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The free energy of the Potts model :

arXiv:hep-th/9303075v1 12 Mar 1993The free energy of the Potts model :from the continuousto the first-order transition region.T. Bhattacharya*, R. Lacaze** and A. MorelService de Physique Th´eorique de Saclay***91191 Gif-sur-Yvette Cedex, FranceABSTRACTWe present a large q expansion of the 2d q-states Potts model free energiesup to order 9 in 1/√q.Its analysis leads us to an ansatz which, in the first-order region, incorporates properties inferred from the known critical regime atq = 4, and predicts, for q > 4, the nth energy cumulant scales as the power(3n/2 −2) of the correlation length.The parameter-free energy distributionsreproduce accurately, without reference to any interface effect, the numerical dataobtained in a simulation for q = 10 with lattices of linear dimensions up to L = 50.The pure phase specific heats are predicted to be much larger, at q ≤10, than thevalues extracted from current finite size scaling analysis of extrema.

Implicationsfor safe numerical determinations of interface tensions are discussed.SPhT-93/022August 2021Submitted for publication to Europhysics Letters* Present address: MS B285, Group T-8, Los Alamos National Laboratory, NM87544, U.S.A.** Chercheur au CNRS*** Laboratoire de la Direction des Sciences de la Mati`ere du CEA

Much effort has been recently devoted to the 2-d Potts model, both withnumerical and analytical techniques. In the former case, the goal was either totest numerical algorithms and criteria for distinguishing first-order from continuoustransitions, or to learn how to extract previously unknown quantities such asthe interface tension.

Although much progress was accomplished, there remainssome unsatisfactory issues such as, for example, slight inconsistencies in finitesize scaling analysis of the energy cumulants close to the transition temperatureβ−1t, and discrepancies between exact results and numerical simulations for theinterface tension. This question is important since only numerical simulations candetermine this quantity in other cases of physical interest such as the 3-d q = 3Potts model or QCD at the deconfinement transition.Analytical works have shown [1] that close to βt, the partition function Z ofthe Potts model, in a box of volume V = L2 with periodic boundary conditions(used all through in this work), is equal to the sum of the ‘partition functions’Zi of the q + 1 pure phases, and a term that falls of exponentially faster in thelinear size of the system.

We shall presently ignore this latter term which contains,amongst others, the interface tension effects, and concentrate on the ith phase freeenergy Fi = ln Zi/V , which is V independent and differentiable many times withrespect to the inverse temperature β at βt.In this letter, we construct explicit formulae for the ordered and disorderedfree energies Fo and Fd of the 2-d q-states Potts model at q > 4. At large q weperform their expansion in power of 1/√q.

At low (q −4) we conjecture theirbehaviour from their known critical behaviour at q = 4. We show that these twodescriptions match in a large intermediate q region.

Then, using the above rigorousresult, we add up the Zi’s so obtained and make absolute predictions on energyprobability densities in very good agreement with numerical data. We show thatthe difficulties encountered in the finite size scaling analysis of numerical data aredue to very large high order cumulants, which has consequences for the extractionof interface tensions.Many properties of the model are known exactly [2].

In particular, it exhibitsa temperature driven phase transition which occurs at βt = ln(√q + 1).Thetransition is second-order for q ≤4, and its critical properties are described, e.g.,by the α and ν indices, which at q = 4 take the common value 2/3. Accordingly,1

the correlation length and the specific heat there diverge asξq=4 ∼| β −βt |−2/3,Cq=4 ∼| β −βt |−2/3 . (1)Hence in the vicinity of β = βt, the ratio C/ξ remains finite at q = 4−.The first-order transition region is q > 4.

There the energies Eo and Ed ofthe ordered and disordered phases respectively are exactly known at βt [3]. Recentworks [4,5] on the largest correlation length at βt have shown a common behaviouras q →4+ξ =18√2 x (1 + O(x−2))withx = exp(π22 ln 12(√q + √q −4)).

(2)These formulae show not only that ξ rapidly diverges as q →4+, but also that theleading behaviour of Eq. (2) is accurate over a very wide range of q values.

Forexample the correction term in Eq. (2) is still of the order of 1% for q as large as 75.The pure phase specific heats Co and Cd are unknown, but their known differencevanishes when q →4+ as x−12 .

