Magnetic properties of a mixed spin-1/2 and spin-3/2 transverse Ising model

Magnetic properties of a mixed spin-

and spin-

transverse Ising model

N. Benayad1, A. Dakhama1, A. Klu¨mper2, J. Zittartz2

1Faculte´ des Sciences Aı¨n Chock, Laboratoire de Physique The´orique, B.P. 5366, Maarif, Casablanca, Morocco 2Institut fu¨r Theoretische Physik, Universita¨t zu Ko¨ln, Zu¨lpicher Strasse 77, D-50937 Ko¨ln, Germany

Received: 21 February 1996 / Revised version: 7 May 1996

Abstract. A theoretical study of a mixed spin-12and spin-32 Ising system with independent transverse fields is present- ed using an effective field method within the framework of a single-site cluster theory. In this approach the effective field equations are derived using a probability distribution method based on the use of generalized van der Waerden identities accounting exactly for the single-site kinematic relations. The effect of the transverse fields on the critical behaviour is studied. The thermal dependence of the lon- gitudinal and transverse components of the magnetization and its higher moments is also studied.

PACS: 05.30.!d; 05.50.#q; 05.70.!a

1. Introduction

Over recent years, there has been considerable interest in the theoretical study of transverse Ising models. The spin- 12system was originally introduced by De Gennes [1] as a relevant model for hydrogen-bonded ferroelectrics such as the KH

2PO

4type. Since then, it has been applied to several physical systems such as cooperative Jahn-Teller systems [2], like DyVO

4, and magnetic materials with strong uniaxial anisotropy in a transverse field [3]. It has been extensively studied by use of various techniques [4—7], including an effective field treatment [8, 9] based on a generalized though approximate Callen-Suzuki rela- tion. Spin-one transverse Ising models [10—15] have re- ceived similar attention as the work on the two-state systems. Ising models with spinShigher than 1 have also been studied by high temperature expansion [16] and the pair approximation with discretized path-integral repres- entation [17]. Very recently, quantum transverse spin-S (S'1) systems have been investigated by the use of the effective field theory based on approximate [18] and exact

Supported by the agreement of cooperation between CNR-Maroc and DFG-FRG

[19] generalized van der Waerden identities. We like to note that the approximate case excludes the study of magnetization moments higher than the quadrupolar moments.

Only few studies have been directed to two-sublattice mixed Ising spin systems in transverse fields (MTIM) [20, 21] described by the Hamiltonian

H"!+

S*,+^T

J*+p;*S;+!C1+

* p9*!C2+ +

S9+, (1) wherepa* andS+a(a,x,z) are components of spin-1

2and spin-S operators at sites i and j, respectively. J

*+ is the exchange interaction,C1andC2are transverse fields, and the first summation is carried out only over nearest-neigh- bour pairs of spins. As far as we know, the above model (1) has only been studied for the caseS"1 [20, 21].

Hamiltonian (1) shows spin reversal symmetry (p;P!p;,S;P!S;,p9P#p9,S9P#S9) which is spontaneously broken below a field dependent critical temperature. In addition to the temperature dependence of the order parameter we are interested in the specific heat in the low-temperature phase of the mixed spin system. For uniform spin systems in a transverse field this quantity is known to be largely independent of the trans- verse field [18]. Therefore it is interesting to investigate this problem for model (1). Although this question was not addressed forS"1 we choose for our study the mixed spin-1

2and spin-3

2transverse Ising model (1). Potentially, the behaviour of half-integral and integral spin systems can be quite different [25]. Therefore the study of this system is worthwhile. To this end, we use an effective method within the framework of a single-site cluster the- ory [22]. The effective field equations are derived using a probability distribution method based on the use of generalized van der Waerden identities [23] that account exactly for the single-site kinematic relations.

The outline of this work is as follows: In Sect. 2 we describe the effective field theory based on a probability distribution method. In Sect. 3 we present our results for the transition temperatures and the longitudinal and transverse moments of the magnetizations for different

transverse field strengths. Also the internal energy and specific heat are presented.

