Mixed spin-1/2 and spin-1 Ising model in transverse random fields

Mixed spin-1=2 and spin-1 Ising model in transverse random elds

N. Benayad

, R. Zerhouni

, A. Klumper

, J. Zittartz

aGroupe de Mecanique Statistique des Transitions de Phases et Phenomenes Critiques, Laboratoire de Physique Theorique, Universite Hassan II, Faculte des Sciences An Chock, BP 5366,

Maarif, Casablanca, Morocco

bInstitut fur Theoretische Physik, Universitat zu Koln, Zulpicher Strasse 77, D-50937 Koln, Germany Received 2 July 1998

Abstract

We investigate the critical properties of a mixed spin-1=2 and spin-1 Ising model in a transverse random eld. In this study, we use two approaches: the pair approximation and the mean eld renormalization group each combined with the discretized path integral representation. It is found that both methods predict quantitatively the same phase diagram. Critical exponents of the system are also calculated. c1999 Elsevier Science B.V. All rights reserved.

PACS: 05.30.-d; 05.50.+q; 05.70.-a

1. Introduction

The eect of quantum uctuations in statistical mechanics models has been investi- gated extensively for the last few decades. The simplest of such systems is the spin-1=2 Ising model in a transverse eld, which was originally introduced by De Gennes [1] as a valuable model for the tunnelling of the proton in hydrogen-bonded ferroelectrics [2]

such as of KH₂PO₄ type. Since then, it has been successfully applied to several phys- ical systems, like the cooperative Jahn-Teller systems [3] (e.g. DyVO₄ and T_bVO₄), the ordering in rare earth compounds with a singlet crystal eld ground state [4], and also some real magnetic materials with strong uniaxial anisotropy in a transverse eld [5]. In the one dimensional case, the model has been solved exactly [6–8]. An im- portant exact result was proved by Suzuki [9] which states that the critical properties of the d-dimensional transverse Ising model at nite temperature (T¿0) are equal to those of the d-dimensional Ising model; and at T= 0, they are equal to those of the

∗Corresponding author. Fax: +49 221 4705159; e-mail: kluemper@thp.uni-koeln.de.

0378-4371/99/$ – see front matter c1999 Elsevier Science B.V. All rights reserved.

PII: S 0378-4371(98)00424-5

(d+ 1)-dimensional Ising model. In two and more dimensions, the model has been studied by high temperature expansions [10], Monte Carlo methods [11], eective eld methods [12,13] and a method of combining the pair approximation with the discretized path integral representation [14,15]. In this latter approach, the noncommutativity of operators in the Hamiltonian is avoided since the quantum Hamiltonian is transformed to a classical one which is treated by an analytical method [16]. It has been found that the transverse Ising model (TIM) has a nite temperature phase transition which can be suppressed to zero temperature where criticality still occurs for a certain value of the transverse eld. Therefore, the TIM serves as a model of quantum critical phenom- ena at zero temperature. In addition to the works on the two state spin systems, the spin-1 transverse Ising models [17–22] and higher spin cases [23–27] have received some attention.

Recently, another problem of growing interest is associated with the transverse ran- dom eld Ising model (TRFIM). Special attention has been devoted to bimodal (two peaks) and trimodal (three peaks) distributions for the transverse random eld. This model has been investigated using dierent approximation schemes, such as the mean eld and mean eld renormalisation group (MFRG) [28], a method of combining the MFRG with the discretized path integral representation (DPIR) [25,29,30] and an approach combining the pair approximation with DPIR [31]. We point out that all transition lines are of second order and the directional randomness of the transverse eld does not change the critical behaviour [32] of the system.

In this work we study the two-sublattice mixed spin-1=2 and spin-S transverse Ising systems described by the Hamiltonian

H=−JX

hiji

_i^zS_j^z−X

i

i_i^x−X

j

jS_j^x; (1)

where _i and S_j (=x; z) are components of spin-1=2 and spin-S operators at sites i and j, respectively. J is the exchange interaction, i and j are transverse elds, and the rst summation is carried out only over nearest-neighbour pairs of spins. The Hamiltonian (1) is of interest because it has less translational symmetry than its single spin counterpart. In the absence of transverse elds (i=j= 0), the system describes a certain type of ferrimagnetism [33]. It has been shown that the MnNi(EDTA)-6H₂O complex is an example of a mixed spin system [34]. The mixed spin-1=2 and spin-1 Ising model has been studied by the renormalization group technique [35,36], by high temperature series expansions [37], in the free fermion approximation [38] and by nite cluster approximation [39]. On the other hand, the inuence of a uniform trans- verse eld (_i=_j ≡ 6= 0) on the transition temperature has been investigated by using dierent approximation schemes, such as the eective-eld theory [26,27,40], discretized path integral representation [22], and the two-spin cluster approximation [22].

