Surface phase transitions of the transverse Ising model with modified surface–bulk coupling

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Surface phase transitions of the

transverse Ising model with modified surface–bulk coupling

T. Lahcini

^a

& N. Benayad

a

Groupe de Mécanique Statistique, Laboratoire de Physique Théorique , Faculté des sciences, Université Hassan II-Aïn Chock , B.P. 5366 Maarif, Casablanca 20100, Morocco

Published online: 28 Apr 2009.

To cite this article: T. Lahcini & N. Benayad (2009) Surface phase transitions of the transverse Ising model with modified surface–bulk coupling, Phase Transitions: A Multinational Journal, 82:3, 197-210, DOI: 10.1080/01411590802575192

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Vol. 82, No. 3, March 2009, 197–210

Surface phase transitions of the transverse Ising model with modified surface–bulk coupling

T. Lahcini and N. Benayad*

Groupe de Me´canique Statistique, Laboratoire de Physique The´orique, Faculte´ des sciences, Universite´ Hassan II-Aı¨n Chock, B.P. 5366 Maarif, Casablanca 20100, Morocco

(Received 15 May 2008; final version received 23 October 2008)

The phase diagram of the three-dimensional transverse Ising system with antiferromagnetic intrasurface J

_S

and surface–bulk J

_?

interactions modified with respect to the ferromagnetic bulk exchange interaction J

_B

is studied by the use of an effective field method in the framework of a single-site cluster theory.

In this investigation, the state equations are derived by means of differential operator technique. Using these coupled equations within the four-layer approximation, it has been shown that the new parameter J

_?

strongly affects the phase diagram obtained recently in the case J

_?

¼ J

B

. Indeed, our calculations reveal some qualitatively interesting features both in the absence and in the presence of the surface and bulk transverse fields.

Keywords: competing interactions; modified surface–bulk coupling; transverse Ising model; surface phase transitions

1. Introduction

Magnetic materials presenting a free surface can, in principle, exhibit interesting conflicts when the surface favours a type of ordering which competes with that favoured by the bulk. This can be the case of magnetic systems when the surface coupling constant J

S

differs in sign from the bulk coupling constant J

B

. This seems to be precisely the case of Cr, which is antiferromagnet with a Ne´el temperature of 312 K. Its (1,0,0) free surface has been investigated [1] using angle-resolved photoelectron spectroscopy, and it presents a ferromagnetic ordering up to 780 50 K.

An example of such system is the simple cubic Ising ferromagnet with a ferromagnetic exchange interaction (J

B

4 0) in the bulk and an antiferromagnetic exchange interaction (J

S

5 0) between surface spins. It has been studied using the mean-field approximation [2], the renormalization group method [3], and Monte Carlo simulation [4,5]. In this case, the surface layer behaves roughly like an Ising antiferromagnet in a (temperature-dependent) field. Below the bulk critical temperature, the bulk is ordered ferromagnetically for all J

S

. For a surface exchange interaction greater than some temperature-dependent value of J

S

, the surface is also in an ordered ferromagnetic state, but for more negative values of J

S

the surface is antiferromagnetic instead. As the temperature increases, the bulk disorders, but

*Corresponding author. Email: noureddine_benayad@yahoo.fr

ISSN 0141–1594 print/ISSN 1029–0338 online ß2009 Taylor & Francis

DOI: 10.1080/01411590802575192 http://www.informaworld.com

Downloaded by [New York University] at 11:20 25 November 2013

for strongly negative J

S

the surface remains ordered up to some higher temperature. The phase boundaries for the bulk and the surface transitions cross at a tetracritical point.

On the other hand, the spin-1/2 Ising model in a transverse field has been studied by the use of various techniques [6–8] including Green’s function treatment [9,10] and effective field method [11,12]. It was originally introduced by De Gennes [13] as a valuable model for the tunnelling of the proton in hydrogen-bonded ferroelectrics [14] such as potassium dihydrogen phosphate (KH

2

PO

4

). Since then, it has been successfully applied to several physical systems, such as cooperative Jahn–Teller systems [15] (like DyVO

4

and TbVO

4

), ordering in rare-earth compounds with a singlet crystal field ground state [16], and also to some real magnetic materials with strong uniaxial anisotropy in a transverse field [17].

