- Email: [email protected]
Ising chain with random short- and long-range interactions
Journal of Magnetism and Magnetic Materials 177 181 (1998) 76 78
~
Jeurnal of
magnetism and magnetic , 4 ~ materials
ELSEVIER
Ising chain with random short- and long-range interactions L.L. Gonqalves*, A.P. Vieira Departamento de F/s/ca, Universidade Federal do Cearh, Campus do Pici, Caixa Postal 6030, 60.451-970 Fortaleza, Cear6, Brazil
Abstract An exact solution is obtained for the 1D lsing model, closed chain, with random short- and long-range interactions satisfying b/modal annealed distributions. The long-range interactions satisfy a strongly correlated distribution, and the distribution of the short-range interactions J~ is explicitly given by ~3(J~) = pO(J i JA) -}- (1 -- P)g)(Ji -- JB). F o r the particular case of uniform ferromagnetic long-range interactions explicit results are presented for the thermodynamic functions. It is shown that for dilute ant/ferromagnetic or for competing short-range interactions the spontaneous magnetization of the system can increase with temperature. It is also shown that the magnetization, as a function of temperature, can exhibit discontinuities, in addition to the usual first- and second-order phase transitions shown by the pure model (p = 0 or 1). ;2 1998 Elsevier Science B.V. All rights reserved. -
-
Keywords." Ising model; One-dimensional systems; Random systems; Phase transitions
In the last decades much effort has been put into the theoretical study of disordered systems. Although many exact results exist for 1D models, our knowledge of higher-dimensional random systems has come mainly from computer simulations or approximation methods, since those models are extremely hard to treat exactly. On the other hand, it is possible to simulate higher-dimensional behaviour in 1D systems by adding a mean-field term to the Hamilton/an. In this paper we use that fact to solve exactly a random-bond 1D Ising model with short- and long-range interactions which, indeed, presents some higher-dimensional features, in particular, the existence of a spontaneous magnetization with a very rich behaviour. In order to treat the model exactly at all temperatures, we have chosen the annealed limit of disorder, since it retains some aspects of the more realistic quenched limit while being more tractable mathematically. The Hamilton/an of the model in its more general form (closed chain) is written as N
H =-
i=1~ Ji~rj~r.i+ 1
1 -- ~j,
N
N
~= 1 lcrjak-- j ~= 1 haj,
(1)
where Jj = njJA + (1 -- nj)JB =- njAJ + JB and I = t l l + (1 --t)12 with t and each nj taking independently the values 0 or 1. By introducing the pseudo-chemical potentials/O,/~1 and #2 in order to adjust the b/modal annealed distributions ~)(Jj) = Pb(JJ JA) q- (1 - p)iJ(Jj - Jn) and ~,)(I) = q16(I - 11) + q26(I - I2), with ql + q2 = 1, the grand partition function is written as (fi = 1/k~T) -
Z =
~
-
e~[~,2~, ,,~+,,t+,~(1-,)-m.
(2)
By performing a decimation on the variables nj, Eq. (2) can be written in the form 2
z
= ru ~ i
(3)
ziQ . 1
with Qi =
2
e [ 2 " i '/~ . . . . . . . (1/U)~'ik
~I,,~,~,+2'~ ,h~,],
(4)
i,rl where zi = exp (tip/),/£ = flJ~ + K', Ii =fili, ff = fih and r and K' are obtained by solving the system z - e ~"~
*Corresponding author. Fax: + 55 85 281 4570; e-mail: [email protected].
magnetic
=
sinh K' sinh( fl AJ - K')'
sinh fl AJ r = z - sinh K'
(5)
As in the non-random model [1], we can express the partition functions Q/ in terms of m/, the
0304-8853/98/$19.00 ~2 1998 Elsevier Science B.V. All rights reserved PII S 0 3 0 4 - 8 8 5 3 ( 9 7 ) 0 0 3 0 6 - 5
L.L. Gonfalves, A,P. Vieira/Journal o f Magnetism and Magnetic Materials 177 18l (1998) 76 78
magnetization of a model with uniform long-range interactions I~: /~. = / ~ + 2[imi.
sinh/7i
(6)
1.0 JA=0, p = 0 . 3
0.8
c
mi = j s i n h Z / 7 i + e 4~' In the thermodynamic limit (N --+ oo ) we obtain In Qi = N(ln 2i -//m/2),
0.6
(7)
a--
d
!
m 0 0.4
~b/i
1
where 0.2
)., = et~[cosh/7i + x/sink2/7/+ e -4~ ]
77
c(= 1.45
b
or- 1.50
c.
c~= 1.52
d--
ct- 1.54
e
ct= 1.57
f ....
ct- 1.63
(8)
This turns the problem into a nearest-neighbour Ising system in an effective strongly correlated random field [2], and, by imposing the conditions q~ = ( t ) and q2 = ( 1 - - t), we get z~ = q~Z/Q~ for i = 1,2. Therefore, the magnetization per lattice site can be written as
0.0
O.O
012
014
0.6
0.8 0
1.0
1.2
1.4
Fig. 1. Spontaneous magnetization mo versus renormalized temperature 0-= k~T./IJl~[ for dilute short-range interactions with p = 0.3 and various values of ~ --- I/Jl~.
