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M. Caselle, F. Gliozzi and S. Vinti

arXiv:hep-th/9304001v1 1 Apr 1993DFTT-12/93March, 1993Condensation of Handlesin the Interface of 3D Ising ModelM. Caselle, F. Gliozzi and S. VintiDipartimento di Fisica Teorica dell’Universit`a di TorinoIstituto Nazionale di Fisica Nucleare,Sezione di Torinovia P.Giuria 1, I-10125 Turin, Italy1AbstractWe analyze the microscopic, topological structure of the interface between domainsof opposite magnetization in 3D Ising model near the critical point.

This interfaceexhibits a fractal behaviour with a high density of handles. The mean area is analmost linear function of the genus.

The entropy exponent is affected by strong finite-size effects.1email address:Decnet= (31890::CASELLE,GLIOZZI,VINTI)Internet=CASELLE(GLIOZZI)(VINTI)@TORINO.INFN.IT

1. IntroductionThe interface in 3D statistical systems above the roughening temperature behaves like afree, fluctuating surface with a topology determined by the boundary conditions.

Recently,new computational algorithms have been applied to these systems [1-6], providing us witha powerful tool for testing our ideas on the behaviour of fluctuating surfaces. Alternatively,these surfaces can be thought of as space-time histories of closed strings which can be usedto describe the infrared behaviour of the dual Z2 gauge theory.In a previous paper [7] we studied the free-energy of the interface as a function ofits shape.

We found rather strong shape effects which are accurately described by thegaussian limit of the Nambu-Goto action.This observation suggests that interface configurations are mostly made of smoothsurfaces subjected to long-wavelength fluctuations which account for the observed finite-size effects. On the other hand, this picture seems to strongly disagree with the processof crumpling, which should take place for random surfaces embedded in a target space ofdimension d ≥1.

In other words, if the interface is described by the Nambu-Goto action,it should take the shape of a branched polymer, rather than that of a smooth surface.In this letter we face this dilemma by studying the interface from a microscopic pointof view. Actually, it turns out that the interface at small scales is much more similar to asponge than to a smooth surface.

In particular there is a strong, almost linear correlationbetween the area and the genus of the surface, indicating the formation of a large number ofmicroscopic handles which is proportional to its area. Yet, the partition function summedover all the genera behaves like that of a smooth toroidal surface: this means that the hugenumber of microscopic handles, produced by the instability toward crumpling, have as thenet effect a simple non-perturbative renormalization of the physical quantities associatedto the surface, according to an old conjecture on string theory [8].1

2. The methodThe first problem to be solved in order to study the microscopic structure of theinterface is to express the 3D Ising model or its dual gauge version as a gas of suitableclosed surfaces.

There are essentially two different ways to do it. One is based on thestrong coupling expansion of the dual gauge model [9] and involves also self-intersectingand non-orientable surfaces.

Here instead we identify the surfaces as the Peierls interfacesof an arbitrary Ising spin configuration on a cubic lattice [10]. By construction these areclosed, orientable surfaces composed of plaquettes of the dual lattice orthogonal to eachfrustrated link.

In this way each plaquette appears at most once in the construction ofthe surface S.Tab. IGraphs for the microscopic reconstruction of the interface.

Each cube is dual to a vertex Vq withq edges of the interface as drawn in fig. 1.

The encircled sites have a spin different from the othersites. Graphs with different vertex decompositions correspond to end-points of contact lines.

Thefirst decomposition is for positive vertices, while the other is for negative vertices.❅❅❜V3❅❅❜❜V4❅❅❜❜❜V5❅❅❜❜2 V3 ∼V6❅❅❜❜2 V3❅❅❜❜❜V3 + V4 ∼V7❅❅❜❜❜❜V4❅❅❜❜❜❜V3 + V5❅❅❜❜❜❜2 V4❅❅❜❜❜❜4 V3 ∼2 V6∼2 V3 + V6❅❅❜❜❜❜V6❅❅❜❜❜3 V3 ∼V6 + V32

A link belonging to S is said to be regular if it glues two plaquettes of S.The onlysingularities which may appear on these surfaces are links glueing four distinct plaquettesof S. These singular links form contact lines of the surface which are sometimes improperlycalled self-intersections. These contact lines are the only potential sources of ambiguitiesin the topological reconstruction of the surface.

