Transverse Ising model with diluted surface and modified surface-bulk coupling

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Transverse Ising Model with Diluted Surface and Modified Surface-Bulk Coupling

T. Lahcini and N. Benayad

^∗

Groupe de Mecanique Statistique, Laboratoire de Physique Theorique, Faculte des Sciences, Universite Hassan II-Ain Chock,

B.P. 5366 Maarif, Casablanca, Morocco (Received June 7, 2008)

The transverse Ising system with a diluted surface and a modified surface-bulk interaction J

_⊥

is studied by the use of an effective field method within the framework of a single- site cluster theory. The surface antiferromagnetic exchange interaction J

S

is distributed randomly and has a concentration p. The new parameter p strongly affects the phase diagram obtained in the pure case (p = 1). Indeed, some qualitatively interesting features have been found, both in the absence and in the presence of the surface transverse field. In particular, we defined a critical value p

^∗

of p, separating two qualitatively different behaviors of the system.

PACS numbers: 75.10.Hk, 75.50.Ee, 75.70.Rf

I. INTRODUCTION

The magnetic order at surfaces is expected to be different from that of bulk materi- als, due to the different coordination number and symmetry of the atoms at the surface.

Indeed, several materials display a different magnetic behavior at the surface [1–3]. These effects have been modeled by a semi-infinite cubic Ising ferromagnet with an intrasurface exchange interaction J _S different from the bulk value J _B . Mean-field theory [4, 5], the cluster variation method [6], effective field theory [7], the finite cluster approximation [8, 9], renormalization group methods [9–13], Monte Carlo techniques [14, 15], and series ex- pansions [16] predict an enriched phase diagram. Indeed, it exhibits four different types of phase transitions associated with the surface. If the ratio R = J _S /J _B is greater than a critical value, R _C , the system may order on the surface at a temperature higher than the bulk, followed by the ordering of the bulk at the bulk transition temperature. These two successive transitions are called the surface and extraordinary transitions, respectively. If R is less than R _C , the system orders at the bulk transition temperature. This is the ordinary phase transition. If R = R _C , the system orders at the bulk transition temperature, but in this case the critical exponents differ from those of the ordinary transition. This is the special phase transition.

An Ising ferromagnet (J _B > 0) with an antiferromagnetic surface (J _S < 0) has also been studied using the mean-field approximation [16], the renormalization group method [17], and Monte Carlo simulation [18,19]. In this case the surface layer behaves roughly

http://PSROC.phys.ntu.edu.tw/cjp 238 c 2009 THE PHYSICAL SOCIETY

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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 in- teraction 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 antiferromag- netic instead. As the temperature increases, the bulk disorders, but 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.

There has been considerable interest in the study of disordered Ising models and their variants, because they have been used to describe many physical situations in different fields of physics. Special attention has been focused on the study of the influence of disorder on critical behavior. Recently, one of us (N.B.) has investigated the influence of the bond dilu- tion on the phase diagram of the three-dimensional transverse Ising system with competing surface and bulk exchange interactions [20]. It has been shown that the surface exhibits some characteristic behavior. In particular, the bond dilution has a significant effect on the surface ordering. In this latter investigation, the surface-bulk coupling has been identified with the bulk exchange interaction J _B .

In order to describe a more realistic situation, we propose, in the present work, to investigate the influence of the bond dilution on the surface phase transitions in the semi- infinite simple cubic transverse Ising ferromagnet with an antiferromagnetic surface and a modified surface-bulk coupling J

⊥

. To this end, we use an effective-field method within the framework of a single-site cluster theory. The state equations are derived using the four-layer approximation. Specially, we focus our attention on the effects of the surface average bond concentration on the phase transitions associated with the surface, both in the absence and in the presence of the surface and bulk transverse fields.

The paper is organized as follows. In Section II, 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 III, the phase diagrams of the system as a function of the dilution parameter are examined and discussed when the surface-bulk coupling and transverse fields are changed. Finally, we comment on our results in Section IV.

