The transverse mixed spin Ising model in a longitudinal random field

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The transverse mixed spin Ising model in a longitudinal random field

N. Benayad , L. Khaya and A. Fathi

Groupe de Mécanique Statistique, Laboratoire de Physique Théorique, Faculté des Sciences Aïn Chock, B.P. 5366 Maarif, Casablanca, Morocco.

The transverse mixed spin Ising system consisting of spin-1/2 and spin-1 with a longitudinal random field is studied within the finite cluster approximation based on single-site cluster theory. The equations are derived using a probability distribution method based on the use of Van der Waerden identities. In the absence of the transverse field, the complete phase diagram in the case of simple cubic lattice is investigated and exhibits interesting behaviours, where the longitudinal field is bimodally and trimodally distributed. The influence of the transverse field on these behaviours are also examined.

I. INTRODUCTION

In the last decades, there has been an interesting number of works dealing with critical behaviour of quantum spin systems. The transverse Ising model is the simplest quantum system and has been introduced to explain the phase transition of hydrogen-bonded ferroelectrics such as KH2PO4¹and other systems (a more detailed application has been reviewed in²) for instance the order-disorder phenomenon with tunnelling effects. It has been extensively studied by the use of various techniques^3-7, including the effective field treatment^8,9 based on a generalized but approximated Callen-Suzuki relation derived by Sà Barreto, Fittipaldi and Zeks. On the otherhand, the spin-one transverse Ising models have been studied^10-14 as well as the Ising models with spin higher than one^15-20.

Recently, attention has been directed to study the magnetic properties of two-sublattice mixed spin Ising system. They are of interest for the following main reasons. They have less translational symmetry than their single spin counterparts, and are well adapted to study a certain type of ferrimagnetism²¹. It has been shown that the MnNi(EDTA)-6H2O complex is an example of mixed spin system²². The mixed Ising spin system consisting of spin-1/2 and spin-1 had been studied by renormalization group technique^23,24, by high-temperature series expansions²⁵, by free-fermion approximation²⁶ and by finite cluster approximation²⁷. A large number of papers have focused on the monoatomic Ising models with a random longitudinal field in the presence of a transverse field, and its properties have been investigated in detail^28,29.

As far as we know, no works have been concerned with the transverse mixed spin Ising model in a random longitudinal field. This system can be described by

H J

_ij _i^z

S H H S

ij j

z

i i i

z

j j j

= − ∑ ^σ − ∑ ^σ − ∑

z

−

-

(

^xj

)

j ix i

∑ S

∑ ⁺

Ω σ

, (1)

where

σ

^αi and

S

^α_j (α = x, z) are components of spin- 1/2 and spin-1 operators at sites i and j, respectively. Jij is the exchange interaction, Ω represents the transverse field, and first summation is carried out only over nearest-neighbour pairs of spins.

The longitudinal random fields Hi are assumed to be independent variables and obey to trimodal probability distribution

( ) ( ) ^{( )}

Q H p H p

i = i + −

δ 1

2 ^×

[

^δ

(

^Hⁱ⁻^H

) (

⁺^δ ^Hⁱ⁺^H

) ]

^{, (2)}

where the parameter p measures the fraction of spins in the system not exposed to the transverse field H. At p=1 or H=0 and Ω=0, the system reduces to the simple mixed spin-1/2 and spin-1 Ising model.

The first purpose of this paper is to investigate the phase diagrams of the transverse mixed spin-1/2 and spin-1 in a longitudinal random field which is bimodally (p=0) and trimodally (p≠0) distributed. The second goal of this work is to study the influence of a transverse field on the obtained phase diagrams. To this end, we use the finite cluster approximation based on single-site cluster theory³⁰. The equations are derived using a probability distribution method based on the use of generalised Van der Waerden identities³¹ that account exactly for the single-site kinematic relations.

Our presentation is as follows: In section 2, we describe the finite cluster approximation based on a probability distribution method. In section 3, the phase diagrams of the system are examined and discussed.

