Nuclear Physics B 644 [FS] (2002) 495–508 www.elsevier.com/locate/npe

Euler–Poincaré characteristic and phase transition in the Potts model on Z2 Philippe Blanchard a , Santo Fortunato b , Daniel Gandolfo a,c,d a Fakultät für Physik, Theoretische Physik and BiBos, Universität Bielefeld, Universitätsstrasse, 25,

D-33615 Bielefeld, Germany b Fakultät für Physik, Theoretische Physik, Universität Bielefeld, Universitätsstrasse, 25,

D-33615 Bielefeld, Germany c PhyMat, Département de Mathématiques, Université de Toulon et du Var, BP 132,

F-83957 La Garde Cedex, France d CPT, CNRS, Luminy case 907, F-13288 Marseille Cedex 9, France

Received 7 May 2002; received in revised form 11 July 2002; accepted 7 August 2002

Abstract Recent results concerning the topological properties of random geometrical sets have been successfully applied to the study of the morphology of clusters in percolation theory. This approach provides an alternative way of inspecting the critical behaviour of random systems in statistical mechanics. For the 2d, q-states Potts model on the square lattice with q
6, intensive and accurate numerics indicates that the average of the Euler characteristic (taken with respect to the Fortuin–Kasteleyn random cluster measure) changes sign at the critical threshold of the magnetization transition.  2002 Elsevier Science B.V. All rights reserved. Keywords: Cluster morphology; Euler–Poincaré characteristic; Phase transition

1. Introduction Recently, new insights in the study of the critical properties of clusters in percolation theory have emerged based on ideas coming from mathematical morphology [1] and integral geometry [2–4]. These mathematical theories provide a set of geometrical and topological measures allowing to quantify the morphological properties of random E-mail addresses: [email protected] (P. Blanchard), [email protected] (S. Fortunato), [email protected] (D. Gandolfo). 0550-3213/02/$ – see front matter  2002 Elsevier Science B.V. All rights reserved. PII: S 0 5 5 0 - 3 2 1 3 ( 0 2 ) 0 0 6 8 1 - 8

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systems. In particular these tools have been applied to the study of random cluster configurations in percolation theory and statistical physics [5–9]. One of these measures is the Euler–Poincaré characteristic χ which is a well-known descriptor of the topological features of geometric patterns. It belongs to the finite set of Minkowski functionals whose origin lies in the mathematical study of convex bodies and integral geometry (see [2–4]). These measures, as we shall explain below, share the following remarkable property: any homogeneous, additive, isometry-invariant and conditionally continuous functional on a compact subset of the Euclidean space Rd can be expressed as a linear combination of the Minkowski functionals. This is the well-known Hadwiger’s theorem [2] of integral geometry which has a wide scope of applications in mathematical physics due its rather general settings. The use of these measures in image analysis [10], problems of shape recognition [1], determination of the large scale structures of the universe [11], modelling of porous media [12], microemulsions [13] and fractal analysis [14] has been a topic of growing interest recently. We also recall that for the problem of bond percolation on regular lattices, Sykes and Essam [15] were able to show, using standard planar duality arguments, that for the case of self-dual matching lattices (e.g., Z2 ), the mean value of the Euler–Poincaré characteristic changes sign at the critical point (this even led them to announce a proof for the value of the critical probability of bond percolation on Z2 ), see also [16]. More recently, Wagner [8] was able to compute the Euler–Poincaré characteristic on the set of all plane regular mosaics (the 11 Archimedean lattices) as a function of the site occupancy probability p ∈ [0, 1]] and showed that a close connection exists between the threshold for site percolation on these lattices and the point where the Euler–Poincaré characteristic (expressed as a function of p) changes sign. The aim of this article is to further investigate the role played by this morphological indicator in statistical physics and to present new results concerning its behaviour in the case of the 2-dimensional Potts model. Namely we present here clear evidence, based on Monte Carlo simulations, that for the 2d Potts model, the Euler–Poincaré characteristic changes sign at the critical point of the thermal transition. Namely we find that for q = 1, . . . , 4 it changes sign continuously at the transition point while, for q = 5, 6 it has a first order transition at the critical point. As far as we know, this is the first example of a discontinuous behaviour of this parameter in a physical model. The paper is organized as follows. In Section 2 we introduce basis facts concerning Minkowski functionals and the necessary definitions for our model, the numerical results are presented in Section 3 followed by some comments and discussion in Section 4.