II. THEORETICAL FRAMEWORK

We consider a three-dimensional simple cubic transverse Ising ferromagnet with a diluted antiferromagnetic surface and a modified surface-bulk coupling. Such a system can be described by the following Hamiltonian:

H = X

<ij>

J _ij ^S s ^z _i s ^z _j − Ω S

X

i

s ^x _i − J

⊥

X

<jk>

s ^z _j s ^z _k − J B

X

<kl>

σ ^z _k σ ^z _l − Ω B

X

l

σ ^x _l , (1)

where σ ^α _i (α = x, z) is the α-component of the spin-1/2 operator at the site i. The first and

second summations are carried out over nearest-neighbor sites and single sites located on

the surface, respectively. The third summation runs over nearest-neighbor sites, one located

on the free surface and the other in the first layer. The fourth and fifth summations run

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over all pairs of remaining nearest-neighbor sites and single sites, respectively. Ω _B and Ω _S represent transverse fields in the bulk and at the surface, respectively. J _ij ^S and J B denote the exchange interactions at the surface and in the bulk, respectively. J

⊥

is the coupling constant between a spin in the surface and its nearest-neighbor in the next layer. J _ij ^S obeys the following probability distribution:

P (J _ij ^S ) = pδ(J _ij ^S − J _S ) + (1 − p)δ(J _ij ^S ), (2) with J _S > 0. The parameter p measures the average bond concentration of the surface.

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

⁰

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

H ₀ = −



 X

j

J _0j σ ^z _j



 σ ^z ₀ − Ω _θ σ ^x ₀ , (3) where θ ≡ S or B, depending on whether the lattice site 0 belongs to the surface or to the bulk, respectively. J _0j ≡ −J _0j ^S if both spins are at the surface layer, J _0j ≡ J

⊥

if one of them is located on the free surface, and J 0j ≡ J B otherwise.

To evaluate the longitudinal and transverse magnetisations for a given configuration of all interactions J _ij , we use a generalized, but approximate, Callen-Suzuki relation [21, 22]:

D O ˆ ₀ E

=

* trace ₀ h

O ˆ ₀ exp (−βH ₀ ) i trace 0 [exp (−βH 0 )]

+

, (4)

derived by S`a barreto et al. [23] for the transverse Ising model. Here, trace ₀ means the partial trace with respect to the site 0, h...i indicates the canonical thermal average and β = 1/k _B T . As pointed out by those authors, 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. We emphasize that in the limit Ω _θ = 0 the Hamiltonian contains only σ _i ^z . Then, relation (4) becomes an exact identity.

The application of (4) for the longitudinal and transverse site magnetisations of the n-th layer leads to the following expressions:

hσ _0n ^z i = A

2E ₀ tanh β

2 E ₀

, (5)

hσ _0n ^x i = Ω _θ

2E 0

tanh β

2 E ₀

, (6)

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where

A = X

j

J oj σ _j ^z , (7)

E 0 = h

(Ω _θ ) ² + A ² i 1/2

. (8)

Introducing the differential operator technique [24], Eqs. (5) and (6) can be written as follows:

hσ _0n ^z i = e ^A

^∇

F _θ (x)

x=0 , (9) hσ _0n ^x i =

e ^A

^∇

G _θ (x)

x=0 , (10) where ∇ = ∂/∂x is a differential operator (defined as e ^A

^∇

F (x) = F (x + A)) and the functions F _θ (x) and G _θ (x) are defined by

F _θ (x) = ^x

2 [ ^(Ω

θ

)

²

+x

²

]

^1/2

× tanh

β 2

h

(Ω _θ ) ² + x ² i 1/2

, (11)

G _θ (x) = ^Ω

²^θ

2 [ (Ω

θ

)

²

+x

²

]

^1/2

× tanh

β 2

h (Ω _θ ) ² + x ² i 1/2

, (12)

By assuming the statistical independence of the lattice sites, that is hσ _i σ _j ...σ _l i = hσ _i i hσ _j i ... hσ _l i, and using the spin-1/2 identity exp (λσ _i ) = cosh(λ/2) + 2σ _i sinh (λ/2), Eqs. (9) and (10) may be written as follows:

hσ _0n ^z i = Y

j

cosh

J _0j 2 ∇

+ 2

σ ^z _j sinh

J _0j 2 ∇

F _θ (x)

x=0

, (13)

hσ _0n ^x i = Y

j

cosh

J _0j 2 ∇

+ 2

σ ^z _j sinh

J _0j 2 ∇

G _θ (x)

x=0

. (14)

Since the surface exchange interaction is randomly distributed, we have to perform the random average of J _0j ^S according to the probability distribution function P

J _ij ^S

given by Eq. (2). The ordering parameters at the surface are then defined by µ ^z _S =

σ ^z _0S and µ ^x _S =

σ ^x _0S

, where the bar denotes the average over disorder. Denoting by µ ^z _n = hσ _0n ^z i and µ ^x _n = hσ ^x _0n i, the ordering parameters of the n-th layer (n ≥ 1), we obtain

µ ^z _n =

q

Y

j=1

a _j + 2µ ^z _j b _j F _θ (x)

x=0

, (15)

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where q is the number of nearest neighbors of site 0, and the coefficients a _j and b _j are given by

a _S = p cosh( ^J ₂

^S

∇) + (1 − p), b _S = −p sinh( ^J ₂

^S

∇),

a

⊥

= cosh( ^J ₂

^⊥

∇), b

⊥

= sinh( ^J ₂

^⊥

∇), a _B = cosh( ^J ₂

^B

∇), b _B = sinh( ^J ₂

^B

∇).