2
m -= ( ~ j ) = Z
qimi,
(9)
i=1
and the Helmholtz free energy per s p i n f i s given by [3] 2
flf = - In r + p In z - ~ qi0n 2i
-
-
(10)
Iimi),
i=1
which, for given field and temperature, is a function ofmx, m2 and z. We can determine the equilibrium states by minimizing the free energy with respect to m~ and m2. From the distribution of the short-range interactions we can write
p = (nj) = x + ya(m),
(11)
where Z
Z
x--cosh(flAJ-
K'), y - - s i n h ( f i A J -
r
r
K'),
(12)
2
G(m) = (ajaj+ l)
=
~,
qiGi(mi)
(13)
i=1
and
_\ li2 2e-4g.(sinh2/7/+ e 4K) Gi(mi) = 1
( cosh~+
)1i2'
(14)
sinh z l T i + e 4Iz
By solving Eqs. (6) and (11) for ml, m2 and K, we can obtain the thermodynamics of the system. F o r the particular case of uniform long-range interactions (I1 = I 2 - = I), which we discuss hereafter, we have m~ = m2 and the sums in Eqs. (9), (I0) and (13) are eliminated. In order to have competition between interactions, we assume JB to be antiferromagnetic (JB < 0), JA to be null or ferromagnetic (JA >~ 0) and the longrange interaction I to be ferromagnetic (I > 0).
The spontaneous magnetization mo is obtained by taking the limit h--*0 in Eq. (9). By looking at the ground-state energy we can show that for T = 0 we have
,no={~
ifl<2,J~i/(l+p),
(15)
if I > 2]JBJ/(1 + p), irrespective of the value of JA >/ 0. These results agree with those obtained for the case JA = --JB > 0 and p = ~ by Paladin et al. in a recent paper [4], where the mo = p states are denoted 'ferrimagnetic'. Their results also show that the spontaneous magnetization can increase with temperature in a certain range. For I < 2[JBI/(I + p) the spontaneous magnetization of the system presents some interesting features, as can be seen in Fig. 1, which shows mo as a function of the reduced temperature 0 - kBT/tJB] for dilute antiferromagnetic short-range interactions and various values of ~ -= I/JB. For ]~l < ~o _~ 1.538 the ground state is ferrimagnetic. For 1.496 < ]~l < ~ the spontaneous magnetization exhibits two discontinuities as the temperature increases. This is a consequence of the competition between interactions and of the random dilution of the antiferromagnetic short-range interactions JB, which favours the ordering of some clusters in the chain due to the action of the (weaker) ferromagnetic mean-field I. The qualitative behaviour of the magnetization in some perovskite-like compounds, such as Z n l ~GaxMn3C (x ~< 1) and C u M n 3 N [5], is similar to that of the present model. The ?,xm isotherms (7 -= h/lJR]) are obtained by solving Eq. (9) for h at a fixed temperature. By looking at the ground-state energy at a given field and JA ~> 0, the T = 0 isotherm can be shown to have the form {~ Iml =
ifl7l<2-(l+p),~,, if I?'1> 2 - (1 + p)lc~l,
(16)
78
L.L. Gonfalves, A.P. Vieira /Journal of Magnetism and Magnetic Materials 177-181 (1998) 76 78
for I~[ < 2/(1 + p), whereas for I~l > 2/1 + p the absolute value of the magnetization is equal to unity. As usual, the sign ofm is the same as the sign ofT. F r o m Eq. (16) we see that the isotherms can exhibit discontinuities at zero and non-zero values of the field, a behaviour which persists at low values of temperature. This work was partially financed by the Brazilian agencies C N P q and F I N E P .
References [-1] A.P. Vieira, L.L. Gonqalves, Condens. Matter Phys. (Ukraine) 5 (1995) 210. [2] L.L. Gonqalves, J. Magn. Magn. Mater. 140 144 (1995) 2653. [3] See, e.g., K. Huang, Statistical Mechanics, 2nd. ed., Wiley, New York, 1987, p. 143. [4] G. Paladin, M. Pasquini, M. Serva, J. Phys. I (France) 4 (1994) 1597. [5] P. L'Heritier, PhD Thesis, INPG, Grenoble, France, 1980.