The purpose of this paper is to investigate the critical behaviour of the mixed spin Ising system in a transverse random eld described by Hamilitonian (1), when the transverse elds are randomly distributed according to the following trimodal

distribution function

Q(i) =p(i) +(1−p)

2 [(i−) +(i+)]; (2)

where the parameter p measures the fraction of spins in the system not exposed to the transverse eld . At p= 1 (or = 0), the system reduces to the simple mixed spin-1=2 and spin-S Ising model.

In the present work, we limit our study to the case S = 1. To this end we use two approaches, the pair approximation (PA) and the mean eld renormalization group (MFRG) each combined with the discretized path integral representation (DPIR). The derived equations in both methods are applicable for arbitrary coordination number z.

The advantage of these methods is the avoidance of noncommutativity of operators as in Hamiltonian (1) due to the transformation to a classical system. The average free energy and the critical transverse elds in the zero temperature limit are analytically calculated.

2. Discretized path integral representation for the mixed spin Ising model in a transverse random ÿeld

Let us rst reformulate the Hamiltonian (1) in the light of DPIR [16]. The quantum spin-1=2 and spin-1, on each site of the - and S-sublattices, will be transformed into P-component vectors i(⁽¹⁾; ⁽²⁾; : : : ; ^(P)) and j(⁽¹⁾; ⁽²⁾; : : : ; ^(P)), respectively;

eventually letting P go to innity. Each component ^(t)(^(t)) is taken to be a classical two-state (three-state) variable^(t)=±1=2(^(t)= 0;±1), in eect creating many copies, or replicas, of the original variables on each site. Thus, the partition function

Z= Tr exp(−H) =X

;

h; |exp(−H)|; i (3)

can be written, in a discretized path integral version, as Z=X

⁽¹⁾

: : : :X

^(P)

X

⁽¹⁾

: : : :X

^(P)

exp



J P

X

hiji

XP t=1

^(t)_{i j}^(t)+X

i

A_i +X

j

A_j



; (4) with

A_i = XP

t=1

(a1^(t)_i ·^(t+1)_i +a2);

A_j = XP

t=1

[b1+b2^(t)_{j j}^(t+1)+b3((^(t)_j )²+ (_j^(t+1))²) +b4((^(t)_j )²(^(t+1)_j )²)]; (5) where

a₁= 2 ln

coth _i

2P

; a₂=1 2ln

1 2sinh

_i P

; b₁= ln

cosh

j

P

; b₂=−ln

tanh j

2P

;

b₃= ln √1

2tanh j

P

; b₄=−ln

tanh j

P

: (6)

Using the above P-component vectors _i and _j, the partition function (4) may be rewritten as

Z=X

1

X

2

· · · ·X

N=2

X

1

X

2

· · · ·X

N=2

exp



J P

X

hiji

_i·_j+X

i

A_i+X

j

A_j



; (7) where A_i and A_j can also be written as

A_i =a1iOˆ i+Pa2;

A_j =Pb1+b2jOˆ j+ 2b3jj+b4²_jOˆ²_j; (8) with

( ˆO)t; t⁰=t; t⁰−1; ( ˆO)P;1= 1; and ²_j= ((⁽¹⁾_j )²;(⁽²⁾_j )²; : : : : : : ;(^(P)_j )²): (9) The form of Eq. (7) suggests that the Hamiltonian Hof the new classical spin system, can be broken up into a reference part H₀, involving only the single-site terms

−H0=X

i

A_i +X

j

A_j ; (10)

and an interaction part V

−V =J P

X

hiji

_i·_j: (11)

The free energy F corresponding to the full Hamiltonian H=H0+V can then be expressed in terms of the reference system free energy F0 and the cumulant expansion in the reference system [41]

−F=−F0+ X∞

n=1

1

n!(−)ⁿCn(V); (12)

with

−F= lnZ ; −F₀= lnZ₀; (13)

and the cumulants given by

C₁(V) =hVi₀; C₂(V) =hV²i₀− hVi²₀; : : : ; (14) h: : : :i₀ denotes the average for the reference system. We have to emphasize that Eq. (12) is not a high temperature series expansion since in path-integral applications, the cumulants themselves depend on=1=T through the temperature dependence of the reference system Hamiltonian. This temperature dependence turns out to be just what is necessary to allow well-dened calculations at all temperatures, including the ground state. It would therefore be better to regard Eq. (12) as an expansion in successively higher orders of uctuations.

