Surface phase diagrams of a random transverse Ising model

P ^HYSICAL J ^OURNAL B

c EDP Sciences

Societ`a Italiana di Fisica Springer-Verlag 2001

Surface phase diagrams of a random transverse Ising model

A. Dakhama, A. Fathi, and N. Benayad

^a

Groupe de M´ecanique Statistique, Laboratoire de Physique Th´eorique, Facult´e des Sciences, Universit´e Hassan II-A¨ın Chock, BP 5366 Mˆaarif, Casablanca, Morocco

Received 6 September 2000 and Received in final form 23 February 2001

Abstract. The effect of surface on the critical behaviour of a ferromagnetic Ising spin model in a random transverse surface and bulk fields is studied by the use of an effective field method within the framework of a single-site cluster theory. The state equations are derived using a probability distribution method based on the use of Van der Waerden identities. The complete phase diagrams are investigated when the transverse fields are bimodally and trimodally distributed. In particular, the influence of the transverse fields, exchange interaction strength ratio and the distribution parameter on the phase diagrams behaviour is examined in detail.

PACS. 05.50.+q Lattice theory and statistics (Ising, Potts, etc.) – 75.10.Hk Classical spin models – 75.50.Gg Ferrimagnetics

1 Introduction

Over recent few decades, there has been considerable in- terest in the theoretical study of the effect of quantum fluctuations in classical spin models. The simplest of such systems is the Ising model in a transverse field. The two state transverse Ising model was originally introduced by De Genne [1] as a valuable model for the tunnelling of the proton in hydrogen-bonded ferroelectrics [2] type. Since then, it has been successfully applied to several physical systems, such as cooperative Jahn-Teller systems [3], or- dering in rare earth compounds with a singlet crystal- field ground state [4] and also to some real magnetic materials with strong uniaxial anisotropy in a transverse field [5]. It has been extensively studied by the use of vari- ous techniques [6–10], including the effective field treat- ment [11, 12] based on a generalized but approximated Callen-Suzuki relation derived by S` a Barreto, Fittipaldi and Zeks. The system shows a phase transition somewhat different from the usual Ising model case. In two or more dimensions it presents a finite-temperature phase tran- sition which can be depressed to zero by increasing the transverse field to a certain critical value.

On the other hand, the problems of surface phase tran- sitions have been investigated for many years, by using a variety of approximations and mathematical techniques.

In particular, the semi-infinite simple cubic spin-1/2 Ising ferromagnet having a free surface has received much atten- tion and has been studied by a various techniques, includ- ing a mean field approximation [13], various effective-field theories [14, 15], series expansion [16], a renormalization group approach [17] and Monte Carlo studies [18, 19].

Many reviews have been written on the subject [20–26].

a e-mail:benayad@facsc-achok.ac.ma

The standard example is the semi-infinite cubic ferromag- netic Ising model, in which the spins on the surface in- teract among one another with an exchange parameter J

S

different from the bulk exchange J

B

. It exhibits dif- ferent types of phase transitions associated with the sur- face; if the ratio R =

_J^JS

B

is greater than a critical value R

c

, the system may order on the surface before it orders in the bulk. The system exhibits two successive transi- tions, namely the surface and bulk phase transitions, as the temperature is lowered. If R

≤

R

c

, the system be- comes ordered at the bulk transition temperature. While the problem of surface phase transitions in classical spin systems seems to be well understood, the corresponding problem in quantum systems is far from revealing all their secrets.

A problem of growing interest is associated with the

random transverse field Ising model (RTFIM). Special at-

tention has been devoted to bimodal (two peaks) and

trimodal distributions for the transverse field. This infi-

nite model has been investigated using different approx-

imate schemes, such as mean-field and mean-field renor-

malization group (MFRG) [27], a method of combining

the MFRG with the discretized path-integral representa-

tion (DPIR) [28–30] and an approach combining the pair

approximation with DPIR [31]. These investigations pre-

dicted a discontinuity in the phase diagram at T = 0, be-

tween the bimodal and the trimodal random distributions

of the transverse field. However, using Suziki-Trotter for-

mula [32], Yokota [33] gave arguments which show that the

above-mentioned discontinuity at the ground state seems

to be an artifact of the mean-field-like approximation. We

point out that all transition lines are of second order and

the directional randomness of the transverse field does not

change the critical behaviour [33] of the system.

