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Spin-12 Ising model in a staggered magnetic field
Volume 140, number 5
PHYSICS LETTERS A
25 September 1989
SPIN-1/2 ISING MODEL IN A STAGGERED MAGNETIC FIELD A. BENYOUSSEF Laboratoire de Magnétisme, Département de Physique, Faculté des Sciences, Av. Ibn Batouta, BP 1014, Rabat, Morocco
T. BIAZ and M. TOUZANI Ecole Normale Supérieure, Takad~oum,BP 5118, Rabat, Morocco Received 17 April 1989; revised manuscript received 25 July 1989; accepted for publication 26 July 1989 Communicated by A.A. Maradudin
A study of the nearest-neighbour ferromagnetic spin-i Ising model in a staggered magnetic field on the square lattice is presented. Using the transfer matrix on finite strips and finite size scaling analysis the critical line is obtained. In this system no tricnticalbehaviour is found.
The ferromagnetic spin-i Ising model with a staggered magnetic field has been studied by the mean field approximation [11; the phase diagram exhibits a first order transition, a second order transition, and a tricritical point. The purpose of this paper is to study this model by a more accurate method, namely the finite size scaling method. The magnetic field is directed up on one row, and down on the next and so on. Hence the Hamiltonian
c~
of the system is H —J ~
~
—H 0
(~ ~
lea
—
~
(1)
,
zeb
where o~=±1 are the Ising spins and J>0, the first sum is over all nearest-neighbours and the two last sums are over spins of different sublattices with opposite sign of the magnetic field H0. This model can • be obtained from a restricted SOS model [21, and is also equivalent to an anisotropic Ising model with uniform magnetic field; indeed this can be obtained by the change of all spins {aI},b to {—a,},b. Within the finite size scaling method [3], we calculate exactly the correlation length of the model on a strip of infinite length and width n, the exact calculation is performed with the transfer matrix. The transfer is taken as shown in fig. I. To preserve for •
258
•
•
~
t)
t)
~4)
((L)
(~
‘ni,
H.
-
H~ 0
-
H
te~~)
-t)
V ~e(’)
0\ o
Fig. 1. The transfer is pe~ormedfrom the sate ~(I) to the state ço(J+ I) with cyclic boundary condition on each row, n is even, and H0 >0 (H0 <0) denotes the sign of the magnetic field.
each strip two independent sublattices only even values for strip widths are considered. The correlation length is given by [3] ‘~“
=
log (2~/2~)
(2)
where 4’~and 2~’~ are the largest and the next largest eigenvalues of the transfer matrix performed on a strip of width n. By using the usual assumption of the finite size scaling,
0375-960 1/89/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 140, number 5
PHYSICS LETTERS A
n’~,,(T~(Ho))=(n+2Y’~+2(T0(Ho)),
25 September 1989
(3)
1.
the critical line is determined in the temperature— magnetic field (T/J—H0/J) plane. It is well known that the convergence of the method is very accurate, so the greatest size we use is 8. The results are shown
inThe fig. critical 2. Nightingale critical exponent line is[4], exponent determined of the correlation Y~,following (Y~=liv, length) thewhere argument along v is the of
H0 (4) j dT
1.
2.
(5)
,~Yr—l
Fig. 3. Critical exponent ~T as a function of H0( T~)for different strip widths n/rn=2/4 (—), 4/6 (- - -), and 6/8 (—‘—~—~—).
and hence YT =
0.
log(n/rn) log fnd~’/dT\ ~md~’ /dT) _______
(6)
where rn=n+2. The values of Y~fordifferent strips widths (2, 4), (4, 6) and (6, 8) are shown in fig. 3. At H0=0 Y~goesto the exact value Y~-=1.0 [5] for increasing strip widths. Using the criterion of the persistence length [6],
T
2.
Tablel The scaled persistence length (~,,/n)calculated on the critical line as determined with sizes n/rn =4/6, and 6/8. H0
n/m=4/6
n/rn=6/8
~/4
~~s/6
!6/6
0.0 0.5 1.0 1.5
0.167127 0.180063 0.216637 0.306830
0.162696 0.170323 0.197078 0.272322
0.162748 0.170473 0.197620 0.274202
0.161180 0.167677 0.191507 0.262013
1.9 1.95 1.97 1.99 1.995
0.780984 1.23815 1.77441 4.02084 6.93639
0.687763 1.09048 1.56305 3.54176 6.1 1068
0.697226 1.10755 1.58992 3.60379 6.25256
0.662202 1.05181 1.51000 3.42234 5.93886
we show that the whole critical line is second order (see table 1). The persistence length given by
i,,,
1.