2. The model We first briefly summarize the basic facts from integral geometry and give the definition of the Minkowski functionals including the Euler–Poincaré characteristic, see [2–4] for more complete expositions. We will then show how to compute the Euler–Poincaré characteristic in the case of a random configuration of sites and bonds

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produced by a Fortuin–Kasteleyn (FK) transformation of the partition function of the Potts model [17]. 2.1. Minkowski functionals The Euler–Poincaré characteristic χ is an additive functional on subsets of Rd such that for A, B ⊂ Rd (see example in Fig. 1) χ(A ∪ B) = χ(A) + χ(B) − χ(A ∩ B) with


χ(A) =

(2.1)

1, if A = ∅, A convex, 0, A = ∅.

The set of Minkowski functionals is then defined by (see, e.g., [5,10])  Wα (A) = χ(A ∪ Eα ) dµ(Ea ), α = 0, . . . , d − 1, Wd (A) = ωd χ(A),

ωd =

π d/2 Γ (1 + d/2)

(2.2) (2.3)

where Eα is an α-dimensional plane in Rd , dµ(Ea ) its density normalized such that, for the d-dimensional ball Bd (r) with radius r, Wα (Bd (r)) = ωd r d−α and ωd is the volume of the unit ball in Rd . Obviously, additivity of the Minkowski functionals is inherited from (2.1), furthermore they are usually conveniently normalized through Mα (A) =

ωd−α Wα (A). ωd ωα

The computation of these normalized functionals in dimensions 1, 2, 3 in terms of the usual geometric measures (length, area, volume, . . . ) is given in Table 1. An important result of integral geometry is Hadwiger’s completeness theorem which asserts that, under not too restrictive and furthermore physically reasonable assumptions, namely: additivity, motion invariance (under translations and rotations) and conditional continuity (which states that any convex body can be smoothly approximated by convex polyhedra),

Fig. 1. Examples of calculation of the Euler–Poincaré characteristic in dimension d = 2 for some combinations of convex subsets of R2 .

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Table 1 Values of the normalized Minkowski functionals for d-dimensional subsets of Rd , d = 1, 2, 3 in terms of geometric measures, L: length, S: area, V : volume, C: circumference, H : integral mean curvature, χd : Euler– Poincaré characteristic d

M0

M1

M2

M3

1 2 3

L S V

χ1 /2 C S

··· χ2 /π H /2π 2

··· ··· 3χ3 /4π

any functional M decomposes as a linear combination of the finite set of Minkowski functionals M(A) =

d 

cν Mν (A),

ν=0

where the cν are real coefficients. This theorem has very important practical consequences, we refer to [2,3,10]. We shall only mention here the principal kinematic formula (see [3,9]) which has been used to estimate the critical thresholds in continuum percolation theory [18,19]. It reads  Mα (A ∪ gB) dg = G

