Effects of random-crystal field on the kinetic decorated Ising model

1,4 2 1,3

1

2 3

4

L

1

L

2

L

1

1

2

L

2

L

1

S >

¹₂

Sr

2

Cu(Re

0.69

Ca

0.3

)O

6

Cu

²⁺

(S =

¹₂

) Cu

³⁺

(S = 1)

J

1

(J

1

> 0) J

2

(J

2

> 0)

∆

_i

∆ ) =

¹₂

[δ(∆ − ∆(1 + α)) + δ(∆ − ∆(1 − α))], α ≥ 0

cos(ωt), H

0

ω = 2πν

1 τ

P

σ

1

, σ

2

, ..., σ

N

P

S

1

, S

2

, ..., S

N

σ σ

1

σ

N

S

1

, S

2

, ..., S

N

W

j

(σ

j

) W

i

(S

_i

) σ

σ

j

σ

j

S S

i

σ S

_i

P

σ

1

, σ

2

, ..., σ

N

P

S

1

, S

2

, ..., S

N

d dξ

σ

1

, σ

2

, ..., σ

N

j

W

j

(σ

j

)

σ

1

, σ

2

, ..., σ

j

, ..., σ

N

+

j

W

_j

( − σ

_j

)

σ

₁

, σ

₂

, ..., − σ

_j

, ..., σ

_N

d dξ

1

, S

2

, ..., S

N

; t) = −

i

[

S_i=Si)

W

i

( →

)]

1

, S

2

, ..., S

i

, ..., S

N

; t)

+

i

W

i

(

→ )[

S_i=Si

1

, S

2

, ..., S

_i

, ..., S

N

; t)]

W

j

(σ

j

) W

i

(S

i

i

)

W

j

(σ

j

) = 1 τ

e

^−β∆Ej

1 + e

^−β∆Ej

,

W

_i

(S

_i

→ S

_i

) = 1 τ

e

^{−βE(Si→Si)}

S_i

e

^{−βE(Si→Si)}

β =

_T¹

∆E

_j

∆E

(S

i

→

S

_i

) T

cmp

T

cmp

< T

c

1 2

1 2

σ

_A

=

¹₂

S

_B

= 1

σ

A

=

¹₂

S

B

= 1

H = − J

₁

<ij>

σ

_i

σ

_j

+ J

₂

<ij>

σ

_i

S

_j

−

i

∆

_i

S

_i²

− H(

i

S

_i

+

j

σ

_j

)

L

1

L

2

L

1

σ

A

=

¹₂

L

2

L

₁

S

_B

= 1

J

1

(J

1

> 0) J

2

(J

2

> 0)

∆

_i

∆ ) =

¹₂

[δ(∆ − ∆(1 + α)) + δ(∆ − ∆(1 − α))], α ≥ 0

cos(ωt), H

0

ω = 2πν

1 τ

P

σ

1

, σ

2

, ..., σ

N

P

S

1

, S

2

, ..., S

N

σ σ

1

σ

N

S

1

, S

2

, ..., S

N

W

j

(σ

j

) W

i

(S

_i

) σ

σ

j

σ

j

S S

i

σ S

_i

P

σ

1

, σ

2

, ..., σ

N

P

S

1

, S

2

, ..., S

N

d dξ

σ

1

, σ

2

, ..., σ

N

j

W

j

(σ

j

)

σ

1

, σ

2

, ..., σ

j

, ..., σ

N

+

j

W

_j

( − σ

_j

)

σ

₁

, σ

₂

, ..., − σ

_j

, ..., σ

_N

d dξ

1

, S

2

, ..., S

N

; t) = −

i

[

S_i=Si)

W

i

( →

)]

1

, S

2

, ..., S

i

, ..., S

N

; t)

+

i

W

i

(

→ )[

S_i=Si

1

, S

2

, ..., S

_i

, ..., S

N

; t)]

W

j

(σ

j

) W

i

(S

i

i

)