3. Pair approximation and path integral methods

Starting from Eq. (1), the eective Hamiltonian for a pair of nearest-neighbor spins i andj in the pair approximation is given by [42]

HII=−J_i^zS_j^z−h^ef_i^z−h^ef_SS_j^z−j_i^x−jS_j^x; (15) where h^ef and h^ef_S are the eective elds acting on sites i and j belonging to - and S-sublattices, respectively. The corresponding pair partition function may be evaluated from

ZII= Tr exp(−HII): (16)

It is dicult to diagonalize directly the Hamiltonian H_II due to the non-commuting operators. However, the DPIR, briey described in Section 2, has been proposed to deal with quantum spin systems [16]. Applying the idea of the DPIR to the pair Hamiltonian HII, Eq. (7) takes the following expression forZII

Z_II=X

i

X

j

exp J

P (_i·_j) +h^ef

P (E·_i) +h^ef_S

P (E·_j) +A_i +A_j

; (17) with E= (1;1; : : : ;1) is a P-component constant vector.

The reference part H_II0 and the interaction part V_II for the case H_II, are given by

−H_II0=h^ef

P (E·_i) +h^ef_S

P (E·_j) +A_i+A_j; −V_II=J

P _i·_j: (18) From Eq. (12) and taking the rst cumulant, the pair partition function may be evalu- ated from

lnZII= ln{Tr exp(−HII0)}+ (−)hVIIi0

= ln

2 cosh

2[(h^ef)²+²_i]¹⁼²

+ ln{1 + 2 cosh[(h^ef_S)²+²_j]¹⁼²} +J

@

@h^ef ln

2 cosh

2((h^ef)²+²_i)¹⁼²

× @

@h^ef_S ln[1 + 2 cosh((h^ef_S)²+²_j)¹⁼²]

: (19)

On the other hand, the one-body eective Hamiltonian is given by

HI=−i_i^x−H^ef_i^z−jS_j^x−H_S^efS_j^z: (20) The corresponding one-body partition function is given by

ZI= Tr exp(−HI)

=

2 cosh

2[(H^ef)²+²_i]¹⁼²

{1 + 2 cosh[(H_S^ef)²+²_j]¹⁼²}: (21) The free energy of the full system is given by the expression [42]

−F=N(1−z)

2 hlnZIi+Nz

2 hlnZIIi; (22)

where z is the number of the nearest-neighbour spins and h: : : : : :i means the average over random eld distribution. Using Eq. (2) we obtain

−F=Nz

2 {p[f(h^ef;0) +g(h^ef_S;0)] + (1−p)[f(h^ef; ) +g(h^ef_S ; )]}

+N(1−z)

2 {p[f(H^ef;0) +g(H_S^ef;0)] + (1−p)[f(H^ef; ) +g(H_S^ef; )]}

+NzJ 2

p²@f(h^ef;0)

@h^ef

@g(h^ef_S;0)

@h^ef_S +p(1−p)

×

@f(h^ef;0)

@h^ef

@g(h^ef_S ; )

@h^ef_S +@f(h^ef; )

@h^ef

@g(h^ef_S;0)

@h^ef_S

+(1−p)²@f(h^ef; )

@h^ef

@g(h^ef_S; )

@h^ef_S

; (23)

where

f(x; y) = ln

2 cosh

2[x²+y²]¹⁼²

; g(x; y) = ln{1 + 2 cosh[x²+y²]¹⁼²}: (24) Let us denote by m andm_S the average magnetizations of spin- and spin-S, respec- tively. In the pair approximation method, the eective elds H^ef; H_S^ef; h^ef, and h^ef_S can be related to m and m_S by the following relations [43]:

H^ef =zJmS; H_S^ef=zJm; h^ef =J(z−1)mS; h^ef_S =J(z−1)m; (25) (i.e. H^ef=_z−1^z h^ef andH_S^ef=_z−1^z h^ef_S ).