α    α Mα−β (B)Mβ (A), β

α = 0, . . . , d,

β=0

where integration is over the group of motions G (i.e., rotations and translations, g = (r, Θ)). This formula is very useful to calculate mean values of Minkowski functionals for random distributions of objects. For instance, it can be applied to the computation of the excluded volume of convex bodies (leading, in case of spherical objects, to Steiner’s formula) which has important applications in continuum percolation theory (see [18]). 2.2. Euler–Poincaré characteristic of 2d cell complexes It is one of the topics of algebraic topology to show that the above general definitions of Minkowski functionals (including Euler–Poincaré characteristic) extend to cell complexes to which random bond configurations on regular lattices belong. The square lattice Z2 can be viewed as a cell-complex L = {L0 , L1 , L2 }, where L0 = Z2 is the set of sites, L1 is the set of bonds and L2 is the set of plaquettes (see [20–23]). To a set of bonds B ⊂ L1 we associate the subcomplex Λ(B) ⊂ L defined as the maximal closed subcomplex containing B. A subcomplex K0 of a complex K is said to be closed if every element of K which precedes (i.e., is less than) some element of K0 is itself an element of K0 . K0 is maximal in the sense that if K0 is a subcomplex of K and K0 is strictly contained in K0 , then K0 is not closed. This subcomplex can be written Λ(B) = Λ0 (B) ∪ Λ1 (B) ∪ Λ2 (B),

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where Λ1 (B) = B, Λ0 (B) = {x ∈ L0 | x ∩ B = ∅}, i.e., it is the set of sites which are endpoints of bonds of B, and   Λ2 (B) = p = [x, y, z, t] ∈ L2 : xy, yz, zt, tx ∈ B . In other words Λ2 (B) is the set of plaquettes all of whose bonds belong to B. We use the notation N 0 (B) ≡ |Λ0 (B)| to denote the number of sites of Λ0 (B), N 1 (B) ≡ |Λ1 (B)| to denote the number of bonds of Λ1 (B) and N 2 (B) ≡ |Λ2 (B)| the number of plaquettes of Λ2 (B). The Euler–Poincaré characteristic of Λ(B) is defined by [21] χ(B) = N 0 (B) − N 1 (B) + N 2 (B).

(2.4)

It satisfies the Euler–Poincaré formula [21] χ(B) = π 0 (B) − π 1 (B) + π 2 (B),

(2.5)

where π 0 (B) and π 1 (B) are respectively the number of connected components and the number of independent cycles of Λ(B). Here π 2 (B) = 0 because Λ(B) is a 2d closed cellcomplex. π 2 (B) characterizes the number of holes that are not homotopy equivalent to a point (see [21]), a situation that can arise only from dimension d = 3. It turns out that this formalism of cell complexes, which has been used at length in order to prove deep results in statistical mechanics [9,22,24–27], is all what we need to compute the Euler–Poincaré characteristic of FK random bond configurations for the Potts model. We produce below the results we get in case of the 2d Potts model but the same formalism extends to higher dimensions. 2.3. Potts model The partition function for the q-states Potts model on a lattice Λ ⊂ Zd at inverse temperature β reads
   Potts Zβ,q (2.6) (Λ) = exp β δ(σi , σj ) , σ

i,j ⊂Λ

where the first sum runs over all configurations σ ⊂ {1, . . . , q}|Λ| , the second one is over each nearest neighbour pair of Potts spins on Λ and δ is the Kronecker symbol. We remind that, whenever q is large enough, in any dimension d  2, this model exhibits a unique (inverse) temperature βc where the mean energy is discontinuous (see [24–26]). In dimension d = 2 for q  5 this is an exact result [28] and it is expected to be true, in d = 3, for q  3 [29]. After performing the (FK) transformation [17], the partition function (2.6) leads to the following random cluster representation 

N 1 (X) N (X) FK eβ − 1 (Λ) = q Λ . Zβ,q (2.7) X

Here the summation is over all graphs X which can be drawn inside the domain Λ and NΛ (X) is the number of connected components of X (including isolated sites). As before,

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Fig. 2. A bond configuration X ⊂ BΛ with 25 sites, 19 bonds and 2 plaquettes.

we shall call N 1 (X) the number of bonds of the configuration X, N 0 (X) the number of sites which are endpoints of a bond in X and N 2 (X) the number of plaquettes of the configuration X, i.e., the set of cells in Λ having 4 occupied bonds on its boundary (see Fig. 2). Formula (2.4) leads to χ(X) = N 0 (X) − N 1 (X) + N 2 (X).

