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The Potts model on a closed cayley tree
Volume 101A, number 5,6
PHYSICS LETTERS
2 April 1984
THE POTTS MODEL ON A CLOSED CAYLEY TREE P.L. CHRISTIANO 1 Departamento de F[sica, Universidade Federal da Paraiba, 58. 000 Jo~o Pessoa Pb., Brasil
and S. GOULART ROSA Jr. 2 Instituto de F(sica e Qufmica de S~o Carlos, Caixa Postal 369, 13.560 S~o Carlos SP, Brasil Received 7 December 1983
The q-state Potts model on the hierarchical lattice introduced by Jelitto is treated exactly by a renormalization group procedure. For q > 2 the correlation function between the top and bottom spins is discontinuous at Tc(q) in the ferromagnetic model and it is zero for all temperatures in the antiferromagnetic case. On the other hand this correlation is continuous at Tc(2) in both ferro and antiferromagnetic lsing models.
In recent years one has witnessed a growing interest in the study o f spin systems on the sub-class of hierarchical lattices [1 ] known as Cayley trees. This interest which was originated because the Bethe approximation is exact on this lattice if surface effects are avoided, has now increased further since new resuits were obtained from the study of the global properties [2] of spin systems on this structure. In 1979 Jelitto [3] has introduced the so-called closed Cayley tree which is formed by joining two identical (open) trees by the spins laying in their surfaces. The main results of the studies of the ferromagnetic Ising model on the closed Cayley tree can be summarized as follows: (i) There is a phase transition even at zero magnetic field [3] and the free energy is non-differentiable [4] at the critical temperature/3cJ = 1/x/~ where 7 is the branching ratio of the tree. (ii) The correlation function between pairs of spins at special sites infinitely apart are non-zero below Tc, while for other pairs of sites the correlation function is non-zero only at T = 0 K. In this letter we study the q-state (ferro and anti-
ferromagnetic) Potts model on a closed Cayley tree with 3, = 2 by a real space renormalization group (RSRG) procedure which is exact and has the advantage of being simpler than the methods used before to treat the ferromagnetic Ising model [3,4]. For reasons which will become clear in a moment we construct the dosed Cayley tree in a slightly different manner by joining the spins at the surfaces of the two open trees via the K bonds as shown in fig. 1 so that Jelitto's construction and the open tree are recovered as special cases (K = ~ and K = 0 respectively). Let us consider the hamiltonian
° -qJ
qK
0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-HoUand Physics Publishing Division)
(1)
where the summations are carded out over all pairs (i1) of nearest neighbor sites on the same open tree and over all pairs ¢ss') on the surface of both trees. Using the representation ?~. = ¢oki, 6o = e 2 n i / q , K i = 0, 1 ..... q - 1, for the q-state Potts spin variable one can write the hamiltonian of any pair of spins at sites (il~ and their correlation function [5] ~ r x q - r ) as q-1
1 CAPES Pre-doctoral Fellow. 2 CNPq Research Fellow.
'~ 6 xsxs' '
c/j = - s
1,
r=l
)
x;
(2) 275
Volume 101A, number 5,6
PHYSICS LETTERS A
2 April 1984
u r = [2u 2Ur_ 1 + (q - 2) u4u 2_ 1]/
p-O
[1 + (q - 1) udur2_ 11 , B
where we have taken uo(K ) as the initial transmissivity in the iterative procedure and u is the transmissivity associated with the bond J. In the (thermodynamic) limit r ~ oo one obtains the following fixed points for the map given by eq. (4):
p-I
J
/
Up* = 0
(5a)
U*+ = (q - 2)/2(q - 1)
\/
±[(q - 2)2u 4 - 4 ( ( / - 1)(1 - 2u2)] 1/2
/2(q - 1)u 2 .
~m,r,3
Fig. 1. The closed Cayley tree with branching ratio "r = 2 and 3-generations.
(q - 2 ) 2 (u~l)) 4 - 4(q - 1) [1 - 2(u(1)) 2] = 0 , (kt3T(1))-i =ln([1 +(q - 1)u~l)]/[1 - u ~ l ) ] ) .
( ~ x/q-r> = (1 - e-qM)/[1 + (q - l) e - q M l ,
(5b)
Recalling that u E (0, 1) if J > 0 and u E (-1/(q - 1), 0) if J < 0 one defines u (1) as the lowest value of u such that u+_ are real, i.e.