2. Effective field theory

The effective field theory we adopt in the study of the transverse mixed spin-1

2and spin-3

2Ising model described by Hamiltonian (1) is based on a single-site cluster theory.

In this method attention is focused on a cluster consisting of just a single selected spin, labelled 0, and the neighbour- ing spins with which it directly interacts. To this end the total Hamiltonian given by (1) is split into two parts, H"H

0#H@, whereH

0includes all terms ofHassociated with the lattice site 0, namely

Hp0"!

A

J0*S;*

B

HS0"!

A

J0+p;+

B

if the lattice site 0 belongs to theporS-sublattice, respec- tively.

The problem consists in calculating the sublattice lon- gitudinal and transverse components of the magnetization and its higher moments. Following Sa` Barreto et al. [8, 9]

we use a set of relations of the type Sg

0pa0T"

T

U

and

Sh0(S0a)/T"

T

U

whereb"1/(k

B¹),a,xorzspecifies the components of the spin operatorsp*aandS+a, andn"1, 2, 3 is the order of the moment; g

0 and h

0represent arbitrary functions of spin variables exceptpa0andS0a. Tr

p0(or Tr

S0) means the partial trace with respect to the p-sublattice site 0 (or S-sublattice site 0), andS2Tdenotes the canonical ther- mal average.

The equations (4) and (5) neglect the fact thatH

0andH@ do not commute. Therefore, they are not exact for an Ising system in a transverse field. Nevertheless, they have been successfully applied to a number of interesting transverse Ising systems. We emphasize that in the Ising limit (C1"C2"0), the Hamiltonian contains only p;* and S;+ terms. Then relations (4) and (5) become exact identities.

a) Order parameters and phase diagrams

Next we have to evaluate the partial traces on the right- hand sides of (4) and (5) over the states of the selected spins labelled 0. To do this, one can either find the eigenstates of H0p and HS0 in a representation in which p; and S; are diagonal, or, more conveniently, one makes use of a coor- dinate rotation [13] which turns the Hamiltonians H0p

andHS0into diagonal form. Forg 0"h

0"1 we obtain ka,Spa0T"Sfa(E

S,C1)T, (6)

m/a,S(S0a)/T"SF/a(E

p,C2)T, (7)

with f;(E

S,C1)"E 2ES

1

tanh

A

B

F1;(Ep,C2)"E 2Ep

2

3 sinh

A

B

A

B

cosh

A

B

A

B

F2;(E

p,C2)" 1 4(E2)2

] 3(3(E

p)2#(bC2)2) cosh

A

B

A

B

cosh

A

B

A

B

(10) F3;(E

p,C2)" E 8(Ep

2)3 ]

3(9(E

p)2#7(bC2)2) sinh

A

B

A

B

cosh

A

B

A

B

(11) with

ES"b ; +*

J0*S;*, E 1"((E

S)2#(bC1)2)1@2, Ep"b+;

+

J0+p;+, E 2"((E

p)2#(bC2)2)1@2,

where z is the nearest-neighbour coordination number.

The corresponding results for the transverse components Sp90TandS(S09)/Tmay be obtained from the longitudinal components by interchangingE

S/bandC1in (6) forSp90T, andE

p/b andC2in (7) forS(S09)/T.

In order to perform the thermal averaging on the right-hand sides of (8)—(11), we expand the functions fa(ES,C1) andF/a(Ep,C2) as finite polynomials ofS;*and p;+ that correctly account for the single-site kinematic relations. This can conveniently be done by employing the generalized van der Waerden operators [23]

fa(E

S,C1)"<

*

O(S)(S;*)fa(E

S,C1), (12)

F/a(E

p,C2)"<

+

O(p)(p;+)Fa/(E

p,C2), (13)

with

O(p)(p;+)"(p;+#1 2)^&&dp^;+",¹2

#(!p;+#1 2)&&&"

dp^;+,^~12 , (14)