(2.8)

this expression will allow us to compute the mean value of the Euler–Poincaré characteristic with respect to the FK measure.

3. Numerical results We have performed Monte Carlo simulations of the 2-dimensional q-state Potts model on the square lattice for q ranging from 2 to 6. We have always simulated the models near the critical temperature, whose value can be exactly determined through the well-known √ formula [29]: βc (q) = J /kTc (q) = log(1 + q). In order to extract a value of the Euler characteristic χ as close as possible to the value at the infinite volume limit, we have taken rather large lattices: for the three models with a continuous transition (q = 2, 3, 4) we arrived at lattice sizes up to 20002. The algorithm we used is the Wolff cluster update [30]. The identification of FK cluster configurations has been performed via the Hoshen– Kopelman algorithm [31]; we always considered free boundary conditions for the cluster labeling. The calculation of the Euler characteristic is done during the cluster labeling procedure and takes basically zero CPU time: the number of active bonds is stored during the main Hoshen–Kopelman sweep of the lattice, the number of sites joined by bonds is trivially obtained right after the clusters are labeled (it is just the total size of the clusters containing at least two sites). The determination of the number of plaquettes is more involved, but it requires only an additional sweep over the elementary squares of the lattice. This procedure is relatively fast since the analysis of a plaquette stops as soon as a “broken” bond is found. The Wolff algorithm is the less efficient the bigger the number q of states. Particularly dramatic is what happens when one passes from the 2-state (Ising) to the 3-state model:

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in the former case, on the 20002 lattice it is enough to perform few updates (5–10) to get uncorrelated configurations for the cluster variables, in the latter one needs about 1000 updates! Because of that, the simulations for q = 3, 4 on the 20002 lattice were very slow, and the relative data could not reach a high statistics. This is also the reason why we used the Hoshen–Kopelman algorithm instead of extracting the information from the single cluster of the update: since for q > 2 basically all the CPU time is taken by the update phase of the program, the analysis of all clusters of a given configuration allows us to improve considerably the statistics of the percolation data without time losses. If χ changes sign at the threshold, on a finite lattice the values measured at each iteration would be distributed around zero, provided the lattice is large enough. Therefore one would see both positive and negative values. For this reason, it is helpful to look at the distribution of χ . The values of the Euler characteristic we shall refer to are meant per lattice site; this has the advantage that we can see if and how the data concentrate around some value by increasing the lattice size. In the following we present separately the results for q = 2, 3, 4 and q = 5, 6. 3.1. Results for the models with a continuous phase transition The first case we consider here is the Ising model. Fig. 3 shows the χ distribution for three different lattice sizes: 5002 , 10002 and 20002 . In each case we have taken 100 000 iterations, measuring the variables of interest every 5 updates. The peak of the distribution shifts towards χ = 0 the larger the lattice. The average values of χ are: χ(5002) = 0.00101(2), χ(10002) = 0.00053(2), χ(20002) = 0.00024(1). We notice that the averages are quite small and decrease sensibly if we go to larger sizes, reducing themselves to about the half when we pass from a lattice to the next one. This approximate linear scaling of χ with the lattice side suggests that the Euler characteristic at the infinite volume limit indeed vanishes.

Fig. 3. Distribution of χ for the FK cluster configurations of the 2d Ising model at the critical point.

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Fig. 4. Distribution of χ for the FK configurations of the 2d, 3-state Potts model at the critical point.

Fig. 5. Distribution of χ for the FK cluster configurations of the 2d, 4-state Potts model at the critical point.