A
(3)
where ( ) stands for the Gibbs average. The Lh.s. of eq. (3), also called the thermal transrnissivity u(J) between the external sites i and/, is the natural variable for the RSRG procedure used here [6]. In particular if we have three spin variables at the sites i, / and k connected in series by the coupling constants Ji/and Jik the equivalent transmissivity U(Jeq ) between the external spins ~/and hk is obtained as the product U(Jeq ) = u(Ji/) u(J/k ). On the other hand if the spins ~/and hi are coupled by Ji/and Kii in parallel the equivalent (dual) transmissivity between i and / is obtained by the product of the (dual) transmissivities uD(Jii) and uD(Ki/), i.e. uD(Jeq) =uD(Ji]) uD(Ku) , where uD(f). = [1 - u ( J ) ] / [ 1 + (q - 1)u(J)]. Using the above composition rules one then replaces every hexagonal cluster of bonds appearing in the seaming of the two trees by its equivalent transmissivity. The systematic implementation of this basic procedure will result in the following recursive relationship for the equivalent transmissivity u r between the top and bottom spins of a closed tree with r-generations:
276
(4)
(6)
The analysis of the stability of the fixed points shows that: (a) For values o f u E (0, u (1)) the fixed points u~ are complex and the paramagnetic fixed point u~ is an attractor, thence the system is in the disordered phase for all values ofu0(K ) E (-1/(q - 1), 1). (b) If U takes values above u (1) but below u(2) = ½x/2 (defined as the value of u(J) at which u* vanish) both u~ and u+ are attractors while u * is a repulsor. The system will be in the ferromagnetic (paramagnetic) phase ff the coupling K between the two open trees is strong (weak) enough so that uo(K ) is larger (smaller)'than the value of the repulsor u*. (c) Ifu (2) < u < 1 both u~. and u * , which is negative on this temperature interval, are attractors while Up is a repulsor, so the correlation (h~ h~ -r) will converge to (u*)u~. displaying a (anti)ferromagnetic ordering if ( K < 0) K > 0 for all J > 0. (d) On the other hand if J ( 0 (but q > 2) then u2(J ")< (Uc(1))2 since u(J) E ( - l / ( q - 1), 0) so that one has again the same situation described in (a). (e) Finally for the particular case q = 2u~ is an attractor in the interval u E (0, u~ 2)) while both u *~ are complex and the system is a paramagnet. Above u~2) both u~ are attractors while Up is a repulsor. A (anti) ferromagnetic ordering is established for all (negative) positive K.
Volume 101A, number 5,6
PHYSICS LETTERS
2 April 1984
u
,)'t
1.0 A.F
Fe
"h, ll K
-
X 2
I'
I
[K
X4 Xb Xb -(q.t) "1 U0
-(q.l) "1
Fig. 2. Schematic plot of the phase diagram of the q-state (q > 2) Potts model on the closed tree. The correlation (AAA~) is discontinous across the line abe displaying a gap A = U~. -- U~. On the segment a-b, A 1 = (q - 2)/2(q - 1). Across be ,x is an decreasing function ofu(K) reaching the mean field value 2ZxI in the limit u(K) --*0. The gap across is ,a = u.~ - u*. The correlation function is continuous across the line dc. The above informations lead to the phase diagram schematically shown in fig. (2). The other correlation functions of interest namely
xV>, and
xV>,
Jellitto. This can be shown using the break and collapse method introduced by Tsallis and Levy [7]. The details of these calculations which are more involved than the one used to prove eqs. (7) will be given elsewhere. The partition function of the system is also evaluated recursively noting that the decimation of the internal spins (X1, ~2, ~3, ~4) in the elementary hexagonal cluster (fig. 3) will generate an effective hamiltonian for the pairs of spins on the top and b o t t o m which is a constant plus a renormalized two-spin term. This partition function can be written as Zclu ster (J , K) = R f Z p a i r ( 2 J 1 ) ,
(h A ~ - r )
(8)
where R 1 = R/~,
are obtained in a similar way by computing the effective transmissivity between the pairs o f sites (AB), (BB) etc .... The calculation o f the first two is accomplished using the expression for the correlation function between the top and b o t t o m spin ( ~ ~ h - r ) , together with the parallel-series composition rules:
= o~[1 + u 4 + ( q -
Fig. 3. The decimation of AI (?,2) in the partition function of the hexagonal cluster will give in the prefaetor R and the effective interaction A"between ht and h 3 (h4). The decimation of each internal spin in the diamond At, h3, h4, hb will result in the prefactor R and the effective interaction J1-
2)ctu4]/[l + (q-
1)a2u 4] , (7a)
R = (q - 2) + e3qJ + e3qK,
Re#qF, = (q _ 1) + e3q (J+K) , / ~ = (q - 2) + e3qJ + eOqK, ,
Re #qll = (q - 1) + e3q(J+~" ) . The systematic implementation of the above decimation procedure of the spins on the surface of the closed tree will lead to the following expression r-1
= ctu[1 + u 2 + (q -- 2)o~u3]/[1 + (q -- l)ot2u 4] , where a = ( ~
x~-r).
