- Email: [email protected]
Exact zero-field susceptibility of the Ising model on a Cayley tree
Journal of Magnetism and Magnetic Materials 177 181 (1998) 185-187
~,~ Journalof amnadgn etlsm magnetic ~ a materials
ELSEVIER
Exact zero-field susceptibility of the Ising rfiodel on a Cayley tree Tatijana Stogi~ a, Borko D. Sto~i~ a'b'*, Ivon P. Fittipaldi b aLaboratory for Theoretical Physics, Institute for Nuclear Sciences, Vin{a, P.O. Box 522, YU-11001 Belgrade, Yugoslavia bLaborat6rio de Fisica Te6rica e Computacional, Departamento de Fisica, Universidade Federal de Pernambuco, 50670-901 Recife-PE, Brazil
Abstract
We establish an exact expression, in closed form, for the zero-field susceptibility of the Ising model on a Cayley tree of arbitrary generation. We use the known exact recursion relations for the partition function to find the corresponding recursion relations for its field derivatives. These relations are then iterated for the zero-field case to yield the expression for the zero-field susceptibility. In the thermodynamic limit, our formula displays behavior in agreement with previous works. © 1998 Elsevier Science B.V. All rights reserved. Keywords: Cayley tree; Ising model; Susceptibility; Zero field
For almost four decades the Ising model on a Cayley tree has been considered one of the rare exactly solved models in statistical mechanics. The first solution was provided by Domb [1], who concluded that the Bethe approximation represents the exact solution for this model, with critical temperature kBT/J = 2/~ln(B + 1) ln(B - 1)], where kB is the Boltzmann constant, J is the nearest-neighbor interaction parameter, and B is the tree branching number (coordination number minus one). Thereafter, the Cayley tree has commonly been referred to in the literature as the Bethe lattice. Fourteen years later, Eggarter 1-2] established exact recursion relations for the partition function, concluding that the Bethe approximation does not represent the exact solution for the bulk. Instead, it applies only to the (arbitrarily small) 'interior'. Simultaneously, in independent works, Heimburg and Thomas [3] and Matsuda [4] established the absence of spontaneous magnetization at any nonzero temperature and the divergence of the zero-field susceptibility in the temperature region below kaT/J = 2/ [ln(,,/B+ 1 ) - l n ( ~ - 1 ) ] . They were followed by Miiller-Hartmann and Zittartz [5], who in turn concluded that the model undergoes a series of phase
*Corresponding author. Fax: + 55 81 271 0359; e-mail: [email protected].
transitions of all orders, the infinite-order transition corresponding to the Bethe solution. All of these works [3-5], however, perform involved approximate analysis to establish the zero-field susceptibility behavior in the thermodynamic limit. Only recently was a new exact expression for the zero-field magnetization established [6]; but, the corresponding exact expression in closed form for the zero-field susceptibility is still lacking. In this work, we establish the exact expression, in closed form, for the zero-field susceptibility of the Ising model on a Cayley tree of arbitrary generation. We use the known exact recursion relations for the partition function, established in Ref. [2], to find the corresponding recursion relations for its field derivatives. These relations are then iterated for the zero-field case, yielding a closed form expression for the zero-field susceptibility. We consider the nearest-neighbor Ising model with the Hamiltonian ovt~=- J ~ (nn)
S i S j - H Z S~,
(1)
i
where J is the coupling constant, H is the external magnetic field, Si = + 1 is the spin at site i, and ( n n ) denotes summation over the nearest-neighbor pairs. Following Eggarter [2], we further consider systems situated on a single n-generation branch of a Cayley tree, composed of B (n - 1)-generation branches connected to a single
0304-8853/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved PII S 0 3 0 4 - 8 8 5 3 ( 9 7 ) 0 0 3 2 9 - 6
186
Z Sto}i6 et al. / Journal of Magnetism and Magnetic Materials 177-181 (1998) 185 187
initial site (for simplicity, hereafter we consider only the tree with B = 2). Thus, the n-generation branch consists of N, = 2" +1 _ 1 spins, the 0-generation branch being a single spin. The recursion relations for the partial partition functions of any two consecutive generation branches are easily derived [2] to be Z++l = y ± l [ x ± l Z + ~- x-T-1Zn ] 2,
