On the mean field solution of the potts model

Volume 99A, number 4

PHYSICS LETTERS

28 November 1983

ON THE MEAN FIELD SOLUTION OF THE POTTS MODEL A. BARACCA 1, M. BELLESI, R. LIVI 1 Dipartimento di Fisica dell'Universitd di Firenze, Largo E. Fermi 2, Florence, Italy

R. RECHTMAN Departamento de Fisica, Facultad de Ciencias, Universidad AutOnoma de M~xico, Mexico D.F., Mexico

and S. RUFFO l Dipartirnento di Fisica dell'Universitd di Pisa, Piazza Torricelli 2 Pisa, Italy

Received 15 August 1983

The complete mean field solution of the q-state Potts model is presented and its equivalence with the exact solution of the infinite range model is shown. The presence of a further singularity of the free energy is evidenced.

The most direct method to obtain some information about the global structure of the phases of a lattice model is the mean field theory. Anyway this approximation is generally unreliable for what concerns critical exponents, exact location of transition temperatures and moreover for recognizing the order of phase transitions. For instance the usual mean field solution of the q-state Potts model [1] predicts a first-order phase transition in any dimension (d) if q > 2 [2], while it is known from the exact results by Baxter in d = 2 [3] that the transition is second order up to q = 4 and becomes first order for q > 4. In spite of the evident drawbacks of this approximation, it is worthwhile to better investigate its real mathematical structure and physical meaning, an investigation which has not been fully completed previously [2]. Such a work is stimulated by the promising results recently obtained concerning a criterion to distinguish between first- and second-order phase transitions based just on the mean field approach [4] : in particular, the appearance of a singular behaviour of the free energy for a temperature lower than the transition temperature of the Potts model is exploited to obtain information on the order of the phase transition. The analysis of this peculiar behaviour is one of the objects of the present paper. A deeper insight in the mean field treatment of the Potts model can be obtained by studying the exact solution of its infinite-range version, whose effective hamiltonian is H = (X/N) ~ i,j 6ai°/ '

(1)

where o i are spin-variables on a hypercubic lattice site i (/= 1 ..... N ) with integer values ranging from 1 to q. The sum over the sites is no longer restricted to nearest neighbour sites, as in the original model. It is known that in the Ising case (q = 2) the mean field solution of the inf'mite range model is exact [5] and is equivalent to the mean field solution of the nearest neighbour model, when the identification ~ = d K is made, K being the inverse temperature. 1 INFN, Sezione di Firenze. 156

0.031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland

Volume 99A, number 4

PHYSICS LETTERS

28 November 1983"

Let us summarize the solution of the model (1), i.e. the exact computation of the free energy density F(~) = lim ( - N -1 lnYr(a ) e H) = Nlira ~ ( - N -1 l n Z ) .

(2)

N ~

Using the following property of the Kronecker 5 function q

5oioj= ~= 5oirbro j ,

(3)

one can write Z = Tr{o} rl-I1 e x p [ ) =

i=1

6ai r

.

(4)

Now it is clear that the linearization of the hamiltonian can be obtained by a gaussian trnaformation: the crucial point is that, in order to perform it in a consistent way, one is forced to introduce q "'magnetic fields" h r. After some straightforward calculations one obtains Z=const× f --oo

= const × _•

(

r=l

dh r Tr{o}exp

r=l dh r e x p N

=

-(N/4X)h 2-

4-~ = hr2 +In

-=

6oirh r

exp(hr)

,

(5)

where the last step is performed by tracing over the configurations the linearized hamiltonian. In the thermodynamic limit (N ~ ~,,) the saddle point method gives directly the free energy density F(X) =

max {Ihr[< o~} 4-~

hr2 - In ~ exp(hr) = . max . f ( h r ) . r=l {Ihrl< ~o)

(6)

The stationary conditions for f(hr) are q )-1 exp(hs)\~.T=1 exp(hr) =hs/2X,

(s = 1 ..... q ) .