The pure three-dimensional Ising model in a transverse field with a modified surface exchange coupling has been studied using Green’s function technique [18], as well as the effects of a random transverse field on the phase diagram of the spin-1/2 semi-infinite simple cubic Ising ferromagnet [19] within the effective field method. The influence of the bulk and surface fields on phase diagrams of transverse Ising films [20,21] has also been investigated. It is worthy of notice here that a modified Ising model in a transverse field within the framework of a Green’s function has been used to describe a variety of ferroelectric thin films [22–24].

Recently, N. Benayad has investigated the influence of the transverse fields on the phase diagram of the spin-1/2 semi-infinite simple cubic Ising ferromagnet with antiferromagnetic surface exchange interaction at the surface [25]. In this case, it has been shown that the surface presents some characteristic behaviours. We notice that in all these surface investigations, the systems include two ferromagnetic exchange interactions J

S

and J

B

at the surface and in the bulk, respectively

In order to describe a ferromagnetic material presenting a strong uniaxial anisotropy with competing bulk and surface ordering placed in a transverse field, we study, in this article, the surface phase transitions in the semi-infinite simple cubic transverse Ising ferromagnet with an antiferromagnetic surface and a modified surface–bulk coupling J

?

. The introduction of this later exchange interaction describes a more realistic situation. In particular, we investigate the effects of the surface–bulk coupling on the phase transitions associated with the surface both in the absence and in the presence of the surface and bulk transverse fields. To this end, we use an effective-field method within the framework of a single-site cluster theory. This latter has been applied to a variety of pure [26–28] and disordered [19,25,29,30] transverse Ising models as well as to mixed spin transverse Ising models [31,32]. This article is organized as follows. In Section 2, we define the model and review the basic points of the effective-field theory with correlations when it is applied to the present model. In Section 3, the phase diagrams of the system as a function of the surface–bulk coupling are examined and discussed when the transverse fields are changed.

Finally, we comment on our results in Section 4.

2. Theoretical framework

We consider a semi-infinite simple cubic transverse Ising ferromagnet with an antiferromagnetic surface and a modified surface–bulk coupling. Such system can be described by the following Hamiltonian:

H ¼ J

S

X

hiji

_i^z

_j^z

S

X

i

_i^x

J

?

X

hjki

^z_j

^z_k

J

B

X

hkli

_k^z

_l^z

B

X

l

_l^x

, ð1Þ

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where

_i

ð ¼ x, z Þ is the -component of spin-1/2 operator at site i. The first and second summations are carried out over nearest-neighbour sites and single sites located on the surface, respectively. The third summation runs over nearest-neighbour sites one located on the free surface and the other in the first layer. The fourth and fifth summations run over all pairs of remaining nearest-neighbour sites and single sites, respectively.

_B

and

_S

represent transverse fields in the bulk and at the surface, respectively. J

S

and J

B

denote the exchange interactions at the surface and in the bulk respectively; whereas J

?

is the coupling constant between a spin in the surface and its nearest-neighbour in the next layer.

The theoretical framework we adopt in the study of the system described by the Hamiltonian (1) is the effective-field theory based on single-site cluster theory. In this approach, attention is focused on a cluster consisting of just a single selected spin, labelled 0, and the neighbouring spins with which it directly interacts. To this end, the total Hamiltonian is split into two parts, H ¼ H

0

þ H

⁰

, where H

0

includes all those terms of H associated with the lattice site 0, namely,

H

0

¼ X

j

J

0j

_j^z

!

₀^z

^x₀

, ð2Þ

where S or B whether the lattice site 0 belongs to the surface or to the bulk, respectively. J

0j

J

S

if both spins are at the surface layer, J

0j

J

?

if one of them is located in the free surface and J

0j

J

B

otherwise. The problem consists in evaluating the longitudinal and transverse site magnetisations. The starting point of our calculation is a generalised, but approximate, Callen–Suzuki relation [33,34]

O ^

0

D E

¼

trace

0

O ^

0

exp ð H

0

Þ h i trace

₀

½ exp ð H

₀

Þ

* +

, ð3Þ

used for transverse Ising model. Here, trace

0

means the partial trace with respect to the site 0, h. . .i indicates the canonical thermal average and ¼ 1/k

B

T. This relation is not exact since H

0

and H

⁰

do not commute. Nevertheless, it has been successfully applied to a number of interesting transverse Ising systems [26–29]. We emphasise that in the limit

¼ 0 the Hamiltonian contains only

_i^z

. Then, relation (3) becomes exact identity.