Thus, the self-consistent equations for m andm_S are therefore obtained by

@

@m(F) = 0 and @

@m_S(F) = 0: (26)

Using Eqs. (26), the second order transition temperature as a function of p and , is determined by

1 T²

p

2 +T(1−p) G tanh

G 2T

p

3 +T(1−p) G

sinh(G=T) (1 + 2 cosh(G=T))

= z² (1−z)² ;

(27) where we introduce the dimensionless parameters

T ≡1=zJ and G≡=z J : (28)

4. Mean ÿeld renormalization group and path integral methods

In this section, we study the critical properties of the model described by Hamiltonian (1) by using a method combining the mean eld renormalization group (MFRG) with the DPIR. For the MFRG, we follow the procedure proposed by Indekeu et al. [44,45] based on the comparison of systems of dierent sizes. We consider two

nite systems (clusters) which have n and n⁰ spins, respectively (n¡n⁰). In this ap- proach, we calculate the order parameter (for both clusters) for symmetry-breaking boundary conditions which, in the mean eld way, simulate the eect of spins in the surrounding innite systems. For the ferromagnetic Ising model, each boundary spin has a xed value equal to b and b⁰ for n- and n⁰-spin clusters, respectively. Calcu- lating the magnetizations per spin m and m⁰ for both clusters, and imposing a scaling relation between such magnetizations corresponding to two dierent length scales, we get

m⁰(J⁰; : : : ; b⁰) =m⁰ (J; : : : ; b); (29) where (J; : : : :) and (J⁰; : : : :) are the coupling parameters for the two rescaled systems, is the scaling factor. If we impose similar scaling relation between b and b⁰, and expanding Eq. (29) for small b andb⁰, we obtain

@

@b⁰{m⁰ (J⁰; : : : : ; b⁰)}|b⁰=0= @

@b{m⁰ (J; : : : : ; b)}|b=0 (30) which is independent of and corresponds to the renormalization recursion rela- tion.

This method has been applied to a variety of systems and good results have been obtained [44–47]. Here, we consider the simplest choice for the clusters, namely n⁰= 1; n= 2 spins, respectively.

Let us introduce K⁰=J⁰ and _i⁰=_i⁰, which are called the scaled reduced tem- perature and the scaled reduced transverse eld, respectively. The reduced Hamiltonian of the single-spin cluster (n⁰= 1) is given by

H₁=−zK⁰_i^zb⁰_S− ⁰_ii^x−zK⁰S_j^zb⁰− _j⁰S_j^x; (31) where b⁰_S (b⁰) is the xed mean value of the z-component of the nearest neighbours of _i^z (S_j^z) and z is the coordination number of the lattice. After diagonalizing H₁, the partition function of the single-spin cluster, for given { ⁰_i; ⁰_j}, is

Z₁= Tr exp(−H₁) =

2 cosh1 2

q(⁰_S)²+ ( ⁰_i)² h

1 + 2 coshq

(⁰)²+ ( ⁰_j)²i

; (32) with

⁰_S=zK⁰b⁰_S; ⁰=zK⁰b⁰:

The corresponding magnetizations per spin read m₁=

@lnZ₁

@⁰_S

0

=

* ⁰_S 2q

(_S⁰)²+ ( ⁰_j)²tanh1 2

q(⁰_S)²+ ( ⁰_j)² +

0

; (33)

m^S₁=

@lnZ1

@⁰

0

=

* q ⁰ (⁰)²+ ( ⁰_j)²

2 sinhq

(⁰)²+ ( ⁰_j)² (1 + 2 coshq

(⁰)²+ ( ⁰_j)²) +

0

: (34)

When the temperature approaches its critical value, b⁰_S and b⁰ become very small and therefore we consider only terms linear in b⁰_S andb⁰. Then, using Eq. (2), the average magnetization of the single-spin cluster is given by

m₁= ⁰_S

2 ⁰_itanh ₀

2i

0=A₁(p⁰; K⁰; ⁰)b⁰_S (35)

m^S₁=

*⁰

j0

2 sinh( ⁰_j) (1 + 2 cosh( _j⁰))

+

0

=B1(p⁰; K⁰; ⁰)b⁰; (36) with

A1(p⁰; K⁰; ⁰) =zK⁰ p⁰

4 + (1−p⁰)tanh( ⁰=2) 2 ⁰

; B1(p⁰; K⁰; ⁰) =zK⁰

2p⁰

3 + (1−p⁰) 2 sinh( ⁰)

0(1 + 2 cosh( ⁰))

;

where p⁰ is the scaled concentration of sites not exposed to the transverse eld.