(7)

It is clear that in general the calculation of the maximum in eq. (6) is responsible for the expected occurrence of non-analiticities of the free energy density. In our case the maximum depends on X and one expects that F(X) = F 1(X),

=F2(a),

for

X < X,

for ~>~,,

so that~ may be identified with the transition point if F I(X) and F2(X) do not connect smoothly in some derivative at X. A straightforward solution of the stationary conditions (7) is

h s = it = 2X/q,

(s = 1 ..... q ) ,

(8)

corresponding to the free energy density F 1(X) = - h / q - In q .

(9) 157

Volume 99A, number 4

PHYSICS LETTERS

28 November 1983

Another trial solution is

hi=[~,

hj='h,

(1'=1 ..... i - 1 , i + 1

..... q ) ,

(10)

which may be interpreted as the breaking of the permutation symmetry in the internal space of the o-variables along the "direction" i. The trial solution (10) gives only two distinct stationary conditions; subtracting them one from the other and introducing the "order parameter" H = (/~ - h)/2X, one obtains (e 2xH - 1)/(e 2xH + q -

1)=H.

(11)

This equation is formally identical to the mean field consistency equation of the Ports model. It admits the solution H = 0 for any value of X, but for k > X = [(q - 1)/(q - 2)] ln(q - 1) another solution appears for H :~ 0 which corresponds to an absolute maximum of the function f(hr). The solutions (8) and (10) reproducethe entire known phenomenology of the mean field solution of the Ports model, that is, a phase transition at k = k with a jump in the order parameter ZX//---2 ln(q - 1), which is zero only in the Ising case. The relation between the coupling constant of the infinite range case and that of the nearest neigh. bour case, K, is X = dK, where d is the dimension of the lattice. In order to perform a complete analysis of the infinite range model, one has to investigate the properties of all the stationary points of the function f(hr) , since in principle one could expect a richer phenomenology. This has been done by a numerical analysis for the cases q = 3 and q = 4. Let us consider first the q = 3 case. The free energy is completely symmetric in the space {hr) and moreover the stationary conditions (7) sum up to give 3 r=l

= 2x.

02)

Therefore one can study the stationary points of the function f in the sub-space {h 1, h2)" The qualitative behaviour of these points is sketched in fig. 1 for typical values of k. For X < X = 2 In 2 the function f shows a single absolute maximum in the symmetric point h I = h 2 = -~X and the appearance of three relative maxima as k approaches ~,: one of them lies on the bisecting line and the others on the straight lines of equations h l + 2 h 2 = 2 k a n d h 2 + 2 h 1 =2X. At k >~ X the three maxima become absolute: X is then interpreted as the first-order phase transition temperature. Nevertheless this behaviour does not cover the entire phenomenology. The present treatment shows that at a higher value of k another singular behaviour of the free energy appears, corresponding to the collapse of four stationary points: the symmetric point becomes a relative minimum, after three relative minima collapse on the symmetric point.

h2

hI ~\

/

",.

"--.X/

\x,

/

'.

/

{a)

h2~\X\

/

/

//

-.

hI

(b)

hI

(e)

hI

Fig. I. The stationary points of the function f(h i, h2) are shown in the (h I, h2) plane for (a) ~ < k, (b) k ~ X < ~, (c) X ~ h. The symbols o, +, o correspond to absolute maxima, relative maxima and relative minima respectively.