In this paper, we are interested in the study of surface phase transitions in the semi-infinite random transverse field Ising model. The purpose is to describe the phase diagrams of the system by the use of an effective field method within the framework of a single-site cluster the- ory. We focus our attention on the effect of the random transverse field on the phase transitions associated with the surface. To our own knowledge it has not been inves- tigated.

The outline of our paper is as follows. In Section 2, we review the basic points of the effective-field theory with correlations (EFT) when it is applied to the present model. In Section 3, the phase diagrams of the system as a function of the random transverse fields are examined and discussed. Finally, we comment on our results in Section 4.

2 Theoretical framework

The semi-infinite random transverse field Ising model is described by the following Hamiltonian

H =

−

J

S

X

hiji

σ

^z_i

σ

^z_j −X

i

(Ω

S

)

i

σ

_i^x

−

J

B

X

hlki

σ

^z_l

σ

^z_k−X

l

(Ω

B

)

l

σ

_l^x

, (1) where σ

^α_i

(α = x, z) is the α-component of spin-1/2 op- erator at site i. The first and second summations are carried out over nearest-neighbour sites and single sites located on the free surface, respectively. The third and fourth summations run over all pairs of remaining nearest- neighbour sites and single sites, respectively. J

S

and J

B

denote the nearest-neighbour exchange interactions at the surface and in the bulk respectively. The transverse fields (Ω

θ

)

i

(θ

≡

S or B) are assumed to be independent vari- ables and obey the trimodal probability distribution

Q((Ω

θ

)

i

) = P

θ

δ((Ω

θ

)

i

) + (1

−

P

θ

) 2

×

[δ((Ω

θ

)

i−

Ω

θ

) + δ((Ω

θ

)

i

+ Ω

θ

)] , (2) where the parameter P

θ

measures the fraction of spins not exposed to the transverse field Ω

θ

. At P

θ

= 1 or Ω

θ

= 0, (for θ

≡

S and θ

≡

B), the system reduces to the simplest semi-infinite Ising model.

The theoretical framework we adopt in the study of the semi-infinite random transverse Ising model (SIRTIM) de- scribed by the Hamiltonian (1), is the effective-field theory based on a single-site cluster theory. In this approach, at- tention is focused on a cluster consisting of just a single selected spin, labelled o, and the neighbouring spins with which it directly interacts. To this end, the total Hamil- tonian is split in two parts, H = H

o

+ H

⁰

, where H

o

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

H

o

=

−



X

j

J

oj

σ

j^z





σ

^zo−

(Ω

θ

)

o

σ

^xo

, (3)

with J

oj ≡

J

S

if both spins are at the surface layer and J

oj≡

J

B

otherwise.

The problem consists in evaluating the longitudinal and transverse site magnetisations for the Ising model in a random transverse field. The starting point of our calcu- lation is a generalized, but approximated, Callen-Suzuki relation [34, 35]

D

O ˆ

o

E

=

*

Tr

o

[ ˆ O

o

exp(

−

βH

o

)]

Tr

o

[exp(

−

βH

o

)]

+

, (4)

derived by S` a Barreto

et al.

[11] for the transverse Ising model. Tr

o

means the partial trace with respect to the site o. As pointed out by those authors, this relation is not exact since H

o

and H

⁰

do not commute. Nevertheless, they have been successfully applied to a number of interesting transverse Ising systems. We emphasize that in the limit ((Ω

θ

)

i

= 0,

∀

i), the Hamiltonian contains only σ

^z_i

. Then, relation (4) becomes exact identity. The application of (4) for the longitudinal and transverse site magnetizations of the nth layer, leads to the following expressions

h

σ

^z_oni

=

A

2E

o

tanh β

2 E

o

, (5)

h

σ

^x_oni

=

(Ω

θ

)

o

2E

o

tanh β

2 E

o

, (6)

with

A =

X

j

J

oj

σ

_j^z

, (7)

E

o

=

((Ω

θ

)

o

)

²

+ A

^21/2

. (8)

Here,

h

...

i

indicates the canonical thermal average and β = 1/k

B

T .