(2)
where Z.+ and Z.- denote the partition functions restricted by fixing the initial spin (connecting the two n-generation branches) into the { + } and { - } position, respectively, and where we have used the notation x - exp(flJ) and y = exp(fiH), with fl = 1/kBT denoting the reciprocal of the product of the Boltzmann constant kB and the temperature T. The usual approach of attempting to establish the field-dependent expression for the partition function in the thermodynamic limit, then finding its field derivatives, and finally taking the limit H - - . 0 yields only approximate solutions for the zerofield susceptibility [3-5]. In order to derive closed-form expressions for zero-field magnetization and susceptibility, we follow the strategy of finding the recursion relations for the field derivatives of the partition function, taking the limit H ~ 0, and only then performing the actual iterations to reach the thermodynamic limit. It is precisely this reversal in the order of taking limits that makes it possible to find the exact solution. By differentiating (2) with respect to field, the recursion relations for the field derivatives of the partition function are easily found to be
symmetry equations, 5Z + 5fl~-
Z, + = Z ; ,
5Z n aflH'
~2Zn+ 52Zn~(flH 2 ) - 8(fill) z'
(5)
are satisfied for a branch of arbitrary generation. Finally, for the zero-field case, it is straightforward to show from Eqs. (2)-(5) that the ratios 5P, -
1 8Z + Z 2 5flH
and
1 52Z +
Y, -
Z+ 8(fill) 2
satisfy the recursion relations, ~.+1 = 1 + 2tJ.,
50o = 1,
,~,+1 = 1 + 4 t Y , + 2 t 2 y
2+2J.,
Jo=
1,
(6)
where t -= tanh(fiJ). Relations (Eq. (6)) can be iterated (by summing the geometric series) to yield closed-form expressions for 5~, and Y-, for arbitrary tree generation n. F o r the restricted magnetization, defined as 1 (m}. ± =
1 5Z.+ - - -
N. z . + ~#H
+ Y.
N.
(7)
'
and for the zero-field susceptibility,
z. = ~
2. e ( f m 2
\ z . ~fli-i/
= ~. J " '
(s)
we thus find. +
~Z++ 8fill 1 _ y ± l' rL -~- (Zn+x±l -}- Zn-XT1)2
=
+
2n+lt n+l - 1 (2 "+1 - 1 ) ( 2 t - 1)'
(9)
and
+ 2(Z+x±i
+1
_}_ZnXT-1)/~Z+
TI~]
eZn
(3)
22n+ 2 t2n+4 fi 2"4 Z, - 2,+1 _ 1 (2t 2 - 1)(2t -- 1) 2 2"+2t n+2
and 52Z++ i
~(flH) ~
+ ~( 2 t - l ) V y±l L(Zn+X-- 1 ~_ ZnX-T-1)2
(SZ+ x±
+4(Z+x ±l + z;xT1)\eflH / 82+ + 2 ~ x
+1
eZn +~-~x
+2(z.+x ±1 + z z ~ ; b \ ~
5Zn
l +~x
2t2--1 +(2t-1)
(2t + 1)~2") 2
2t 2 -
"
(10)
Eq. (9) is identical to the one obtained in Ref. [6] using a different approach, while the general expression (Eq. (10)) for the zero-field susceptibility has not yet been reported in the literature. F o r n>> 1, Eq. (10) reduces to
g l"~
)
fi(t + 1)2/[(i - 2t2)];
gl ~2 )
Xn = ~+~(~Tx ~ (4)
Starting from a single spin (0th generation branch), for which we have Z g = y ± l , 5 Z g / S f l H = + _ y ± l , and ~2Z~/SflH2 = y± ~, it is easily shown by mathematical induction using Eqs. (2)-(5), that for zerofield (y = 1) the
2t 2 < 1,
fl(2f2)n+z/[(2t 2 __ 1)(2t -- 1)2]; 2t 2 > 1,
(11)
which is in agreement with the findings of Refs. [3 5]. To be more precise, performing an approximate analysis, this asymptotic behavior of X was correctly deduced for 2t 2 < 1 in Refs. [3, 5], and its divergence for 2t 2 > 1 was noted in all three works [3 5], although the exact general limiting expression given in Eq. (11) was not established. In conclusion, in this work we have derived the exact expression for the zero-field susceptibility of the Ising
Sto}i~ et aL /Journal of Magnetism and Magnetic Materials 17~181 (1998) 185-187
model on a Cayley tree, representing one of the few exact known solutions of this kind. Considering the simplicity of our derivation, it seems curious that this problem has evaded solution for almost four decades of continued interest.
This work was supported in part by the C N P q (Brazilian Agency).
187
References [1] [2] [3] [4] [5]
C. Domb, Adv. Phys. 9 (1960) 145; 81 (1984) 3088. T.P. Eggarter, Phys. Rev. B 9 (1974) 2989. J. von Heimburg, H. Thomas, J. Phys. C 7 (1974) 3433. H. Matsuda, Prog. Theor. Phys. 5"1 (1974) 1053. E. Miiller-Hartmann, J. Zittartz, Phys. Rev. Lett. 33 (1974) 893. [6] R. Mdlin, J.C. Angl6s d'Auriac, P. Chandra, B. Dougot , preprint, cond-mat/9509035.
