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Disordered magnet near the percolation threshold
Volume 103A, number 5
PHYSICS LETTERS
9 July 1984
DISORDERED MAGNET N E A R THE PERCOLATION THRESHOLD ~r Michael J STEPHEN Serm Physws Laboratory, Rutgers Umverszty, l~scataway, NJ 08854, USA Recewed 2 March 1984 Revased manuscrapt recewed 16 Apral 1984
A disordered Ports magnet containing a random mixture of ferromagnetic exchange constants Ja and Jb (Ja "~Jb) near the percolation threshold is considered The scahng form for the free energy contams two crossover exponents Duahty arguments m two dtmenslons show that these exponents are equal They are also shown to be equal to umty m d = 6 - e dimensions to order e
In this letter we consider a disordered Potts magnet near the percolation threshold On each site of a regular lattice we place a q component spin X~ Nearest neighbor luke and unluke spins t a n d / h a v e an interaction energy - e t ! and zero respectively The eu are assumed to randomly take on two values e a and e h (e a "~ % ) with probabxhtles 1 - p and p respectwely This model is the magnetac analog of the random resistor network [1] with two types of conductors o a and o b (ora "~ % ) Two special cases are o f lmterest (a) e a = 0, e b > 0 This case has been stu&ed extensively near the percolation threshold by a variety of methods* 1 For p < Pc the magnetization is zero and the model can exhibit ferromagnetism when an infinite cluster of ferromagnetic bonds forms above Pc (b) e a > 0, e b = ~ Thts is the magnetic analog o f the resistor network with a maxture of normal and superconducting bonds The magnetic case does not appear to have been studied For p < Pc the lnfimte exchange constants form fimte clusters and T c is finite, the model being ferromagnetic for T < T c and paramagnetlc for T > T c As the percolation threshold is approached from below T c goes to mfimty The phase diagram is sketched m hg l a Above Pc an mfmlte cluster of mfimte exchange constants exists and the model as ferromagnetic at all temperatures Supported m part by the National Science Foundation under Grant No DMR-81-0615 t .1 For a review see ref [2]
0
.To/ kT ~
ORDER
0~
tel
P
!
. . . . . .
DS IORDER\ 0
a)
0
b)
Fag 1 (a) Phase dmgram for a random mtxture of infinite and finite exchange constants (b) Cntmal surface for a random magnet with two exchange constants The percolation points are 0 and 0' 270
0 375-9601/84/$ 03 00 © Elsevier Science Publishers B V (North-Holland Physics Pubhshlng Division)
Volume 103A, number 5
PHYSICS LETTERS
9 July 1984
This problem is of physical interest In several ways (1) Most reahzatlons o f a dilute magnet will actually have a weak but non-zero coupling for the bonds that In case (a) are set exactly equal to zero (a 0 Case (b) can be approximated by a material in which the ground state of the ions is non magnetic ( J = 0) but there IS a magnetic state above the ground state ( J > 0) Such a situation is quite common in salts of the rare earths Pr and Tb In this case the concentration p of magnetic ions would be temperature dependent In two dimensions this model on a square lattice is self dual The dual o f a bond (= is
Ke ee/kT)
e x p ( - K~) = [exp(Ke) - 1] / [exp(Kq) + q - 1] Let Ka, b = rara* = rbr
(1)
ea,b/kTand introduce the variables Fa, b = [exp(Ka,b) -
1]/x/q The dual relationship is
(2)
-- 1
On the square lattice the free energy per site satisfies F ( F a, 1 - p, Fb, p) = F(Fa*, 1 - p, F~, p) + const
(3)
For p = 1 the critical line is F b = 1 and for p = 0 It is F a = 1 and F a = F b --- 1 is a critical line for all p A sketch of the cntlcal surface is given in fig l b We expect a transition characteristic of the Potts model from an ordered to a disordered state on this surface It is second order for q < 4 Along the hnes 1-'a = 0, F b = co percolation behavior will be observed at the percolation point 0 (or 0') In the vicinity o f this point where F a ~ < 1 and F g 1 x/q e x p ( - Kb) "~ 1 we expect scaling behavior which describes the crossover from Potts-hke behavior to percolation behavior We write the singular part of the free energy in the form (r 0 ~ p - Pc)