(7b)
The remaining correlation functions are all zero at all temperatures, as in the q = 2 case studied by
~ ~ Zr =z"~=0 RT+il AAX
exp(flq2Jr 8XAX~) '
(9)
where 2J r is the renormalized coupling constant between the top and b o t t o m spins o f the closed tree with r-generation. The free energy per site is 277
Volume 101A, number 5,6
PHYSICS LETTERS
2 April 1984
References
1 /r-1
-~f=~r(i~__O
?ilnRi+l
+ l n { q t ( q - I ) + exp(flq2J,)l}) ,
(lO)
_/
where 3'/= 2 r - i is the n u m b e r of spins on the (r - i)th generation N r = 2(2 r+l - 1) is the total n u m b e r o f spins o n the tree and
R r = expO3qJr) [2eaqJ + (q - 2)] + ef3qJ[ef3qJ + 2(q -- 2)] -- 3(q -- 1 ) .
(11)
In the ( t h e r m o d y n a m i c ) l i m i t r -+ oo the last term in eq. ( 1 0 ) v a n i s h e s and the sequence { Rr} will converge to a singular f u n c t i o n at the critical temperature.
278
[1] A.N. Beker and S. Ostlund, J. Phys. C12 (1979) 4961; M. Kaufman and R.B. Griffiths, Phys. Rev. B24 (1981) 496. [2] E. MtiUer-Hartmann and J. Zittartz, Phys. Rev. Lett. 33 (1974) 893. [3] R.J. Jelitto, Physica 99A (1979) 268. [4] M.L. Glasser and M.K. Goldbetg, Physica 117A (1983)670; J.E. Krizan, P.F. Barth and M.L. Glasser, Physica 119A (1983) 230. [5] L. Mittag and M.J. Stephen, J. Phys. A7 (1974) LI09. [6] A.P. Young and R.B. Stinchcombe, J. Phys. C9 (1976) 4419. [7] C. Tsallis and S.V. Levy, Phys. Rev. Lett. 47 (1981) 950.
PHYSICS LETTERS
2 April 1984
THE POTTS MODEL ON A CLOSED CAYLEY TREE P.L. CHRISTIANO 1 Departamento de F[sica, Universidade Federal da Paraiba, 58. 000 Jo~o Pessoa Pb., Brasil
and S. GOULART ROSA Jr. 2 Instituto de F(sica e Qufmica de S~o Carlos, Caixa Postal 369, 13.560 S~o Carlos SP, Brasil Received 7 December 1983
The q-state Potts model on the hierarchical lattice introduced by Jelitto is treated exactly by a renormalization group procedure. For q > 2 the correlation function between the top and bottom spins is discontinuous at Tc(q) in the ferromagnetic model and it is zero for all temperatures in the antiferromagnetic case. On the other hand this correlation is continuous at Tc(2) in both ferro and antiferromagnetic lsing models.
In recent years one has witnessed a growing interest in the study o f spin systems on the sub-class of hierarchical lattices [1 ] known as Cayley trees. This interest which was originated because the Bethe approximation is exact on this lattice if surface effects are avoided, has now increased further since new resuits were obtained from the study of the global properties [2] of spin systems on this structure. In 1979 Jelitto [3] has introduced the so-called closed Cayley tree which is formed by joining two identical (open) trees by the spins laying in their surfaces. The main results of the studies of the ferromagnetic Ising model on the closed Cayley tree can be summarized as follows: (i) There is a phase transition even at zero magnetic field [3] and the free energy is non-differentiable [4] at the critical temperature/3cJ = 1/x/~ where 7 is the branching ratio of the tree. (ii) The correlation function between pairs of spins at special sites infinitely apart are non-zero below Tc, while for other pairs of sites the correlation function is non-zero only at T = 0 K. In this letter we study the q-state (ferro and anti-
ferromagnetic) Potts model on a closed Cayley tree with 3, = 2 by a real space renormalization group (RSRG) procedure which is exact and has the advantage of being simpler than the methods used before to treat the ferromagnetic Ising model [3,4]. For reasons which will become clear in a moment we construct the dosed Cayley tree in a slightly different manner by joining the spins at the surfaces of the two open trees via the K bonds as shown in fig. 1 so that Jelitto's construction and the open tree are recovered as special cases (K = ~ and K = 0 respectively). Let us consider the hamiltonian
° -qJ
qK
0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-HoUand Physics Publishing Division)
(1)
where the summations are carded out over all pairs (i1) of nearest neighbor sites on the same open tree and over all pairs ¢ss') on the surface of both trees. Using the representation ?~. = ¢oki, 6o = e 2 n i / q , K i = 0, 1 ..... q - 1, for the q-state Potts spin variable one can write the hamiltonian of any pair of spins at sites (il~ and their correlation function [5] ~ r x q - r ) as q-1