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Volume 99A, number 4

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28 November 1983

This value, X = ½q, corresponds to a further non-analiticity of the free energy. In the usual mean field solution of the nearest neighbour Ports model this value ~ corresponds to the appearance of a relative minimum of the free energy for negative values of the order parameter, which has been interpreted in a general criterion to distinguish between first- and second-order transitions [4]. In that case one could question that this singular behaviour was obtained for "unphysical" values of the order parameter. On the contrary, in the present approach the true symmetry of the model has been considered, by analysing the complete space of the variational parameters (hr} without choosing a priori a particular order parameter breaking the symmetry in a "direction" at the beginning of the mean field derivation of the free energy. In a sense, the symmetry along one direction is naturally broken: the maximum condition on the function f(hr) chooses spontaneously a particular direction. Let us notice that the usual mean field solution can be recovered in the q = 3 case with the restriction h 1 + 2h 2 = 2X. Using this condition, which corresponds to a breaking of the symmetry in one direction, on derives a function of a single "order parameter", which is reproduced in fig. 2 for the relevant values of X. Another interesting observation is that the section of the function f(hl, h2) along the plane h I + h 2 = 4 X repro_3 duces the typical evolution of the free energy of an Ising system with a second order phase transition at Xc - ~. The numerical analysis repeated for the q = 4 case yields the same features. A first order phase transition takes 3 In 3 by a breaking of the symmetry in the (hr) space along one direction and moreover another sinplace at X = ~ gular behaviour of the free energy appears at X = 2, where the symmetric point h r = ½~ becomes a minimum. On the basis of the present analytical and numerical results one can infer that similar features hold for any value o f q . The last step for a complete understanding of these results is to answer the following question, which kind of mean field approximation for the nearest neighbour Potts model does reproduce the features of the solution of the infinite range model we have just analyzed? One can reasonably expect that q variational parameters B r must be introduced in the mean field approximation from the beginning. Using the convexity inequality for the exponential function ((e A )/> e(A>), one can estimate the partition function Zp as follows: Zp(K) = Tr(a } e x p ( K

~6o.a.I>~ZM.F.(Br,K )

(i/>

t I/

=[Tr{a}exp(qr~=iBr~i ~oir)]exp(K(~ii>8oio/-~r Br~i8oir)Br,

(13)

f, Fig. 2. Sections of the surface f(h 1, h2) along the plane h 1 + 2h2 = h for (a) X ~ ~., (b) h ~< ~., (c) h = h, (d) ~ > X ~> ~., (e) ~ = h, (f) h > ~'. The origin is the symmetric point h I = h 2 =32-X.

159

Volume 99A, number 4

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28 November 1983

where ( )Br stands for the mean value over the statistical ensemble defined by the single point action A M.F. = ~r Br Y~i 8air" The corresponding free energy density is FM.F. = N -1 In ZM.F. = ln(Y~r B r ) + d K Y~r e x p ( 2 B r ) / [ ~ r exp(Br)] 2 _ ~r Br exp(Br)/Er exp(Br) •

(14)

The best estimate of the true free energy density is obtained by maximizing the r.h.s, of eq. (13) as a function of the B r; the stationary conditions are expressed by q equations. In a completely similar way as in the saddle point solution of the infinite r a n ~ model the solution B r = B for all r holds for any value of K and the symmetry b r a k i n g condition B i = B, B! = B, (1" = 1 ..... i - 1, i + 1, ..., q) yields the consistency equation (11) w i t h H = B - B and X = dK. In this case too at K = q / 2 d one finds the singular behaviour already stressed in the infinite range case. We may summarize the main conclusions of our analysis as follows: (1) the correct way of deriving the mean field approximation of the Potts model requires the introduction of q variational parameters without breaking the symmetry at the beginning; (2) this mean field solution eliminates some inconsistencies in the analysis of the singular behaviour of the free energy at ~ = dK = ½q. We acknowledge helpful discussions with A. Maritan, A. Vulpiani, F. Fucito and G. Parisi.

References [1] [2] [3] [4] [5]

R.B. Potts, Proc. Cambridge Philos. Soc. 48 (1952) 106. L. Mittag and M.J. Stephen, J. Phys. A7 (1974) L109. R.J. Baxter, J. Phys. C6 (1973) L445. R. Livi, A. Maritan, S. Ruffo and A. Stella, Phys. Rev. Lett. 50 (1983) 459. M. Kac, Statistical physics, phase transitions and superfluidity, Vol. 1 (Gordon and Breach, New York, 1968) p. 241.

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