Ka/x/q
Fs,ng ~ Ir0 [ 2 - a f ( F a / I r 0 I~'l, Pb Ir0l ~')
(4)
Two crossover exponents ~"and ~'1 have been introduced Duality Imphes that ~'1 = ~"in two dimensions A similar relation has been obtained in two dimensions for resistor networks [3,1] a is a percolation exponent On the critical surface the function f is singular lri order to reproduce Potts behavior In order to develop a mean field theory and e expansion we introduce n replicas of the Potts model and consider the generating function
Z=[eXP~nnK,l~6x,~,xm+h~6xt(~,l)] , 1,(~
(5)
C
where a = 1, , n IS a replica index and we have included a field acting on spins in state 1 The square brackets with subscnpt c indicate an average over all configurations This average leads to a pair hamlltonian which we define by [v = p)]
p/(1 -
exp(-Htl)=exp(Ka ~J6x" h e , to, ,c~]l+°exp(Kb~ 6hio,,xl~)
(6)
As K b N K a we have
-Hq=Ka~'x,c,,x,c~+ln[l+uexp(Kb~6X,~,h,c~)] The log term can be expanded as follows
In[l+ o
=
exp(Kb~g~,c~,~,a)]=~(--1)l+lolexp(Kbl~,htcoX, a 1 - - -)- ~ - - - / =
~( 1)/+1 / =~(_ t--~-v l-I (1 + [exp(Kb/) -- 1]6x,~,x/a) l=1 1=1 a
1),+1o / ~ l
(
exp(Kbl)-I 1 +q-- ] +
q~l
r
-r)
exp(Kbl)r=1 X'aXia
'
(7) 271
Volume 103A, number 5
PHYSICS LETTERS
9 July 1984
where we have used a representation in which the Xt~ take on values equal to the q roots of unity so that q-1 ~X,x,=q-1 (1+~ ~.r~.'-r 1 r=l l and we have omitted terms which vanish as n + 0 On expanding the product in (7) and using K b >> 1, the pair hamlltoman can be written -Hq =A 1 ~ ar
r
-r r l r 2 a<3
where A k = - l n ( 1 - p) + (Ka/q)6k, 1 - koq e x p ( - Kb)
(9)
The pair hamlltonlan (8) shows that m the mean held approximation the crmcal line is zA 1 = 1 where z is the number o f nearest neighbors The percolation point is - z ln(1 - Pc) = 1 so that close to Pc the critical hne is [z/(1 - Pc)] (P - Pc) + ZKa/q - [ZPc/(1 - Pc)] q e x p ( - Kb) = 0
(10)
F o r K b = oo this is o f the form shown in fig 1 RegardlngK a and e x p ( - Kb) as independent variables eq (10) also mdmates that m mean field theory ~"= ~'1 = 1 In order to obtain the e expansion for the critical exponents we go over into a continuum form by applying the H u b b a r d - S t r a t o n o v l t c h transformation to (8) The continuum hamlltonlan can be written
- u = ½1P, Izq~[ 2(r01 + q 2 ) + ½ ~ qar
qr
~ IZ(qr)el2(r02 + q 2 ) + O ( z 3 ) , c~<13
(11)
where rOk ~ (-Pc -- p)/(1 -- Pc) -- ( g a / q ) S k , 1 + kqo e x p ( - Kb) and z}r) etc are conjugate to the xra in (8) A sxmllar h a m d t o m a n has been obtained by Stephen and Grest [4] in the Ismg case and Dasgupta et al [5] m the Potts case These authors have determined the recurslon relations for the rOk to first order m e = 6 - d In the limit n -+ 0 these recurslon relations show that all the rOk scale in the same way which Implies that ~"= ~'1 = 1 + O(e 2) A simple argument based on the S k a l - S h k l o v s k l [6] and de Gennes [7] nodes and links model can be given for the exponent ~'1 Below Pc the large couphng constants make up fxmte clusters Some of these clusters link up and form the infinite cluster at Pc Let us focus attention on those bonds which eventually link these clusters to form the infinite cluster Let II(p) be the probability that one of these bonds is occupied by an infinite couphng constant II(Pc) = 1 so that II(p) = 1 - (Pc - P)II'(Pc) The average inverse coupling constant is [ K - 1] c = (Pc - p ) I I ' / K a + 1 - (Pc - p ) I I ' / K b ~ (Pc -- P)/Ka which shows that the crossover exponent ~'1 = 1 I am grateful to Professors A Conlgho and H E Stanley for a useful conversation I wish to thank J Straley for several suggestions and for fig l b [1] J.P Straley, Phys. Rev B15 (1977) 5733. [2] D Stauffer Phys. Rep 54 (1979) 1, J.W. Essam, Rep Prog Phys 43 (1980) 833 [3] A M Dykne, Zh Eksp Teor Flz 59 (1971) 110 [Sov Phys JETP 32 (1971) 63] [4] M J Stephen and G S Grest, Phys Rev Lett 38 (1977) 567 [5] C Dasgupta, A B. Hams and T.C Lubensky, Phys Rev B17 (1978) 1375 [6] A S Ska/and B I. Shklovsku, FIz Tekh Poluprov 8 (1974) 1586 [Sov. Phys Semlcond 8 (1974) 1029] [7] P G de Gennes, J Phys (Paris) Lett 37 (1976) L1 272