It is worth mentioning that this approximation, in spite of its simplicity since it neglects corrections between different sites, leads to quite satisfactory results. Eqs. (15) take the form:

-for the surface (n ≡ S)

u _S = [a _S + 2v _S b _S ] ⁴ × [a

⊥

+ 2v ₁ b

⊥

] F _S (x)| _x=0 , (16) v _S = [a _S + 2u _S b _S ] ⁴ × [a

⊥

+ 2u ₁ b

⊥

] F _S (x)| _x=0 ; (17) -for the first layer (n = 1)

u ₁ = [a

⊥

+ 2v _S b

⊥

] [a _B + 2v ₁ b _B ] ⁴ × [a _B + 2v ₂ b _B ] F _B (x)| _x=0 , (18) v ₁ = [a

⊥

+ 2u _S b

⊥

] [a _B + 2u ₁ b _B ] ⁴ × [a _B + 2u ₂ b _B ] F _B (x)| _x=0 ; (19) -for any layer n ≥ 2

u _n = [a _B + 2v _n

−

1 b _B ] [a _B + 2v _n b _B ] ⁴ × [a _B + 2v _n+1 b _B ] F _B (x)| _x=0 , (20) v _n = [a _B + 2u _n

−

1 b _B ] [a _B + 2u _n b _B ] ⁴ × [a _B + 2u _n+1 b _B ] F _B (x)| _x=0 . (21) Here u n and v n are the two sublattice longitudinal magnetisations of the n-th layer, respec- tively. Eqs. (16)–(21) can be written in the following form:

-for the surface (n ≡ S)

u S = A 1 v S + A 2 v 1 + A 3 v ² _S v 1 + A 4 v _S ³ + A 5 v ⁴ _S v 1 ,

v _S = A ₁ u _S + A ₂ u ₁ + A ₃ u ² _S u ₁ + A ₄ u ³ _S + A ₅ u ⁴ _S u ₁ ; (22) -for the first layer (n = 1)

u ₁ = B ₁ v _S + B ₂ (4v ₁ + v ₂ ) + B ₃ v _S v ₁ (2v ₂ + 3v ₁ ) + B ₄ v ₁ ² (2v ₁ + 3v ₂ ) + B ₅ v _S v ₁ ³ (v ₁ + 4v ₂ ) + B ₆ v ⁴ ₁ v ₂ , v ₁ = B ₁ u _S + B ₂ (4u ₁ + u ₂ ) + B ₃ u _S u ₁ (2u ₂ + 3u ₁ ) +

B ₄ u ² ₁ (2u ₁ + 3u ₂ ) + B ₅ u _S u ³ ₁ (u ₁ + 4u ₂ ) + B ₆ u ⁴ ₁ u ₂ ;

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-for any layer n ≥ 2

u _n = C ₁ (v _n

−

1 + 4v _n + v _n+1 ) +

C ₂ v _n (2v _n

−

1 v _n+1 + v _n (3v _n

−

1 + 2v _n + 3v _n+1 )) + C ₃ v _n ³ (4v _n

−

1 v _n+1 + v _n

−

1 v _n + v _n v _n+1 ) ,

v _n = C ₁ (u _n

−

1 + 4u _n + u _n+1 ) +

C ₂ u _n (2u _n

−

1 u _n+1 + u _n (3u _n

−

1 + 2u _n + 3u _n+1 )) + C ₃ u ³ _n (4u _n

−

1 u _n+1 + u _n

−

1 u _n + u _n u _n+1 ) .

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The coefficients A _i (i = 1–5), B _i (i = 1–6) and C _i (i = 1–3) are listed in the Appendix.

The bulk longitudinal magnetization u _B is determined by setting u _n

−

1 = u _n = u _n+1 (v _n

−

1 = v _n = v _n+1 ) in Eq. (24). Thus, u _B is the solution of the equation

u _B = 6C ₁ u _B + 10C ₂ (u _B ) ³ + 6C ₃ (u _B ) ⁵ . (25)

III. PHASE DIAGRAMS AND DISCUSSION

In order to investigate the system described by the Hamiltonian (1), especially the influence of the disorder on surface transitions, we have to solve the coupled Eqs. (22)–(24).