~,~ Journalof amnadgn etlsm magnetic ~ a materials
ELSEVIER
Exact zero-field susceptibility of the Ising rfiodel on a Cayley tree Tatijana Stogi~ a, Borko D. Sto~i~ a'b'*, Ivon P. Fittipaldi b aLaboratory for Theoretical Physics, Institute for Nuclear Sciences, Vin{a, P.O. Box 522, YU-11001 Belgrade, Yugoslavia bLaborat6rio de Fisica Te6rica e Computacional, Departamento de Fisica, Universidade Federal de Pernambuco, 50670-901 Recife-PE, Brazil
Abstract
We establish an exact expression, in closed form, for the zero-field susceptibility of the Ising model on a Cayley tree of arbitrary generation. We use the known exact recursion relations for the partition function to find the corresponding recursion relations for its field derivatives. These relations are then iterated for the zero-field case to yield the expression for the zero-field susceptibility. In the thermodynamic limit, our formula displays behavior in agreement with previous works. © 1998 Elsevier Science B.V. All rights reserved. Keywords: Cayley tree; Ising model; Susceptibility; Zero field
For almost four decades the Ising model on a Cayley tree has been considered one of the rare exactly solved models in statistical mechanics. The first solution was provided by Domb [1], who concluded that the Bethe approximation represents the exact solution for this model, with critical temperature kBT/J = 2/~ln(B + 1) ln(B - 1)], where kB is the Boltzmann constant, J is the nearest-neighbor interaction parameter, and B is the tree branching number (coordination number minus one). Thereafter, the Cayley tree has commonly been referred to in the literature as the Bethe lattice. Fourteen years later, Eggarter 1-2] established exact recursion relations for the partition function, concluding that the Bethe approximation does not represent the exact solution for the bulk. Instead, it applies only to the (arbitrarily small) 'interior'. Simultaneously, in independent works, Heimburg and Thomas [3] and Matsuda [4] established the absence of spontaneous magnetization at any nonzero temperature and the divergence of the zero-field susceptibility in the temperature region below kaT/J = 2/ [ln(,,/B+ 1 ) - l n ( ~ - 1 ) ] . They were followed by Miiller-Hartmann and Zittartz [5], who in turn concluded that the model undergoes a series of phase
*Corresponding author. Fax: + 55 81 271 0359; e-mail: [email protected].
transitions of all orders, the infinite-order transition corresponding to the Bethe solution. All of these works [3-5], however, perform involved approximate analysis to establish the zero-field susceptibility behavior in the thermodynamic limit. Only recently was a new exact expression for the zero-field magnetization established [6]; but, the corresponding exact expression in closed form for the zero-field susceptibility is still lacking. In this work, we establish the exact expression, in closed form, for the zero-field susceptibility of the Ising model on a Cayley tree of arbitrary generation. We use the known exact recursion relations for the partition function, established in Ref. [2], to find the corresponding recursion relations for its field derivatives. These relations are then iterated for the zero-field case, yielding a closed form expression for the zero-field susceptibility. We consider the nearest-neighbor Ising model with the Hamiltonian ovt~=- J ~ (nn)
S i S j - H Z S~,
(1)
i
where J is the coupling constant, H is the external magnetic field, Si = + 1 is the spin at site i, and ( n n ) denotes summation over the nearest-neighbor pairs. Following Eggarter [2], we further consider systems situated on a single n-generation branch of a Cayley tree, composed of B (n - 1)-generation branches connected to a single
0304-8853/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved PII S 0 3 0 4 - 8 8 5 3 ( 9 7 ) 0 0 3 2 9 - 6
186
Z Sto}i6 et al. / Journal of Magnetism and Magnetic Materials 177-181 (1998) 185 187
initial site (for simplicity, hereafter we consider only the tree with B = 2). Thus, the n-generation branch consists of N, = 2" +1 _ 1 spins, the 0-generation branch being a single spin. The recursion relations for the partial partition functions of any two consecutive generation branches are easily derived [2] to be Z++l = y ± l [ x ± l Z + ~- x-T-1Zn ] 2,
(2)
where Z.+ and Z.- denote the partition functions restricted by fixing the initial spin (connecting the two n-generation branches) into the { + } and { - } position, respectively, and where we have used the notation x - exp(flJ) and y = exp(fiH), with fl = 1/kBT denoting the reciprocal of the product of the Boltzmann constant kB and the temperature T. The usual approach of attempting to establish the field-dependent expression for the partition function in the thermodynamic limit, then finding its field derivatives, and finally taking the limit H - - . 0 yields only approximate solutions for the zerofield susceptibility [3-5]. In order to derive closed-form expressions for zero-field magnetization and susceptibility, we follow the strategy of finding the recursion relations for the field derivatives of the partition function, taking the limit H ~ 0, and only then performing the actual iterations to reach the thermodynamic limit. It is precisely this reversal in the order of taking limits that makes it possible to find the exact solution. By differentiating (2) with respect to field, the recursion relations for the field derivatives of the partition function are easily found to be
symmetry equations, 5Z + 5fl~-
Z, + = Z ; ,
5Z n aflH'
~2Zn+ 52Zn~(flH 2 ) - 8(fill) z'
(5)
are satisfied for a branch of arbitrary generation. Finally, for the zero-field case, it is straightforward to show from Eqs. (2)-(5) that the ratios 5P, -
1 8Z + Z 2 5flH
and
1 52Z +
Y, -
Z+ 8(fill) 2
satisfy the recursion relations, ~.+1 = 1 + 2tJ.,
50o = 1,
,~,+1 = 1 + 4 t Y , + 2 t 2 y
2+2J.,
Jo=
1,
(6)
where t -= tanh(fiJ). Relations (Eq. (6)) can be iterated (by summing the geometric series) to yield closed-form expressions for 5~, and Y-, for arbitrary tree generation n. F o r the restricted magnetization, defined as 1 (m}. ± =
1 5Z.+ - - -
N. z . + ~#H
+ Y.