H0
fer matrix, 1is a measure of the free energy between (7) where
0.
______________________
0.
I
2.
Fig. 2. Line of critical points T~(Ho)for different strip widths n/m=2/4 (—), 4/6 (- - -) and 6/8 (—~—~—~—).
~
is the third largest eigenvalue of the trans-
the different phases which coexist at the first order transition line. It is calculated along the critical line (as obtained by (3)). If the scaled persistence length (~0/n)decreases when n increases the transition is second order, but if it is an increasing function of n the transition is first order. 259
Volume 140, numberS
PHYSICS LETTERS A
It can be seen from fig. 3 that for nonvanishing H0 there is a crossing point at a given value of H0. Before this crossing point the exponent decreases with increasing size, while after, it increases with increasing size. This feature together with the behaviour of the persistence length on the critical line suggests that for the thermodynamic limit the exponent is presumably the same over the whole critical line and it is equal to the exact value at JI~=0 (say 1.0). In conclusion, we have performed a phenomenological finite size scaling analysis of the spin-i Ising model under a staggered magnetic field. This study shows that the predictions of the mean field approximation are not correct. One of the authors (M.T.) has carried out a part of the numerical computation in the Dipartimento
260
25 September 1989
di Fisica, Università di Genova, Italy with the financial support of the “I.C.T.P. Programme for training and research in Italian Laboratories, Tneste, Italy”.
References [l]J.M. KincaidandE.G.D. Cohen, Phys. Rep. 22(1975) 57. [2] M. Touzani and M. Wortis, Phys. Rev. B 36 (1987) 3598. [3] M.N. Barber, in: Phase transitions and critical phenomena, eds. C. Domb and J. Lebowitz (Academic Press, New York, 1984), and references therein. [4J P. Nightingale, J. Appl. Phys. 53 (1982) 7927. [5] P. Nightingale, Physica A 83 (1976) 561. [6]P.D.Beale,Phys.Rev.B33 (1986) 1717; M.G. Grynberg and H. Ceva, Phys. Rev. B 36 (1987) 7091: P.A. Rikvold, W. Kinzel, J. Gunton and K. Kaski, Phys. Rev. B 28 (1983) 2686.
PHYSICS LETTERS A
25 September 1989
SPIN-1/2 ISING MODEL IN A STAGGERED MAGNETIC FIELD A. BENYOUSSEF Laboratoire de Magnétisme, Département de Physique, Faculté des Sciences, Av. Ibn Batouta, BP 1014, Rabat, Morocco
T. BIAZ and M. TOUZANI Ecole Normale Supérieure, Takad~oum,BP 5118, Rabat, Morocco Received 17 April 1989; revised manuscript received 25 July 1989; accepted for publication 26 July 1989 Communicated by A.A. Maradudin
A study of the nearest-neighbour ferromagnetic spin-i Ising model in a staggered magnetic field on the square lattice is presented. Using the transfer matrix on finite strips and finite size scaling analysis the critical line is obtained. In this system no tricnticalbehaviour is found.
The ferromagnetic spin-i Ising model with a staggered magnetic field has been studied by the mean field approximation [11; the phase diagram exhibits a first order transition, a second order transition, and a tricritical point. The purpose of this paper is to study this model by a more accurate method, namely the finite size scaling method. The magnetic field is directed up on one row, and down on the next and so on. Hence the Hamiltonian
c~
of the system is H —J ~
~
—H 0
(~ ~
lea
—
~
(1)
,
zeb
where o~=±1 are the Ising spins and J>0, the first sum is over all nearest-neighbours and the two last sums are over spins of different sublattices with opposite sign of the magnetic field H0. This model can • be obtained from a restricted SOS model [21, and is also equivalent to an anisotropic Ising model with uniform magnetic field; indeed this can be obtained by the change of all spins {aI},b to {—a,},b. Within the finite size scaling method [3], we calculate exactly the correlation length of the model on a strip of infinite length and width n, the exact calculation is performed with the transfer matrix. The transfer is taken as shown in fig. I. To preserve for •
258
•
•
~
t)
t)
~4)
((L)
(~
‘ni,
H.