Introducing the differential operator technique [36], equations (5) and (6) can be written as follows

h

σ

^zoni

= e

^A∇

F

θ

(x)

|^x=0

, (9)

h

σ

_on^x i

=

e

^A∇

G

θ

(x)

|^x=0

, (10) where

∇

= ∂/∂x is a differential operator (defined by e

^A∇

F(x) = F (x + A)) and functions F

θ

(x) and G

θ

(x) are defined by

F

θ

(x) = x

2 [((Ω

θ

)

o

)

²

+ x

²

]

^1/2

×

tanh β

2 [((Ω

θ

)

_o

)

²

+ x

²

]

^1/2

, (11)

G

θ

(x) = ((Ω

θ

)

o

)

²

2 [((Ω

θ

)

o

)

²

+ x

²

]

^1/2

×

tanh β

2 [((Ω

θ

)

o

)

²

+ x

²

]

^1/2

. (12)

By assuming the statistical independence of the lattice sites, that is

h

σ

i

σ

j

...σ

li ' h

σ

ii h

σ

ji

...

h

σ

li

and using the spin-1/2 identity exp(λσ

i

) = cosh(λ) + σ

i

sinh(λ), equa- tions (9) and (10) may be written as follows

h

σ

_on^z i

=

Y

j

cosh(J

oj∇

) + σ

_j^z

sinh(J

oj∇

)

×

F

θ

(x)

|x=0

, (13)

h

σ

on^x i

=

Y

j

cosh(J

oj∇

) + σ

j^z

sinh(J

oj∇

)

×

G

θ

(x)

|^x=0

. (14) Since the transverse fields are randomly distributed, we have to perform the random average of (Ω

θ

)

o

according to the probability distribution function Q((Ω

θ

)

o

) given by (2). The ordering parameters µ

^z_n

and µ

^x_n

are then de- fined by µ

^z_n

=

hh

σ

_on^z iir

and µ

^x_n

=

hh

σ

^x_oniir

, where

h

...

ir

de- notes the transverse field average. Furthermore, the func- tions F

θ

(x) and G

θ

(x) must be replaced by

F

θ

(x) =

Z

Q((Ω

θ

)

o

)F

θ

(x)d(Ω

θ

)

o

, (15)

G

θ

(x) =

Z

Q((Ω

θ

)

o

)G

θ

(x)d(Ω

θ

)

o

. (16) Therefore, the longitudinal magnetization of the nth layer is given by

µ

^z_n

=

Yz j=1

cosh(J

oj∇

) + µ

^z_j

sinh(J

oj∇

)

×

F

θ

(x)

|^x=0

, (17) where z is the number of nearest-neighbours of the site o.

It is worth mentioning that this approximation, in spite of its simplicity since it neglects correlations between dif- ferent sites, leads to quite satisfactory results. Applying equation (17) to our system we have

-for the surface

(n = 1)

µ

^z_S

= [cosh(J

S∇

) + µ

^z_S

sinh(J

S∇

)]

⁴

×

[cosh(J

B∇

) + µ

^z2

sinh(J

B∇

)] F

S

(x)

|^x=0

(18)

-for any layer

n

≥

2.