N.
(7)
'
and for the zero-field susceptibility,
z. = ~
2. e ( f m 2
\ z . ~fli-i/
= ~. J " '
(s)
we thus find
~Z++ 8fill 1 _ y ± l' rL -~- (Zn+x±l -}- Zn-XT1)2
=
+
2n+lt n+l - 1 (2 "+1 - 1 ) ( 2 t - 1)'
(9)
and
+ 2(Z+x±i
+1
_}_ZnXT-1)/~Z+
TI~]
eZn
(3)
22n+ 2 t2n+4 fi 2"4 Z, - 2,+1 _ 1 (2t 2 - 1)(2t -- 1) 2 2"+2t n+2
and 52Z++ i
~(flH) ~
+ ~( 2 t - l ) V y±l L(Zn+X-- 1 ~_ ZnX-T-1)2
(SZ+ x±
+4(Z+x ±l + z;xT1)\eflH / 82+ + 2 ~ x
+1
eZn +~-~x
+2(z.+x ±1 + z z ~ ; b \ ~
5Zn
l +~x
2t2--1 +(2t-1)
(2t + 1)~2") 2
2t 2 -
"
(10)
Eq. (9) is identical to the one obtained in Ref. [6] using a different approach, while the general expression (Eq. (10)) for the zero-field susceptibility has not yet been reported in the literature. F o r n>> 1, Eq. (10) reduces to
g l"~
)
fi(t + 1)2/[(i - 2t2)];
gl ~2 )
Xn = ~+~(~Tx ~ (4)
Starting from a single spin (0th generation branch), for which we have Z g = y ± l , 5 Z g / S f l H = + _ y ± l , and ~2Z~/SflH2 = y± ~, it is easily shown by mathematical induction using Eqs. (2)-(5), that for zerofield (y = 1) the
2t 2 < 1,
fl(2f2)n+z/[(2t 2 __ 1)(2t -- 1)2]; 2t 2 > 1,
(11)
which is in agreement with the findings of Refs. [3 5]. To be more precise, performing an approximate analysis, this asymptotic behavior of X was correctly deduced for 2t 2 < 1 in Refs. [3, 5], and its divergence for 2t 2 > 1 was noted in all three works [3 5], although the exact general limiting expression given in Eq. (11) was not established. In conclusion, in this work we have derived the exact expression for the zero-field susceptibility of the Ising
Sto}i~ et aL /Journal of Magnetism and Magnetic Materials 17~181 (1998) 185-187
model on a Cayley tree, representing one of the few exact known solutions of this kind. Considering the simplicity of our derivation, it seems curious that this problem has evaded solution for almost four decades of continued interest.
This work was supported in part by the C N P q (Brazilian Agency).
187
References [1] [2] [3] [4] [5]
C. Domb, Adv. Phys. 9 (1960) 145; 81 (1984) 3088. T.P. Eggarter, Phys. Rev. B 9 (1974) 2989. J. von Heimburg, H. Thomas, J. Phys. C 7 (1974) 3433. H. Matsuda, Prog. Theor. Phys. 5"1 (1974) 1053. E. Miiller-Hartmann, J. Zittartz, Phys. Rev. Lett. 33 (1974) 893. [6] R. Mdlin, J.C. Angl6s d'Auriac, P. Chandra, B. Dougot , preprint, cond-mat/9509035.