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A new symmetry of the Potts model on the Bethe lattice
PHYSICS LETTERS A
Physics Letters A 162 (1992) 232—235 North-Holland
A new symmetry of the Potts model on the Bethe lattice ES. de Aguiar Departamento de Fisica, Universidade Federal do Amazonas, 69000 Manaus, AM., Brazil
and S. Goulart Rosa Jr. Inslituto de FIsica e Quimica de São Car/os, Universidade de São Paulo, 13560 São Car/os, S.P., Brazil Received 23 September 1991; revised manuscript received 2 December 1991; acceptedfor publication 2 December 1991 Communicated by A.A. Maradudin
We show that the free energy functional of the p-state Potts model in the Bethe—Peierls approximation is mapped by the combined operations of dilation and inversion of the magnetic field and of the order parameter, on the free energy of the p’( =p/ (p— 1) )-state model. The study of this functional in the interval pe (1, 2) is sufficient to determine completely the thermodynamic behavior of the system for all p> 2.
The standard Potts model [11 was proposed in 1952 as a generalization of the Ising model. It is prescribed that each Potts spin variable can assume p states such that a pair of spins interacting with a ferromagnetic coupling will display p ground state configurations if the spins are in the same state and p(p— 1) excited state configurations if the spins are in different states. The ground state degeneracy is removed completely if and only if a positive symmetry breaking field, which favors one of the p states, is turned on. The presence of a negative field is capable only of reducing by one the ground state degeneracy. Its presence yields several thermodynamics consequences such as residual entropy, occurrence of metastable states [21 as well as convexity violations [31.Therefore the study of the field dependence is essential to obtain a clear and complete understanding of this model. In this Letter we show that, by the combination of the operations of dilation and inversion of the magnetic field as well as of the order parameters, the properties ofthe p-state model in the Bethe—Peierls approximation are mapped on those of the p’ ( =p/ (p— 1) )-state model. By defining the system on a Cayley tree we obtain a precise prescriplion to describe the system behavior in this approx232
imation, particularly the sign effects of the symmetry breaking field [21. Using the Mittag—Stephen [41 representation for the Potts variable we can write the system Hamiltonian on the Cayley tree as p—I
*= —pJ ~ ~
~ 2,~Ay—”—pH ~ ~O
i
~ q=O
.~‘— i
~
~
(1)
—
where the first summation is over all pairs of nearest neighbor sites in the tree, the second (third) sum is over all sites in the interior (surface) of the tree, and Ag is the ghost spin. The edges representing the interactions of the ghost with the other spins change drastically the most relevant topological feature of the usual open loopless tree by transforming it into the closed asymmetric Cayley tree (CAT); fig. 1. (We observe that by freezing the ghost spin in any one of its p states, one recovers the Hamiltonian for the system on the open tree in the presence of an external inhomogeneous magnetic field.) Besides the presence of loops it is worthy of mention that the CAT is composed of closed branches which are hier-
0375-960 1/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
Volume 162, number 3
PHYSICS LETTERS A
10 February 1992
alent dual transmissivity 1d
t~’(Jeq)
is the product
(J,~)td (K1~).
Applying now these composition rules to the
~
\
/
“\\
\\
/ I
\
\
\ \\
i
I
/ /
I / /
I ~ \~i~/ \
ii.
‘ /
/
\ \ I / // I
a spin in the (n+ 1 )th shell. This renormalization group transformation, from now on called Bethe— Peierls (BP) map, is given by
~
/ I I i \
transmissivities in the elementary generating cluster of the closed asymmetric branch onethe obtains the equivalent transmissivity X~ between ghost and
/
X~~1=B(X~,t,H,p,y)
//
/
l—G(X~,t,H,p,y) l+(p—l)G(X~,t,H,p,y)’
i /
/
(3)
n=O, 1, where
Fig. I. A closed asymmetric Cayley branch with coordination number y= 3 and with two shells (generations). The branch with one more shell is obtained by replacingeach dashed edge by the elementary cluster (cut diamond). This cluster is formed by two solid edges representing the coupling J, two dashed edges representing the renormalized field H~andby the dot dashed edge (bare
G (X~,t, H, p, y) = exp (
magnetic field H) connecting the ghost spin with the top spin in the diamond. The BP map given by eqs. (3)—(5) is obtained by calculating the effective transmissivity between the top and ghost spins in the cut diamond. The CAT is formed by connecting the top sites ofthree branches to an extra (central) site.