Let us now examine the case q = 3. In Fig. 4 we again plot the χ distribution for the same three lattice sizes we have considered for the Ising model. Since we collected a different number of measurements for the different lattices, for a real comparison of the distributions we needed to renormalize the total number of measurements on each lattice to the same value: we decided to renormalize all data sets to the number of measurements on the 5002 lattice (10 000). The distributions are broader than the Ising ones but they appear almost exactly centered at χ = 0. The average values are in fact much smaller than before: χ(5002) = 0.00024(4), χ(10002) = 0.00012(3), χ(20002) is zero within errors. We then deduce that also for q = 3, χ = 0 at the critical point. To complete our analysis we studied the case q = 4. In Fig. 5 we present a comparison of the χ distributions for two lattice sizes, 8002 and 20002 . There is a clear shift of the center of the distribution towards zero when one goes from the smaller to the larger lattice. The average values of the Euler

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characteristic in the two cases are χ(8002 ) = −0.00145(8) and χ(20002) = −0.0006(2). Also here there is no apparent convergence to some value, even if the lattices are rather large: |χ| reduces itself to less than its half by changing the lattice size. From all this we also deduce that the Euler characteristic of the FK clusters of the 2-dimensional 4-state Potts model vanishes at criticality. We also found that the variance 1χ of the χ distribution for the Ising model scales as L−0.95 ≈L−1 , where L is the lattice side; for the 3-state Potts model it seems that 1χ scales as L−0.78 . We notice that, in the Ising case, if the correct scaling behaviour of 1χ goes indeed as the inverse lattice side, the fluctuations of the Euler characteristic would behave like the energy ones. But the fluctuations of the energy are proportional to the specific heat of the system, and the L−1 behaviour is a consequence of the fact that the specific heat of the 2D Ising model diverges logarithmically at criticality (α = 0). That could mean that, in the Ising case, the fluctuation of the Euler characteristic diverges with temperature like the specific heat. This impression is confirmed by the results on the 3-state Potts model: since in that case α/ν = 2/5, the scaling behaviour of the energy fluctuations with L goes like L−4/5 = L−0.8 , in excellent agreement with our numerical estimate. 3.2. Results for the models with a first order phase transition For q > 4 the 2d q-state Potts model undergoes a first order phase transition, i.e., the thermal variables vary discontinuously at the critical threshold. The magnetization, for instance, makes a jump, varying from zero to a non-zero value. Because of that, we expect that the cluster configurations change abruptly at the critical point, and that the cluster variables exhibit as well discontinuities. In particular, the Euler characteristic may jump from a value to another. We analyze here the 5- and 6-state Potts models. In both cases we have performed simulations on three lattices: 1002 , 2002 and 3002 . In Figs. 6 and 7 we compare the distribution histograms on the 3002 lattice of the magnetization M and the Euler characteristic χ at three different temperatures near Tc . We define the magnetization by taking the excess of sites in the majority spin state (per lattice site) with respect to the value 1/q in the paramagnetic phase, when all spin states are equally distributed. Therefore we always measure M > 0 and that removes the degeneracy of the magnetization states due to the Z(q) symmetry of the Hamiltonian. In this way, if one finds a double peak structure in some temperature range, one can suspect that the transition is discontinuous. Looking at the magnetization histograms of the figures one clearly sees the spontaneous symmetry breaking by reducing the temperature. The double peak structure of M suggests that the transition is first order, as it is known. The corresponding histograms of the Euler characteristic show a perfectly analogous pattern. To check whether the transition is indeed first-order, we determined the temperature βH at which the two peaks of the Euler characteristic distribution are equally high. For 5-state Potts we found βH (1002) = 1.17343, βH (2002) = 1.17405 and βH (3002) = 1.17422; for 6-state Potts βH (1002) = 1.23763, βH (2002) = 1.23804 and βH (3002) = 1.23812. Successively we analyzed the scaling of the hump between the two peaks with the linear dimension L of the lattice. The height of the hump χm decreases with L according to the law log(χm ) ∝ −L2 , which is