We shall see shortly how to remove themin a simple, consistent way. A vertex of the surface S is dual to an elementary cube ofthe lattice of the Ising configurations.

According to the distribution of spins inside thecube, there are twelve distinct vertices as listed in Table I. Some of them include singularlinks and may be interpreted as the coalescence of distinct, regular vertices.

Only the end-points of contact lines can be decomposed into two or more inequivalent ways: for instancethe first graph in the second row of the table can be thought either as the coalescence oftwo vertices with three edges, or as a single vertex with six edges and the singular link issplitted accordingly into two different ways as drawn in figure 1.bfeca❅❅❅❅❞❞d↔adbcfeadbcfeadfebc→Fig. 1Example of decodification of a graph of Table I.

This cube has six frustrated links, labelled bya, b, . .

. , f, which are dual to six plaquettes forming a vertex with a singular link.It can beconsidered either as a 6-vertex, or as two 3-vertices.This choice depends on the sign of themagnetization of the cube, which can be in this case ±4.Choosing arbitrarily this splitting procedure for each end-point may generate globalobstructions in the surface reconstruction, due to a constraint which links together thesplitting of two end-points connected by a contact line.

In order to formulate this con-3

straint, it is useful to attribute a sign to each singular vertex according to the sign of themagnetization of the corresponding elementary cube. Then, working out some explicitexample, it is easy to convince oneself that there are no global obstructions if one choosesdifferent splittings for vertices of different sign.

More precisely, the only constraint to befulfilled in the replacement of each singular link with a pair of regular ones is that twoend-points connected by a contact line must be splitted in the same way if they havethe same sign, or in the opposite way if their sign is different. The two decompositionsappearing for singular vertices in table I correspond precisely to the two possible signsof the vertex.

The only case in which the surface reconstruction is not immediate is thefirst graph of the last row of tab. I, corresponding to a vertex with six singular lines: thethree possible decompositions cannot be selected by the sign of its magnetization (whichis zero), but by the signs of the end-points of the six contact lines.The main consequence of the above construction is that we have obtained a simple ruleto assign unambiguously to each Ising configuration a set of self-avoiding closed randomsurfaces.

Notice that this fact agrees with a result of David [11] who found that a gas ofself-avoiding surfaces in a special three-dimensional lattice belongs to the same universalityclass of the Ising model.We may now evaluate the genus h of each self-avoiding surface through the EulerrelationF −E + V = χ(S) = 2 −2h(1)where F is the number of faces (plaquettes), E the number of edges, V the number ofvertices and χ the Euler characteristic; the genus h gives the number of handles. Denotingby Nq the number of vertices of coordination number q, we have obviously, according totable I,V = N3 + N4 + N5 + N6 + N7 ; 4F = 2E = 3N3 + 4N4 + 5N5 + 6N6 + 7N7 ,(2)4

which gives at once an even simpler expression for the genus of this kind of surfaces4(V −F) = N3 −N5 −2N6 −3N7 = 8 −8h . (3)3.

ResultsWe are interested on the topology of the interface between two macroscopic domains ofopposite magnetization. For this reason we consider very elongated lattices with periodicboundary conditions in the two short directions (denoted in the following by L) andantiperiodic boundary conditions in the long direction (denoted by Lz).

This forces theformation of an odd number of interfaces in the Lz direction. We then isolate one of theseinterfaces by reconstructing all the spin clusters of the configuration, keeping the largestone and flipping the others.

Note, as a side remark, that for the values of β, L and Lzwe have studied, typical configurations contained only one macroscopic cluster, besidesa huge number of microscopic ones. It is now easy to evaluate area of the interface bycounting simply the number of frustrated links of this cleaned configuration.The growth of this area as a function of the size of the lattice gives a first descriptionof the fractal behaviour of the interface: in a set of Monte Carlo simulations near thecritical point on elongated lattices L2 × Lz with Lz ≥120 and 8 ≤L ≤16 (see table II)we found that the mean area < F > of the interface is a strongly varying function of thetransverse size L, well parametrized by the following power law< F >= κ LdH,(4)with dH ∼3.7 and κ ∼0.47 .