~
archial lattices. As a consequence of this feature it is possible to develop a simple and exact real space normalization group procedure to study this system. In order to obtain the renormalization group transformation let us define first the thermal transmissivity t(J~) and its dual transmissivity td(Ju):
trized by the reduced temperature t(J), by the external field H, by the coordination number y, which determines its degree, and by the number of states of the spin variable p.
1 _td(Ju) I +(p— l)td(J,,) t(J,~)E(—l/(p—l),
1).
t(J0)=exp(flpJ,
),
X
=
(
—
1 tx~ )I~~I 1 + (~ ) tx~
/JpH)
—
—
(4)
‘
1 —exp( —flpH~) 1 + (P 1) exp ( flpH~) —
—
x0
(5)
H~is the effective renormalized field acting upon a
spin in the nth shell. Thence the BP map is a rational recursion relationship connecting the effective fields in two consecutive shells of the tree. It is parame-
Proposition 1. Let B(X, t, H, p, y) be the BP map of the p-state Potts model at the reduced temperacoordination ture tin an external number fieldHput y. Then the onfollowing a Cayley tree relationwith ship holds:
(2)
The thermal transmissivity t(J0) is the most natural variable because of its simple composition rules [5]. For instance if three spins at sites i,j and k are connected in series by the coupling constants J~,,~ the equivalent transmissivity t(Jeq) between the external spins is the product t(J0)t(J,k). If on the other hand we have two spins at sites iandj which are coupled in parallel by the coupling J~,and K,, the equiv-
B(X,t,H,P,y)=—(p’—l)B(X’,t,H’,p’,y), where (p— l)(p’— 1
) 1,
H
x= (a’— 1 )X’.
—
(6)
(p’— l)H’,
—
Proof The operations of dilation and magnetic field reversal, H—p (p’ 1 )H’ , and of dilation and re—
—
233
Volume 162, number 3
PHYSICS LETTERS A
version of the order parameter, X—~ (p eq. (4) give that G(X,t,H,p,y)=G~(X’,t,H’,p’,y). —
—
1 )X’ , in
10 February 1992
which is obtained from eq. (4) observing that G(X’, t, H, p, X) is the dual of X*, yields that
(7) pØ(X~,t,p, y)
The result then follows directly from the replacementofGbyG~ in eq. (3). Remark 1. The scaling of the magnetic field is a consequence of writing the number of states p as a pre-factor in the Hamiltonian. This has been done so that all quantities transform in the same way. Remark 2. By keeping X fixed and transforming instead the reduced temperature t = (p’ 1) t’ we obtain a similar relationship between the maps, e.g. —
—
B(X,t,H,p,y)=—(p’—1)B(X,t’,H’,p’,y)
.
(8)
=pØ(x*~,t, p’, y) +ln(p’— 1) +/3p’H’.
(12)
This completes the proof. Remark 3. In the ferromagnetic phase X* and X*~ are the stable fixed points obtained with different sign initial conditions. Remark 4. The result expressed in eq. (12) is also valid for the antiferromagnetic model whose free energy density is given by BØ(X’~,X~,i,p,y)
The proof follows with minor modifications noting that the self-consistency equation relates now XT and
Proof The free energy functional of the p-state ferromagnetic model in the Bethe—Peierls approximation is given by
Remark 5. Since the transformation (p— 1) X (p’— 1 )= 1 maps the interval (1, 2) into the interval
—
flØ(X*, t, p, y) =
—
(y— 1) ln( 1 _tX*)
+ln(1 _X*)+~(y_2) +~yln(l—t)—lnp,
(9)
1 G(X*, t, H, p, ‘) (10) (p— 1)G(X*, t, H, p~ Now the substitutions X” = (p’ 1) X*~,H= (p’ )H’ together with relationship —
—
1+
—
—
—
— 1
—
1 +
exp( fl~’H’)(
(p’ 1) tx*’) l+tX*~
Y
~
[—
(y_ 1) ln( 1 —tX~)+ln( 1 —Xfl]
i=
+ ~(y_-2) ln[ 1 + (p— 1 )tXTXfl +~yln(l—t)—lnp.
(13)
x~.