The application of (3) for the longitudinal and transverse site magnetisations of the n-th layer leads to the following expressions:

_0n^z

¼ A 2E

0

tanh 2 E

0

, ð4Þ

_0n^x

¼

2E

0

tanh 2 E

0

, ð5Þ

where

A ¼ X

j

J

_oj

^z_j

, ð6Þ

E

0

¼ ð Þ

²

þA

² 1=2

: ð7Þ

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Introducing the differential operator technique [35], Equations (4) and (5) can be written as follows:

^z_on

¼ e

^Ar

F

ðxÞ

_x¼0

i

ð8Þ

^x_on

¼ e

^Ar

G ðxÞ

_x¼0

i

ð9Þ where r ¼ @=@x is a differential operator (defined as e

^Ar

F x ð Þ ¼ F x ð þ A Þ) and functions F (x) and G (x) are defined by

F

ð Þ ¼ x x

2 ð

Þ

²

þ x

² 1=2

tanh

2 ð

Þ

²

þx

² 1=2

, ð10Þ

G

ð Þ ¼ x

²

2 ð Þ

²

þx

² 1=2

tanh

2 ð

Þ

²

þx

² 1=2

, ð11Þ

We assume the statistical independence of the lattice sites, that is h

i

j

. . .

l

i ¼ h

_i

ih

_j

i h i. This means that the used effective field method is designed to treat all

_l

spin self-correlations exactly while still neglecting correlations between different spins.

Thus, using the spin-1/2 identity exp ð

i

Þ ¼ coshð=2Þ þ 2

i

sinh ð =2 Þ, Equations (8) and (9) may be written as follows:

_0n^z

¼ Y

j

cosh J

_0j

2 r

þ 2 D E

^z_j

sinh J

_0j

2 r

F

ð Þ x

x¼0

, ð12Þ

^x_0n

¼ Y

j

cosh J

0j

2 r

þ 2 D E

_j^z

sinh J

0j

2 r

G ð Þ x

x¼0

: ð13Þ

Denoting the ordering parameters by

^z_n

¼

_0n^z

and

^x_n

¼

_0n^x

, the longitudinal magnetisation of the surface and the n-th layer is given by

^z_n

¼ Y

^q

j¼1

a

_j

þ 2

^z_j

b

_j

h i

F

ð Þ x

x¼0

, ð14Þ

where q is the number of nearest neighbours of site 0 and the coefficients a

j

and b

j

are given by

a

S

¼ cosh J

S

2 r

, b

S

¼ sinh J

S

2 r

,

a

?

¼ cosh J

?

2 r

, b

?

¼ sinh J

?

2 r

,

a

B

¼ cosh J

B

2 r

, b

B

¼ sinh J

B

2 r

:

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Equation (14) takes the following forms:

. for the surface (n S)

^z_{S 1}

¼ a

S

þ 2

^z_{S 2}

b

S

4

a

?

þ 2

^z_{1 2}

b

?

F

S

ð Þ x

x¼0

,

^z_{S 2}

¼ a

S

þ 2

^z_{S 1}

b

S

4

a

?

þ 2

^z_{1 1}

b

?

F

S

ð Þ x

x¼0

,

ð15Þ

. for the first layer (n ¼ 1)

^z_{1 1}

¼ a

?

þ 2

^z_{S 2}

b

?

a

B

þ 2

^z_{1 2}

b

B

4

a

B

þ 2

^z_{2 2}

b

B

F

B

ð Þ x

x¼0

,

^z_{1 2}

¼ a

?

þ 2

^z_{S 1}

b

?

a

_B

þ 2

^z_{1 1}

b

_B 4

a

_B

þ 2

^z_{2 1}

b

_B

F

_B

ð Þ x

x¼0

,

ð16Þ

. for any layer n 2

^z_{n 1}

¼ a

B

þ 2

^z_n1

2

b

B

a

B

þ 2

^z_{n 2}

b

B

4

a

B

þ 2

^z_nþ1

2

b

B

F

B

ð Þ x

x¼0

,

^z_{n 2}

¼ a

B

þ 2

^z_n1

1

b

B

a

B

þ 2

^z_{n 1}

b

B

4

a

B

þ 2

^z_nþ1

1

b

B

F

B

ð Þ x

x¼0

,

ð17Þ

Here ð

^z_n

Þ

₁

and ð

^z_n

Þ

₂

are the two sublattice longitudinal magnetisations of the n-th layer, respectively. Expanding the right-hand sides of (15)–(17), we obtain

. for the surface (n S)