As a consequence, a sizeable fraction of the lattice is invadedby the interface, and a small increasing of the transverse lattice section implies a rapidgrowth of the area of the interface, so we have to take very elongated lattices in order toavoid wrapping of the interface around the antiperiodic direction, which would give riseto unwanted finite volume effects.5

Each interface S of area F of an arbitrary Ising configuration contributes to the par-tition function simply with a term e−βF; then we can define the following generatingfunctional ZZ(β) =XFX# surfaces of area Fe−βF =XFXhZh(F),(5)where Zh(F) is the partition function for a surface of area F and genus h. Since we aredealing with macroscopic surfaces, we can consistently assume that their multiplicity iswell described by the asymptotic behaviour of the entropy of large random surfaces [12],which yieldsZh(F)F →∞∼F bχ−1eµcFe−βF = Fbχ−1e−µF,(6)where, using the terminology of two-dimensional quantum gravity (2DQG), µ is the cos-mological constant and the exponent bχ is a function of the Euler characteristic χ. In thecontinuum theory of 2DQG this exponent can be evaluated exactly for the coupling toconformal matter of central charge c ≤1 [13].

The result isbχ = −b2χ = b(h −1),(7)withb = 25 −c +p(1 −c)(25 −c)12. (8)b is a monotonically decreasing function of c for c ≤1 and becomes complex for c > 1,where the surface get crumpled and these formulas lose any physical meaning.

It is knownthat for c = 1 there are logarithmic corrections to eq. (6), while for c > 1 it is only knownthat the number of surfaces of given area is exponentially bounded [12].

It turns out thateq. (6) fits well to our numerical data.We can evaluate the exponent bχ for the interface by measuring the mean area at fixedgenus.

Indeed definingZh(µ) =XFZh(F),(9)6

we have< F >h= −∂∂µ log Zh(µ) ∼bχµ. (10)In figure 2 we report a typical outcome of this analysis.

Actually < F >h is dominatedby a linear term, like in eq. (7), but there is also a small, negative quadratic contributionwhich probably accounts for the self-avoidness constraint: when the number of handlesh is very large the excluded volume effects (proportional to h2) become important.

Thedata are well fitted by< F >h= aµ + bµh + dµh2 . (11)The slope b can be called the entropy exponent, and a is related to the strings susceptibilityby a = γs −2.

Comparing eq. (11) with eq.

(7) we have of course, for c ≤1, d = 0 andb = −a ≡2 −γs . (12)Since we are dealing with surfaces of very high genus h ∼102, we can evaluate b/µand d/µ ∼−10−4b/µ very accurately, while the constant a is affected by large errors andcannot used to evaluate γs, nevertheless we shall argue shortly that for our surfaces eq.

(12)is no longer true.Combining eq. (10) with other physical quantities at fixed genus it is possible to evaluatethe cosmological constant µ as a function of β and of the lattice size.

In particular, usingthe derivative ∂Zh(µ)∂h, we get easily the following equationlog(1/µ) = −ψ(bχ)+ < log F >,(13)where ψ denotes the logarithmic derivative of the Euler Γ function. This formula fits wellthe numerical data for large areas and we used it to evaluate µ.

The results are reportedin table II as well as the entropy exponent b defined in eq. (11) .It turns out that bis affected by rather strong finite size effects, being a decreasing function of the latticesize L an hence of the area.

Similar finite size effects have been observed for the stringsusceptibility exponent in the planar quantum gravity coupled to c ≥1 matter [14]. Note7

however that, if one assumes eq. (12), one should conclude that γs exceeds the theoreticalupper bound γs = 12 [15].

A possible way out is that eq. (12) is not fulfilled by this kindof surfaces, indeed the argument leading to such upper bound assumes h = 0 and givesno restriction on b.

On the other hand there is a renormalization group argument [16]showing clearly that eq. (12) is justified only for c ≤1.Tab.