µ

^z_n

=

cosh(J

B∇

) + µ

^z_n−1

sinh(J

B∇

)

×

[cosh(J

B∇

) + µ

^z_n

sinh(J

B∇

)]

⁴

×

cosh(J

B∇

) + µ

^z_n+1

sinh(J

B∇

)

×

F

B

(x)

|^x=0

. (19) Thus, neglecting high-order terms in the magnetiza- tion near the critical temperature, linearization of equa- tions (18) and (19) leads to

µ

^zS

= A

1

µ

^z_S

+ A

2

µ

^z2

, (20)

µ

^z_n

= B

0

µ

^z_n−1

+ 4µ

^z_n

+ µ

^z_n+1

, (21)

where, A

1

= 1

4

2F

S

( J

B

2 + 2J

S

)

−

2F

S

J

B

2

−

2J

S

+ 4F

S

J

B

2 + J

S

+ 4F

S

J

B

2

−

J

S

, (22)

A

2

= 1 16

2F

S

J

B

2 + 2J

S

+ 2F

S

J

B

2

−

2J

S

+ 12F

S

×

J

B

2

+ 8F

S

J

B

2 + J

S

+ 8F

S

J

B

2

−

J

S

, (23)

B

0

= 1

32 [2F

B

(3J

B

) + 8F

B

(2J

B

) + 10F

B

(J

B

)] . (24) Now we assume that µ

^z_n+1

= Aµ

^z_n

,

i.e.

the magnetization µ

^z_n

of each layer decreases exponentially into the bulk.

Equation (20) reduces to

AA

2

+ A

1

= 1. (25) Using equation (21), the decay parameter A is given by

A =

(1

−

4B

0

)

−

[(6B

0−

1)(2B

0−

1)]

^1/2

/2B

0

. (26)

3 Results and discussions

Now we are in a position to investigate, within the above theoretical framework, the surface phase diagrams of the semi-infinite system described by the Hamiltonian (1). As already mentioned, when Ω

θ

increases from zero, the tran- sition temperature T

c

falls from its value in the Ising model and reaches zero at a critical value of the transverse field depending on the value of P

θ

. The bulk critical tempera- ture is determined by setting µ

^z_n−1

= µ

^zn

= µ

^zn+1

in equa- tion (21);

i.e.

6B

0

= 1; namely

2F

B

(3J

B

) + 8F

B

(2J

B

) + 10F

B

(J

B

) = 16/3. (27)

3.1 The infinite system

First, we study the infinite system (bulk properties) for the square lattice (z = 4) and simple cubic lattice (z = 6).

In Figures 1a and 1b, we represent the phase diagrams

in the (T, Ω

θ

) plane for various values of P

θ

in the case

of the square (θ

≡

S) and simple cubic (θ

≡

B) lattices,

respectively. When the transverse field is bimodally dis-

tributed (P

θ

= 0), the critical temperature decreases grad-

ually from its value T

c

(Ω

θ

= 0), to vanish at some critical

value Ω

^c_θ

(Ω

_S^c

/J = 1.375 for z = 4 and Ω

_B^c

/J = 2.352 for

z = 6). As shown in the figures, when we consider a tri-

modal random transverse field distribution (i.e. P

θ6

= 0) ,

a finite critical transverse field Ω

_θ^c

also exists for relatively

(a)

0 .0 2 .0 4 .0 6 .0 8 .0 1 0 .0

0 .0 0 .5 1 .0 1 .5

P_B= 0 .2 .4 .6 .7 .8

k

_B

T /J

_B 1

Ω_B

/J

_B

.5

(b)

Fig. 1. The phase diagram inT-Ωθ plane of the Ising system in a random transverse field, (a) on square lattice (θ≡S and z= 4), and (b) on simple cubic lattice (θ≡B andz= 6). The number accompanying each curve denotes the value ofPθ.

small values of P

θ

. So, we conclude that the thermody- namic properties of the system are continuous between the two distributions. It is worthy of notice here that the Ising model in the trimodal random transverse field (2) has been investigated within the mean-field-like approxima- tions [31] and the mean-field renormalization group [27].

These studies showed a discontinuity in the ground state phase diagram between the trimodal ( P

θ

1) and the bimodal (P

θ

= 0) random transverse field distributions.