PHYSICS LETTERS
9 July 1984
DISORDERED MAGNET N E A R THE PERCOLATION THRESHOLD ~r Michael J STEPHEN Serm Physws Laboratory, Rutgers Umverszty, l~scataway, NJ 08854, USA Recewed 2 March 1984 Revased manuscrapt recewed 16 Apral 1984
A disordered Ports magnet containing a random mixture of ferromagnetic exchange constants Ja and Jb (Ja "~Jb) near the percolation threshold is considered The scahng form for the free energy contams two crossover exponents Duahty arguments m two dtmenslons show that these exponents are equal They are also shown to be equal to umty m d = 6 - e dimensions to order e
In this letter we consider a disordered Potts magnet near the percolation threshold On each site of a regular lattice we place a q component spin X~ Nearest neighbor luke and unluke spins t a n d / h a v e an interaction energy - e t ! and zero respectively The eu are assumed to randomly take on two values e a and e h (e a "~ % ) with probabxhtles 1 - p and p respectwely This model is the magnetac analog of the random resistor network [1] with two types of conductors o a and o b (ora "~ % ) Two special cases are o f lmterest (a) e a = 0, e b > 0 This case has been stu&ed extensively near the percolation threshold by a variety of methods* 1 For p < Pc the magnetization is zero and the model can exhibit ferromagnetism when an infinite cluster of ferromagnetic bonds forms above Pc (b) e a > 0, e b = ~ Thts is the magnetic analog o f the resistor network with a maxture of normal and superconducting bonds The magnetic case does not appear to have been studied For p < Pc the lnfimte exchange constants form fimte clusters and T c is finite, the model being ferromagnetic for T < T c and paramagnetlc for T > T c As the percolation threshold is approached from below T c goes to mfimty The phase diagram is sketched m hg l a Above Pc an mfmlte cluster of mfimte exchange constants exists and the model as ferromagnetic at all temperatures Supported m part by the National Science Foundation under Grant No DMR-81-0615 t .1 For a review see ref [2]
0
.To/ kT ~
ORDER
0~
tel
P
!
. . . . . .
DS IORDER\ 0
a)
0
b)
Fag 1 (a) Phase dmgram for a random mtxture of infinite and finite exchange constants (b) Cntmal surface for a random magnet with two exchange constants The percolation points are 0 and 0' 270
0 375-9601/84/$ 03 00 © Elsevier Science Publishers B V (North-Holland Physics Pubhshlng Division)
Volume 103A, number 5
PHYSICS LETTERS
9 July 1984
This problem is of physical interest In several ways (1) Most reahzatlons o f a dilute magnet will actually have a weak but non-zero coupling for the bonds that In case (a) are set exactly equal to zero (a 0 Case (b) can be approximated by a material in which the ground state of the ions is non magnetic ( J = 0) but there IS a magnetic state above the ground state ( J > 0) Such a situation is quite common in salts of the rare earths Pr and Tb In this case the concentration p of magnetic ions would be temperature dependent In two dimensions this model on a square lattice is self dual The dual o f a bond (= is
Ke ee/kT)
e x p ( - K~) = [exp(Ke) - 1] / [exp(Kq) + q - 1] Let Ka, b = rara* = rbr
(1)
ea,b/kTand introduce the variables Fa, b = [exp(Ka,b) -
1]/x/q The dual relationship is
(2)
-- 1
On the square lattice the free energy per site satisfies F ( F a, 1 - p, Fb, p) = F(Fa*, 1 - p, F~, p) + const
(3)
For p = 1 the critical line is F b = 1 and for p = 0 It is F a = 1 and F a = F b --- 1 is a critical line for all p A sketch of the cntlcal surface is given in fig l b We expect a transition characteristic of the Potts model from an ordered to a disordered state on this surface It is second order for q < 4 Along the hnes 1-'a = 0, F b = co percolation behavior will be observed at the percolation point 0 (or 0') In the vicinity o f this point where F a ~ < 1 and F g 1 x/q e x p ( - Kb) "~ 1 we expect scaling behavior which describes the crossover from Potts-hke behavior to percolation behavior We write the singular part of the free energy in the form (r 0 ~ p - Pc)
Ka/x/q
