suzuki,Prog. Theor. Phys.51, 1992 (1974).

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Ising model with triplet interaction Algebraic approach

II. Exact mapping equivalence with the zero-field eight-vertex model

The factorized form of the partition function:

Z = X

{σ_i,j}

Y

i,j S

X

S_i,j=−S

exp(−βHi,j) = X

{σ_i,j}

Y

i,j

ω(σ_i,j, σ_i+1,j, σ_i+1,j+1, σ_i,j+1)

The effective Boltzmann factorωcan be replaced via the star-square transformation:

ω=

S

X

n=−S

exp(βDn²) coshh

βn(J1σi,jσi+1,j+J2σi+1,jσi+1,j+1

+J3σi+1,j+1σi,j+1+J4σi,j+1σi,j)i

=R0exp(βR1σi,jσi+1,j+1+βR2σi+1,jσi,j+1+βR4σi,jσi+1,jσi+1,j+1σi,j+1).

The schematic representation of the generalized star-square transformation:

s_{i , j} Si , j s_i₊₁_{, j}

J₁ J2

J3

J4

s_{i , j+1} s_i_{+1 +1}_{, j}

s_{i , j} s_i₊₁_{, j}

R₁ R4

R2

s_{i , j+1} s_i_{+1 +1}_{, j}

SST

The ’self-consistency’ condition unambiguously determines the mapping parameters:

R0= (ω1ω3ω5ω7)^1/4, βR1= ln ω1ω7

ω3ω5

, βR2= ln ω1ω5

ω3ω7

, βR4= 4 ln ω1ω3

ω5ω7

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Ising model with triplet interaction Algebraic approach

II. Exact mapping equivalence with the zero-field eight-vertex model

The generalized star-square transformation [1-4]:

ω=

S

X

n=−S

exp(βDn²) coshh

βn(J1σi,jσi+1,j+J2σi+1,jσi+1,j+1

+J3σ_i+1,j+1σ_i,j+1+J4σ_i,j+1σ_i,j)i

=R0exp(βR1σi,jσi+1,j+1+βR2σi+1,jσi,j+1+βR4σi,jσi+1,jσi+1,j+1σi,j+1), satisfies the ’self-consistency’ condition on assumption that:

R0= (ω1ω3ω5ω7)^1/4, βR1= ln ω1ω7

ω3ω5

, βR2= ln ω1ω5

ω3ω7

, βR4= 4 ln ω1ω3

ω5ω7

. The star-square transformation establishes an exact mapping relation with the partition function of the spin-1/2 Ising model on two interpenetrating square lattices with the pair interactionsR1andR2, which are coupled together by means of the quartic interactionR4.

Z(β, J1, J2, J3, J4, D) =R^2N₀ Z8−vertex(β, R1, R2, R4).

This is nothing but the Ising representation of the zero-field eight-vertex model [5,6].

[1] M. E. Fisher,Phys. Rev.113, 969 (1959).

[2] O. Rojas, J. S. Valverde, S.M. de Souza,Physica A388, 1419 (2009).

[3] J. Streˇcka,Phys. Lett. A374, 3718 (2010).

[4] J. Streˇcka,On the Theory of Generalized Algebraic Transformations, LAP, Saarbrucken, 2010.

[5] F. Y. Wu,Phys. Rev. B4, 2312 (1971).

[6] L. P. Kadanoff, R. J. Wegner,Phys. Rev. B4, 3989 (1971).

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The most interesting results

Mixed spin-(1/2,S) Ising model on a centered square lattice

H=−J1

▽

X

i,j

⊳

X

i,j

Si,jσi+1,jσi+1,j+1−J3

△

X

i,j

Si,jσi,j+1σi+1,j+1

−J4

⊲

X

i,j

Si,jσi,jσi,j+1−DX

i,j

S_i,j²

J1

J2

J3

J4

si , j=±1/2 S =i , j -S, -S+1,...,S

Model A:J≡J1=J2=J3=J4

Model B:J≡J1, J^′≡J2=J3=J4

Model C:J≡J1=J3, J^′≡J2=J4

Model D:J≡J1=J2, J^′≡J3=J4

J=J1=J2=J3=J4

MODEL A

J=J1=J3, ´=J J2=J4

MODEL C

J=J1, ´=J J2=J3=J4

MODEL B

J=J1=J2, ´=J J3=J4

MODEL D

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Critical behaviour of Model A and Model B

Model A(J≡J1=J2=J3=J4) : ω1=

S

X

n=−S

exp(βDn²) cosh (βnJ), ω3=ω5=ω7=

S

X

n=−S

exp(βDn²).

Critical condition:

ω1= 3ω3 ⇒

S

X

n=−S

exp(βcDn²) cosh (βcnJ) = 3

S

X

n=−S

exp(βcDn²).

Critical exponents:

tan(µ/2) =p

ω1/ω3=√

3 ⇒ α=α^′= 1/2, β= 3/32, ν=ν^′= 3/4, γ=γ^′= 21/16.