These properties are of course in accordance withthe point ( q = 4, β = βt ) being a second-order transition point, and lead one tospeculate that Cd ∼Co diverges as q →4+, possibly in such a way that the ratioC/ξ is finite on both sides of q = 4.First we shall take advantage of the fact that, for the correlation length andlatent heat, the ”small q −4 ” region extends in practice up to large q values,and start from the opposite end. We compute the free energy of the model inthe framework of a large q expansion, extrapolate down in q as far as we can,and analyze the resulting energy cumulants as functions of x in an intermediateq-value region.

Nice regularities emerge, among which a smooth behaviour of C/xis ascertained.The large q expansion of the ordered free energy Fo (that for Fd in the dis-ordered phase follows from duality), was obtained through the Fortuin-Kasteleyn[6] representation of the Potts model partition functionZ =XX(eβ −1)lqn(3)where X is any configuration of bonds on a cubic lattice, l its number of bondsand n its number of connected components, or clusters of sites ( two sites bound toeach other belong to the same cluster, an isolated site is a cluster). The completely2

ordered configuration corresponds to n = 1 and l = 2V . So the partition functioncan be reorganized as an expansion in q−12 about this configuration:Zo = q(eβ −1)2VXl≥0,n≥0Nl,n(V )(eβ −1√q)−lqn−l2 ,(4)where Nl,n(V ) is the number of configurations in a volume V with l removed bondsand n + 1 clusters.

The enumeration of all the Nl,n(V ) such that (l −2n) ≤Myields an expansion of Zo to order M. Details will be given elsewhere [7]. Toany given finite order M, a large enough volume V can be chosen to eliminateall boundary terms, so that all configurations retained correspond to disorderedislands in a bulk ordered phase.

We check that the sum in Eq. (4) exponentiatesin V up to terms of order M + 1, defining a series for Fo truncated beyond orderM.

We have computed up to order M = 9, including terms up to N49,20(V ). Thisseries, whose first terms can be compared to existing low temperature series [8],provides us with similar series for the kth derivative with respect to β, F (k)o. Atβ = βt the k = 0 (free energy) and k = 1 (internal energy) series match the exactresults [3] up to M = 9.

The k = 2 and 3 cases giveF (2)o= 16q + 34q3/2 + 114q2 + 254q5/2 + 882q3 + 1944q7/2 + 6128q4+ 13550q9/2 ,(5)F (3)o= −64q −430q3/2 −2654q2−12186q5/2 −57018q3−224732q7/2−888024q4−3164682q9/2. (6)Let us make a few comments on these expansions.

(i) F (2)ogives the specific heat Co = β2t F (2)o. The F (3)oseries gives – ⟨(E −Eo)3⟩,the first odd moment of the energy distribution in this phase.

(ii) All terms in each series have the same sign, F (2) being of course positive whileF (3) < 0 means that E > Eo is favoured with respect to E < Eo. (iii) The coefficients are fastly increasing with the order, the more so for largervalues of k.This confirms the expectation that huge energy fluctuationsare associated with the large correlation length, of order x in Eq.

(2) whenq decreases towards q = 4.This is expected to have a direct impact onthe numerical analysis of these models, large values of the high cumulantsinvalidating the commonly used two gaussian formula [9].To proceed with a quantitative analysis of F (2) and F (3) as functions of q,we conjecture for these quantities an essential singularity at q = 4 as ξ has and3

construct Pad´e approximants for the series of ln F (k) instead of F (k), i.e. take asestimates of F (k)F (k)est = expPad´e(ln F (k))(7)The result of this construction for F (2) and F (3)/F (2) as functions of x for q =30, 20, 15, 10, 8, 7 and 6 is summarized in Fig.

1 as a log-log plot. The error barsare rough estimates of the uncertainties resulting from the higher order terms andhave been obtained by varying the degrees of the numerator and denominatorof the Pad´e approximant.