W

j

(σ

j

) = 1 τ

e

^−β∆Ej

1 + e

^−β∆Ej

,

W

_i

(S

_i

→ S

_i

) = 1 τ

e

^{−βE(Si→Si)}

S_i

e

^{−βE(Si→Si)}

β =

_T¹

∆E

_j

∆E

(S

i

→

S

_i

) T

cmp

T

cmp

< T

c

1 2

1 2

σ

_A

=

¹₂

S

_B

= 1

σ

A

=

¹₂

S

B

= 1

H = − J

₁

<ij>

σ

_i

σ

_j

+ J

₂

<ij>

σ

_i

S

_j

−

i

∆

_i

S

_i²

− H(

i

S

_i

+

j

σ

_j

)

L

1

L

2

L

1

σ

A

=

¹₂

L

2

L

₁

S

_B

= 1

M

σ

M

S

M

σ

= 1 2π

2π

0

m

σ

(ξ)dξ

M

S

= 1 2π

2π

0

m

S

(ξ)dξ

( ˙ m =

^dm_dt

, m) m

σ

m

S

1 2

,

( ˙ m =

^dm_dt

, m)

α α = 0.3

(F

¹

2

, P ) (P, P ) (t

_tc

= 0.41, h

_tc

= 1.51) F

¹

1 2

2

α = 1.2

(t

tc

= 0.47, h

tc

= 1.65) (F

¹

2

, F

−1

2

) α = 1.2

(F

¹

2

, F

₋1

)

(t

tc

= 0.69, h

tc

= 1.66) F

−1

2

F

₋1

M =

⁻₂¹

(F

¹

2

, P )

∆E

_j

= 2σ

j

(+J

1

<i>

σ

i

− J

2

<i>

S

i

+ H)

∆E

(S

i

→ S

_i

) = − (S

_i

− S

i

)( − J

2

<j>

σ

j

+ H) − (S

_i²

− S

_i²

)∆

i

∆E

_j

∆E

(S

i

→ S

_i

) σ

j

S

i

W

i

(S

i

→ S

_i

) S

i

W

i

(S

i

→ S

_i

) W

i

(S

_{i i}

σ

j

S

i

Ω dm

σ

dξ = − m

σ

+ 1

2 tanh 1

2T [4J

1

m

σ

− 4J

2

m

S

+ H

0

cosξ]

Ω dm

S

dξ = − m

S

− 2 sinh

_T¹

(2J

2

m

σ

− H

0

cos ξ) 2 cosh

_T¹

(2J

2

m

σ

− H

0

cos ξ) + e

^−∆Ti

, Ω = τ ω ξ = ωt m

σ

=< σ > m

S

=< S >

Ω dm

S

dξ = − m

_S

−

2 sinh

_T¹

(2J

2

m

σ

− H

0

cos ξ)

2 cosh

_T¹

(2J

2

m

σ

− H

0

cos ξ) + e

^−∆Ti

P (∆

_i

)d∆

_i

Ω dm

S

dξ = − m

S

− 2 sinh

_T¹

(2J

2

m

σ

− H

0

cos ξ)

2 cosh

_T¹

(2J

2

m

σ

− H

0

cos ξ) + e

^{−∆(1+α)T}

− 2 sinh

_T¹

(2J

2

m

σ

− H

0

cos ξ) 2 cosh

_T¹

(2J

2

m

σ

− H

0

cos ξ) + e

^{−∆(1T−α)}

α = 0 m

σ

=

¹₂

, m

S

= 0

m

σ

=

¹₂

, m

S

= − 1 0 < α < 1 m

σ

= 0, m

S

= 0 (m

σ

=

¹₂

, m

σ

=

⁻₂¹

) (m

σ

=

¹₂

, m

σ

=

⁻¹₂

) (m

σ

=

¹₂

, m

S

=

⁻¹₂

)

(m

σ

=

¹₂

, m

S

= − 1) r + d(1 − α) r + d(1 + α)

α > 1 (m

σ

=

¹₂

, m

S

=

⁻₂¹

) (m

σ

=

1

2

, m

S

= − 1) (m

σ

=

¹₂

, m

S

=

⁻¹₂

) (m

σ

=

¹₂

, m

σ

=

⁻¹₂

)

r + d(1 − α) = 0 r + d(1 + α) = 0

M

σ

M

S

M

σ

= 1 2π

2π

0

m

σ

(ξ)dξ

M

S

= 1 2π

2π

0

m

S

(ξ)dξ

( ˙ m =

^dm_dt

, m) m

σ

m

S

1 2

,

( ˙ m =

^dm_dt

, m)

