The interface free energy: Comparison of accurate Monte Carlo results for the 3D Ising model with effective interface models
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The interface free energy: Comparison of accurateMonte Carlo results for the 3D Ising model witheffective interface models
Michele Caselle a , Martin Hasenbusch b and Marco Panero ca Dipartimento di Fisica Teorica dell’Universit`a di Torino and I.N.F.N.,Via Pietro Giuria 1, I-10125 Torino, Italye–mail: [email protected] b Dipartimento di Fisica dell’Universit`a di Pisa and I.N.F.N.,Largo Bruno Pontecorvo 3, I-56127 Pisa, Italye–mail:
[email protected] c Institute for Theoretical Physics, University of Regensburg,93040 – Regensburg, Germanye–mail: [email protected]
Abstract
We provide accurate Monte Carlo results for the free energy of in-terfaces with periodic boundary conditions in the 3D Ising model. Westudy a large range of inverse temperatures, allowing to control correc-tions to scaling. In addition to square interfaces, we study rectangularinterfaces for a large range of aspect ratios u = L /L . Our numeri-cal results are compared with predictions of effective interface models.This comparison verifies clearly the effective Nambu-Goto model upto two-loop order. Our data also allow us to obtain the estimates T c / √ σ = 1 . m / √ σ = 3 . and R + = f σ = 0 . Introduction
Interfaces play an important rˆole in various fields of natural sciences. In softcondensed matter physics, in chemistry and in biology, interfaces separatingtwo different media, for instance two different magnetization domains, ortwo different fluids, or a fluid and its vapour, are studied. The properties ofsuch interfaces might be described by a unique effective model such as thecapillary wave model [1].Our motivation to study interfaces originates from the theory of highenergy physics. An interface with given boundary conditions can be asso-ciated with the world-sheet of a fluctuating flux tube in the confinementregime of a gauge theory. For intermediate and long distances between thesources, the relevant degrees of freedom for a system of confined quarks aresupposed to be independent of the short distance gauge interaction, andmight be modelled by string fluctuations ( effective string picture ).The simplest set-up for a numerical study of interfaces is provided bythe Ising spin model on a simple cubic lattice. Its duality with respect tothe Z gauge model [2] maps the ordered phase to the confined regime.The classical Hamiltonian of the Ising spin model reads: H ( { J } , { h } , { s } ) = − X h xy i J h xy i s x s y − X x h x s x , s x ∈ { , − } , (1)where x = ( x , x , x ) is a site of the lattice, and h xy i denotes a pair ofnearest neighbours on the lattice. Here and in the following, the latticespacing a is set to 1, and we shall always consider the case of a vanishingexternal field h x = 0, ∀ x . The site coordinates run over 0 ≤ x i ≤ L i − i ∈ { , , } label the three directions.In the case of periodic boundary conditions we take J h xy i = 1 for alllinks h xy i . Anti-periodic boundary conditions, say, in the direction 0, can beimplemented imposing J h xy i = − x = ( L − , x , x ) and y = (0 , x , x ),and J h xy i = 1 otherwise.The partition function is obtained as the sum over all configurations { s } of the Boltzmann factor: Z { J } ( β ) = X { s } exp ( − βH ( { J } , { s } )) , (2)where β = 1 / ( k B T ) is the inverse of the temperature of the three-dimensionalclassical spin model. 1he goal of our work is to study an interface between the phases ofpositive and negative magnetization in the low-temperature regime of thespin model — which corresponds to the confining regime of the gauge theory.Such an interface can be forced into the system by appropriate boundaryconditions. For instance, one could constrain the spins at x = 0 to take thevalue − x = L − L effects are smallerand better understood than for Dirichlet boundary conditions.In recent works [3–12] we studied interfaces with Dirichlet boundaryconditions in one direction and periodic boundary conditions in the otherdirection: via duality, this corresponds to a Polyakov loop correlator in thegauge model.The comparison with the Nambu-Goto effective string model resulted inunexpected discrepancies at subleading orders. While finite L corrections,in the direction with periodic boundary conditions, are