IIMean area < F >, entropy exponent b, cosmological constant µ and renormalizedcosmological constant µR of the interface in a set of elongated lattices of shape L2 × Lz atβ −βc = 0.0059 .L2 × Lz< F >bµµR82 × 1201057(10)0.73(20)0.0141(39)0.00122(4)102 × 1202434(19)0.61(15)0.0097(24)0.000613(15)122 × 1804719(47)0.30(3)0.0047(5)0.000347(4)142 × 2408506(103)0.19(3)0.0026(5)0.000208(3)162 × 32014803(249)0.11(2)0.0015(2)0.000125(5)It is also possible to define a renormalized cosmological constant µR by consideringthe sum over all the genera Z(F) = Ph Zh(F) . It turns out that for large areas thedistribution of surfaces is accurately described by an exponential fall offof the typeZ(F)F →∞∼ce−µRF,(14)as shown in fig.3.

Comparing this equation with eq. (6), we argue that the sum over all thetopologies produces as net effect an effective interface which behaves as a smooth surfacewith a different cosmological constant.

Actually the renormalization effect is very large:µR is about one order of magnitude smaller than the unrenormalized quantity µ(seetable II). This phenomenon seems the two-dimensional analogue of a quantum gravityeffect described by Coleman [17], who argued that the sum over topologies has the effectof making the cosmological constant vanishing small.In conclusion, we have found a simple way to generate high-genus self-avoiding random8

surfaces. The most interesting question is of course whether their study will lead to newinsights in the physics of random surfaces.Acknowledgements.

We thank A.Pelissetto and A. Sokal for sending us their veryefficient program of cluster reconstruction, and D. Boulatov, V. Kazakov and A. Migdalfor enlightening discussions.References[1] H.Gausterer, J.Potvin, C.Rebbi and S.Sanielevici, preprint BU-HEP-92-16. [2] S.Klessinger and G.M¨unster, Nucl.

Phys. B386 (1992) 701.

[3] B.Berg, U.Hansmann and T.Neuhaus, preprint BI-TP 92/10, June 1992. [4] H.Meyer-Ortmanns and T.Trappenberg, J. Stat.

Phys. 58 (1990) 185.

[5] M.Hasenbusch and K.Pinn, preprint MS-TPI-92-24, September 1992. [6] M.Hasenbusch, preprint KL-TH 16/92, September 1992.

[7] M.Caselle, F.Gliozzi and S.Vinti, Phys. Lett.

B302 (1993) 74. [8] M. Ademollo, A. D’Adda, P. Di Vecchia, F. Gliozzi, E. Napolitano and S.Sciuto, Nucl.Phys.

B94 (1975) 221. [9] C.Itzykson, Nucl.

Phys. B210 [FS6] (1982) 477; A.Casher, D.F¨orster and P. Widney,Nucl.

Phys. B251 [FS13] (1985) 29; A.R.Kavalov and A.Sedrakyan Nucl.

Phys. 285[FS19] (1987) 264; J.Ambjørn, A.Sedrakyan and G. Thorleifsson, preprint NBI-HE-92-80, December 1992.

[10] M.Karowski and H.J.Thun, Phys. Rev.

Lett. 54 (1985) 2556; M.Karowski, J.Phys.

A19 (1986) 3375. [11] F.David, Europhys.

Lett.9 (1989) 575.9

[12] J.Ambjørn, B.Durhuus and J. Fr¨olich, Nucl. Phys.

B257 [FS14] (1985) 433. [13] J.Distler and H.Kawai, Nucl.

Phys. B321 (1989) 509; F.David, Mod.

Phys. Lett.

A3(1988)1651. [14] J.Ambjørn, S.Jain and G.Thorleifsson, preprint NBI-HE-93-4, February 1993.

[15] J.Ambjørn, B.Durhuus, J. Fr¨olich and P.Orland, Nucl. Phys.

B270 [FS16] (1986)457. [16] E.Br´ezin and J.Zinn-Justin, Phys.

Lett. B288 (1992) 54.

[17] S.Coleman, Nucl. Phys.

B310 (1988) 643.Figure CaptionsFigure 2 The mean area of the interface as a function of the genus in a lattice of size122 × 240 at β −βc = 0.0044 .Figure 3 The multiplicity of interface configurations summed over all the topologies as afunction of the area in a lattice of size 142 × 240 at β −βc = 0.0059. The lower setof data is the multipicity a fixed genus h = 70.

The stright line is the fit to eq. (14).10

This figure "fig1-1.png" is available in "png" format from:http://arxiv.org/ps/hep-th/9304001v1

This figure "fig2-1.png" is available in "png" format from:http://arxiv.org/ps/hep-th/9304001v1

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