^z_{S 1}

¼ A

1

^z_{S 2}

þA

2

^z_{1 2}

þA

3

^z_S ²₂

^z_{1 2}

þA

4

^z_S ³₂

þA

5

^z_S ⁴₂

^z_{1 2}

,

^z_{S 2}

¼ A

₁

^z_{S 1}

þA

₂

^z_{1 1}

þA

₃

^z_S ²₁

^z_{1 1}

þA

₄

^z_S ³₁

þA

₅

^z_S ⁴₁

^z_{1 1}

,

ð18Þ

. for the first layer (n ¼ 1)

^z_{1 1}

¼ B

1

^z_{S 2}

þB

2

4

^z_{1 2}

þ

^z_{2 2}

þ B

3

^z_{S 2}

^z_{1 2}

2

^z_{2 2}

þ3

^z_{1 2}

þ B

₄

^z₁ ²₂

2

^z_{1 2}

þ3

^z_{2 2}

þ B

₅

^z_{S 2}

^z₁ ³₂

^z_{1 2}

þ4

^z_{2 2}

þ B

6

^z₁ ⁴₂

^z_{2 2}

,

^z_{1 2}

¼ B

₁

^z_{S 1}

þB

₂

4

^z_{1 1}

þ

^z_{2 1}

þ B

₃

^z_{S 1}

^z_{1 1}

2

^z_{2 1}

þ3

^z_{1 1}

þ B

4

^z₁ ²₁

2

^z_{1 1}

þ3

^z_{2 1}

þ B

5

^z_{S 1}

^z₁ ³₁

^z_{1 1}

þ4

^z_{2 1}

þ B

₆

^z₁ ⁴₁

^z_{2 1}

,

ð19Þ

. for any layer n 2

^z_{n 1}

¼ C

1

^z_n1

2

þ4

^z_{n 2}

þ

^z_nþ1

2

þ C

₂

^z_{n 2}

2

^z_n1

2

^z_nþ1

2

þ

^z_{n 2}

3

^z_n1

2

þ2

^z_{n 2}

þ3

^z_nþ1

2

þ C

3

^z_n ³₂

4

^z_n1

2

^z_nþ1

2

þ

^z_n1

2

^z_{n 2}

þ

^z_{n 2}

^z_nþ1

2

,

^z_{n 2}

¼ C

1

^z_n1

1

þ4

^z_{n 1}

þ

^z_nþ1

1

þ C

₂

^z_{n 1}

2

^z_n1

1

^z_nþ1

1

þ

^z_{n 1}

3

^z_n1

1

þ2

^z_{n 1}

þ3

^z_nþ1

1

þ C

3

^z_n ³₁

4

^z_n1

1

^z_nþ1

1

þ

^z_n1

1

^z_{n 1}

þ

^z_{n 1}

^z_nþ1

1

,

ð20Þ

where the coefficients A

i

(i ¼ 1 5), B

i

(i ¼ 1 6) and C

i

(i ¼ 1 3) are given in the Appendix.

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We note that the bulk longitudinal magnetisation

^z_B

is determined by setting

^z_n1

¼

^z_n

¼

^z_nþ1

in Equation (20). Thus,

^z_B

is the solution of the following equation

^z_B

¼ 6C

₁

^z_B

þ 10C

₂

^z_B

3

þ6C

₃

^z_B

5

: ð21Þ

3. Phase diagrams and discussions

In order to investigate the phase diagrams of the system described by the Hamiltonian (1), we have to solve the coupled Equations (18)–(20). However, we are unable to solve them analytically. Even if we use a numerical method, they must be terminated at a certain layer.

Note that, as n goes to infinity, the magnetisation

^z_n

should approach the bulk value

^z_B

. For this purpose, let us assume that the magnetisations remain unaltered after the third layer, that is,

^z₃

¼

^z₄

¼ ¼

^z_B

, which may be called the four-layer approximation.

We have to mention here that three-layer and four-layer approximations have been used for a variety of semi-infinite Ising systems [25,36–38] and its has been shown that such approximations give a reasonable results.

3.1. Bulk and surface order–disorder transition temperatures

First, let us evaluate the order–disorder transition temperatures for the bulk and the surface ordering based on the four-layer approximation. By solving Equation (21), the bulk critical temperature k

B

T

C

/J

B

is determined by the equation

1 ¼ 6C

1

: ð22Þ

k

B

T

C

/J

B

is a function of

B

. In zero transverse field (

B

¼ 0), comparing with the Monte Carlo value 1.128 [39], we remark that the obtained result 1.268 improves the mean-field value 1.500. One should also note that the critical transverse field

c

(T

C

¼ 0) is 2.352J

B

, to be compared with the mean-field result 3J

B

.