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Critical behaviour of Model A and Model B

Model A(J≡J1=J2=J3=J4) : ω1=

S

X

n=−S

exp(βDn²) cosh (βnJ), ω3=ω5=ω7=

S

X

n=−S

exp(βDn²).

Critical condition:

ω1= 3ω3 ⇒

S

X

n=−S

exp(βcDn²) cosh (βcnJ) = 3

S

X

n=−S

exp(βcDn²).

Critical exponents:

tan(µ/2) =p

ω1/ω3=√

3 ⇒ α=α^′= 1/2, β= 3/32, ν=ν^′= 3/4, γ=γ^′= 21/16.

Model B(J≡J1, J^′≡J2=J3=J4) : ω1=

S

X

n=−S

4 J+ 3J^′

, ω3=ω5=ω7=

S

X

n=−S

4 J−J^′

.

Critical condition:

ω1= 3ω3 ⇒

S

X

n=−S

exp(βcDn²) cosh βcn

4 J+ 3J^′

= 3

S

X

n=−S

4 J−J^′

.

Critical exponents:

tan(µ/2) =p

ω1/ω3=√

3 ⇒ α=α^′= 1/2, β= 3/32, ν=ν^′= 3/4, γ=γ^′= 21/16.

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Critical behaviour of Model C

Model C(J≡J1=J3, J^′≡J2=J4) :

ω1 =

S

X

n=−S

2 J+J^′

,

ω3 =

S

X

n=−S

2 J−J^′

,

ω5 = ω7=

S

X

n=−S

exp(βDn²).

J = J

1

= J

3

, ´= J J

2

= J

4

MODEL C

Critical conditionω1=ω3+ 2ω5:

S

X

n=−S

2 J+J^′

=

S

X

n=−S

2 J−J^′

+ 2

S

X

n=−S

exp(βcDn²).

Critical exponents:

tanµ 2

= rω1ω3

ω²₅ =

sω3(ω3+ 2ω5)

ω₅² ⇒

α = α^′= 2−π

µ, β= π

16µ, ν=ν^′= π

2µ, γ=γ^′=7π 8µ.

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Critical behaviour of Model D

Model D(J≡J1=J2, J^′≡J3=J4) :

ω1 =

S

X

n=−S

2 J+J^′

,

ω3 = ω5=

S

X

n=−S

exp(βDn²),

ω7 =

S

X

n=−S

2 J−J^′

.

J = J

₁

= J

₂

, ´= J J

₃

= J

₄

MODEL D

Critical conditionω1= 2ω3+ω7:

S

X

n=−S

2 J+J^′

=

S

X

n=−S

2 J−J^′

+ 2

S

X

n=−S

exp(βcDn²).

Critical exponents:

tanµ 2

= rω1

ω7

=

s2ω3+ω7

ω7 ⇒

α = α^′= 2−π

µ, β= π

16µ, ν=ν^′= π

2µ, γ=γ^′=7π 8µ.

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The most interesting results Model A

Critical behaviour of the isotropic model A

-2 -1 0 1 2

0.0 0.5 1.0 1.5 2.0

-2 -1 0 1 2

0.0 0.4 0.8 1.2 1.6

S = 1 S = 2 S = 3

a) b) kB Tc

/J

kB Tc

/J

S = 1/2

D / J D / J

S = 3/2 S = 5/2 J = J

1 = J

2 = J

3 = J

4

J = J 1

= J 2

= J 3

= J 4

critical temperature monotonically decreases with decreasing the single-ion anisotropy integer vs. half-odd-integer spinsS: fundamental difference in their critical behaviour the spin system with integer-valued spinsSis always disordered forD/J <−1.0 the isotropic model A withJ≡J1=J2=J3=J4 has constant critical exponents:

α=α^′= 1/2, β= 3/32, ν=ν^′= 3/4, γ=γ^′= 21/16, δ= 15, η= 1/4

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The most interesting results Model A

Critical behaviour of the anisotropic models with S = 1/2

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.1 0.2 0.3

S = 1/2 J' = J

2 = J

4

J' = J 3

= J 4 J'=J

2 =J

3

=J 4

a) b) kB

Tc

/J

S = 1/2

J' / J J' / J

J' = J 2

= J 4

J' = J 2

= J 3

= J 4 J' = J

3

= J 4 or

critical temperature monotonically increases with increasing the interaction ratioJ^′/J the model B withJ≡J1, J^′≡J2=J3=J4has constant critical exponents:

α=α^′= 1/2, β= 3/32, ν=ν^′= 3/4, γ=γ^′= 21/16, δ= 15, η= 1/4 the critical exponentαof the model C withJ≡J1=J3,J^′≡J2=J4monotonically decreases with the interaction ratioJ^′/J

the critical exponentαof the model D withJ≡J1=J2,J^′≡J3=J4 monotonically increases with the interaction ratioJ^′/J