II. METHOD

The theoretical framework to be used in the study of the system described by the Hamiltonian (1), is the finite cluster approximation (FCA), based on a single- site cluster theory. In this method, attention is focused on a cluster consisting of just a single selected spin, M.J. CONDENSED MATTER VOLUME 3, NUMBER 1 1 JULY 2000

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σ0(S0), and the neighbouring spins with which it directly interacts. We split the total Hamiltonian (1) into two parts, H=H_o+H', where H_o includes all parts of H associated with the lattice site o. In the present system, H0 takes the form

H

_o

J S

_oj

j zj H oz

ox σ

= − ∑  σ

  

  −

+ 0

Ωσ

, (3)

H

^S_o

J

_oi

S S

i iz H oz

ox

= − ∑ 





+

 −

σ

0

Ω

, (4) whether the lattice site o belongs to σ or S-Sublattice, respectively.

Now, the problem consists in evaluating the sublattice longitudinal and transverse components of the magnetization and its quadrupolar moments.

Following Sà Barreto et al^8,9, the starting point of our approach, in the framework of the single-site cluster theory, is the set of the following identities

( )

σ σ β

β

α σ

σ o σ

o o

o

Tr H

o

= −

− exp

exp

, (5) and

( ) ( ) ( )

( )

S

Tr S H

Tr H

o

n S o

n

oS

S So

o

α

β

= β −

− exp

exp

^{, (6)}

where β=1/T, α =x or z specifies the components of the spin operators

σ

^α_i ^and

S

^α_j , and n =1, 2 correspond to the magnetization and the quadrupolar moment, respectively. Trσo (or TrSo) means the partial trace with respect to the σ-sublattice site o (or S-sublattice site o) and <...> denotes the canonical thermal average.

The equations (5) and (6) neglect the fact that Ho and H' do not commute. Therefore, they are not exact. Nevertheless, they have been successfully applied to a number of interesting transverse Ising systems. We emphasize that in the Ising limit (Ω =0), the Hamiltonian contains only

σ

iz

and

S

^z_j and then, relations (5) and (6) become exact identities.

To calculate

σ

^αo and

( ) ^S

^o^α ⁿ one has first to diagonalize the single-site Hamiltonians

H

_o^σ and

H

^S_o, respectively. We obtain

^σ

^α^o

⁼ ^{f E H}

^α

(

^S

^,

⁰

^, ^Ω )

, (7)

( ) ^S

^o^α ⁿ

= ^{F E H}

ⁿ^α

(

σ

,

₀

, Ω )

, (8) with

( )

f E H E H

E

z

E

S

, , (

S

)

0 0

tanh

1

2 2

Ω = + 

 



β

, (9)

( ) [ ^{( )} _{( )} ]

F E H E H

E

E E

z

1 0

0 2

2 2

2

σ σ β 1 2

, , ( ) sinh

Ω = + cosh

+ ^{, (10)}

( ) ( ) [ ^{( )} ( ) ]

F E H

E E

z

2 0

2 2

2

1

σ

1 2 , , βΩ

Ω = cosh

+





 

( ) ( )

( ) ^{( )}

[ ( ) ]

+ + +

+







2

1 2

0

2 2

2

E H E

E

σ β βΩ cosh

cosh , (11)

( )

f E H

E

x

E

S

,

₀

, tanh

1

2 2

Ω Ω

= 

 



, (12)

( ) ( )

[ ( ) ]

F E H

E

E E

x

1 0

2

2 2

2

σ

, , 1 2 sinh Ω Ω cosh

= +

^{, (13)}

( ) ( )

F E H

E

x

2 0

2 2

1

σ

, , Ω =

( ) ( ( ) ( ) ) ^{( )}

[ ( ) ]

× + + + +

+

E H E H E

E

σ

β

0

βΩ

σ

β

2 2

0 2

2

2 1 2

cosh cosh

(14) and

E

_S

J S

_oi

i z

iz

= β ∑

= 1

,

( ) ^{( )}

( )

E

₁

= E

_S

+ β H

₀ ²

+ βΩ

^{2 1 2}^/ ^,

E J

_oi

i z

iz

σ

= β σ

∑

=

1 ,

( ) ( )

( )

E

₂

= E

_σ

+ β H

₀ ²

+ βΩ

^{2 1 2}^/ ^,

where z is the nearest-neighbour coordination number of the lattice.

The next step is to carry out the averaging over the longitudinal random field to be denoted by <...>r.