In order to obtain the surface order–disorder critical temperature, we have to linearise Equations (18)–(20). Thus neglecting higher-order terms in the magnetisation near the critical temperature and within the four-layer approximation, we obtain

^z_{S 1}

¼ A

1

^z_{S 2}

þA

2

^z_{1 2}

,

^z_{S 2}

¼ A

₁

^z_{S 1}

þA

₂

^z_{1 1}

, ð23Þ

^z_{1 1}

¼ B

₁

^z_{S 2}

þB

₂

4

^z_{1 2}

þ

^z_{2 2}

,

^z_{1 2}

¼ B

1

^z_{S 1}

þB

2

4

^z_{1 1}

þ

^z_{2 1}

, ð24Þ

^z_{2 1}

¼ C

₁

^z_{1 2}

þ4

^z_{2 2}

þ

^z_B

2

,

^z_{2 2}

¼ C

1

^z_{1 1}

þ4

^z_{2 1}

þ

^z_B

1

, ð25Þ

The surface order–disorder critical temperature, when the bulk is disordered (

^z_B

¼ 0), is analytically obtained through a determinantal equation. In Figures 1 and 2, we represent

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the critical line of antiferromagnetic-paramagnetic surface transition (S) in the (k

B

T/J

B

, J

S

/J

B

) plane. In all cases, when the ratio R ¼ J

S

/J

B

is greater than a critical value R

C

, the surface may antiferromagnetically order at a temperature higher than the bulk critical temperature. As seen from these figures, R

C

depends on the values of R

?

¼ J

?

=J

_B

and the transverse fields. First, in the absence of transverse fields (

_S

¼

_B

¼ 0), Figure 1 shows the influence of the surface–bulk coupling J

?

on the order-disorder surface critical temperature. Thus, some surface transition lines corre- sponding to different values of R

?

are plotted. As clearly seen, we note that for a given value of the ratio R ¼ J

S

/J

B

(greater than a critical value), the surface critical temperature decreases with increasing value of R

?

. Second, we specially examine the effects of the surface transverse field

S

on the surface transition for selected values of R

?

. In Figure 2(a)–(c), corresponding to R

?

¼ 0.5, R

?

¼ 1 and R

?

¼ 2 respectively, we represent the variation of the antiferromagnetic–paramagnetic surface transition (S) with

S

. From these figures, we point out that the domain where the surface is antiferromagnetic (when the bulk is disordered) becomes less and less large when

S

increases and disappears when

S

exceeds its critical value

^c_S

=J

S

¼ 1:375 (

^c_S

=J

S

does not sensitively depend on R

?

). The above surface transition seems to be independent of the bulk transverse field. One also notes that the critical ratio R

_C

¼ ðJ

_S

=J

_B

Þ

_C

of the surface and bulk exchange interactions depends on the values of R

?

,

S

and

B

as is shown in Figure 3. From this latter, on the one hand we note that, for a given value of

B

, R

C

is an increasing function of R

?

and

S

. On the other hand for weak values of

S

/J

S

, R

C

depends on R

?

and this dependence

0.0 1.0 2.0 3.0

J

_S

/J

_B

0.0

1.0 2.0 3.0

k

B

T / J

B

Ω

_S

/J

_S

= Ω

_B

/J

_B

= 0.

R

_⊥

= 8.

6. 7.

5.

4.

3.

2.

.5 1.

.25

BP SP

BF SAF BF

SF

BP SAF (S)

(O) (Sp) (E)

(L)

(L)

(S) (S)

(L)

Figure 1. Phase diagram in T J

_S

/J

_B

plane of the semi-infinite transverse Ising model with modified surface–bulk coupling when

S

¼

B

¼ 0. The number accompanying each curve denotes the value of R

_?

.

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0.0 1.0 2.0 3.0

0.0 1.0 2.0 3.0

J_S/J_B

J_S/J_B

J_S/J_B 0.0

1.0 2.0 3.0

kBT/JBkBT/JBkBT/JB

R_⊥= 0.5

R_⊥= 1

R_⊥= 2

Ω_S/J_S= 0.

0.5

Ω_B/J_B= 0.