The lines represent our prejudices F (2) ≃Cst x andF (3)/F (2) ≃Cst x3/2 (see below), where the constants are fixed by the q = 10values. It is clear that the general trend of both quantities is well reproduced overan astonishingly large range by such simple forms.As a by-product of this study we obtain analytical estimates for the orderedphase specific heat, which are compared to existing numerical data in Table 1.4

qCanaloCexpoRef205.362(3)5.2(2)[10]1018.06(4)12.7(3)[11]∼18[11]10.7(1.0)[12]837.5(4)23.(3.)[12]771.3(1.0)47.5(2.5)[13]50.(10. )[11]44.4(2.2)[14]Table 1The ordered phase specific heat, our prediction compared to numerical estimatesAt q = 20 our prediction is in good agreement with the numerical estimate.In contrast it strongly disagrees at q ≤10 with the value of Co obtained from afinite size analysis of the maximum of the specific heat measured in the coexistenceregime.

However it agrees at q = 10 with the estimate obtained in [11] at β = βt,in accordance with the rigorous statements of [1].The large q expansion analysis supports the idea that not only does the cor-relation length in a pure phase diverge at β = βt, q →4+, but also that theassociated fluctuations imply divergences of the energy cumulants. For examplewe obtain F (3)o∼−1800 at q = 10 (see Fig.

1). Moreover the internal energyfluctuations behave in a way consistent with C/ξ being finite at q = 4+ as it isknown to be at q = 4−.

We then propose the following ansatz:There exists a region of q > 4 where the free energies around β = βt reflectaccurately the scaling properties associated with the second-order point lyingat q = 4, β = ln 3 and characterized by the corresponding critical indices αand ν.Specifically, according to the known value 2/3 of α, we parametrize F (2)o(β) atq = 4 and for β →(βt)+ asF (2)o(β) = A (β −βt)−2/3. (8)5

The higher derivatives are trivially deduced and in their expressions we replace(β −βt) by ( Cstξ−3/2 ) from Eq. (1) and, boldly continuing above q = 4 atβ = βt, reexpress F (p+2)oas a function of x via Eq.

(2) to getF (p+2) ≡F (p+2)o(βt) = A (−)p Γ(2/3 + p)Γ(2/3)(Bx)1+3p/2,(9)where the constant B takes into account proportionality constants. Thus we getthe following representation for the ordered phase free energyFo(β) = F(βt) −Eo(β −βt) +∞Xn=2(β −βt)n F (n)n!

(10)where we introduced the known linear term (F (1) = −Eo), and took F (n) as givenby Eq. (9) for n > 1.

Note that all the odd cumulants are negative, as we foundto be the case for F (3) from the large q expansion. A similar expression holds forFd, starting from Eq.

(8) with (β −βt) →(βt −β), and replacing Eo by Ed inEq. (10).It is easy to sum the series Eq.

(10). For later convenience, we introducescaled temperature (v), energy (ǫ) and length (S) variablesv = (β −βt)(Bx)3/2,ǫ = E(Bx)1/23A,S2 = (Bx)23A(11)and end up with the following compact resultFo(β) = F(βt) + 1S2−34 −(ǫo + 1)v + 34(1 + v)4/3(12)This equation is the central result of this letter.We claim that, although weneglected all regular and less singular contributions to Eq.

(8), Eq. (12) summarizesaccurately, over a wide range of q > 4 values, all the properties of the model.Let us justify this statement by comparing the predictions of Eq.

(12) forthe energy distribution to data [11] taken at q = 10 with various L values. Thisdistribution is obtained by inverse Laplace transform of the partition functionPV (E) = N1Z β0+i∞β0−i∞dβhq exp[V Fo(β)] + exp[V Fd(β)]iexp[V E(β −βt)](13)where Fd follows from Eq.

(12) by v →−v and ǫo →ǫd (this is consistent withduality up to terms of order x−3/2 as compared to 1) and with N1 a factor ensuring6

the probability normalization. Trading (β −βt) for v and E for ǫ of Eq.

(11), wegetP oV (E) = N2Z vo+i∞vo−i∞dv exp(LS )2[(ǫ −ǫo −1)v + 34(1 + v)4/3)](14)Details on the computation of this integral will be given in [7] and here we limitourselves to short remarks.i) As a function of ǫ−ǫo, P oV depends on q and L only through the scaled volume(L/S)2.ii) Any v0 > −1 is suitable and the integral converges exponentially.7

iii) At large (L/S), a saddle point method can be valuable. However (L/S) isnot large in practice and there exists an energy value slightly above Eo wherethe saddle point value reaches v = −1 (metastability point).iv) Actual computation requires numerical values of A and B.