described well by theeffective theory, the finite L corrections, in the direction of the Dirichletboundary conditions, show unexpected deviations.In order to further investigate this issue, we pick up again the work oninterfaces with periodic boundary conditions in both directions. In [13–15] such a comparison had been performed for square interfaces L = L ;in these studies, the numerical values of the interface tension were takenfrom [7].In the present work, our results for the interface free energy allow for anindependent determination of the interface tension, which is computed intechnically quite a different way with respect to [7]; the consistency of thetwo results provides a non-trivial check of their validity.We obtain results for a large range of the inverse temperature β , allowingto study possible scaling corrections. Furthermore, we also compute theinterface free energy for L = L for a large range of u = L /L : this enablesus to compare with the non-trivial dependence on u , which is predicted bythe effective interface models.Finally, the results for the interface tension are also used in combinationwith a series analysis of the second moment correlation length in the hightemperature phase. This yields a precise estimate of the universal amplituderatio: R + = f σ , (3)where the amplitudes are defined by σ ≃ σ ( − t ) µ and ξ ≃ f t − ν . Here, σ is the interface tension, ξ the second moment correlation length in thehigh temperature phase, t = ( T − T c ) /T c the reduced temperature and ν ,2 = 2 ν the critical exponents of the correlation length and the interfacetension, respectively. The result for R + can be compared e.g. with resultsobtained from experiments on binary mixtures.We also update the estimate for: m / √ σ , (4)where now the error is dominated by the estimate of the mass m ofthe 0 ++ glueball. Note that under duality the interface tension of the Isingspin model is equal to the string tension of the Z gauge model and theexponential correlation length in the low temperature phase of the Ising spinmodel is equal to the inverse mass of the 0 ++ glueball in confined phase ofthe Z gauge model. While there is no direct experimental particle physicsrelevance of this result, it is interesting for theoretical reasons to compare m / √ σ obtained from different gauge theories. Finally we also provide anupdated estimate of the finite temperature transition T c / √ σ . Note that herewe refer to the temperature of the two-dimensional quantum field theory. Itstemperature is given by T = 1 / ( aL ) and should not be confused with thetemperature of the three-dimensional classical system defined above. In thefollowing we shall denote the critical value of L by N t ; i.e. T c = 1 / ( aN t ).The content of this paper is the following: In section 2 we define theinterface free energy for finite interface area L × L and finite transverse sizeof the system L . Next, in section 3, we briefly summarize the predictionsfor the dependence of the interface free energy on ( L , L ), according toan effective string-like description. In section 4 we present our numericalmethod to compute the interface free energy. Our results for square and forrectangular interfaces are presented in section 5, while section 6 contains ourresults for the universal amplitude ratios. A summary and our conclusionsare given in section 7. The numerical integration methods are presented inthe appendix A. The basic quantity that we shall determine numerically is the ratio betweenthe partition functions of the system with anti-periodic Z a and periodicboundary conditions Z p . The purpose of this section is to provide a definitionof the interface free energy in terms of this ratio.The ratio Z a /Z p can be expressed in terms of the eigenvalues λ n x of the3ransfer matrix and the parity p n x = ± Z a Z p = P n P x = s,a p n x λ L n x P n P x = s,a λ L n x . (5)For L ≫ ξ , where ξ = − / ln( λ s /λ s ) is the bulk correlation length or theinverse of the mass of the theory, the partition function ratio in eq. (5) isdominated by the largest eigenvalues λ s and λ a : Z a Z p ≃ λ L s − λ L a λ L s + λ L a = 1 − ( λ a /λ s ) L λ a /λ s ) L . (6)In this regime, the so-called tunneling mass: m t = − ln( λ a /λ s ) (7)can thus be obtained from: m t = − L ln (cid:18) − Z a /Z p Z a /Z p (cid:19) . (8)Now let us relate the ratio of partition functions with the