The reduced Hamiltonian for the two-spin cluster can be written as H2=−K⁰_i^zS_j^z− i_i^x− jS_j^x−X

k6=j

Kik_i^zbS−X

k6=i

KkjS_j^zb; (37) where K=J and i(orj)=i(orj). The spin_i^z interacts directly with S_j^z through the term −K_i^zS_j^z, and _i^z (S_j^z) interacts with (z−1) nearest-neighbour spins through the term −K_i^zbS (−KS_j^zb). The diagonalization of this reduced Hamiltonian is not so simple. Here again, the two-spin cluster’s partition function Z₂ is also manageable by the use of the DPIR formula. So, from Eq. (12) and taking the rst cumulant, we obtain

lnZ2= ln

2 cosh1

2[²_S+ ²_i]¹⁼²

+ ln{1 + 2 cosh[²+ ²_j]¹⁼²}

+J S

[²_S+ ²_i]¹⁼²

tanh1

2[²_S+ ²_i]¹⁼²

×

[²+ ²_j]¹⁼²

( sinh[²_S+ ²_j]¹⁼² 1 + 2 cosh[²_S+ ²_j]¹⁼²

)

; (38)

with S= (z−1)KbS; = (z−1)Kb.

The corresponding average magnetizations per spin read m₂=

@lnZ₂

@S

=

* S

2[²_S+ ²_i]¹⁼²tanh 1

2[²_S+ ²_i]¹⁼²

+J

[²+ _j²]¹⁼²

sinh[²+ ²_j]¹⁼² (1 + 2 cosh[²+ ²_j]¹⁼²)

×

( 2 i

[ ²_i +²_S]³⁼²tanh 1

2[ _i²+²_S]¹⁼²

+ ²_S 2[ _i²+²_S]

1−tanh² 1

2[ _i²+_S²]¹⁼²

; (39)

m^S₂=

@lnZ₂

@

=

*

[²+ ²_j]¹⁼²

2sinh[²+ _j²]¹⁼²

(1 + 2 cosh[²+ _j²]¹⁼²)+J S

[²_S+ ²_i]¹⁼²

×tanh 1

2[²_S+ ²_i]¹⁼²

×

( 2 j

[²+ _j²]³⁼²

sinh[²+ ²_j]¹⁼² (1 + 2 cosh[²+ ²_j]¹⁼²) + ²

[²+ _j²]

(cosh[²+ ²_j]¹⁼²+ 2) (1 + 2 cosh[²+ ²_j]¹⁼²)²

) +

: (40)

SincebandbS approach zero in the vicinity of the critical temperatureTc, we consider in Eqs. (39) and (40), only terms linear in b andb_S. Then, using Eq. (2), the average magnetizations of the two-spin cluster are given by

m₂=A₂(p; K; )b_S+B₂(p; K; )b; (41)

m^S₂=C₂(p; K; )b+B₂(p; K; )b_S; (42)

with

A₂(p; K; ) = (z−1)K p

4 + (1−p)tanh( =2) 2

; B2(p; K; ) = (z−1)K²

p²

6 +p(1−p)

sinh( )

2 (1 + 2 cosh( ))+ 1

3 tanh( =2)

+(1−p)²sinh( )tanh( =2)

2(1 + 2 cosh( ))

; (43)

C2(p; K; ) = (z−1)K 2p

3 + (1−p) 2 sinh( ) (1 + 2 cosh( ))

:

According to the idea of the two-clusters MFRG [44,45], the magnetizations (m₁; m^S₁) and (m₂; m^S₂) are assumed to scale as the symmetry breaking elds (b⁰; b⁰_S) and (b; bS), i.e.

m₁=m₂; m^S₁=m^S₂ ; (44a)

b⁰=b; b⁰_S=bS: (44b)

We thus obtain the following recursion relation:

[A₁(p⁰; K⁰; ⁰)−A₂(p; K; )][B₁(p⁰; K⁰; ⁰)−C₂(p; K; )] = [B₂(p; K; )]²: (45) Although we have three parameters p; K and , we did not obtain, as in standard renormalization theory, three recursion relations. This is based in our technique which is phenomenological renormalization. So, from the single equation (45), we study its xed point for the case K⁰=K ≡K_c; p⁰=p and ⁰= . The reduced critical tem- perature K_c⁻¹ is therefore determined by the equation

K_c² p

3 + (1−p) sinh( ) (1 + 2 cosh( ))

p

2 + (1−p)tanh( =2)

= 1

(z−1)² : (46) Using the dimensionless parameters T andG given by the relations (28), the obtained xed point equation (46) coincides exactly with the Eq. (27). Thus, both methods we used in the investigation of the mixed spin-1=2 and spin-1 Ising system in a transverse random eld, predict quantitatively the same phase diagram.

5. Results and discussions

We are interested in studying the phase diagram and other critical properties. In the beginning, it is instructive to discuss some special limiting cases. At p= 1 (or = 0), we recover the simple mixed spin-1/2 and spin-1 Ising model. From Eq. (46), its critical temperature is given by

K_c⁻¹(= 0) = (z√−1)

6 : (47)

For the square lattice (z= 4), we have K_c⁻¹(= 0) = 1:224 which improves over the value of the mean eld approximation 1.633 [48] and the nite cluster approximation 1.298 [49], and is in good agreement with the Monte Carlo result 0.976 [50]. The second case corresponds to the bimodal distribution (p= 0). The xed point equation (46) changes into

K_c2

sinh( )tanh( =2) (1 + 2 cosh( )) = 1

(z−1)² : (48)

For the square lattice (z= 4; p= 0), the results are similar to those we found for a uniform transverse eld [51], see Fig. 1. The critical temperature decreases gradually

Fig. 1. Phase diagram of the mixed spin-1/2 and spin-1 Ising model in a transverse random eld for the square lattice corresponding to dierent values of the parameterp.

from its value K_c⁻¹(=0)=1:224, to vanish at some critical valuec. From Eq. (48), _c is given by

c

J =(z√−1)

2 : (49)

We also have solved numerically Eq. (46). The phase diagrams in (K⁻¹; ) plane are plotted in Fig. 1 for various values of p. As expected, the transition temperature increases with increasing value of p for xed values of the transverse eld. As seen in the gure, the transition temperature decreases with increasing transverse eld but does not vanish for any value of p6= 0, i.e. there is no critical transverse eld_c. We thus nd a crossover from the trimodal distribution (p1) to the bimodal distribution (p= 0) indicating a discontinuity between these two cases in the ground state phase diagram. This means that the thermodynamic properties of the mixed spin Ising model in a random transverse eld, described by Hamiltonian (1), are discontinuous between the bimodal and the trimodal random eld distributions.

Besides the above complete phase diagram obtained from Eqs. (27) or (46), we can also use the recursion equation (45) to calculate the critical exponents Y_T andY, using the following formula [52]:

@R⁰

@R

FP

=L^YR; (50)

Table 1

Special critical temperature (Kc⁻¹), critical transverse eld (c=J), and critical ex- ponentsYT andY for the square (z= 4) and simple cubic (z= 6) lattices

z Kc⁻¹(= 0) YT c

J(Kc⁻¹= 0) Y

4 1.224 0.6438 2.12 0.6438

6 2.041 0.6671 3.53 0.6671

whereR can beK or ; L= (n=n⁰)^1=d is the rescaling factor and the label FP indicates that the derivative is taken at the xed point. d is the dimensionality of the system and n and n⁰ are the respective numbers of spins of the two clusters.

In Table 1 we list the values of the reduced critical temperature K_c⁻¹, critical trans- verse eld _c=J, and the critical exponents Y_T and Y for dierent lattices from Eqs.

(47),(49), and (50). We note that for the case of the square lattice (d=z=2 = 2), the obtained values of Y_T and Y are equal to those calculated for the monoatomic Ising models in a transverse eld [25,53].

Acknowledgements

This work was supported by the agreement of cooperation between CNR (Morocco) and DFG (Germany). We want to thank both organizations.

One of us (N.B) acknowledges the hospitality of the Institut fur Theoretische Physik der Universitat zu Koln.

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