Ω_B/J_B= 1.5 Ω_B/J_B= 0.

Ω_B/J_B= 1.5

Ω_B/J_B= 0.

Ω_B/J_B= 1.5

1.0 1.3 BP

SP

BF SAF

BF SAF

BF SAF

SAFBP

BP SAF BF

SF (O)

(E) (Sp)

(S)

(L)

(S)

(S) (S)

(L) (L)

0.0 1.0 2.0 3.0

Ω_S/J_S= 0.

Ω_S/J_S= 0.

0.5

1.0

1.3 BPSP

BFSF (L)

(S)

(O) (Sp)

(E)

(L) (L)

(S)

(S) (S)

0.0 1.0 2.0 3.0

0.0 1.0 2.0 3.0

0.5

1.0 1.3 BPSP

BFSF (L)

(S)

(O) (Sp)

(E) (S)

(S) (S)

(L) (L) (a)

(b)

(c)

SAFBP

Figure 2. Phase diagram in T J

_S

/J

_B

plane of the semi-infinite transverse Ising model with modified surface bulk coupling when

B

¼ 0 (solid lines) and

B

/J

B

¼ 1.5 (dashed lines) for (a) R

?

¼ 0.5, (b) R

_?

¼ 1 and (c) R

_?

¼ 2. The number accompanying each curve denotes the value of

S

/J

S

.

Downloaded by [New York University] at 11:20 25 November 2013

becomes more significant for large values of the surface–bulk coupling. Furthermore, when

S

/J

S

approaches its critical value

^c_S

=J

S

, the order at the surface is more disturbed by the surface transverse field. Therefore, the critical ratio R

C

does not depend sensitively on R

?

.

3.2. Other surface transitions

The system under investigation here undergoes other surface transitions. In fact, the steps described before do not lead to the whole phase diagram. Indeed, any two nearest neighbours on the surface interact via an antiferromagnetic coupling. In the absence of the transverse fields (

S

¼

B

¼ 0) and at the ground state of the Hamiltonian (1), the surface makes a first-order transition from antiferromagnetically ordered state for R > R

?

=4 to ferromagnetically ordered state for R < R

?

=4. In order to obtain the remaining phases and transitions, we must solve numerically the coupled Equations (18)–(20) within the four- layer approximation scheme. The analysis of these equations leads to interesting surface phenomena. The surface behaviours, when the bulk is ordered, and their dependences on the surface–bulk coupling and transverse fields are also represented in Figures 1 and 2.

In addition to the previous phases:

SP, BP: surface and bulk paramagnetic,

SAF, BP: surface antiferromagnetic and bulk paramagnetic, two other phases are identified:

SF, BF: surface and bulk ferromagnetic,

SAF, BF: surface antiferromagnetic and bulk ferromagnetic.

As is seen from Figures 1 and 2, the above phases are separated by different transition lines. Among them, we find all critical lines obtained in the semi-infinite simple cubic

1.5 2.0 2.5 3.0

R_⊥ (J_S/J_B)_c

Ω_B/J_B = 0.

1.

0.

Ω_S/ J_S= 1.25 Ω_B/J_B= 1.5

.5 0.

.5 1.

1.25

0 2 4 6 8 10

Figure 3. Dependence of the critical ratio R

c

¼ (J

S

/J

B

)

c

as a function R

_?

, with

B

¼ 0 (solid lines) and

_B

¼ 1.5 (dashed lines). The number accompanying each curve denotes the value of

S

/J

S

.

Downloaded by [New York University] at 11:20 25 November 2013

ferromagnetic Ising model [40–42]. Extending the accepted terminology used in the semi- infinite Ising systems [19,41,43], these transitions correspond to the surface (S), the ordinary (O), the extraordinary (E) and the special (Sp) transitions. Note that the critical exponents of the special transition differ from those of ordinary transition. In addition to these transitions, when the bulk is ferromagnetically ordered, the surface exhibits, at finite temperature, a second-order transition line (L) from the ferromagnetic state (SF, BF) to the antiferromagnetic state (SAF, BF). In Figure 1, we show the effects of the surface–bulk coupling on these transition lines. Here, we find similar phases and transitions to those obtained for the transverse Ising system with competing surface and bulk exchange interactions (J

?

¼ J

B

) done by one of us (N. Benayad) [25]. It is worthy of notice here that the above transition line (L) has also been found in other Ising models with competing surface and bulk interactions [2,3].