Yokota [33] discussed this result and, using the Suzuki- Trotter formula [32], he showed that the above-mentioned discontinuity may be an artifact of the mean-field-like ap- proximation. In the present work, we have not found any discontinuity in the phase diagram at T = 0 (Figs. 1a,b) between the trimodal and the bimodal random trans- verse field distributions. Thus, our calculation agrees with

Table 1. The critical temperature Tc, the critical transverse field Ω^c_θ and the critical valueP_θ^∗ for the square (z = 4) and simple cubic (z= 6) lattices.

z Tc/J(Ωθ= 0) Ωθ^c/J(Pθ= 0) Pθ^∗

4 0.7727 1.3755 0.6651

6 1.2683 2.3528 0.5333

Yokota’s prediction. This is due to the fact that we have used a method which treats correctly auto-spin correla- tions, while neglecting correlations only between spins on different sites; whereas in the mean-field approximation all correlations are neglected. Moreover, the appearance of such a finite critical value Ω

_θ^c

for P

'

0, can be explained by the existence of a small cluster of zero transverse field sites which, at the ground state, can not keep order in the system for any Ω. On the other limit, if only a small fraction of spins is exposed to Ω

_θ

(i.e. P

'

1), the cluster of zero transverse field sites includes nearly all sites of the lattice. Because of the size of such a cluster, it is reasonable to expect that the long range order (LRO) persists at low temperature for any value of the transverse field strength.

From the above limit behaviours obtained for P

'

0 and P

'

1, one can reasonably expect that there appears a critical value P

_θ^∗

of the concentration of zero transverse field sites (P

_S^∗

= 0.6651 for z = 4 and P

_B^∗

= 0.5333 for z = 6) showing two qualitatively different behaviours of the system which depend on the range of P

θ

. Indeed, for 0

≤

P

θ

< P

_θ^∗

, the cluster of zero transverse field sites is small and hence the order, at T = 0, is destroyed beyond a finite critical value Ω

_θ^c

of Ω

θ

. But for P

_θ^∗≤

P

θ≤

1, such a cluster is sufficiently large to keep order in the system at very low temperature, even in the limit of infinitely large values of the transverse field. Thus, we conclude that the existence of a finite critical value Ω

_θ^c

at the ground state, is related to the size of the cluster of zero transverse field sites. These qualitative expectations are indeed verified by our quantitative calculations (Fig. 1b). We have to point out here that the above observed behaviour is similar to that found in our recent study of the mixed spin Ising model in a transverse random field [37]. We note that the all transitions are of second-order and for a fixed value of Ω

θ

, the critical temperature is an increasing function of P

θ

. In Table 1, we give the corresponding values of the critical transition temperature T

c

when Ω

θ

= 0, the critical transverse field Ω

_θ^c

when P

θ

= 0, and the critical value P

_θ^∗

.

3.2 The semi-infinite system

In this section, let us examine the effect of a random

transverse field especially on the surface ordering in the

semi-infinite cubic ferromagnetic Ising model. This can

completely be determined from equation (25). The phase

diagram, in the (k

B

T /J

B,

J

S

/J

B

) plane, of the semi-

infinite random transverse field Ising model is represented

in Figures 2a and 2b for P

B

= 0.4 and P

B

= 0.8,

respectively. The phase diagrams are investigated for

0 .0 1 .0 2 .0 3 .0 4 .0 5 .0

J

_S

/J

_B

0 .0 1 .0 2 .0 3 .0 4 .0

k

_B

T /J

_B

P_B= 0 .4 Ω_B/J_B = 1 .0 Ω_S/J_S = 1 .0

1 .

P_S= 0 . .2

.4 .6

.8

B P S P

B P S F

(O ) (S p ) (E )

(S )

B F S F

(a)

0 .0 1 .0 2 .0 3 .0 4 .0 5 .0

J

_S

/J

_B

0 .0 1 .0 2 .0 3 .0 4 .0

k

_B

T /J

_B

P_B= 0 .8 Ω_B/J_B = 1 .0 Ω_S/J_S = 1 .0

P_S= 1 .

P_S= 0 . .2

.4 .6

.8

B P S P

B P S F

B F S F

(b)

Fig. 2.The phase diagrams inT-JS/JBplane for fixed values of the transverse fields (ΩB/JB=ΩS/JS = 1), with (a)PB= 0.4, and (b)PB= 0.8. The number accompanying each curve denotes the value ofPS.