(2, oo) the study of the p-state Potts model in the interval 2) is sufficient to determine the properties ofpe the(1,model for all other values ofp>2. For
ln[l + (p— 1 )tX*2]
where /3_1= k~T,X’P= X~( t, H, P, y) is the solution of the self-consistency equation [21 (this equation is obtained from the BP map by letting n-+oo, with the initial condition H 0=H~),i.e. —
=
2
Theorem 1. The local properties ofthep-state Potts model defined on the Cayley tree with coordination number y, at the reduced temperature t(J), in the presence of an external field H are mapped on the properties of the model with p’ (=p/ (p— 1)) states at the same temperature tin the presenceofthe scaled and inverted field H’= (p— 1 )H.
1
—
p=l theBPmapbecomesapolynomialmapandthe transformation is not applicable. This case has been analysed in detail elsewhere [6,7]. In order to obtain the well-known result that the free energy of the Ising model (p=2) is an even function in the field strength, i.e. that the Ising free energy is invariant under the transformations of scaling and inversion we must add to eq. (9) the contribution /J(H+J). These terms correspond to the energy zero shift resulting from writing the Hamiltonian as a sum of Kronecker deltas instead ofa sum of products of pairs of spin variables as is done traditionally in the definition of the Ising Hamiltonian.
1-Xe’ (11) = l+(p~_l)X*~’ 234
Remark 6. Our results are independent ofthe value
Volume 162, number 3
PHYSICS LETTERS A
of the parameter y. In particular in the limit of infinite coordination number one can reobtain from eqs. (3) and (9) the Bragg—Williams map and the mean-field expression for the free energy functional [21. The proof that the symmetry also exists in this limit follows the same steps as presented above.
10 February 1992
(2] F.S. de Aguiar, L.B. Bernardes and S. Goulart Rosa Jr., J. Stat. Phys. 64 (1991) 673. 131 Griffiths Gujrati, J. Stat. Phys. 30 (1983) 563. [4] R.B. L. Mittag and and M.J.PD. Stephen, J. Math. Phys. 12 (1971) 441. [5] C. Tsallis and S.V.F. Levy, Phys. Rev. Lett. 47 (1981) 950. [6] F.S. de Aguiar, F.A. Bosco, A.S. Martinez and S. Goulart Rosa Jr., J. Stat. Phys. 58 (1990) 1231. [7] F.S. de Aguiar and S. Goulart Rosa Jr., Phys. Lett. A 143 (1990) 186.
References [1] RB. Potts, Proc. CambridgePhilos. Soc. 48 (1952) 106.
235
Physics Letters A 162 (1992) 232—235 North-Holland
A new symmetry of the Potts model on the Bethe lattice ES. de Aguiar Departamento de Fisica, Universidade Federal do Amazonas, 69000 Manaus, AM., Brazil
and S. Goulart Rosa Jr. Inslituto de FIsica e Quimica de São Car/os, Universidade de São Paulo, 13560 São Car/os, S.P., Brazil Received 23 September 1991; revised manuscript received 2 December 1991; acceptedfor publication 2 December 1991 Communicated by A.A. Maradudin
We show that the free energy functional of the p-state Potts model in the Bethe—Peierls approximation is mapped by the combined operations of dilation and inversion of the magnetic field and of the order parameter, on the free energy of the p’( =p/ (p— 1) )-state model. The study of this functional in the interval pe (1, 2) is sufficient to determine completely the thermodynamic behavior of the system for all p> 2.
The standard Potts model [11 was proposed in 1952 as a generalization of the Ising model. It is prescribed that each Potts spin variable can assume p states such that a pair of spins interacting with a ferromagnetic coupling will display p ground state configurations if the spins are in the same state and p(p— 1) excited state configurations if the spins are in different states. The ground state degeneracy is removed completely if and only if a positive symmetry breaking field, which favors one of the p states, is turned on. The presence of a negative field is capable only of reducing by one the ground state degeneracy. Its presence yields several thermodynamics consequences such as residual entropy, occurrence of metastable states [21 as well as convexity violations [31.Therefore the study of the field dependence is essential to obtain a clear and complete understanding of this model. In this Letter we show that, by the combination of the operations of dilation and inversion of the magnetic field as well as of the order parameters, the properties ofthe p-state model in the Bethe—Peierls approximation are mapped on those of the p’ ( =p/ (p— 1) )-state model. By defining the system on a Cayley tree we obtain a precise prescriplion to describe the system behavior in this approx232
imation, particularly the sign effects of the symmetry breaking field [21. Using the Mittag—Stephen [41 representation for the Potts variable we can write the system Hamiltonian on the Cayley tree as p—I
*= —pJ ~ ~
~ 2,~Ay—”—pH ~ ~O
i
~ q=O
.~‘— i
~
~
(1)
—
where the first summation is over all pairs
0375-960 1/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
Volume 162, number 3
PHYSICS LETTERS A
10 February 1992
alent dual transmissivity 1d
t~’(Jeq)
is the product
(J,~)td (K1~).