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The most interesting results Model B

Critical behaviour of the anisotropic model B

-2 -1 0 1 2

0.0 0.5 1.0 1.5

-2 -1 0 1 2

0.0 0.2 0.4 0.6 0.8 1.0

S = 3/2 1.0

0.5 J' / J = 2.0

0.25 1.5

a) b)

kB

Tc

/J

kB

Tc

/J

S = 1

D / J D / J

J' / J = 2.0 J' = J

2 = J

3 = J

4

J' = J

2 = J

3 = J

4

1.0

0.5

0.25 1.5

critical temperature monotonically decreases with decreasing the single-ion anisotropy integer vs. half-odd-integer spinsS: fundamental difference in their critical behaviour the model B withJ≡J1, J^′≡J2=J3=J4has constant critical indices regardless of the spin magnitudeS, the interaction ratioJ^′/J and the single-ion anisotropyD/J:

α=α^′= 1/2, β= 3/32, ν=ν^′= 3/4, γ=γ^′= 21/16, δ= 15, η= 1/4

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The most interesting results Model C

Critical behaviour of the anisotropic model C: integer spins S

-1 0 1 2

0.50 0.55 0.60 0.65 0.70

-1 0 1 2

0.0 0.1 0.2 0.3 0.4 0.5 0.6

S = 1 J' / J = 0.25

0.5

0.75

1.0

a) b) kB

Tc

/J

S = 1

D / J D / J

J = J 1

= J 3

, J' = J 2

= J 4

J = J 1

= J 3

, J' = J 2

= J 4

J' / J = 1.0

0.75

0.5

0.25

critical temperature monotonically decreases with decreasing the single-ion anisotropy the model C exhibits continuously varying critical exponents withD/JandJ^′/J the critical exponentαfor the model C withJ≡J1=J3,J^′≡J2=J4

monotonically increases with increasing the single-ion anisotropy for integer spinsS

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The most interesting results Model C

Critical behaviour of the anisotropic model C: half-odd-integer spins S

-2 -1 0 1 2

0.50 0.55 0.60 0.65 0.70

-2 -1 0 1 2

0.0 0.2 0.4 0.6 0.8

S = 3/2

J' / J = 0.25

0.5

0.75

1.0

a) b) kB

Tc

/J

S = 3/2

D / J D / J

J' / J = 1.0 J = J

1 = J

3 , J' = J

2 = J

4

J = J

1 = J

3 , J' = J

2 = J

4

0.75

0.5

0.25

critical temperature monotonically decreases with decreasing the single-ion anisotropy the model C exhibits continuously varying critical exponents withD/JandJ^′/J the critical exponentαfor the model C withJ≡J1=J3,J^′≡J2=J4exhibits non-monotonous dependence with a minimum for half-odd-integer spinsS

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The most interesting results Model D

Critical behaviour of the anisotropic model D: integer spins S

-1 0 1 2

0.35 0.40 0.45 0.50

-1 0 1 2

0.0 0.1 0.2 0.3 0.4 0.5 0.6

S = 1

J' / J =1.0

0.75

0.5

0.25

a) b) kB

Tc

/J

S = 1

D / J D / J

J = J 1

= J 2

, J' = J 3

= J 4

J = J

1 = J

2 J' = J

3 = J

4

J' / J =1.0

0.75

0.5

0.25

critical temperature monotonically decreases with decreasing the single-ion anisotropy the model D exhibits continuously varying critical exponents withD/J andJ^′/J the critical exponentαfor the model D withJ≡J1=J3,J^′≡J2=J4

monotonically decreases with increasing the single-ion anisotropy for integer spinsS

(26)

The most interesting results Model D

Critical behaviour of the anisotropic model D: half-odd-integer spins S

-2 -1 0 1 2

0.35 0.40 0.45 0.50

-2 -1 0 1 2

0.0 0.2 0.4 0.6 0.8

S = 3/2 J' / J = 1.0

0.75

0.5

0.25

a) b) kB

Tc

/J

S = 3/2

D / J D / J

J' / J = 1.0 J = J

1 = J

2 , J' = J

3 = J

4

J = J 1

= J 2

, J' = J 3

= J 4

0.75

0.5

0.25

critical temperature monotonically decreases with decreasing the single-ion anisotropy the model D exhibits continuously varying critical exponents withD/J andJ^′/J the critical exponentαfor the model D withJ≡J1=J3,J^′≡J2=J4 exhibits non-monotonous dependence with a maximum for half-odd-integer spinsS

(27)

Conclusions

Concluding remarks

Mixed spin-(1/2, S) Ising model with the triplet interaction on a centered square lattice

exactly solvable model due to a precise mapping equivalence with Baxter’s zero-field eight-vertex model

strong-universal critical behaviour of the isotropic model A and the anisotropic model B

weak-universal critical behaviour of the anisotropic models C and D the dependence of critical exponents on the interaction parameters is fundamentally different for the models with the half-odd-integer and integer spins