624

O(S)(S;*)"1

48[!3!2S;*#12(S;*)2#8(S;*)3]^&&dS;*^",³2

#1

16[9#18S;*!4(S;*)2!8(S;*)3]^&&dS^;*",¹2

#1

16[9!18S;*!4(S;*)2#8(S;*)3]^&&dS^;*",^~12

#1

48[!3#2S;*#12(S;*)2!8(S;*)3]^&&dS^;*",^~32

(15) wheredoA,!is a forward Kronecker delta-function substitu- ting any operator A to the right by its eigenvaluea. In order to carry out the average over all spin configurations involved in (6) and (7), we have to deal with multiple spin correlation functions. In this work, we consider the simplest approximation by neglecting correlations between quant- ities pertaining to different sites. Doing this we find Sfa(E

S,C1)T"<Z

*/1

`32

+ S^;*/^~32

P(S;*) fa(E

S,C1), (16)

SF/a(E

p,C2)T" Z

<

+/1

`12

+ p;+/^~12

Q(p;+) F/a(E

p,C2), (17) with

P(S*Z)"

`32

+ -/^~32

a(l)dS^Z*,l, (18)

Q(p+Z)"

`12

+ ,/^~12

b(k)dp^Z+,,, (19) where

a(^$3 2 )"1

48[!3G2m1;#12m2;$8m3;], (20) a(^$1

2 )"1

16[9$18m1;!4m2;G8m3;], (21) b(^$1

2 )"(1

2$k;). (22)

Thus, using the probability distributions we obtain the following set of coupled equations forkaandm/a

ka"

`32

+ -¹/^~32

2

`32

+ -^;/^~32

C

a(l*)

D

]fa(S;*(l 1),S2;(l

2),2,S;;(l

;)), (23)

m/a"

`12

+ ,¹/^~12

2

`12

+ ,^;/^~12

C

b(k+)

D

]F/a(p;*(k1),p2;(k2) ,2,p;;(k;)), (24) with S;*(l)"l and p;*(k)"k. We like to note that these equations can be solved directly by numerical iteration without further algebraic calculations. This treatment has successfully been used in the study of other systems [24].

Since the value ofSand the total number of loopszare relatively large, the combined sum in (23) extends over a large number, (2S#1)z, of terms, leading to quite long computational times, particularly near second-order phase transitions. Therefore, it is advantageous to carry

out further algebraic manipulations on (16) and (17) employing the differential operator technique

fa(E

S,C1)"Exp(E SD

9) fa(x,C1)D9?0, (25)

F/a(E

p,C2)"Exp(E pD

9)F/a(x,C2)D9?0. (26) or the integral representation

fa(E

S,C1)":dxd(x!E

S) fa(x,C1), (27) F/a(E

p,C2)":dxd(x!E

p)F/a(x,C2), (28) with the delta-function

d(x)"1

2n:dye*:9. (29)

Choosing the differential operator approach we obtain from (16), (17), (25) and (26)

ka"

`32

+ l/^~32

a(l) elbJD9

Z

fa(x,C1)D9/0, (30)

m/a"

`12

+ ,/^~12

b(k) e,bJD9

Z

F/a(x,C2)D9/0. (31)

Using the multinomial expansion we find ka" +Z

/1/0 Z~/1

/2+/0

Z~/1~/2

/3+/0

3~Z`/<sup>2</sup>`/³16~ZC;/1CZ~//2 ¹CZ~//3 ¹~/² ](!3#2m1;#12m2;!8m3;)/1

](9!18m1;!4m2;#8m3;)/2

](9#18m1;!4m2;!8m3;)/3

](!3!2m1;#12m2;#8m3;)Z~/1~/2~/3

]fa

A

B

and m/a"+Z

//0

C/;

A

B

A

B

A

B

(33) whereC1/are the binomial coefficientsn!/p!(n!p)!. The iteration process of these equations becomes suitable for the study of the present system even in the vicinity of the critical temperature.

b) Internal energy and specific heat

Next we calculate the internal energy º of the system described by Hamiltonian (1)