Applying now these composition rules to the
~
\
/
“\\
\\
/ I
\
\
\ \\
i
I
/ /
I / /
I ~ \~i~/ \
ii.
‘ /
/
\ \ I / // I
a spin in the (n+ 1 )th shell. This renormalization group transformation, from now on called Bethe— Peierls (BP) map, is given by
~
/ I I i \
transmissivities in the elementary generating cluster of the closed asymmetric branch onethe obtains the equivalent transmissivity X~ between ghost and
/
X~~1=B(X~,t,H,p,y)
//
/
l—G(X~,t,H,p,y) l+(p—l)G(X~,t,H,p,y)’
i /
/
(3)
n=O, 1, where
Fig. I. A closed asymmetric Cayley branch with coordination number y= 3 and with two shells (generations). The branch with one more shell is obtained by replacingeach dashed edge by the elementary cluster (cut diamond). This cluster is formed by two solid edges representing the coupling J, two dashed edges representing the renormalized field H~andby the dot dashed edge (bare
G (X~,t, H, p, y) = exp (
magnetic field H) connecting the ghost spin with the top spin in the diamond. The BP map given by eqs. (3)—(5) is obtained by calculating the effective transmissivity between the top and ghost spins in the cut diamond. The CAT is formed by connecting the top sites ofthree branches to an extra (central) site.
~
archial lattices. As a consequence of this feature it is possible to develop a simple and exact real space normalization group procedure to study this system. In order to obtain the renormalization group transformation let us define first the thermal transmissivity t(J~) and its dual transmissivity td(Ju):
trized by the reduced temperature t(J), by the external field H, by the coordination number y, which determines its degree, and by the number of states of the spin variable p.
1 _td(Ju) I +(p— l)td(J,,) t(J,~)E(—l/(p—l),
1).
t(J0)=exp(flpJ,
),
X
=
(
—
1 tx~ )I~~I 1 + (~ ) tx~
/JpH)
—
—
(4)
‘
1 —exp( —flpH~) 1 + (P 1) exp ( flpH~) —
—
x0
(5)
H~is the effective renormalized field acting upon a
spin in the nth shell. Thence the BP map is a rational recursion relationship connecting the effective fields in two consecutive shells of the tree. It is parame-
Proposition 1. Let B(X, t, H, p, y) be the BP map of the p-state Potts model at the reduced temperacoordination ture tin an external number fieldHput y. Then the onfollowing a Cayley tree relationwith ship holds:
(2)
The thermal transmissivity t(J0) is the most natural variable because of its simple composition rules [5]. For instance if three spins at sites i,j and k are connected in series by the coupling constants J~,,~ the equivalent transmissivity t(Jeq) between the external spins is the product t(J0)t(J,k). If on the other hand we have two spins at sites iandj which are coupled in parallel by the coupling J~,and K,, the equiv-
B(X,t,H,P,y)=—(p’—l)B(X’,t,H’,p’,y), where (p— l)(p’— 1
) 1,
H
x= (a’— 1 )X’.
—
(6)
(p’— l)H’,
—
Proof The operations of dilation and magnetic field reversal, H—p (p’ 1 )H’ , and of dilation and re—
—
233
Volume 162, number 3
PHYSICS LETTERS A
version of the order parameter, X—~ (p eq. (4) give that G(X,t,H,p,y)=G~(X’,t,H’,p’,y). —
—
1 )X’ , in
10 February 1992
which is obtained from eq. (4) observing that G(X’, t, H, p, X) is the dual of X*, yields that
(7) pØ(X~,t,p, y)
The result then follows directly from the replacementofGbyG~ in eq. (3). Remark 1. The scaling of the magnetic field is a consequence of writing the number of states p as a pre-factor in the Hamiltonian. This has been done so that all quantities transform in the same way. Remark 2. By keeping X fixed and transforming instead the reduced temperature t = (p’ 1) t’ we obtain a similar relationship between the maps, e.g. —
—
B(X,t,H,p,y)=—(p’—1)B(X,t’,H’,p’,y)
.