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Fig. 6. Distribution histograms of the magnetization M and the Euler characteristic χ for the 2-dimensional 5-state Potts model at three different temperatures. The lattice size is 3002 . The behaviour of χ is driven by M.

the typical behaviour at a first order phase change. As the result is valid for the 5-state and the 6-state Potts model, it is likely to be valid also for q > 6, when the discontinuity of M at the threshold is sharper. Looking at both figures we remark that the centers of the peaks of χ look approximately symmetric with respect to zero. If this symmetry exists, it would be an interesting feature, and at the moment we have no arguments to justify it. In order to determine with some accuracy the values of χ in the two coexisting phases we would need to increase considerably the size of the lattice, but the required computer time would increase dramatically for the reasons we explained at the beginning of this section.

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Fig. 7. Distribution histograms of the magnetization M and the Euler characteristic χ for the 2-dimensional 6-state Potts model at three different temperatures. The lattice size is 3002 . The behaviour of χ is driven by M.

4. Conclusions This work clearly indicates that the Euler–Poincaré characteristic χ is indeed an important indicator of the phase transition for the 2d Potts model, to the extent of the cases (q = 2, . . . , 6) studied here. For this model, it reveals that the topology of cluster configurations has a deep meaning concerning criticality. This result is not trivial. Considering the 2d Ising model, the Euler–Poincaré characteristic computed from the physical clusters of +/− spins changes sign near Tc but definitely not at the critical point. For the 11 plane mosaics, Wagner has found [8] that χ changes sign close to (but not right at) the critical percolation point. For several models with continuous spins, a similar behaviour as the one described in this paper holds. We know already that

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the modification of the clusters topology induced by the FK representation has non-trivial consequences on the critical behaviour of spin systems [32–34]. The fact that χ changes sign at Tc for the 2d Ising model in FK representation can be understood heuristically in the following way. From Tc up to T = ∞ the system is in its disordered phase and the only excitations one can get in the FK-bond representation are made of isolated bonds (the probability to see any plaquette vanishes exponentially). Applying the Euler–Poincaré formula (2.5), one sees that χ behaves like π 0 (X) times a term of the order of the volume of the system. However, from Tc down to T = 0, the system is in its ordered phase and the corresponding FK-configuration is (with high probability) made of O(1) connected bond components. Missing bonds constitute the excitations and their number scales with the volume of the system so, using again (2.5), one gets that χ behaves like −π 1 (X) times a term of the order of the volume of the system. This explains in the case of the Ising model the change of sign of χ at Tc . For Potts spins, when the transition is first order (q  3), a similar argument holds in FK representation at the critical point. The striking phenomenon is the vanishing of the Euler–Poincaré characteristic at Tc when the transition is second order. Of primary interest is of course to understand how to relate the Euler–Poincaré characteristic to the order parameter of the phase transition. This is probably not a simple task and has not been done so far, even for models when an exact formulae can be derived for the Euler–Poincaré characteristic (see [5]). Scaling properties and critical exponent of this quantity are also subjects of great interest. Another important question concerns the critical behaviour in gauge models. For example a similar study could provide some insights concerning the deconfining transition in SU(N ) gauge theory. Indeed, some works [35] tend to indicate that this transition could be probed by percolation of some physical clusters related to color fields in lattice QCD. These models have been thoroughly investigated in the past and the tools coming from algebraic topology have been of primary importance to uncover profound duality results concerning their phase structure [23,36]. Other spin systems have to be investigated in order to see whether this property of the Euler–Poincaré characteristic is shared by models with continuous symmetries such as the (X–Y )-model or the Widom–Rowlinson model. How does χ behave in spin glasses for example?

Acknowledgements We are indebted to H. Wagner who initiated our interest for the topics developed in this paper and to J. Ruiz for fruitful discussions. Financial support from the BiBoS Research Center (University of Bielefeld), TMR network ERBFMRX-CT-970122 and the DFG under grant FOR 339/1-2 are gratefully acknowledged.

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