Indeed, the state Equations (18)–(20) show interesting features of the thermal behaviours of the surface magnetisations. First, in zero surface and bulk transverse fields (

S

¼

B

¼ 0), Figure 4(a) shows the temperature dependence of ð

^z_S

Þ

₁

, ð

^z_S

Þ

₂

,

^z_S

¼

¹₂

ðð

^z_S

Þ

₁

þ ð

^z_S

Þ

₂

Þ and

^z_B

. They exhibit many kinds of typical behaviours depending on the value of the ratio R ¼ J

S

/J

B

. For R greater than a (R

?

-dependent) critical value R

C

(R ¼ 2 in Figure 4a), and at low temperature, the system is in the BFSAF phase. With the increase of T ð

^z_S

Þ

₁

, jð

^z_S

Þ

₂

j and

^z_B

curves decrease. As seen from this figure, the bulk disorders (

^z_B

¼ 0) at the bulk transition temperature but the surface sublattice magnetisations vanish at a temperature higher than the bulk. These two successive transitions correspond to the transitions (E) and (S) in Figures 1 and 2, as shown in Section 3.1. For

^R₄^?

< R < R

C

(R ¼ 1 in Figure 4a), the surface sublattice magnetisations take their saturation values ð

^z_S

Þ

₁

¼ 1=2 and ð

^z_S

Þ

₂

¼ 1=2 at T ¼ 0 K which means that the surface is antiferromagnetic and the bulk is ferromagnetic (SAFBF). We note that in this latter phase, when T 6¼ 0, ð

^z_S

Þ

₁

and ð

^z_S

Þ

₂

have both a finite total magnetisation

^z_S

and a finite staggered magnetisation. When the temperature is increased from zero, ð

^z_S

Þ

₁

and

ð

^z_S

Þ

₂

decrease as long as the system is still in SAFBF phase. Then, at a critical temperature, the sublattice magnetisation ð

^z_S

Þ

₂

changes its sign, undergoing a second- order transition (L) (Figures 1 and 2) from the SAFBF phase to SFBF phase. In this case the total surface magnetisation

^z_S

passes through a maximum, then decreases and vanishes at the bulk transition temperature. This latter transition corresponds to the ordinary transition (O) as shown in Figures 1 and 2. Finally, when R <

^R₄^?

(R ¼ 0.25 in Figure 4a), the bulk promotes its order to the surface in such a way that the bulk and the surface are either paramagnetic or ferromagnetic. Thus, the surface and bulk magnetisations take their saturation values at T ¼ 0 K and vanish at the bulk transition temperature. This transition corresponds to the ordinary transition denoted (O) as shown in Figures 1 and 2.

Second, the influence of the transverse fields on the phase diagram is also examined.

In particular, their effects on the transition line (L) is clearly shown in Figure 2 for selected values of the ratio R

?

. Thus, the transition line (L) is sensitive to the surface transverse field for any fixed value of R

?

. At low temperatures, this sensitivity becomes more pronounced when the surface–bulk coupling is relatively stronger (R

?

4 1); whereas for weaker R

?

(R

?

5 1) the transition (L) is less influenced by surface transverse field.

Furthermore, we note that the effects of the bulk transverse field on the transition (L) become more and more significant with increasing values of R

?

.

Furthermore, on the one hand we remark that, for a given value of R

?

and when the

S

increases, the domain of the phase BFSAF becomes less and less large and disappears when

S

exceeds its critical value

^c_S

=J

S

. This means that, beyond

^c_S

=J

S

, the surface and bulk exhibit an ordinary transition from BPSP to BFSF phases at the bulk

Downloaded by [New York University] at 11:20 25 November 2013

transition temperature. On the other hand, and as we can point out from Figure 2, the location of the transition line (L) depends qualitatively and quantitatively on the strength of R

?

, specially at very low temperatures. In order to clarify this situation, we plot in Figure 5 the phase diagram in R

?

R plane at different temperatures (below k

B

T

C

/J

B

).

4. Conclusions

In this article, we have investigated the surface phase diagrams of the semi-infinite simple- cubic spin-1/2 transverse Ising ferromagnet with an antiferromagnetic interaction

−0.5 0.0 0.5

1.

R= 2.5 .1

1.

2.5

1. 2.5

0.0 0.4 0.8 1.2 1.6 2.0

k_BT/J_B

k_BT/J_B

0.0 0.4 0.8 1.2 1.6 2.0

−0.5 0.0 0.5

1. R = 2.

R_⊥= 2 , Ω_B/J_B= Ω_S/J_S= 0

.25 1. 2.