º"!N 2bSE

pS0;T!N

2 C1Sp90T!N

2 C2SS09T, (34) whereNis the total number of magnetic atoms. In order to evaluate SE

pS0;T, we substitute h 0"E

p in (5) with

ka" Z + /¹/0

Z~/1

+ /²/0

Z~/1~/2

+ /³/0

/1

+ -¹/0

/1~-1

+ ,¹/0

/2

+ -²/0

/2~-2

+ ,²/0

/3

+ -³/0

/2~-3

+ ,³/0

Z~/1~/2~/3

+ -⁴/0

Z~/1~/2~/3~/4

+ ,⁴/0

(!1)-¹`-<sup>3</sup>`,²`,<sup>4</sup> ]2~4;`,¹`,<sup>2</sup>`,³`,<sup>4</sup>`3(-¹`-<sup>2</sup>`-³`-<sup>4</sup>). 3/<sup>1</sup>~/<sup>3</sup>~,<sup>1</sup>`2,²`2,³~,⁴~-¹~-⁴

]C;/¹C;~//² 1C;~//³ 1~/2C/-¹1C/,1¹~-1C/-²2C/,2²~-2C/-²3C/,3³~-3C;~/-⁴ 1~/2~/3

]C;~/,^{4 1}~/²~/³~-⁴. (m1;),1`,2`,3`,4. (!1#4m2;);~/2~/3~,1~,4~-1~-4

](9!4m2;)/2`/3~,2~,3~-2~-3. (m3;)-1`-2`-3`-4.fa

A

B

and m/a"+;

//0

;~/+ ,¹/0

+/ ,²/0

C/;C;~/,¹ C/,²

A

B

A

B

n"1 anda"z. Then it can be written as SE

pS0;T"SU(E

p,C2)T, (35)

whereU(E

p,C2)"E pF1;(E

p,C2). Following the same pro- cedure as for the evaluation ofkaandm/a, namely using the generalized van der Waerden operator (13) and the single- site probability distribution functionQ(p;+) given by (19), we find

SE

pS0;T"

`12

+ ,¹/^~12

2

`12

+ ,^;/^~12

C

b(k*)

D

]U(p;*(k1),p2;(k2),2,p;;(k;)). (36) withp;*(k)"k. Employing the differential operator tech- nique (36) can be written in the form:

SE

pS0;T"bJ 2

+Z //0

C/;

A

B

A

B

]F1;

A

B

Finally, for evaluating the internal energyº we have to usekaandm/a(a"x,zandn"1, 2, 3) which quantities are given by (32) and (33).

Using the expression ofº(¹,C1,C2), we determine the magnetic contribution to the specific heat via the standard relation

C"Lº(¹,C1,C2)

L¹ (38)

3. Results and discussions

First, we investigate the phase diagram of the system. At high temperature the longitudinal magnetizations k;,m1; and m3; are zero. Below a transition temperature ¹

# we have spontaneous ordering,k;O0,m1;O0,m3;O0, while the corresponding transverse magnetization momentsk9, m19 and m39 are unequal zero at all temperatures. To calculate¹

#it is preferable to expand the right-hand sides of (32) and (33) with respect to m1; andm3;(ork;). Doing this we find

For thez-components (a"z), they can be written in the following form

k;"A(bJ,C1,m2;)m1;#B(bJ,C1,m2;)m3;#2, (41) m1;"A

1(bJ,C2)k;#B

1(bJ,C2) [k;]3#2, (42) m3;"A

3(bJ,C2)k;#B

3(bJ,C2) [k;]3#2, (43) whereA,B,A

*,B

*(i"1, 2,2) are obtained from (39) and (40) by choosing the appropriate corresponding combina- tions of indices k

* and l

* (i"1, 2,2). Retaining only terms linear ink;,m1;andm3;, the transition temperature is obtained from the equation

1"A(bJ,C1,m2#; ) .A1(bJ,C2)#B(bJ,C1,m2#; ) .A3(bJ,C2), (44) where m2#; is the solution of equation (40) for k;P0, namely

m2#;" ; + //0

C/;

A

B

A

B

Figure 1 shows the phase diagrams in (¹,C) space for the case of a uniformly applied transverse field C1"C2"C and coordination numbers appropriate for the honey- comb (z"3), square (z"4) and simple cubic (z"6) latti- ces. WhenC"0, the present system reduces to the two- sublattice mixed spin-1

2and spin-3

2Ising system discussed in [25]. As is seen from the figure, the critical temperature decreases monotonously from its value in the mixed Ising system,¹

#(C"0), to vanish at some critical value of the transverse field strength. For various coordination num- bers the values¹

#(C"0) andC#are collected in Table 1.