In order to perform the thermal averaging on the right-hand side of eqs (7) and (8), we expand the

(3)

functions

^{f E H}

^α

(

^S

^,

⁰

^, ^Ω )

^and

^{F E H}

ⁿ^α

(

σ

,

₀

, Ω )

^as

finite polynomials of

S

^z_j_and

σ

_i^z, respectively, that correctly account for the single-site kinematic relations.

This can conveniently be done by employing the Van der Waerden operators³¹

( )

^{( )}

( ) ⁽ ⁾

f E H

_S

O

^S

S f E H

j zj

α

,

₀

, Ω = ∏

α S

,

₀

, Ω

^{, (15)}

( ) ( ) ⁽ ⁾

F E H_n O F E H

i iz

α n

σ σ α

σ σ

, ₀,Ω =

∏

^{( )} , ₀,Ω ^{, (16)} where

( )

O

^{( )}σ _i^z _i^z _i^z, / _i^z _i^z, /

σ σ

σ =  σ + δ σ δ

 

 + −  +

 



→ →

−

1 2

1

1 2

2

1 2, (17)

( ) ( )

O

( )^S

S

^z_j

S

^z_j _j^z Szj

(S )

,

=  + +



1

→

2

δ

1

₂ ¹ ( ⁻ ^S

^z^j

⁺ ^{(S )}

^j^z ²

)

^→

^δ

^S^z^j^,⁻¹

^{+ −} ( ¹ ^{(S )}

^z^j ²

)

^→

^δ

^S^z^j^,⁰

^ _

^{, (18)}

where ^→

δ

A a, is a forward Kronecker delta-function substituting any operator A to the right by its eigenvalue a. In order to carry out the thermal and configurational averaging, we have to deal with correlation functions. In this work, we consider the simplest approximation by neglecting correlations between quantities pertaining to different sites.

f E H

^α

( ,

_s ₀

, Ω =

^P ^z^j ^{f E H}

( )

j S z

S zj

(S ) , ,

=−

+

=

∑

∏

^











1 1

1 α 0 Ω , (19)

F E H

_n

R

_i^z

i z

iz

α σ

σ

( , , ) ( )

/ / 0

1 2 1 2

1

Ω = 

 





 



=−

+

=

∑

∏

( )

× F E H

_n^α _σ

,

₀

, Ω

, (20) with

( ) ^{( )}

P S

^z_j

a I

I

I S Ij

=

z

=

=−

+

∑

1

0 1 1 1

2 1

δ

_, 1, (21)

( ) ( )

R

_i^z

b K

_K

K

K ⁱ

σ = δ

_σz

=

=−

+

∑ ∑

1

0 1 1 2 1 2

1 2

1

/ , /

, (22) where

( ) ( )

a ± 1 = 1 ± m

^z_j

+ m

^z_j

2

¹ 2 , (23)

( ) ( )

a 0 = − 1 m

^z_j₂ ^,

(24)

b  ±

_i^z

 

 =  ±

 



1 2

1 2 µ

, (25)

with, z

z i i

= σ

µ

^,

^m

^z^jn

⁼ ( ) ^S

^{z n}^j ^.

Since the longitudinal field is randomly distributed, we have to perform the random average over Hi according to the probability distribution function Q(Hi) given by

Eq.(2). The ordering parameters

µ

^α^and

m

^α_n are then defined as

µ

^α⁼

µ

^α_i

rand

m

_n^α=

m

_nj

r α .

Thus, using the probability distributions, we obtain the following set of coupled equations for

µ

^α^and

m

^α_n

( )

µ

^α

= 

  

 

=

=−

+

=−

+

∑ ∏

∑ ^...

_I _j^z

^{a I}

^j

I _z 1 1

1

1 1

1

(26)

× f á ( S 1 ^z ^(I ¹ ^),..., ^S ^z ^z ^(I ^z ^), ^p, ^H, ^Ù )

^,

( )

m

_n

b K

_i

i z

K

K z

α

= 







=

=−

+

=−

+

∑ ^... ∑ ∏

/ /

1 1 2 1 2

1 2 1 2

1

( ^ó ^(K ^),..., ^ó ^(K ^),p,H, ^Ù )

F

z 1 ^zz ^z

á 1

×

n , (27)

with

^{S I}

zj

( ) = ^I

^et

σ

iz

( ) k = k

^and

µ

^αi and

m

^α_nj_in

eqs.(23)-(25) are replaced by

µ

^α^and

m

^α_n respectively.