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 magnetisations u _n and v _n should approach the bulk values u _B . For this purpose, let us assume that the magnetisations remain unaltered after the third layer, that is

u ₃ = u ₄ = ... = u _B v ₃ = v ₄ = ... = v _B , which may be called the four-layer approximation.

Using Eq. (25), the bulk order-disorder critical temperature k _B T _C /J _B , as a function of Ω _B , is determined by the equation

1 = 6C ₁ . (26)

In zero transverse field (Ω _B = 0), comparing with the Monte Carlo value 1.128 [25], one notes that the obtained result 1.268 improves the mean-field value 1.500. One should also note that the critical transverse field Ω _c (T _C = 0) is 2.352J _B , to be compared with the mean-field result 3J _B .

III-1. Effects of the dilution parameter on the surface order-disorder transition temperature

In order to obtain the surface order-disorder critical temperature, we have to lin- earize Eqs. (22)–(24). Neglecting higher-order terms in the magnetization near the critical temperature and within the four-layer approximation, we obtain

u S = A 1 v S + A 2 v 1 ,

v _S = A ₁ u _S + A ₂ u ₁ , (27)

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u 1 = B 1 v S + B 2 (4v 1 + v 2 ) ,

v ₁ = B ₁ u _S + B ₂ (4u ₁ + u ₂ ) , (28)

u ₂ = C ₁ (v ₁ + 4v ₂ + v _B ) ,

v 2 = C 1 (u 1 + 4u 2 + u B ) . (29)

When the bulk is disordered (u _B = v _B = 0), the surface order-disorder critical temperature is analytically obtained through a determinantal equation. In Figs. 1, we represent, in the (k _B T /J _B , J _S /J _B ) plane, the effects of the dilution parameter p on the critical line of antiferromagnetic-paramagnetic surface transition (S). For given values of the transverse fields, we note that for a ratio R = J _S /J _B greater than a critical value R _C (when it exists), the surface may antiferromagnetically order at a temperature higher than the bulk critical temperature. As clearly seen in Figs. 1 corresponding to different values of R

⊥

, the surface transition lines (S) seem to be not very sensitive to the ratio R

⊥

= J

⊥

/J B . Moreover, as can be expected, we point out that the domain where the surface is antiferromagnetic (when the bulk is disordered) becomes less and less large when Ω _S increases or/and p decreases. One can note that, in addition to its dependence on R

⊥

and Ω _S , the critical ratio R _C = (J _S /J _B ) _C depends also on p. These dependencies are plotted in Figs. 2. As seen from these figures, and for given values of R

⊥

and Ω _S , R _C varies weakly with p for p close to 1; but it becomes very sensitive to p when it approaches its (R

⊥

, Ω _S )-dependent critical value p

^∗

. For selected values of R

⊥

and Ω _S in Fig. 2, the values of p

^∗

are indicated in Table I.

TABLE I: p

^∗

values for selected values of R

_⊥

and Ω

S

in Fig. 2.

Ω

B

/J

B

= 0

Ω

S

/J

S

0 0.5 1

R

_⊥

1 3 5 1 3 5 1 3 5

p

^∗

0.424 0.392 0.338 0.577 0.557 0.516 0.815 0.791 0.743 Ω

B

/J

B

= 1.5

Ω

S

/J

S

0 0.5 1

R

_⊥

1 3 5 1 3 5 1 3 5

p

^∗

0.423 0.387 0.329 0.576 0.555 0.513 0.814 0.790 0.741

III-2. Influence of the dilution parameter on the ferromagnetic- antiferromagnetic surface transition

Note that, for the pure system (p = 1), any two nearest neighbors on the surface inter-

act via an antiferromagnetic coupling. In the absence of the transverse fields (Ω _S = Ω _B =0)

and at the ground state of the Hamiltonian (1), the surface makes a first-order transition

from an antiferromagnetically ordered state for R > ^R ₄

^⊥

to a ferromagnetically ordered

state for R < ^R ₄

^⊥

. In order to obtain the remaining phases and transitions, when the bulk

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0 2 4 6 0.0

1.0 2.0 3.0 4.0 5.0

R_⊥= 0.5 Ω

Ω Ω

0 2 4 6

0.0 1.0 2.0 3.0 4.0 5.0

R_⊥= 2.

Ω

Ω Ω

0 2 4 6

0.0 1.0 2.0 3.0 4.0 5.0

R_⊥= 1.