For comparison, we have also derived the phase diagrams by a simple mean-field approximation (MFA)

ka"fa(ESMFA,C1), (46) m/a"F/a(EpMFA,C2), (47) where f;andF;/are defined by (8—11) with

ES"ESMFA"bJ ; +

*/1SS

*T and E

p"EpMFA"bJ ; + +/1Sp+T,

626

Fig. 1. Phase diagrams in¹!Cplane for uniformly applied trans- verse fields (C1"C2"C) for coordination numbersz"3, 4, 6. Solid (dashed) lines correspond to results obtained from effective field theory (mean field theory)

Table 1

Coordination kB¹

#(C"0) C#/J

number

MFA EFT Exact [26] MFA EFT

Z"3 1.678 1.235 0.919 2.599 1.813

Z"4 2.237 1.794 — 3.465 2.675

Z"6 3.355 2.908 — 5.197 4.394

and f9andF9/are given by

f9"f;(C1,ESMFA), (48) F9/"F;/(C2,EpMFA). (49) As shown in the table and Fig. 1, the effective field theory remarkably improves the MFA results, since the former method neglects only correlations between different spin variables.

Furthermore, it is interesting to investigate the phase diagrams of the system with transverse fields C1 and C2 taking different values. The variation of the critical tem- perature with the transverse fieldC2/J, keepingC1fixed, is obtained from the solution of (44). Results for the case of the honeycomb lattice (z"3) are shown in Fig. 2. As expected, the critical field strength C#2 above which the longitudinal components of magnetizations vanish is greater (lower) than C# if the fixed value of C1 is lower (greater) than C#"1.813J. Therefore the critical phase boundary line (dashed line) exhibits a large deviation from that of uniformly applied fields (solid line). We like to note that the same comments can be made if we investigate the phase diagrams in the¹!C1plane for different values of the transverse fieldC2.

Fig. 2. The phase diagram in ¹!C2 plane for the honeycomb lattice (z"3) for different fixed transverse fieldsC1(dashed lines):

(a) C1"0, (b) C1"J, (c) C1"1.5J, (d) C1"2J, (e) C1"3J. The solid line indicates the results for uniformly applied transverse fields (C1"C2"C)

Fig. 3. The temperature dependence of longitudinal magnetizations (k;,m;1,M

;) for the honeycomb lattice (z"3) in zero transverse field and in uniformly applied fields (C1"C2"C)

Next, we investigate the temperature dependence of the longitudinal and transverse components of the sublat- tice magnetizationskaand m1a (a"x,z) as well as those corresponding to the total magnetizationM

a defined by Ma"(ka#m1a)/2. In addition, we study the behaviour of the higher moments, namelym/awithn"2, 3 The thermal

Fig. 4. The temperature dependence of the transverse magnetiz- ationsk9andm91aandM

9bfor the honeycomb lattice (z"3) in uniformly applied transverse fields (C1"C2"C)

dependence ofkaandm/amay be obtained from (23, 24) or (32, 33). In Fig. 3, we show typical results for the longitu- dinal magnetizationsk;,m1;andM

;, for the case of zero transverse fields (C1"C2"C"0) and uniformly applied transverse fields (C1"C2"C"J). The results are shown for a honeycomb lattice, but those for other lattices exhibit the same qualitative behaviour. The results for the trans- verse components k9, m19 and M

9 are shown in Fig. 4 wherek9"m19"M

9"0 forC"0. Thus it is clear, and expected, that the transverse fieldCdecreases the values of k;,m1;andM