Ω

B

/J

B

= Ω

S

/J

S

= 1, with different values of P

S

. As shown in these figures, three physically different phases are identified where the domains of existence depend on the ratio R = J

S

/J

B

. These phases are indicated on the phase diagram by the following symbols:

SP, BP: both Surface and Bulk are paramagnetic;

SF, BP: Surface ferromagnetic and Bulk paramagnetic;

SF, BF: both Surface and Bulk are ferromagnetic.

The above phases are separated by different phase transi- tions associated with the surface. They correspond to the surface (S), the ordinary (o), the extraordinary (E), and the special point (Sp) transitions. For the system under study, we extend the accepted terminology used in the classical semi-infinite Ising systems [13, 16, 38, 39]. On the other hand, Figure 2a shows the influence of the random transverse field on the pure surface behaviour (P

S

= 1).

Thus, some surface transition lines corresponding to vari- ous values of P

S

are plotted. As clearly seen from the ob- tained phase diagrams, we note that for given values of Ω

θ

and the ratio R = J

S

/J

B

(greater than a critical value), the surface critical temperature decreases with P

S

, and therefore the SFBP phase domain becomes less and less large when the concentration of sites not exposed to Ω

S

decreases. We also point out that the

^J_J^S

B

-component of the special point transition weakly varies when 0

≤

P

S≤

1.

In order to examine the effect of the surface transverse field strength on the surface phase transition, we plot the phase diagram of the system for different values of Ω

S

when P

S

takes separately the values 0.4 and 0.8. These latter values of P

S

are, respectively, less and greater than the critical value P

_S^∗

. As mentioned in the previous section and Figure 3a, for P

S

= 0.4 (< P

_S^∗

), the long range order disappears at a critical value of Ω

S

, while for P

S

= 0.8 (> P

_S^∗

), the system is ordered at very low temperature for any value of Ω

S

. This means that the surface transverse field strength will have different effects on the surface tran- sition which depend on the values of P

S

. Phase diagrams for P

S

= 0.4 and P

S

= 0.8 are shown in Figures 3a and 3b, respectively. From these two figures, we notice that for small values of the surface transverse field strength, the behaviour of the surface transition line is qualitatively in- dependent of the range of P

S

. On the other hand, for large values of the surface transverse field, the situation changes considerably and depends whether P

S

is less or greater than P

_S^∗

. Indeed, for instance P

S

= 0.8 (P

S

> P

_S^∗

), the sur- face transition still exists even for very large values of the surface transverse field strength Ω

S

(the asymptotic case Ω

S

=

∞

is shown in Fig. 3b). But for 0 < P

S

< P

_S^∗

, the surface transition and the BPSF phase disappear when the transverse field strength is larger than a critical value (for instance Ω

S

/J

S

= 2.73 for P

S

= 0.4 see Fig. 3a). We note that the critical ratio R

c

= (J

S

/J

B

)

c

depends on the con- centrations P

S

and P

B

and the transverse fields strengths Ω

S

and Ω

B

. Now, we focus on the effect of the concentra- tion P

S

on the critical value R

c

. In Figure 4, we represent the critical ratio R

c

as a function of the surface transverse field Ω

S

when P

S

= P

B

takes different values. The bulk transverse field is fixed to be Ω

B

/J

B

= 1. We notice that the critical ratio R

c

= (J

S

/J

B

)