α and F n α

f

are given by

( ) ( ) ( )

f x p H

^α

, , , Ω = ∫ Q H

_o

f x H

^α

,

⁰

, Ω dH

_o^{, (28)}

( ) ( ) ( )

F x p H

_n^α

, , , Ω = ∫ Q H

_o

F x H

_n^α

,

_o

, Ω dH

_o^{, (29)}

We like to note that these equations can be solved directly by numerical iteration without further algebraic calculations. This treatment has successfully been used in the study of other systems³². Since the total number of loops 2z is relatively large, the combined sums in (26) and (27) extend over large numbers ( [2(2S+1)]z and [2(2σ+1)]z, respectively) of terms, leading to quite long computational time, particularly near second-order phase transition. Therefore, it is advantageous to carry out further algebraic manipulations on Eqs (19) and (20) employing the differential operator technique.

Thus,using

18 N. BENAYAD , L. KHAYA AND A. FATHI 3

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f E H^α( _S, _o, )Ω =Exp E D f X H( _S _X) ( ,^α _o, )Ω _X_→_o^{, (30)} F E H_n _o Exp E D F X H_X _n _o

X o α( σ, , )Ω = ( σ ) α( , , )Ω _→ ^{, (31)} we obtain from Eqs (19) and (20)

( )

µ

^α

=  β

  

 

=−

∑

+

^{a I} ^{I JD}

^x I

z

1 1

1

exp( )

^× ^{f x p H}

^α

( ^{, , ,} ^Ω )

_{x o}₌ , (32)

^m

ⁿ

^{b K} ( ) ^{K JD}

^x

K

z

α

=  β

  

 

=−

+

∑

¹ ¹

1 2 1 2

1

exp( )

/

^× ^{F x p H}

ⁿ^α

( ^{, , ,} ^Ω )

_{x o}₌ , (33)

which can be reduced to

µ

^α

=  + β



1 2 ( m

¹^z

m

²^z

)exp( JD

_x

) + − + − + − 



1 2 ( m

₁^z

m

₂^z

)exp( JD

_x

) ( 1 m

₂^z

)

z

β ^× ^{f x p H}

^α

( ^{, , ,} ^Ω )

_{x o}₌

^,

(34)

m JD

n

z x

α

µ β

=  +

 ( 1 )exp( )

2 2

+ − − 

( 1 ) exp( ) 

2 µ

_z

β 2

x

JD

z

( )

× F x p H

_n = x o

α

, , , Ω

, (35)

Using the multinomial expansion, we find

µ

β

α

= + − +

× − −

− −

=

−

= −

− −

∑

∑ ²

1 1 2

1 2 2 0

1

1 1 2

1 2

1

1 2

1

2

1 2 n n n

z n

n z z n z z

n o z

z z n n

C C m m m m

m f J n n p H

( ) ( )

( ) ( ( ), , , ) Ω

(36)

( )

m C

F J z n p H

n z

n z n z z n

n o z

n x o

α

µ µ

β

= + −

× −

−

=

∑

₁ ¹ ¹ ¹

1 2

1 2 2 1

( ) ( )

( ), , , Ω

(37)

where

C

_n^p are the binomial coefficients n!/[p!(n-p)!].

The iteration process of these equations becomes suitable for the study of the present system even in the

vicinity of the critical temperature. Eqs. (36) and (37) can be written in the form

µ

^α

=

^{− −}

=

−

=

∑ ∑ ∑

−

∑ ²

¹

2 2

1 1

2 1

1

1 1 2

0 0 0 0

n n i

n

i n

n z n

n z

z n

n

C C

n

^× ^{C C}

ⁱⁿ¹¹ ⁿⁱ²²

( ) ⁻ ¹

ⁱ²

( ) ( ) ^m

¹^{z i}¹⁺ⁱ²

^m

^{z n}² ¹⁺ⁿ²⁻ⁱ¹⁻ⁱ²

( ) ( )