1 CAPES Pre-doctoral Fellow. 2 CNPq Research Fellow.
'~ 6 xsxs' '
c/j = - s
1,
r=l
)
x;
(2) 275
Volume 101A, number 5,6
PHYSICS LETTERS A
2 April 1984
u r = [2u 2Ur_ 1 + (q - 2) u4u 2_ 1]/
p-O
[1 + (q - 1) udur2_ 11 , B
where we have taken uo(K ) as the initial transmissivity in the iterative procedure and u is the transmissivity associated with the bond J. In the (thermodynamic) limit r ~ oo one obtains the following fixed points for the map given by eq. (4):
p-I
J
/
Up* = 0
(5a)
U*+ = (q - 2)/2(q - 1)
\/
±[(q - 2)2u 4 - 4 ( ( / - 1)(1 - 2u2)] 1/2
/2(q - 1)u 2 .
~m,r,3
Fig. 1. The closed Cayley tree with branching ratio "r = 2 and 3-generations.
(q - 2 ) 2 (u~l)) 4 - 4(q - 1) [1 - 2(u(1)) 2] = 0 , (kt3T(1))-i =ln([1 +(q - 1)u~l)]/[1 - u ~ l ) ] ) .
( ~ x/q-r> = (1 - e-qM)/[1 + (q - l) e - q M l ,
(5b)
Recalling that u E (0, 1) if J > 0 and u E (-1/(q - 1), 0) if J < 0 one defines u (1) as the lowest value of u such that u+_ are real, i.e.
A
(3)
where ( ) stands for the Gibbs average. The Lh.s. of eq. (3), also called the thermal transrnissivity u(J) between the external sites i and/, is the natural variable for the RSRG procedure used here [6]. In particular if we have three spin variables at the sites i, / and k connected in series by the coupling constants Ji/and Jik the equivalent transmissivity U(Jeq ) between the external spins ~/and hk is obtained as the product U(Jeq ) = u(Ji/) u(J/k ). On the other hand if the spins ~/and hi are coupled by Ji/and Kii in parallel the equivalent (dual) transmissivity between i and / is obtained by the product of the (dual) transmissivities uD(Jii) and uD(Ki/), i.e. uD(Jeq) =uD(Ji]) uD(Ku) , where uD(f). = [1 - u ( J ) ] / [ 1 + (q - 1)u(J)]. Using the above composition rules one then replaces every hexagonal cluster of bonds appearing in the seaming of the two trees by its equivalent transmissivity. The systematic implementation of this basic procedure will result in the following recursive relationship for the equivalent transmissivity u r between the top and bottom spins of a closed tree with r-generations:
276
(4)
(6)
The analysis of the stability of the fixed points shows that: (a) For values o f u E (0, u (1)) the fixed points u~ are complex and the paramagnetic fixed point u~ is an attractor, thence the system is in the disordered phase for all values ofu0(K ) E (-1/(q - 1), 1). (b) If U takes values above u (1) but below u(2) = ½x/2 (defined as the value of u(J) at which u* vanish) both u~ and u+ are attractors while u * is a repulsor. The system will be in the ferromagnetic (paramagnetic) phase ff the coupling K between the two open trees is strong (weak) enough so that uo(K ) is larger (smaller)'than the value of the repulsor u*. (c) Ifu (2) < u < 1 both u~. and u * , which is negative on this temperature interval, are attractors while Up is a repulsor, so the correlation (h~ h~ -r) will converge to (u*)u~. displaying a (anti)ferromagnetic ordering if ( K < 0) K > 0 for all J > 0. (d) On the other hand if J ( 0 (but q > 2) then u2(J ")< (Uc(1))2 since u(J) E ( - l / ( q - 1), 0) so that one has again the same situation described in (a). (e) Finally for the particular case q = 2u~ is an attractor in the interval u E (0, u~ 2)) while both u *~ are complex and the system is a paramagnet. Above u~2) both u~ are attractors while Up is a repulsor. A (anti) ferromagnetic ordering is established for all (negative) positive K.