(8)
=pØ(x*~,t, p’, y) +ln(p’— 1) +/3p’H’.
(12)
This completes the proof. Remark 3. In the ferromagnetic phase X* and X*~ are the stable fixed points obtained with different sign initial conditions. Remark 4. The result expressed in eq. (12) is also valid for the antiferromagnetic model whose free energy density is given by BØ(X’~,X~,i,p,y)
The proof follows with minor modifications noting that the self-consistency equation relates now XT and
Proof The free energy functional of the p-state ferromagnetic model in the Bethe—Peierls approximation is given by
Remark 5. Since the transformation (p— 1) X (p’— 1 )= 1 maps the interval (1, 2) into the interval
—
flØ(X*, t, p, y) =
—
(y— 1) ln( 1 _tX*)
+ln(1 _X*)+~(y_2) +~yln(l—t)—lnp,
(9)
1 G(X*, t, H, p, ‘) (10) (p— 1)G(X*, t, H, p~ Now the substitutions X” = (p’ 1) X*~,H= (p’ )H’ together with relationship —
—
1+
—
—
—
— 1
—
1 +
exp( fl~’H’)(
(p’ 1) tx*’) l+tX*~
Y
~
[—
(y_ 1) ln( 1 —tX~)+ln( 1 —Xfl]
i=
+ ~(y_-2) ln[ 1 + (p— 1 )tXTXfl +~yln(l—t)—lnp.
(13)
x~.
(2, oo) the study of the p-state Potts model in the interval 2) is sufficient to determine the properties ofpe the(1,model for all other values ofp>2. For
ln[l + (p— 1 )tX*2]
where /3_1= k~T,X’P= X~( t, H, P, y) is the solution of the self-consistency equation [21 (this equation is obtained from the BP map by letting n-+oo, with the initial condition H 0=H~),i.e. —
=
2
Theorem 1. The local properties ofthep-state Potts model defined on the Cayley tree with coordination number y, at the reduced temperature t(J), in the presence of an external field H are mapped on the properties of the model with p’ (=p/ (p— 1)) states at the same temperature tin the presenceofthe scaled and inverted field H’= (p— 1 )H.
1
—
p=l theBPmapbecomesapolynomialmapandthe transformation is not applicable. This case has been analysed in detail elsewhere [6,7]. In order to obtain the well-known result that the free energy of the Ising model (p=2) is an even function in the field strength, i.e. that the Ising free energy is invariant under the transformations of scaling and inversion we must add to eq. (9) the contribution /J(H+J). These terms correspond to the energy zero shift resulting from writing the Hamiltonian as a sum of Kronecker deltas instead ofa sum of products of pairs of spin variables as is done traditionally in the definition of the Ising Hamiltonian.
1-Xe’ (11) = l+(p~_l)X*~’ 234
Remark 6. Our results are independent ofthe value
Volume 162, number 3
PHYSICS LETTERS A
of the parameter y. In particular in the limit of infinite coordination number one can reobtain from eqs. (3) and (9) the Bragg—Williams map and the mean-field expression for the free energy functional [21. The proof that the symmetry also exists in this limit follows the same steps as presented above.
10 February 1992
(2] F.S. de Aguiar, L.B. Bernardes and S. Goulart Rosa Jr., J. Stat. Phys. 64 (1991) 673. 131 Griffiths Gujrati, J. Stat. Phys. 30 (1983) 563. [4] R.B. L. Mittag and and M.J.PD. Stephen, J. Math. Phys. 12 (1971) 441. [5] C. Tsallis and S.V.F. Levy, Phys. Rev. Lett. 47 (1981) 950. [6] F.S. de Aguiar, F.A. Bosco, A.S. Martinez and S. Goulart Rosa Jr., J. Stat. Phys. 58 (1990) 1231. [7] F.S. de Aguiar and S. Goulart Rosa Jr., Phys. Lett. A 143 (1990) 186.
References [1] RB. Potts, Proc. CambridgePhilos. Soc. 48 (1952) 106.
235