c

of the surface and bulk exchange interactions increases weakly with increasing P

S

for small values of the surface transverse field Ω

S

, while

it is very sensitive to the values of P

S

, for large values of

Ω

S

. The two qualitatively different behaviours related to

P

_θ^∗

and observed in the infinite square (Fig. 1a) and sim-

ple cubic (Fig. 1b) lattices, have some consequences on

semi-infinite system. Indeed, when 0

≤

P

S

< P

_S^∗

, as seen

in Figure 4, there is no critical ratio R

c

of the exchange

interactions beyond a critical value of Ω

S

/J

S

which de-

pends on the values of the distribution parameter P

S

. This

means that there is no surface, extraordinary and special

transitions; and therefore the phase BPSF disappears at a

P

S

-dependent critical value of the surface transverse field

strength Ω

^c_S

. The system exhibits only one kind of transi-

tion which is the ordinary transition. On the other hand,

when P

S

> P

_S^∗

, the critical ratio R

c

is weakly dependent

0 .0 2 .0 4 .0 6 .0 8 .0 1 0 .0

J

_S

/J

_B

0 .0 1 .0 2 .0 3 .0 4 .0 5 .0

k

_B

T /J

_B

P_S= P_B= 0 .4 Ω_B/J_B = 1 .0

Ω_S/J_S = 0 . 1 . 1 .5 2 .

2 .5

2 .7

(a)

0 .0 3 .0 6 .0

J

_S

/J

_B 9 .0

0 .0 1 .0 2 .0 3 .0 4 .0 5 .0

k

_B

T /J

_B

P_S= P_B= 0 .8

Ω_B/J_B = 1 .0 ^Ω_S/J_S = 0 . 2 . 4 .

6 .

"

B P S P

B P S F

B F S F

(b)

Fig. 3. The phase diagrams inT-JS/JB for a fixed value of the bulk transverse fieldΩB/JB= 1, with (a)PB=PS= 0.4, and (b)PB=PS= 0.8. The number accompanying each curve denotes the value of the surface transverse fieldΩS.

on the values of the surface transverse field strength Ω

S

. Moreover, all transitions (surface, extraordinary, special and ordinary) may exist in the system for any value of Ω

S

/J

S

(see also Fig. 3b for Ω

S

/J

S

=

∞

).

The dependence of the critical ratio R

c

with the sur- face transverse field is represented in Figure 5, when the transverse fields Ω

S

/J

S

and Ω

B

/J

B

are distributed with different concentrations (P

S 6

= P

B

). In this case, we ob- tain similar behaviours to those observed in Figure 4. In particular, for values of P

S

greater than a Ω

B−

dependent critical value P

_S^∗

, the surface transition still exists for any value of Ω

S

. The critical ratio R

c

and P

_S^∗

are weakly sen- sitive to the bulk transverse field distribution parameter P

B

. In contrast with the case P

S

= P

B

(Fig. 4), R

c

is always a decreasing function of P

S

for any fixed value of Ω

S

. R

c

also depends on the value of the bulk transverse

0 .0 4 .0 Ω_S

/J

_S

0 .0 2 .0 4 .0

(J

_S

/J

_B

)

_c

Ω_B/J_B = 1

.2 .4 .6

P_S= P_B= 0 0 .8

Fig. 4.The variation of the critical ratio of the exchange inter- actions (JS/JB)cwith the surface transverse field strength for a fixed value of the bulk transverse fieldΩB/JB= 1. The number accompanying each curve denotes the value ofPB=PS.

0 .0 2 .0 Ω_S

/J

_S 4 .0

0 .0 2 .0 4 .0

(J

_S

/J

_B

)

_c

P_S= 0 .

Ω_B/J_B =^∞ & P_{B = 0 .6} Ω_B/J_B = 1 .0 & PB = 0 .6 Ω_B/J_B = 1 .0 & PB = 0 .2

0 .2 0 .4

0 .6

0 .8

P_S= 0 . 0 .2

0 .4 0 .6 0 .8

Fig. 5. The dependence of the critical ratio of the exchange interactions (JS/JB)c as a function of the surface transverse field strength forΩB/JB= 1, withPB= 0.2 (solid lines) and PB= 0.6 (dashed lines). The broken lines are for infinite bulk transverse field (ΩB=∞) withPB= 0.6. The number accom- panying each curve denotes the value ofPS.

field Ω

B

. But its behaviours obtained for large Ω

B

and for P

_B^∗

< P

B

< 1 are qualitatively similar such that the bulk can be ordered below its critical temperature. This is shown in Figure 5 for the asymptotic value (Ω

B

=

∞

).