Ω Ω Ω

FIG. 1: Phase diagram of the diluted semi-infinite transverse Ising model with a modified surface

bulk coupling for Ω

B

=0, with Ω

S

/J

S

=0 (solid lines) and Ω

S

/J

S

=0.5 (dashed lines), when (a)

R

_⊥

=0.5, (b) R

_⊥

=1, and (c) R

_⊥

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

p. The phase boundaries terminate at a critical value (J

S

J

B

)

C

, when p > p

^∗

.

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Ω Ω

⊥

Ω Ω

⊥

FIG. 2: Dependence of the critical ratio R

C

of the exchange interactions (J

S

/J

B

)

C

as a function of the surface dilution parameter p for R

_⊥

=1 (solid lines), R

_⊥

=3 (dotted lines), and R

_⊥

=5 (dashed lines), with (a) Ω

B

/J

B

=0 and (b) Ω

B

/J

B

=1.5. The number accompanying each curve denotes the value of Ω

S

/J

S

.

is ordered (u _B 6=0, v _B 6=0), we must solve numerically the coupled Eqs. (22)–(24) within the four-layer approximation scheme. The effects of the dilution parameter on surface behav- iors (when the bulk is ordered), for given values of the surface-bulk coupling and transverse fields, are represented in Figs. 1. In addition to the previous phases, SP, BP : surface and bulk paramagnetic, and SAF, BP : surface antiferromagnetic and bulk paramagnetic, the analysis of Eqs. (22)–(24) identifies two other phases:

SAF, BF : surface antiferromagnetic and bulk ferromagnetic, and SF, BF : surface and bulk ferromagnetic.

As is shown in Fig. 1, the above phases are separated by different transition lines.

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TABLE II: p

^∗

and p

^c

values for selected values of R

_⊥

and Ω

S

in Fig. 1.

Ω

B

/J

B

= 0

Ω

S

/J

S

0 0.5

R

_⊥

0.5 1 2 0.5 1 2

p

^∗

0.4273 0.4242 0.4116 0.5771 0.5768 0.5757

p

^c

0.3929 0.3927 0.3922 0.5753 0.5751 0.5748

Using the accepted terminology used for semi-infinite Ising models [26, 27, 28], these transitions correspond to the surface (S), the ordinary (O), the extraordinary (E), the special (Sp), and (L) transitions. These phase boundaries terminate at a critical ratio R _C = (J _S /J _B ) _C , when it exists. At the (L) transition, the surface exhibits, at finite temperature, a second-order transition from the ferromagnetic state (SF, BF ) to the an- tiferromagnetic state (SAF, BF ). Note that such a transition (L) has been found in Ising models with competing surface and bulk interactions [16, 17, 24, 27].

Let us now investigate the influence of the dilution parameter p on the transition line (L). In Figs. 1, plotted for selected values of R

⊥

and Ω _S , we show that p has a significant effect on the transition line (L). For given values of R

⊥

and Ω _S , the domain of the phase BF SAF becomes less and less large with decreasing values of the dilution parameter p, and disappears when p is less than a (R

⊥

, Ω _S )-dependent critical value p ^c (for selected values of R

⊥

and Ω _S in Fig. 1, the values of p ^c are indicated in Table II). This means that for p < p ^c the bulk promotes its order to the surface in such a way that the bulk and the surface are both paramagnetic or ferromagnetic. On the other hand, as mentioned in Section 3.1, the critical value p

^∗

of p indicates two qualitatively different behaviors of the surface, which depend on the range of p. Thus, for 1> p > p

^∗

, the critical ratio (J S /J B ) C exists, and beyond this value the surface exhibits a surface transition at a temperature higher than the bulk. For p

^∗

> p > p ^c , there is no critical ratio of the surface and bulk exchange interactions. This means that there is no surface, extraordinary, and special transitions;

and therefore the phase SAF BP disappears at a (R

⊥

, Ω _S )-dependent critical value p

^∗

of the surface dilution parameter (for selected values of R

⊥

and Ω _S in Fig. 1, the values of p

^∗

are indicated in Table II). In this case, the system exhibits only two kinds of transitions, namely the ordinary (O) and (L) transitions. As can be expected, the critical value p

^∗

depends on the surface-bulk coupling and the surface transverse field. Its variation with R

⊥

for selected values of Ω S is plotted in Fig. 3. From this figure, we note that the location of p

^∗

does not depend essentially on the bulk transverse field.