Volume 101A, number 5,6
PHYSICS LETTERS
2 April 1984
u
,)'t
1.0 A.F
Fe
"h, ll K
-
X 2
I'
I
[K
X4 Xb Xb -(q.t) "1 U0
-(q.l) "1
Fig. 2. Schematic plot of the phase diagram of the q-state (q > 2) Potts model on the closed tree. The correlation (AAA~) is discontinous across the line abe displaying a gap A = U~. -- U~. On the segment a-b, A 1 = (q - 2)/2(q - 1). Across be ,x is an decreasing function ofu(K) reaching the mean field value 2ZxI in the limit u(K) --*0. The gap across is ,a = u.~ - u*. The correlation function is continuous across the line dc. The above informations lead to the phase diagram schematically shown in fig. (2). The other correlation functions of interest namely
xV>, and
xV>,
Jellitto. This can be shown using the break and collapse method introduced by Tsallis and Levy [7]. The details of these calculations which are more involved than the one used to prove eqs. (7) will be given elsewhere. The partition function of the system is also evaluated recursively noting that the decimation of the internal spins (X1, ~2, ~3, ~4) in the elementary hexagonal cluster (fig. 3) will generate an effective hamiltonian for the pairs of spins on the top and b o t t o m which is a constant plus a renormalized two-spin term. This partition function can be written as Zclu ster (J , K) = R f Z p a i r ( 2 J 1 ) ,
(h A ~ - r )
(8)
where R 1 = R/~,
are obtained in a similar way by computing the effective transmissivity between the pairs o f sites (AB), (BB) etc .... The calculation o f the first two is accomplished using the expression for the correlation function between the top and b o t t o m spin ( ~ ~ h - r ) , together with the parallel-series composition rules:
= o~[1 + u 4 + ( q -
Fig. 3. The decimation of AI (?,2) in the partition function of the hexagonal cluster will give in the prefaetor R and the effective interaction A"between ht and h 3 (h4). The decimation of each internal spin in the diamond At, h3, h4, hb will result in the prefactor R and the effective interaction J1-
2)ctu4]/[l + (q-
1)a2u 4] , (7a)
R = (q - 2) + e3qJ + e3qK,
Re#qF, = (q _ 1) + e3q (J+K) , / ~ = (q - 2) + e3qJ + eOqK, ,
Re #qll = (q - 1) + e3q(J+~" ) . The systematic implementation of the above decimation procedure of the spins on the surface of the closed tree will lead to the following expression r-1
= ctu[1 + u 2 + (q -- 2)o~u3]/[1 + (q -- l)ot2u 4] , where a = ( ~
x~-r).
(7b)
The remaining correlation functions are all zero at all temperatures, as in the q = 2 case studied by
~ ~ Zr =z"~=0 RT+il AAX
exp(flq2Jr 8XAX~) '
(9)
where 2J r is the renormalized coupling constant between the top and b o t t o m spins o f the closed tree with r-generation. The free energy per site is 277
Volume 101A, number 5,6
PHYSICS LETTERS
2 April 1984
References
1 /r-1
-~f=~r(i~__O
?ilnRi+l
+ l n { q t ( q - I ) + exp(flq2J,)l}) ,
(lO)
_/
where 3'/= 2 r - i is the n u m b e r of spins on the (r - i)th generation N r = 2(2 r+l - 1) is the total n u m b e r o f spins o n the tree and
R r = expO3qJr) [2eaqJ + (q - 2)] + ef3qJ[ef3qJ + 2(q -- 2)] -- 3(q -- 1 ) .
(11)
In the ( t h e r m o d y n a m i c ) l i m i t r -+ oo the last term in eq. ( 1 0 ) v a n i s h e s and the sequence { Rr} will converge to a singular f u n c t i o n at the critical temperature.
278
[1] A.N. Beker and S. Ostlund, J. Phys. C12 (1979) 4961; M. Kaufman and R.B. Griffiths, Phys. Rev. B24 (1981) 496. [2] E. MtiUer-Hartmann and J. Zittartz, Phys. Rev. Lett. 33 (1974) 893. [3] R.J. Jelitto, Physica 99A (1979) 268. [4] M.L. Glasser and M.K. Goldbetg, Physica 117A (1983)670; J.E. Krizan, P.F. Barth and M.L. Glasser, Physica 119A (1983) 230. [5] L. Mittag and M.J. Stephen, J. Phys. A7 (1974) LI09. [6] A.P. Young and R.B. Stinchcombe, J. Phys. C9 (1976) 4419. [7] C. Tsallis and S.V. Levy, Phys. Rev. Lett. 47 (1981) 950.