1.

2.

(m_S)₁ (m_S)₂ m_S mB z

mSmBz,

z z

z z

(m_S)₁ (m_S)₂ m_S m_B z

z z z (mS)1,2mSmBz,z(mS)1,2z

(a)

(b) R_⊥= 2 , Ω_B/J_B= 0, Ω_S/J_S= 1.0

Figure 4. Temperature dependences of ð

^z_S

Þ

₁

, ð

^z_S

Þ

₂

, and

^z_S

for the system with (a)

_S

¼

_B

¼ 0 and (b)

S

/J

S

¼ 1,

B

¼ 0. The number accompanying each curve denotes the value of R. The dashed line corresponds to the bulk magnetisation curve

^z_B

.

Downloaded by [New York University] at 11:20 25 November 2013

(J

S

5 0) on the surface and a modified surface–bulk coupling J

?

. To do this, we have used an effective-field method within the framework of a single-site cluster theory. In this approach, we have derived the state equations using the differential operator technique and neglecting correlations between different sites. Using these coupled equations within the four-layer approximation, rich phase diagram have been discovered and some significant results have been obtained.

Let us summarise by stating the main results of this investigation. We identified four physically different phases separated by five transition lines. In particular, on the one hand, we have shown that when the bulk is disordered and the ratio R ¼ J

S

/J

B

is greater than a (R

?

¼ J

?

/J

B

-dependent) critical value R

C

, the surface may antiferromagnetically order at a temperature (surface transition) higher than the bulk. On the other hand, when the bulk is ferromagnetically ordered, the surface may undergo, at finite temperature, a second-order phase transition (L) from the ferromagnetic order to the antiferromagnetic one. Its location in the phase diagram depends on the strength of the surface–bulk coupling J

?

and the surface transverse field

S

.

References

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0.0 0.4 0.8 1.2 1.6 2.0

R_⊥ R_⊥

J_S/J_B

Ω_S/J_S= Ω_B/J_B=0.

0.

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k_BT/J_B=1.2

BF SF BF SAF

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0 2 4

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0.

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k_BT/J_B=1.

0.8

BF SF BF SAF

0.6

0.4

3 1

(b)

0 1 2 3 4

(a)

Figure 5. Variation of the ratio of the exchange interactions R¼ J

S

/J

B

as a function of R

_?

with (a) (

_S

¼ 0,

_B

¼ 0) and (b) (

_S

/J

_S

¼ 1,

_B

/J

_B

¼1.5). The number accompanying each curve denotes the value of temperature.

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Appendix

The coefficients A

i

(i ¼ 1 5), B

i

(i ¼ 1 6) and C

i

(i ¼ 1 3) in Equations (18)–(25) are defined by A

₁

¼ 8a

³_S

b

_S

a

?

F

_S

ð Þ x

_x¼0

,

A

₂

¼ 2a

⁴_S

b

_?

F

_S

ð Þ x

_x¼0

, A

3

¼ 48a

²_S

b

²_S

b

?

F

S

ð Þ x

_x¼0

, A

₄

¼ 32a

_S

b

³_S

a

?

F

_S

ð Þ x

_x¼0

, A

₅

¼ 32b

⁴_S

b

?

F

_S

ð Þ x

_x¼0

,

ðA1Þ

B

₁

¼ 2a

⁵_B

b

?

F

_B

ð Þ x

_x¼0

, B

₂

¼ 2a

?

a

⁴_B

b

_B

F

_B

ð Þ x

_x¼0

, B

₃

¼ 16a

³_B

b

?

b

²_B

F

_B

ð Þ x

_x¼0

, B

₄

¼ 16a

_?

a

²_B

b

³_B

F

_B

ð Þ x

_x¼0

, B

5

¼ 32a

B

b

?

b

⁴_B

F

B

ð Þ x

_x¼0

, B

₆

¼ 32a

?

b

⁵_B

F

_B

ð Þ x

_x¼0

,

ðA2Þ

C

₁

¼ 2a

⁵_B

b

_B

F

_B

ð Þ x

_x¼0

, C

2

¼ 16a

³_B

b

³_B

F

B

ð Þ x

_x¼0

, C

₃

¼ 32a

_B

b

⁵_B

F

_B

ð Þ x

_x¼0

:

ðA3Þ

Downloaded by [New York University] at 11:20 25 November 2013