Furthermore, as we can point out from Figs. 1, the location of the transition (L)

depends qualitatively and quantitatively on the strength of R

⊥

and p, especially at very

low temperatures. In order to clarify this situation, we plot in Fig. 4 the phase diagram in

the R

⊥

− R plane at different temperatures less than the bulk critical temperature.

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0 10 20 30 0.0

0.2 0.4 0.6

⊥ Ω Ω

Ω Ω

FIG. 3: Variation of the critical value p

^∗

as a function of the ratio R

_⊥

= J

_⊥

/J

B

, when Ω

B

= 0 (solid lines) and Ω

B

/J

B

= 1 (dashed lines). The number accompanying each curve denotes the value of Ω

S

/J

S

.

IV. CONCLUSIONS

In this work, we have studied the effects of the surface bond dilution on the phase diagram of the semi-infinite simple-cubic spin-1/2 transverse Ising ferromagnet with an antiferromagnetic interaction on the surface and a modified surface-bulk coupling J

⊥

. The surface exchange interaction J _S is distributed randomly and has a concentration p. 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 coupled state equations using the differential operator technique and the four-layer approximation.

Let us summarize by stating the main results of this investigation. We found that the surface dilution parameter p has a qualitative and quantitative influence on the surface behavior. Thus, some characteristic behavior for the surface magnetism has been found.

In particular, for a given value of the surface transverse field, we defined a J

⊥

-dependent

critical value p

^∗

of p separating two qualitatively different behaviors of the system depending

on the range of p. For p greater than p

^∗

, the critical ratio R C = (J S /J B ) C exists and the

system exhibits the five different phase transitions cited in Section 3; among them the

surface order-disorder transition at a temperature higher than the bulk. However, for p less

than p

^∗

, R C does not exist, which means that the surface undergoes only (i) the ordinary

transition at the bulk transition temperature and (ii) a second-order transition from the

antiferromagnetic state to the ferromagnetic state. We have to note that, for given values

of R

⊥

= J

⊥

/J B and the surface transverse field Ω S /J S , the location of this latter transition

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0 1 2 3 4 0.0

0.5 1.0 1.5 2.0 2.5

⊥

Ω Ω

FIG. 4: Dependence of the ratio of the surface and bulk exchange interactions (R = J

S

/J

B

) as a function of R

_⊥

, with Ω

B

= Ω

S

=0 when p =0.8. The number accompanying each curve denotes the value of the temperature, T /J

B

.

depends essentially on the values of the ratio of surface and bulk interactions R and the surface dilution parameter p.

V. APPENDIX

The coefficients A _i (i = 1–5), B _i (i = 1–6) and C _i (i = 1–3) in Eqs. (22)–(29) are defined by

A ₁ = 8a ³ _S b _S a

⊥

F _S (x) x=0 , A ₂ = 2a ⁴ _S b

⊥

F _S (x)

x=0 , A ₃ = 48a ² _S b ² _S b

⊥

F _S (x)

x=0 , A ₄ = 32a _S b ³ _S a

⊥

F _S (x)

x=0 , A 5 = 32b ⁴ _S b

⊥

F S (x)

x=0 ,

B ₁ = 2a ⁵ _B b

⊥

F _B (x) x=0 , B ₂ = 2a

⊥

a ⁴ _B b _B F _B (x)

x=0 , B ₃ = 16a ³ _B b

⊥

b ² _B F _B (x)

x=0 , B ₄ = 16a

⊥

a ² _B b ³ _B F _B (x)

x=0 , B 5 = 32a B b

⊥

b ⁴ _B F B (x)

x=0 , B 6 = 32a

⊥

b ⁵ _B F B (x)

x=0 ,

(13)

C ₁ = 2a ⁵ _B b _B F _B (x) x=0 , C ₂ = 16a ³ _B b ³ _B F _B (x)

x=0 , C ₃ = 32 a _B b ⁵ _B F _B (x)

x=0 .

References

∗

Electronic address: noureddine_benayad@yahoo.fr

[1] K. Binder, in Phase Transitions and Critical Phenomena, Vol. 8, eds. C. Domb, J. L. Lebowitz, (Academic Press, New York, 1983), p.

[2] H. W. Diehl, in Phase Transitions and Critical Phenomena, Vol. 10, eds. C. Domb, J. L.

Lebowitz, (Academic Press, New York, 1986), p.

[3] T. Kaneyoshi, Introduction to Surface Magnetism, (CRC Press, Boca Raton, FL, USA, 1991).

[4] D. L. Mills, Phys. Rev. B 3, 3887 (1971).

[5] T. C. Lubensky and M. H. Rubin, Phys. Rev. B 12, 3885 (1975); B 11, 4533 (1975).