× − 1 m

₂^{z z n}^{− −}¹ ⁿ²

f

^α

β J n (

₁

− n

₂

), , , p H Ω

, (38) and

m

_n ^{z i} ⁱ

i n

i z n

n z

α

=

^{− + +}

=

−

=

∑ ∑

∑ ²

¹ ²

2 1

1 1

1 0 0 0 ^× ⁻

( )

⁺

=

∑

− ^{C C}^zⁿ ⁱ^{z n}^Cⁱⁿ ^{z i} ⁱ n

z n

1 1 1

1

2 1 2

2 0 1

µ

×  −

 

F J 

z n p H

nα

β

2 ( 2

₁

), , , Ω

, (39) At high temperature, the longitudinal

magnetizations µz and mz are disordered. Below a transition temperature Tc , they order (µz

≠

^{0 and}

mz

≠

0 ) while the corresponding transverse magnetizations µx and mx are expected to be ordered at all temperatures. To calculate Tc, we substitute m₁^z and m₂^z in (38) with their expression taken form (39).

For the z-components (α=z), Eqs. (38) and (39) can be written in the following form

( )

µ

^z

= A

1

β J p H , , , , Ω m m

^z2 1^z

( )( )

+ B

₁

β J p H , , , , Ω m

^z₂

m

₁^z ³

+ ....

, (40)

(5)

( )

m

₁^z

= A

₂

β J p H , , , Ω µ

^z

⁺ ^B

²

( ^β ^{J p H} ^{, , ,} ^Ω ) ( ) ^µ

^z ³

⁺ ^....

, (41)

( )

m

₂^z

= A

₃

β J p H , , , Ω

⁺ ^B

³

( ^β ^{J p H} ^{, , ,} ^Ω ) ( ) ^µ

^z ²

⁺ ^....

, (42) a = A1A2 and b = A1B2 + B1(A2)³

where Ai, Bi ,... (i=1,2,3) are obtained from eqs (38) and (39) by choosing the appropriate corresponding combinations of indices ij (j=1,2). Then, we obtain an equation for µz of the form

µ

^z

= a µ

z

+ b µ

³z

....

, (43) within this approximation. As usual, the second-order transition is determined by the condition a(T,p,H,Ω)=1.

We note here that at this transition, the transverse sublattice magnetizations µx and mx keep in fact finite values which are given by

where

( )

µ

^x c

n n n

z n

n z

z n

z n n

c

z n n

T T

ⁿ

C C m

( = ) =

^{− −}

=

−

= −

∑

+

∑ ²

¹

2 1

1

1 1

2 1 2

0 0

2

^{× −} ( ¹ ^m

²^z^c

)

^{z n}^{− −}¹ ⁿ²

^f

^x

( ^β ^{J n} ⁽

¹

⁻ ⁿ

²

^{), , ,} ^{p H} ^Ω )

, (44)

m T

^x

T

_c ^{z n}

C

_zⁿ

n z

1 0

2

¹ ¹

1

( = ) =

^{− −}

∑

=

^× ^F

ⁿ^α

^ _ ^β ₂ ^J ⁽ ^z ⁻ ² ^{n p H}

¹

^{), , ,} ^Ω ^ _

(45) with

m

^c_c

C

_zⁿ ^{z n}

n z 2

0

1 1

1

= 2

^{− −}

∑

=

×  −

 

F J 

z n p H

z

2

β 2 2

1

( ), , , Ω

, (46) The magnetization µz in the vicinity of the second-order transition line is given by

µ

_z

a b

2

1 = −

.

(47) The right-hand side of (47) must be positive. If this is not the case, the transition is of the first-order and in the (T,H) plane (for given values of p and Ω), the point at which a=1 and b=0 characterizes the tricritical point.

III . RESULTS AND DISCUSSIONS First, let us investigate the phase diagram of the system in the absence of the transverse field (Ω=0). We are interested in studying the effect of the longitudinal random field on the mixed spin-1/2 and spin-1 Ising model. In figure 1, we represent the phase diagram in the (T,H) plane for various values of p (the fraction of the system not exposed to the longitudinal field H) when Ω=0. As seen from this figure, the tricritical points (black points) appear only for relatively small values of p (0

≤

p<0.2899). In This range of p, the critical line ends in a tricritical point. Moreover, we note the existence of two critical values of p : p1*

=0.2899 and p2*=0..5259. The former shows that for p>p1^*, all transitions are of the second order, while the second one p2* indicates two qualitatively different behaviours of the system which depend on the range of p.