α α = 0.3

(F

¹

2

, P ) (P, P ) (t

_tc

= 0.41, h

_tc

= 1.51) F

¹

1 2

2

α = 1.2

(t

tc

= 0.47, h

tc

= 1.65) (F

¹

2

, F

−1

2

) α = 1.2

(F

¹

2

, F

₋1

)

(t

tc

= 0.69, h

tc

= 1.66) F

−1

2

F

₋1

M =

⁻₂¹

(F

¹

2

, P )

0 0.5 1 1.5 2 2.5 0

0.2 0.4 0.6 0.8 1 1.2

1.4 r=0.4α=1.2d=−0.9

(F1/2,F−1/2)

(P,P)

h

t

α = 1.2

0 0.5 1 1.5 2

0 0.2 0.4 0.6 0.8 1 1.2

1.4 r=0.1α=0.3d=−1

(F1/2,P)

(P,P)

h

t

α = 0.3

0 0.5 1 1.5 2 2.5 0

0.2 0.4 0.6 0.8 1 1.2

1.4 r=0.4α=1.2d=−0.9

(F1/2,F−1/2)

(P,P)

h

t

α = 1.2

0 0.5 1 1.5

−0.1 0 0.1 0.2 0.3 0.4 0.5 0.6

h=0.5r=0.1d=−1alpha=0.3

T

M

0 0.5 1 1.5

−0.1 0 0.1 0.2 0.3 0.4 0.5 0.6

h=1.5r=0.1d=−1alpha=0.3

T

M

(m

σ

, m

S

) α = 0.3

˙ m

S

, m

S

m

σ

= 0 m

S

= 0

α = 1.2 M

σ

M

S

F

¹

2

, F

−1

T

_c

T

_cmp 2

T

cmp

F

¹

2

, F

−1 2

F

¹

2

, F

−1

F

−1 2

2

F

¹

2

( ˙ m = 0, m = 0)

0 0.5 1 1.5 2

0 0.2 0.4 0.6 0.8 1 1.2 1.4

r=0.4α=1.2d=1

(F1/2,F−1)

(P,P)

h

t

α = 1.2

(F

¹

2

, F

−1

2

) (F

¹

2

, F

₋1

)

m(ξ) ( ˙ m =

dm

dt

, m) α = 0.3

M

_σ

M

σ

M

σ

=

¹₂

M

S

M

S

= 0

L

₂

L

₂

α = 0.3

m

_σ

(ξ) m

_S

(ξ) ( ˙ m =

^dm_dt

, m) m

σ

m

S

m ˙ = ( ˙ m

σ

, m ˙

S

)

F

¹

2

m

σ

=

¹₂

, m

S

= 0

0 0.5 1 1.5

−0.1 0 0.1 0.2 0.3 0.4 0.5 0.6

h=0.5r=0.1d=−1alpha=0.3

T

M

0 0.5 1 1.5

−0.1 0 0.1 0.2 0.3 0.4 0.5 0.6

h=1.5r=0.1d=−1alpha=0.3

T

M

(m

σ

, m

S

) α = 0.3

˙ m

S

, m

S

m

σ

= 0 m

S

= 0

α = 1.2 M

σ

M

S

F

¹

2

, F

−1

T

_c

T

_cmp 2

T

cmp

F

¹

2

, F

−1 2

F

¹

2

, F

−1

F

−1 2

2

F

¹

2

( ˙ m = 0, m = 0)

0 0.5 1 1.5

−0.5

−0.4

−0.3

−0.2

−0.1 0 0.1 0.2 0.3 0.4

h=0.1r=0.4d=−0.9alpha=1.20.5

T

M

0 0.5 1 1.5

−0.3

−0.2

−0.1 0 0.1 0.2 0.3 0.4

h=0.5r=0.4d=−0.9alpha=1.20.5

T

M

0 0.5 1 1.5

−0.2

−0.1 0 0.1 0.2 0.3 0.4

h=1.7r=0.4d=−0.9alpha=1.20.5

T

M

(m

σ

, m

S

) α = 1.2

0 100 200 300

−2

−1 0 1 2

ξ −1−0.5 0 0.5 1

−0.5 0 0.5 1 1.5x 10⁻⁶

m

dm/t

0 100 200 300

−2

−1 0 1 2

ξ

m

−0.5 0 0.5 1

−1

−0.5 0 0.5 1

m

dm/t

0 100 200 300

−2

−1 0 1 2

ξ −0.2−0.5 0 0.5

−0.1 0 0.1 0.2

m

dm/t

0 100 200 300

−2

−1 0 1 2

ξ

m

−0.5 0 0.5

−1

−0.5 0 0.5 1

m

dm/t

(a) (b)