[6] J. M. Sanchez and J. L. Moran-Lopez, in Magnetic Properties of Low Dimensional Systems, eds. L. M. Flicov and J. L. Moran-Lopez (Springer, Berlin, 1986). J. L. Moran-Lopez and J.

Transverse Ising model with diluted surface and modified surface-bulk coupling

Transverse Ising Model with Diluted Surface and Modified Surface-Bulk Coupling

T. Lahcini and N. Benayad

Groupe de Mecanique Statistique, Laboratoire de Physique Theorique, Faculte des Sciences, Universite Hassan II-Ain Chock,

B.P. 5366 Maarif, Casablanca, Morocco (Received June 7, 2008)

The transverse Ising system with a diluted surface and a modified surface-bulk interaction J

is studied by the use of an effective field method within the framework of a single- site cluster theory. The surface antiferromagnetic exchange interaction J

of p, separating two qualitatively different behaviors of the system.

PACS numbers: 75.10.Hk, 75.50.Ee, 75.70.Rf

I. INTRODUCTION

The magnetic order at surfaces is expected to be different from that of bulk materi- als, due to the different coordination number and symmetry of the atoms at the surface.

An Ising ferromagnet (J B > 0) with an antiferromagnetic surface (J S < 0) has also been studied using the mean-field approximation [16], the renormalization group method [17], and Monte Carlo simulation [18,19]. In this case the surface layer behaves roughly

http://PSROC.phys.ntu.edu.tw/cjp 238 c 2009 THE PHYSICAL SOCIETY

OF THE REPUBLIC OF CHINA

II. THEORETICAL FRAMEWORK

We consider a three-dimensional simple cubic transverse Ising ferromagnet with a diluted antiferromagnetic surface and a modified surface-bulk coupling. Such a system can be described by the following Hamiltonian:

H = X

<ij>

J ij S s z i s z j − Ω S

X

i

s x i − J

X

<jk>

s z j s z k − J B

X

<kl>

σ z k σ z l − Ω B

X

l

σ x l , (1)

where σ α i (α = x, z) is the α-component of the spin-1/2 operator at the site i. The first and

second summations are carried out over nearest-neighbor sites and single sites located on

the surface, respectively. The third summation runs over nearest-neighbor 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-neighbor sites and single sites, respectively. Ω B and Ω S represent transverse fields in the bulk and at the surface, respectively. J ij S and J B denote the exchange interactions at the surface and in the bulk, respectively. J

is the coupling constant between a spin in the surface and its nearest-neighbor in the next layer. J ij S obeys the following probability distribution:

P (J ij S ) = pδ(J ij S − J S ) + (1 − p)δ(J ij S ), (2) with J S > 0. The parameter p measures the average bond concentration of the surface.

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

H 0 = −



 X

j

J 0j σ z j



 σ z 0 − Ω θ σ x 0 , (3) where θ ≡ S or B, depending on whether the lattice site 0 belongs to the surface or to the bulk, respectively. J 0j ≡ −J 0j S if both spins are at the surface layer, J 0j ≡ J

if one of them is located on the free surface, and J 0j ≡ J B otherwise.

To evaluate the longitudinal and transverse magnetisations for a given configuration of all interactions J ij , we use a generalized, but approximate, Callen-Suzuki relation [21, 22]:

D O ˆ 0 E

=

* trace 0 h

O ˆ 0 exp (−βH 0 ) i trace 0 [exp (−βH 0 )]

+

, (4)

derived by S`a barreto et al. [23] for the transverse Ising model. Here, trace 0 means the partial trace with respect to the site 0, h...i indicates the canonical thermal average and β = 1/k B T . As pointed out by those authors, this relation is not exact since H 0 and H

do not commute. Nevertheless, it has been successfully applied to a number of interesting transverse Ising systems. We emphasize that in the limit Ω θ = 0 the Hamiltonian contains only σ i z . Then, relation (4) becomes an exact identity.