At q = 10 we getA = .193 and B = .386 from the large q expansion results (at other q values,slight changes have to be made according to Fig. 1 ).v) At q = 10 the length scale is S ∼60, nearly 6 times the correlation length, sothat for current values of L the ratio (L/S) is hardly of order 1 !For the above reasons, we compute the integral Eq.

(14) numerically. The8

results are shown together with the data of [11] on Fig. 2 for L = 16, 20, 24, andon Fig.

3 for L = 36, 44, 50. Remembering that the continuous curves are absolutepredictions without any free parameter, it is quite striking to see how such a simpleansatz as Eq.

(14) yields good results. Because (L/S) is not large, they are verydifferent from what an asymptotic expansion would give.

For example Epeak −Eobehaves effectively as ∼1/L over a wide range of L values, whereas the asymptoticexpectation (saddle point method in Eq. (13)) isL2(Eo −Epeak) = −F (3)o/(2F (2)o) ∼15, 100, 350, 1000atq = 20, 10, 8, 7.Marked discrepancies only appear at the external edges of the ordered (left)and disordered (right) peaks.

In particular the theoretical curve levels out undulyat the bottom right of Fig. 2.

This is a spurious effect : the tail of the orderedpeak contributes more than the disordered phase. However it appears at a neg-ligibly small level at larger L’s, being asymptotically of order exp[−c L2].

As Lis increased (Fig. 3), the agreement between predictions and data becomes betterand better at nearly all values of E, but around the dip between the peaks.

Thereindeed mixed phase contributions with percolating interfaces should finally winover the pure phase contributions. We consider the small departure of the theo-retical curve below the data points at L = 50 as an evidence for the emergence ofmixed phase contributions.

Since the dip region is often used for the determinationof the interface tension σ because strip configurations eventually yield an energyindependent plateau in PV (E) with [15]2 σ ≃−1L ln PV (Edip)PV (Epeak),(15)the value and L dependence of the right hand side of Eq. (15) are interestingissues.

Although in our construction this quantity diverges as L asymptotically,we unexpectedly find it roughly constant for L ≤50, at a value around .11, notvery far from the exact value 1/ξd = 0.95 [5]. This casts some doubt on attemptsto determine σ from Eq.

(15) [13,14] unless the plateau in E is seen [16,17]Summarizing, we have shown, for the 2-d Potts model at q > 4, up to largeq values, that the bulk properties are accurately described close to βt by freeenergies Fo(β) and Fd(β) inferred by a simple ansatz from the known propertiesof the second-order point at q = 4.Definite expressions for Fo(β) and Fd(β)were obtained by matching this ansatz to a calculation of their large q expansion,achieved at order (1/√q)9. Our main result is expressed in Eq.

(12) and illustrated9

by the adequacy of the corresponding energy distribution to explain most of thefeatures observed in numerical simulations at finite volume.In the first-order region, each phase knows little about the existence of theother ones, and rather feels the effectively close critical point q = 4, β = βt, fromwhich it inherits nice scaling properties. This is so as long as the linear size ofthe box is smaller than the length scale S, proportional to the correlation lengthξ, but many times larger ( S = 5 to 7ξ in the range q = 7 to 20).Only atV ≥S2 asymptotics takes place, so that interface tension effects can show up andbe measured.Similar ideas could be applied to other situations.

One such situation is the3-d Potts model at q ≥3 where the first-order regime could also be influenced bythe second-order point q3d situated between q = 2 (Ising) and q = 3. Then onemight get some information on this critical point from numerical studies at q ≥3.Another interesting case is QCD at finite temperature; although the transition isfirst-order, universality might be nevertheless invoked to relate its behaviour tothat of the 3-d 3 states Potts model, both models sharing the same (universal)behaviour at a ”close” point of parameter space.We are aware of the fact that our ansatz describes the pure phase free energiesas analytic functions of β at βt, which they probably are not, in the same way asin field driven first-order transitions the zero-field point is an essential singularityof the free energy [18].

This question deserves further discussion, but we believethat our ansatz, although not ‘analytically correct’, takes into account the mostsignificant features of the model.It is a pleasure to thank R. Balian and A. Billoire for illuminating discussions.We also acknowledge useful conversations with B. Grossmann and T. Neuhaus.10

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