;. In Fig. 5 results are shown for longitudinal and transverse quadrupolar moments for zero transverse field and for C1"C2"C"J. For small value of C, m2; (m29) decreases (increases) rapidly from its zero temper- ature value till the transition temperature¹

# is reached.

Above¹

#the quadratic moments vary slightly and tend to their high temperature asymptotic value5

4. In Fig. 6 the temperature dependence of the longitudinal and trans- verse higher order momentsm39andm3;is depicted. From

Fig. 5. The temperature dependence of the longitudinal and trans- verse quadrupolar moments (m;2andm92) for the honeycomb lattice (z"3) in zero transverse field and in uniformly applied transverse field (C1"C2"C)

Fig. 6. The temperature dependence of the longitudinal and trans- verse component of the magnetization momentsmz3andmx3for the honeycomb lattice (z"3) in zero transverse field and in uniformly applied transverse fields (C1"C2"C)

these figures, it is clear that the longitudinal components of all odd moments of the sublattice magnetizations van- ish at the same transition temperature. At low temper- ature the corresponding transverse components vary slightly with temperature, pass through a cusp at the second order transition temperature, and then fall off

628

Fig. 7. Reduced longitudinal and transverse total magnetizations M;andM9as functions of reduced temperature for the honeycomb lattice (z"3) in uniformly applied transverse fieldC1"C2"C"J.

Solid and dashed lines denote the effective-field (EFT) and mean- field approximation (MFA) results, respectively

Fig. 8. Reduced longitudinal m;2 and transversem92 quadrupolar magnetizations as functions of reduced temperature for the honey- comb lattice (z"3) in uniformly applied transverse field C1"C2"C"J. Solid and dashed lines denote the effective-field (EFT) and mean-field approximation (MFA) results, respectively

keeping a finite value in the whole temperature range. In comparison to corresponding results of the mixed spin-1 and spin-1 transverse Ising model we would like to note2 that the findings for the present mixed spin-1

2and spin-3 system are quite similar. 2

In order to complete the comparison between our EFT results and the MFA predictions, we have depicted in Figs. 7—9 the temperature dependence of the reduced

Fig. 9. Reduced longitudinal and transverse components of the magnetizationm;3andm93as functions of reduced temperature for the honeycomb lattice (z"3) in uniformly applied transverse field C1"C2"C"J. Solid and dashed lines denote the effective-field (EFT) and mean-field approximation (MFA) results, respectively

Fig. 10. The temperature dependence of the internal energyºand the magnetic specific heatCfor the honeycomb lattice (z"3) in zero transverse field and in uniformly applied transverse fields (C1"C2"C)

total magnetizationsM

;andM

9and their reduced higher momentsm/a(a"x,zandn"2, 3) in a uniform transverse field C1"C2"C. As is seen from the figures, the EFT results are quantitatively different from the MFA behav- iour. For instance, in the ordered phase and for¹near¹

#, M9andm39 are increasing functions with temperature in contrast to the opposite behaviour predicted by the MFA.

We like to note that such a behaviour ofM

9obtained by

the EFT is also found by other sophisticated methods in the spin-1

2transverse Ising model [27, 28].

In Fig. 10 the temperature dependence of the internal energyºand the magnetic specific heatCis depicted for the honeycomb lattice for zero transverse field (C1"C2"C"0) and for uniformly applied transverse field (C1"C2"C) fixed at some typical values. As is seen from the figure the absolute value of the internal energy of the systems increases with increasing strength of the trans- verse field showing the same qualitative behaviour as in the pure spin Ising model [29] and the transverse Ising models in [18]. Finally, the magnetic contribution to the specific heat is remarkably depressed by increasing the transverse field strengthC. It is also clear that the ampli- tude of the jump at the transition temperature gradually decreases with increasing value ofC. We like to note that at low temperatures the magnetic specific heat of the mixed spin system investigted here and presumably of all mixed spin systems (1) is sensitive to the strength of the transverse fieldCin contrast to the insensitive behaviour observed in the pure spin transverse Ising systems [18].

One of us (N.B) acknowledges the hospitality of the Institut fu¨r Theoretische Physik der Universita¨t zu Ko¨ln.

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