Fs,ng ~ Ir0 [ 2 - a f ( F a / I r 0 I~'l, Pb Ir0l ~')
(4)
Two crossover exponents ~"and ~'1 have been introduced Duality Imphes that ~'1 = ~"in two dimensions A similar relation has been obtained in two dimensions for resistor networks [3,1] a is a percolation exponent On the critical surface the function f is singular lri order to reproduce Potts behavior In order to develop a mean field theory and e expansion we introduce n replicas of the Potts model and consider the generating function
Z=[eXP~nnK,l~6x,~,xm+h~6xt(~,l)] , 1,(~
(5)
C
where a = 1, , n IS a replica index and we have included a field acting on spins in state 1 The square brackets with subscnpt c indicate an average over all configurations This average leads to a pair hamlltonian which we define by [v = p)]
p/(1 -
exp(-Htl)=exp(Ka ~J6x" h e , to, ,c~]l+°exp(Kb~ 6hio,,xl~)
(6)
As K b N K a we have
-Hq=Ka~'x,c,,x,c~+ln[l+uexp(Kb~6X,~,h,c~)] The log term can be expanded as follows
In[l+ o
=
exp(Kb~g~,c~,~,a)]=~(--1)l+lolexp(Kbl~,htcoX, a 1 - - -)- ~ - - - / =
~( 1)/+1 / =~(_ t--~-v l-I (1 + [exp(Kb/) -- 1]6x,~,x/a) l=1 1=1 a
1),+1o / ~ l
(
exp(Kbl)-I 1 +q-- ] +
q~l
r
-r)
exp(Kbl)r=1 X'aXia
'
(7) 271
Volume 103A, number 5
PHYSICS LETTERS
9 July 1984
where we have used a representation in which the Xt~ take on values equal to the q roots of unity so that q-1 ~X,x,=q-1 (1+~ ~.r~.'-r 1 r=l l and we have omitted terms which vanish as n + 0 On expanding the product in (7) and using K b >> 1, the pair hamlltoman can be written -Hq =A 1 ~ ar
r
-r r l r 2 a<3
where A k = - l n ( 1 - p) + (Ka/q)6k, 1 - koq e x p ( - Kb)
(9)
The pair hamlltonlan (8) shows that m the mean held approximation the crmcal line is zA 1 = 1 where z is the number o f nearest neighbors The percolation point is - z ln(1 - Pc) = 1 so that close to Pc the critical hne is [z/(1 - Pc)] (P - Pc) + ZKa/q - [ZPc/(1 - Pc)] q e x p ( - Kb) = 0
(10)
F o r K b = oo this is o f the form shown in fig 1 RegardlngK a and e x p ( - Kb) as independent variables eq (10) also mdmates that m mean field theory ~"= ~'1 = 1 In order to obtain the e expansion for the critical exponents we go over into a continuum form by applying the H u b b a r d - S t r a t o n o v l t c h transformation to (8) The continuum hamlltonlan can be written
- u = ½1P, Izq~[ 2(r01 + q 2 ) + ½ ~ qar
qr
~ IZ(qr)el2(r02 + q 2 ) + O ( z 3 ) , c~<13
(11)
where rOk ~ (-Pc -- p)/(1 -- Pc) -- ( g a / q ) S k , 1 + kqo e x p ( - Kb) and z}r) etc are conjugate to the xra in (8) A sxmllar h a m d t o m a n has been obtained by Stephen and Grest [4] in the Ismg case and Dasgupta et al [5] m the Potts case These authors have determined the recurslon relations for the rOk to first order m e = 6 - d In the limit n -+ 0 these recurslon relations show that all the rOk scale in the same way which Implies that ~"= ~'1 = 1 + O(e 2) A simple argument based on the S k a l - S h k l o v s k l [6] and de Gennes [7] nodes and links model can be given for the exponent ~'1 Below Pc the large couphng constants make up fxmte clusters Some of these clusters link up and form the infinite cluster at Pc Let us focus attention on those bonds which eventually link these clusters to form the infinite cluster Let II(p) be the probability that one of these bonds is occupied by an infinite couphng constant II(Pc) = 1 so that II(p) = 1 - (Pc - P)II'(Pc) The average inverse coupling constant is [ K - 1] c = (Pc - p ) I I ' / K a + 1 - (Pc - p ) I I ' / K b ~ (Pc -- P)/Ka which shows that the crossover exponent ~'1 = 1 I am grateful to Professors A Conlgho and H E Stanley for a useful conversation I wish to thank J Straley for several suggestions and for fig l b [1] J.P Straley, Phys. Rev B15 (1977) 5733. [2] D Stauffer Phys. Rep 54 (1979) 1, J.W. Essam, Rep Prog Phys 43 (1980) 833 [3] A M Dykne, Zh Eksp Teor Flz 59 (1971) 110 [Sov Phys JETP 32 (1971) 63] [4] M J Stephen and G S Grest, Phys Rev Lett 38 (1977) 567 [5] C Dasgupta, A B. Hams and T.C Lubensky, Phys Rev B17 (1978) 1375 [6] A S Ska/and B I. Shklovsku, FIz Tekh Poluprov 8 (1974) 1586 [Sov. Phys Semlcond 8 (1974) 1029] [7] P G de Gennes, J Phys (Paris) Lett 37 (1976) L1 272