(c) (d)

α = 0.3

( ˙ m =

^dm_dt

, m)

α = 1.2

M

σ

M

S 1

2

T

c

( ˙ m, m)

α = 1.2 F

₋₁

F

¹

2

M

σ

M

S

0 0.5 1 1.5

−0.5

−0.4

−0.3

−0.2

−0.1 0 0.1 0.2 0.3 0.4

h=0.1r=0.4d=−0.9alpha=1.20.5

T

M

0 0.5 1 1.5

−0.3

−0.2

−0.1 0 0.1 0.2 0.3 0.4

h=0.5r=0.4d=−0.9alpha=1.20.5

T

M

0 0.5 1 1.5

−0.2

−0.1 0 0.1 0.2 0.3 0.4

h=1.7r=0.4d=−0.9alpha=1.20.5

T

M

(m

σ

, m

S

) α = 1.2

0 0.5 1 1.5

−1.5

−1

−0.5 0

0.5h=0r=0.4d=1alpha=1.2

0 0.5 1 1.5

−0.3

−0.2

−0.1 0 0.1 0.2 0.3 0.4

0.5h=0.9r=0.4d=1alpha=1.2

0 0.5 1 1.5

−0.1 0 0.1 0.2 0.3 0.4 0.5

0.6h=1.7r=0.4d=1alpha=1.2

(m

σ

, m

S

) α = 1.2

0 100 200 300

−1.5

−1

−0.5 0 0.5 1 1.5

ξ −2−1 0 1

−1.5

−1

−0.5 0 0.5 1 1.5

2x 10⁻⁷

m

dm/dt

0 100 200 300

−1.5

−1

−0.5 0 0.5 1 1.5

ξ

m

−0.5 0 0.5

−1

−0.5 0 0.5 1

m

dm/dt

0 100 200 300

−1.5

−1

−0.5 0 0.5 1 1.5

ξ −0.4−0.5 0 0.5

−0.3

−0.2

−0.1 0 0.1 0.2 0.3

m

dm/dt

0 100 200 300

−1.5

−1

−0.5 0 0.5 1 1.5

ξ

m

−0.5 0 0.5

−1

−0.5 0 0.5 1

m

dm/dt

(a) (b)

(c) (d)

α = 1.2

( ˙ m =

^dm_dt

, m)

0 0.5 1 1.5

−1.5

−1

−0.5 0

0.5h=0r=0.4d=1alpha=1.2

0 0.5 1 1.5

−0.3

−0.2

−0.1 0 0.1 0.2 0.3 0.4

0.5h=0.9r=0.4d=1alpha=1.2

0 0.5 1 1.5

−0.1 0 0.1 0.2 0.3 0.4 0.5

0.6h=1.7r=0.4d=1alpha=1.2

(m

σ

, m

S

) α = 1.2

´ eel

´ eorie

´ e

0 100 200 300

−2

−1 0 1 2

ξ

m

−0.2 0 0.2

−0.4

−0.2 0 0.2 0.4

m

dm/dt

0 100 200 300

−2

−1 0 1 2

ξ

m

−5 0 5

x 10⁻⁴

−3

−2

−1 0 1 2x 10⁻⁵

m −20 100 200 300

−1 0 1 2

ξ

m

−0.05 0 0.05

−0.2

−0.1 0 0.1 0.2

m

dm/dt

0 100 200 300

−2

−1 0 1 2

ξ −2−1 0 1

−1 0 1 2

m

dm/dt

0 100 200 300

−2

−1 0 1 2

ξ

m

−0.5 0 0.5

−2

−1 0 1 2

m

dm/dt

(a) (b)

(c) (d) (e)

α = 1.2

( ˙ m =

^dm_dt

, m)

(F

¹₂

, F

₋₁

) (F

¹

2

, F

−1 2

) (F

¹

2

, 0)

´ eel

´ eorie

´

e

U ˜

U ˜

U ˜