-
H~ 0
-
H
te~~)
-t)
V ~e(’)
0\ o
Fig. 1. The transfer is pe~ormedfrom the sate ~(I) to the state ço(J+ I) with cyclic boundary condition on each row, n is even, and H0 >0 (H0 <0) denotes the sign of the magnetic field.
each strip two independent sublattices only even values for strip widths are considered. The correlation length is given by [3] ‘~“
=
log (2~/2~)
(2)
where 4’~and 2~’~ are the largest and the next largest eigenvalues of the transfer matrix performed on a strip of width n. By using the usual assumption of the finite size scaling,
0375-960 1/89/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 140, number 5
PHYSICS LETTERS A
n’~,,(T~(Ho))=(n+2Y’~+2(T0(Ho)),
25 September 1989
(3)
1.
the critical line is determined in the temperature— magnetic field (T/J—H0/J) plane. It is well known that the convergence of the method is very accurate, so the greatest size we use is 8. The results are shown
inThe fig. critical 2. Nightingale critical exponent line is[4], exponent determined of the correlation Y~,following (Y~=liv, length) thewhere argument along v is the of
H0 (4) j dT
1.
2.
(5)
,~Yr—l
Fig. 3. Critical exponent ~T as a function of H0( T~)for different strip widths n/rn=2/4 (—), 4/6 (- - -), and 6/8 (—‘—~—~—).
and hence YT =
0.
log(n/rn) log fnd~’/dT\ ~md~’ /dT) _______
(6)
where rn=n+2. The values of Y~fordifferent strips widths (2, 4), (4, 6) and (6, 8) are shown in fig. 3. At H0=0 Y~goesto the exact value Y~-=1.0 [5] for increasing strip widths. Using the criterion of the persistence length [6],
T
2.
Tablel The scaled persistence length (~,,/n)calculated on the critical line as determined with sizes n/rn =4/6, and 6/8. H0
n/m=4/6
n/rn=6/8
~/4
~~s/6
!6/6
0.0 0.5 1.0 1.5
0.167127 0.180063 0.216637 0.306830
0.162696 0.170323 0.197078 0.272322
0.162748 0.170473 0.197620 0.274202
0.161180 0.167677 0.191507 0.262013
1.9 1.95 1.97 1.99 1.995
0.780984 1.23815 1.77441 4.02084 6.93639
0.687763 1.09048 1.56305 3.54176 6.1 1068
0.697226 1.10755 1.58992 3.60379 6.25256
0.662202 1.05181 1.51000 3.42234 5.93886
we show that the whole critical line is second order (see table 1). The persistence length given by
i,,,
1.
H0
fer matrix, 1is a measure of the free energy between (7) where
0.
______________________
0.
I
2.
Fig. 2. Line of critical points T~(Ho)for different strip widths n/m=2/4 (—), 4/6 (- - -) and 6/8 (—~—~—~—).
~
is the third largest eigenvalue of the trans-
the different phases which coexist at the first order transition line. It is calculated along the critical line (as obtained by (3)). If the scaled persistence length (~0/n)decreases when n increases the transition is second order, but if it is an increasing function of n the transition is first order. 259
Volume 140, numberS
PHYSICS LETTERS A
It can be seen from fig. 3 that for nonvanishing H0 there is a crossing point at a given value of H0. Before this crossing point the exponent decreases with increasing size, while after, it increases with increasing size. This feature together with the behaviour of the persistence length on the critical line suggests that for the thermodynamic limit the exponent is presumably the same over the whole critical line and it is equal to the exact value at JI~=0 (say 1.0). In conclusion, we have performed a phenomenological finite size scaling analysis of the spin-i Ising model under a staggered magnetic field. This study shows that the predictions of the mean field approximation are not correct. One of the authors (M.T.) has carried out a part of the numerical computation in the Dipartimento
260
25 September 1989
di Fisica, Università di Genova, Italy with the financial support of the “I.C.T.P. Programme for training and research in Italian Laboratories, Tneste, Italy”.
References [l]J.M. KincaidandE.G.D. Cohen, Phys. Rep. 22(1975) 57. [2] M. Touzani and M. Wortis, Phys. Rev. B 36 (1987) 3598. [3] M.N. Barber, in: Phase transitions and critical phenomena, eds. C. Domb and J. Lebowitz (Academic Press, New York, 1984), and references therein. [4J P. Nightingale, J. Appl. Phys. 53 (1982) 7927. [5] P. Nightingale, Physica A 83 (1976) 561. [6]P.D.Beale,Phys.Rev.B33 (1986) 1717; M.G. Grynberg and H. Ceva, Phys. Rev. B 36 (1987) 7091: P.A. Rikvold, W. Kinzel, J. Gunton and K. Kaski, Phys. Rev. B 28 (1983) 2686.