~
Jeurnal of
magnetism and magnetic , 4 ~ materials
ELSEVIER
Ising chain with random short- and long-range interactions L.L. Gonqalves*, A.P. Vieira Departamento de F/s/ca, Universidade Federal do Cearh, Campus do Pici, Caixa Postal 6030, 60.451-970 Fortaleza, Cear6, Brazil
Abstract An exact solution is obtained for the 1D lsing model, closed chain, with random short- and long-range interactions satisfying b/modal annealed distributions. The long-range interactions satisfy a strongly correlated distribution, and the distribution of the short-range interactions J~ is explicitly given by ~3(J~) = pO(J i JA) -}- (1 -- P)g)(Ji -- JB). F o r the particular case of uniform ferromagnetic long-range interactions explicit results are presented for the thermodynamic functions. It is shown that for dilute ant/ferromagnetic or for competing short-range interactions the spontaneous magnetization of the system can increase with temperature. It is also shown that the magnetization, as a function of temperature, can exhibit discontinuities, in addition to the usual first- and second-order phase transitions shown by the pure model (p = 0 or 1). ;2 1998 Elsevier Science B.V. All rights reserved. -
-
Keywords." Ising model; One-dimensional systems; Random systems; Phase transitions
In the last decades much effort has been put into the theoretical study of disordered systems. Although many exact results exist for 1D models, our knowledge of higher-dimensional random systems has come mainly from computer simulations or approximation methods, since those models are extremely hard to treat exactly. On the other hand, it is possible to simulate higher-dimensional behaviour in 1D systems by adding a mean-field term to the Hamilton/an. In this paper we use that fact to solve exactly a random-bond 1D Ising model with short- and long-range interactions which, indeed, presents some higher-dimensional features, in particular, the existence of a spontaneous magnetization with a very rich behaviour. In order to treat the model exactly at all temperatures, we have chosen the annealed limit of disorder, since it retains some aspects of the more realistic quenched limit while being more tractable mathematically. The Hamilton/an of the model in its more general form (closed chain) is written as N
H =-
i=1~ Ji~rj~r.i+ 1
1 -- ~j,
N
N
~= 1 lcrjak-- j ~= 1 haj,
(1)
where Jj = njJA + (1 -- nj)JB =- njAJ + JB and I = t l l + (1 --t)12 with t and each nj taking independently the values 0 or 1. By introducing the pseudo-chemical potentials/O,/~1 and #2 in order to adjust the b/modal annealed distributions ~)(Jj) = Pb(JJ JA) q- (1 - p)iJ(Jj - Jn) and ~,)(I) = q16(I - 11) + q26(I - I2), with ql + q2 = 1, the grand partition function is written as (fi = 1/k~T) -
Z =
~
-
e~[~,2~, ,,~+,,t+,~(1-,)-m.
(2)
By performing a decimation on the variables nj, Eq. (2) can be written in the form 2
z
= ru ~ i
(3)
ziQ . 1
with Qi =
2
e [ 2 " i '/~ . . . . . . . (1/U)~'ik
~I,,~,~,+2'~ ,h~,],
(4)
i,rl where zi = exp (tip/),/£ = flJ~ + K', Ii =fili, ff = fih and r and K' are obtained by solving the system z - e ~"~
*Corresponding author. Fax: + 55 85 281 4570; e-mail: [email protected].
magnetic
=
sinh K' sinh( fl AJ - K')'
sinh fl AJ r = z - sinh K'
(5)
As in the non-random model [1], we can express the partition functions Q/ in terms of m/, the
0304-8853/98/$19.00 ~2 1998 Elsevier Science B.V. All rights reserved PII S 0 3 0 4 - 8 8 5 3 ( 9 7 ) 0 0 3 0 6 - 5
L.L. Gonfalves, A,P. Vieira/Journal o f Magnetism and Magnetic Materials 177 18l (1998) 76 78
magnetization of a model with uniform long-range interactions I~: /~. = / ~ + 2[imi.
sinh/7i
(6)
1.0 JA=0, p = 0 . 3
0.8
c
mi = j s i n h Z / 7 i + e 4~' In the thermodynamic limit (N --+ oo ) we obtain In Qi = N(ln 2i -//m/2),
0.6
(7)
a--
d
!
m 0 0.4
~b/i
1
where 0.2
)., = et~[cosh/7i + x/sink2/7/+ e -4~ ]
77
c(= 1.45
b
or- 1.50
c.
c~= 1.52
d--
ct- 1.54
e
ct= 1.57
f ....
ct- 1.63
(8)
This turns the problem into a nearest-neighbour Ising system in an effective strongly correlated random field [2], and, by imposing the conditions q~ = ( t ) and q2 = ( 1 - - t), we get z~ = q~Z/Q~ for i = 1,2. Therefore, the magnetization per lattice site can be written as
0.0
O.O
012
014
0.6
0.8 0
1.0
1.2
1.4
Fig. 1. Spontaneous magnetization mo versus renormalized temperature 0-= k~T./IJl~[ for dilute short-range interactions with p = 0.3 and various values of ~ --- I/Jl~.