0.0 4.0 8.0

0.0 0.5 1.0 1.5 2.0 2.5

P=.0 T_c/J

H/J .2

.7 .15

.25

.6

.45 .5

.55

.52 .53

.95 1

.8

.

.35

Ω=0.

. .. .

FIG.1: The phase diagram in T-Η plane of the mixed spin- 1/2 and spin-1 Ising system in a random longitudinal field on simple cubic lattice (z=6). The number accompanying each curve denotes the value of p.

Thus, for p<p2*, the system exhibits at the ground state a phase transition at a finite critical value Hc of H. But for p2*<p<1, there is no critical longitudinal field, and therefore, at very low temperature, the ordered state is stable for any value of the longitudinal field strength.

As is expected, we can see in Fig.1 that for a fixed value of Η, the critical temperature is an increasing function of p.

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0.0 0.5 1.0 1.5 2.0 2.5 0.0

0.5 1.0 1.5 2.0 2.5

3 2

1 Ω=0.

Tc/J

H/J p=0.2

. .

FIG.2(a) : The phase diagram in T-H plane of the transverse mixed spin-1/2 and spin-1 Ising system in random longitudinal field on simple cubic lattice (z=6) with p=0.2. The number accompanying each curve denotes the value of Ω/J.

0.0 1.0 2.0 3.0 4.0

0.0 1.0 2.0 3.0

1 3 2

Tc/J

H/J Ω=.0 p=.45

FIG.2(b) : The phase diagram in T-H plane of the transverse mixed spin-1/2 and spin-1 Ising system in random longitudinal field on simple cubic lattice (z=6) with p=0.45. The number accompanying each curve denotes the value of Ω/J.

On the other hand, it is important to investigate the influence of a transverse field on the above three behaviours of the system. First, let us investigate the effect of Ω on the phase diagram, when it shows a tricritical point. The typical phase diagrams are depicted in Fig.2(a) by selecting an appropriate value of p (p=0.2) and plotting various transition lines when the strength of the transverse field Ω takes different values. It is seen that the system keeps a tricritical behaviour only when Ω is relatively small. The H- and T-components of the tricritical point decreases, with increasing transverse field. When Ω/J

>

1, the tricritical behaviour disappears and therefore all transitions are always of second order for any value of the longitudinal field. As shown in Fig.2(b), the transverse field does

not change qualitatively the phase diagram when p belongs to the range p1*<p< p2*

0.0 2.0 4.0 6.0 8.0

0.0 1.0 2.0 3.0

3 2 1

Ω=.0 T_c /J

H/J .5

p=.52

FIG.2(c) : The phase diagram in T-H plane of the transverse mixed spin-1/2 and spin-1 Ising system in random longitudinal field on simple cubic lattice (z=6) with p=0.52. The number accompanying each curve denotes the value of Ω/J.

As seen in Fig.1, a reentrant phenomenon may be observed for a relatively small range of p. Fig.2(c) shows that the system keeps a reentrant behaviour only when the transverse field is relatively small. In the absence of the transverse field and when p>p2*, the system does not exhibit a phase transition at zero temperature. In Fig.2(d) we plot the influence of

Ω

on such phenomenon. As seen from this figure, the transverse field lead to two qualitatively different behaviours of the system which depend on the range of

Ω

. For instance for p=.6, when 0

≤

^Ω/J

<

1.136, the system keeps its qualitatively behaviour when

Ω

= 0 ; while, for Ω/J

>

^{1.136, the}

system exhibits at the ground state a phase transition at finite critical value Hc of H.

0.0 2.0 4.0 6.0 8.0 10.0

0.0 0.5 1.0 1.5 2.0 2.5

Ω=0.

2

1 Tc/J

H/J p=.6

1.136 3

FIG.2(d) : The phase diagram in T-H plane of the transverse mixed spin-1/2 and spin-1 Ising system in random longitudinal field on simple cubic lattice (z=6) with p=0.6. The number accompanying each curve denotes the value of Ω/J.

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The transverse mixed spin Ising model in a longitudinal random field