Finally, in order to complete our investigation, we study with more details the variation of the critical ra- tio R

c

= (J

S

/J

B

)

c

of the exchange interactions with the bulk transverse field Ω

B

. As we have noted before, the effect of the bulk transverse field is to lower the value of the critical temperature and, at the same time, R

c

moves to lower values. Eventually, when the transverse fields are bimodally distributed (P

S

= P

B

= 0), R

c

reaches a min- imum value at the critical value Ω

^c_B

(P

B

= 0) = 2.3528J

B

as seen in Figure 6. This result is similar to that obtained in surface behaviour of the transverse Ising model [40, 41].

For trimodal case, results are shown in the same figure for

0 .0 4 .0 8 .0Ω_B

/J

_B

0 .0 1 .0

(J

_S

/J

_B

)

_c

Ω_S/J_S = 1

P_S= P_B= 0

.8

.2 .3 .3 6

1 .

.6

.4 .4 2

Fig. 6.The variation of the critical ratio of the exchange inter- actions (JS/JB)cwith the bulk transverse field strength for a fixed value of the surface transverse fieldΩS/JS= 1. The num- ber accompanying each curve denotes the value ofPB=PS.

various values of the distribution parameters P

S

and P

B

when the surface transverse field is kept fixed (Ω

S

/J

S

= 1). Here again, it exists a minimum of R

c

which coincides with the critical value of the bulk transverse field Ω

^c_B

(P

B

).

From Figures 4 and 6, we note that the surface and the bulk transverse fields act on R

c

= (J

S

/J

B

)

c

in opposite ways. Above Ω

_B^c

, even though the infinite bulk transition temperature goes to zero, the system can still exhibit a phase transition for sufficiently large values of R = J

S

/J

B

. As shown in the figure, when we consider a trimodal ran- dom fields distribution (i.e. P

S

= P

B 6

= 0), the minimum becomes more and more large with increasing P

B

to disap- pear when P

B

is greater than the critical value P

_B^∗

= 0.533.

Thus, for P

_B^∗

< P

B ≤

1, there is no critical bulk transverse field and therefore, there is no minimum of the critical ra- tio R

c

of the exchange interactions in that range of P

B

since the ordered state is stable at very low temperature, for any values of the transverse field strengths.

4 Conclusions

In this paper, we have focused on the surface phase dia- grams of the random transverse spin-1/2 Ising system in the semi-infinite simple cubic lattice with a (001) free sur- face. The surface and bulk transverse fields are bimodally and trimodally distributed. 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 Van der Waerden identities accounting exactly for the single-site kinematic relations.

For the infinite Ising system (bulk properties), we have investigated the variation of the critical temperature T

c

with the transverse fields Ω

S

and Ω

B

on square lattice (z = 4) and simple cubic lattice (z = 6) respectively for various values of P

S

and P

B

(P

S

and P

B

measure the frac- tion of spins not exposed to Ω

S

and Ω

B

, respectively).

We have defined a critical value P

^∗

of P (P

^∗

= 0.6651 for

z = 4 and P

^∗

= 0.5333 for z = 6) separating two quali- tatively different behaviours of the system depending on the range of P . Thus, for 0

≤

P < P

^∗

, the system exhibits at the ground state a phase transition at a finite critical value Ω

c

of Ω. But for P

^∗ ≤

P

≤

1, there is no critical transverse field, and therefore, at very low temperature, the ordered state is stable for any value of the transverse field strength.

For the semi-infinite system, we have investigated the phase diagrams for different values of the distribution pa- rameters P

S

and P

B

of the surface and the bulk trans- verse fields respectively, and found some interesting fea- tures. In particular, and for P

S

= P

B

, we have found that for 0

≤

P

B

< P

_B^∗

, the critical ratio of the exchange interactions R

c

= (J

S

/J

B

)

c

reaches a minimum value.

This minimum disappears when P

_B^∗

belongs to the range P

_B^∗ ≤

P

B ≤

1 since the ordered state is stable for any value of the bulk transverse field strength. We have also noted that the surface and the bulk transverse fields act on R

c

= (J

S

/J

B

)

c

in opposite ways.

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