2
m -= ( ~ j ) = Z
qimi,
(9)
i=1
and the Helmholtz free energy per s p i n f i s given by [3] 2
flf = - In r + p In z - ~ qi0n 2i
-
-
(10)
Iimi),
i=1
which, for given field and temperature, is a function ofmx, m2 and z. We can determine the equilibrium states by minimizing the free energy with respect to m~ and m2. From the distribution of the short-range interactions we can write
p = (nj) = x + ya(m),
(11)
where Z
Z
x--cosh(flAJ-
K'), y - - s i n h ( f i A J -
r
r
K'),
(12)
2
G(m) = (ajaj+ l)
=
~,
qiGi(mi)
(13)
i=1
and
_\ li2 2e-4g.(sinh2/7/+ e 4K) Gi(mi) = 1
( cosh~+
)1i2'
(14)
sinh z l T i + e 4Iz
By solving Eqs. (6) and (11) for ml, m2 and K, we can obtain the thermodynamics of the system. F o r the particular case of uniform long-range interactions (I1 = I 2 - = I), which we discuss hereafter, we have m~ = m2 and the sums in Eqs. (9), (I0) and (13) are eliminated. In order to have competition between interactions, we assume JB to be antiferromagnetic (JB < 0), JA to be null or ferromagnetic (JA >~ 0) and the longrange interaction I to be ferromagnetic (I > 0).
The spontaneous magnetization mo is obtained by taking the limit h--*0 in Eq. (9). By looking at the ground-state energy we can show that for T = 0 we have
,no={~
ifl<2,J~i/(l+p),
(15)
if I > 2]JBJ/(1 + p), irrespective of the value of JA >/ 0. These results agree with those obtained for the case JA = --JB > 0 and p = ~ by Paladin et al. in a recent paper [4], where the mo = p states are denoted 'ferrimagnetic'. Their results also show that the spontaneous magnetization can increase with temperature in a certain range. For I < 2[JBI/(I + p) the spontaneous magnetization of the system presents some interesting features, as can be seen in Fig. 1, which shows mo as a function of the reduced temperature 0 - kBT/tJB] for dilute antiferromagnetic short-range interactions and various values of ~ -= I/JB. For ]~l < ~o _~ 1.538 the ground state is ferrimagnetic. For 1.496 < ]~l < ~ the spontaneous magnetization exhibits two discontinuities as the temperature increases. This is a consequence of the competition between interactions and of the random dilution of the antiferromagnetic short-range interactions JB, which favours the ordering of some clusters in the chain due to the action of the (weaker) ferromagnetic mean-field I. The qualitative behaviour of the magnetization in some perovskite-like compounds, such as Z n l ~GaxMn3C (x ~< 1) and C u M n 3 N [5], is similar to that of the present model. The ?,xm isotherms (7 -= h/lJR]) are obtained by solving Eq. (9) for h at a fixed temperature. By looking at the ground-state energy at a given field and JA ~> 0, the T = 0 isotherm can be shown to have the form {~ Iml =
ifl7l<2-(l+p),~,, if I?'1> 2 - (1 + p)lc~l,
(16)
78
L.L. Gonfalves, A.P. Vieira /Journal of Magnetism and Magnetic Materials 177-181 (1998) 76 78
for I~[ < 2/(1 + p), whereas for I~l > 2/1 + p the absolute value of the magnetization is equal to unity. As usual, the sign ofm is the same as the sign ofT. F r o m Eq. (16) we see that the isotherms can exhibit discontinuities at zero and non-zero values of the field, a behaviour which persists at low values of temperature. This work was partially financed by the Brazilian agencies C N P q and F I N E P .
References [-1] A.P. Vieira, L.L. Gonqalves, Condens. Matter Phys. (Ukraine) 5 (1995) 210. [2] L.L. Gonqalves, J. Magn. Magn. Mater. 140 144 (1995) 2653. [3] See, e.g., K. Huang, Statistical Mechanics, 2nd. ed., Wiley, New York, 1987, p. 143. [4] G. Paladin, M. Pasquini, M. Serva, J. Phys. I (France) 4 (1994) 1597. [5] P. L'Heritier, PhD Thesis, INPG, Grenoble, France, 1980.