The application of (4) for the longitudinal and transverse site magnetisations of the n-th layer leads to the following expressions:

hσ 0n z i = A

2E 0 tanh β

2 E 0

, (5)

hσ 0n x i = Ω θ

2E 0

tanh β

2 E 0

, (6)

where

A = X

j

J oj σ j z , (7)

E 0 = h

(Ω θ ) 2 + A 2 i 1/2

. (8)

Introducing the differential operator technique [24], Eqs. (5) and (6) can be written as follows:

hσ 0n z i = e A

F θ (x)

x=0 , (9) hσ 0n x i =

e A

G θ (x)

x=0 , (10) where ∇ = ∂/∂x is a differential operator (defined as e A

An Ising ferromagnet (J _B > 0) with an antiferromagnetic surface (J _S < 0) has also been studied using the mean-field approximation [16], the renormalization group method [17], and Monte Carlo simulation [18,19]. In this case the surface layer behaves roughly

J _ij ^S s ^z _i s ^z _j − Ω S

s ^x _i − J

s ^z _j s ^z _k − J B

σ ^z _k σ ^z _l − Ω B

σ ^x _l , (1)

where σ ^α _i (α = x, z) is the α-component of the spin-1/2 operator at the site i. The first and

over all pairs of remaining nearest-neighbor sites and single sites, respectively. Ω _B and Ω _S represent transverse fields in the bulk and at the surface, respectively. J _ij ^S and J B denote the exchange interactions at the surface and in the bulk, respectively. J

is the coupling constant between a spin in the surface and its nearest-neighbor in the next layer. J _ij ^S obeys the following probability distribution:

P (J _ij ^S ) = pδ(J _ij ^S − J _S ) + (1 − p)δ(J _ij ^S ), (2) with J _S > 0. The parameter p measures the average bond concentration of the surface.

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

H ₀ = −

J _0j σ ^z _j

 σ ^z ₀ − Ω _θ σ ^x ₀ , (3) where θ ≡ S or B, depending on whether the lattice site 0 belongs to the surface or to the bulk, respectively. J _0j ≡ −J _0j ^S if both spins are at the surface layer, J _0j ≡ J

To evaluate the longitudinal and transverse magnetisations for a given configuration of all interactions J _ij , we use a generalized, but approximate, Callen-Suzuki relation [21, 22]:

D O ˆ ₀ E

* trace ₀ h

O ˆ ₀ exp (−βH ₀ ) i trace 0 [exp (−βH 0 )]

derived by S`a barreto et al. [23] for the transverse Ising model. Here, trace ₀ means the partial trace with respect to the site 0, h...i indicates the canonical thermal average and β = 1/k _B T . As pointed out by those authors, this relation is not exact since H ₀ and H

do not commute. Nevertheless, it has been successfully applied to a number of interesting transverse Ising systems. We emphasize that in the limit Ω _θ = 0 the Hamiltonian contains only σ _i ^z . Then, relation (4) becomes an exact identity.

hσ _0n ^z i = A

2E ₀ tanh β

2 E ₀

hσ _0n ^x i = Ω _θ

2 E ₀

J oj σ _j ^z , (7)

(Ω _θ ) ² + A ² i 1/2

hσ _0n ^z i = e ^A

F _θ (x)

x=0 , (9) hσ _0n ^x i =

e ^A

G _θ (x)

x=0 , (10) where ∇ = ∂/∂x is a differential operator (defined as e ^A

F (x) = F (x + A)) and the functions F _θ (x) and G _θ (x) are defined by

F _θ (x) = ^x

2 [ ^(Ω

(Ω _θ ) ² + x ² i 1/2

G _θ (x) = ^Ω

h (Ω _θ ) ² + x ² i 1/2

By assuming the statistical independence of the lattice sites, that is hσ _i σ _j ...σ _l i = hσ _i i hσ _j i ... hσ _l i, and using the spin-1/2 identity exp (λσ _i ) = cosh(λ/2) + 2σ _i sinh (λ/2), Eqs. (9) and (10) may be written as follows:

hσ _0n ^z i = Y

J _0j 2 ∇

σ ^z _j sinh

J _0j 2 ∇

F _θ (x)

hσ _0n ^x i = Y

J _0j 2 ∇

σ ^z _j sinh

J _0j 2 ∇

G _θ (x)

Since the surface exchange interaction is randomly distributed, we have to perform the random average of J _0j ^S according to the probability distribution function P

J _ij ^S

given by Eq. (2). The ordering parameters at the surface are then defined by µ ^z _S =

σ ^z _0S and µ ^x _S =

σ ^x _0S

, where the bar denotes the average over disorder. Denoting by µ ^z _n = hσ _0n ^z i and µ ^x _n = hσ ^x _0n i, the ordering parameters of the n-th layer (n ≥ 1), we obtain

µ ^z _n =

a _j + 2µ ^z _j b _j F _θ (x)

where q is the number of nearest neighbors of site 0, and the coefficients a _j and b _j are given by

a _S = p cosh( ^J ₂

∇) + (1 − p), b _S = −p sinh( ^J ₂

= cosh( ^J ₂