phenomenologicalpicture of interfaces separating the phases of positive and negative magneti-sation. We assume that, to the leading approximation, the free energy of aninterface is proportional to its area. Hence, for finite L , in the L , L → ∞ limit, there is only one interface in the system with anti-periodic bound-ary conditions and none in the system with periodic boundary conditions.Based on this scenario, the interface free energy is naturally defined as: F (1) s = − ln( Z a /Z p ) + ln L , (9)where the ln L term takes into account the “entropy” due to the fact thatthe interface can be located at any point in the x -direction. Note that for finite L , L the value of F (1) s depends on L and in par-ticular, the limit L → ∞ is not finite. This last problem is related to the Eq. (5) can be justified as it follows: In the basis of slice configurations Σ, the matrixassociated with anti-periodic boundary conditions is given by P Σ ′ , Σ = δ Σ ′ , − Σ , where − Σmeans that all spins in the slice are flipped. Since the external field h is vanishing, thetransfer matrix commutes with P Σ ′ , Σ ; furthermore, P squares to the identity, thereforeit has eigenvalues p n x = ±
1. We label eigenvectors with p n x = 1 by x=s and those with p n x = − λ n x is decreasing with increasing n . In principle, one might also add a further ln 2 term, to take into account that thepositive magnetization domain can be realized on the left-hand side of the interface andthe negative one on its right-hand side, or vice versa . L , it is favoured by the entropy to createadditional pairs of interfaces.The presence of additional pairs of interfaces can be addressed in thedilute gas approximation. I.e. we assume that the interaction of two inter-faces is short ranged and that the average distance between interfaces is largecompared with the range of the interaction. For n separate, non-interactingand indistinguishable interfaces with the free energy F s one obtains: Z I = X n n ! L n exp( − nF s ) = X n n ! exp[ − n ( F s − ln L )] . (10)The sum runs over non-negative even integers in the case of periodic bound-ary conditions, and positive odd integers in the case of anti-periodic bound-ary conditions, and the n ! factor takes into account that the interfaces areindistinguishable. Hence: Z a Z p = P ∞ m =0 1(2 m +1)! exp[ − (2 m + 1)( F s − ln L )] P ∞ m =0 1(2 m )! exp[ − m ( F s − ln L )]= tanh { exp[ − ( F s − ln L )] } . (11)The solution of this equation with respect to F s provides us with a seconddefinition of the interface free energy: F (2) s = ln L − ln (cid:18)
12 ln 1 + Z a /Z p − Z a /Z p (cid:19) . (12)Upon comparison between eq. (12) and eq. (8), the tunneling mass m t canbe expressed in terms of the interface free energy as: m t = 2 exp( − F (2) s ) , (13)which confirms that the definition F (2) s , in contrast to F (1) s , has a finite,meaningful L → ∞ limit. This limit is well approximated for L ≫ ξ .All our simulations are done in this regime. Note that for L ≪ ξ t , i.e. Z a /Z p close to zero, F (1) s is a good approximation of F (2) s . For most of oursimulations, this condition is satisfied. In quantum gauge theory, the low-energy behaviour of a confined pair ofstatic sources at a distance r might be described by an effective string. In5he confining regime, the flux lines between the two sources are squeezedinto a thin tube, which might be idealized as a uni-dimensional object.The long-distance properties of the system are dominated by the transversefluctuations of this tube; in this regime, the excitation spectra of the fieldsin the interior of the tube are expected to be much higher-lying.Under this assumption, the properties of the system are described througha string partition function, obtained integrating over the possible world-sheetconfigurations. Each of them has the topology of a cylinder, and contributesa Boltzmann-like factor, whose exponent is given by an effective string ac-tion.In principle, the functional form of the latter is unknown, however it canbe constrained, by requiring that it satisfies certain self-consistency proper-ties, and that it yields the correct physical limit for large distances r betweenthe two sources.This approach underlies the models that have been proposed by Polchin-ski and Strominger in the 1990’s [16]: S eff = 14 π Z dτ + dτ − (cid:20) a ( ∂ + X · ∂ − X )+ (cid:18) D − (cid:19) ( ∂ X · ∂ − X )( ∂ + X · ∂ − X )( ∂ + X · ∂ − X ) + O ( r − ) (cid:21) , (14)(in which τ ± are light-cone world-sheet coordinates, and a is a length scalerelated to the string tension) and by L¨uscher, Symanzik and Weisz alreadyat the beginning of the 1980’s [17, 18] (see also [19, 20] for more recentdevelopments): S eff = σrL + µL + S + S + S + . . . , with: S = 12 Z d ξ ( ∂ a h∂ a h ) , (15)(where L denotes the length of the closed world-lines of the static sources, σ is the string tension, and µ is a coefficient associated to a perimeter-liketerm). The construction of a string action in the form of eq. (14) allows ageneric conformally invariant world-sheet QFT, with the coefficient of thevarious terms fixed by anomaly cancellation in any dimension D . The actioncan be built by converting the path integral for the collective coordinates ofthe underlying field theory to covariant form. The X field is unconstrained(and the model still represents a generic interface in D dimensions), but theterm appearing in the second line of eq. (14) takes the Polyakov determinantin conformal gauge into account. Eq. (14) yields an effective string modelwhich can be expanded around the long-string vacuum. Poincar´e invariance6onstrains the O ( r − ) term of the string spectrum to the form it has in theNambu-Goto model [21–24] (for the definition of the Nambu-Goto modelsee below).In eq. (15), the S term describes a conformal model, while the other,higher-dimensional, S n terms are responsible for the string self-interactions.In this case, the variable h represents a vector with D − S , which isexpected to be a “boundary term” [19], is actually forbidden by open-closedstring duality [20].Among the main implications of the effective string model, we mentionthe existence of a negative, O ( r − ) correction to the asymptotic linear po-tential (the L¨uscher term, which is a Casimir effect), and the logarithmicgrowth of the square width of the flux tube [25–27]. Both aspects are re-lated to the fact that, at leading order, the effective string fluctuations in a D -dimensional target space can be modelled as ( D −
2) free, massless bosons.Since the infinite number of terms appearing on the right-hand sides ofeq. (14) and eq. (15) are not known a priori , it is, in general, not possibleto work out all-order predictions for the observables of interest.Alternatively, one might use an explicit ansatz on the functional form ofthe effective string action (consistent with the constraints mentioned above,and compatible with the other effective models at its lowest orders): thisallows to address the complete mathematical calculation for the expectationvalues of the physical observables, and to perform an all-order comparisonwith the numerical results.A natural choice for the effective string action (for any world-sheet ge-ometry) is the area of the string world-sheet itself. For the case of a closedinterface, it can be expressed introducing the ξ coordinates over the inter-face, and the g αβ metric induced by the embedding in the target space: S = σ Z d ξ p det g αβ (16)(the string tension σ has energy dimension 2).Eq. (16) has a natural interpretation in the context of string theory,where it represents the (Euclidean space formulation of) the model dueto Nambu and Goto [28, 29], describing the relativistic quantum dynamicsof a purely bosonic string. Although it is well-known that this model isaffected by an anomaly (breakdown of rotational symmetry out of the criticalspace-time dimension D = 26) and is non-renormalizable (because it is non-polynomial), it has been studied as a possible effective description of the7ow-energy dynamics of confining gauge theories. The reason is that, inthe infra-red regime, the lowest-lying degrees of freedom associated witha confining flux tube are transverse fluctuations, and are expected to bemodeled by a bosonic string-like dynamics; in that case, σ represents theasymptotic value of the string tension. In particular, the geometry of aninterface with periodic boundary conditions in both directions would beassociated to the description of a torelon, i.e. a string winding around acompact target space, that has been studied in ref. [30].A number of implications for this effective description of confinementhave been derived theoretically and tested numerically in the literature [31–51].On the other hand, in a condensed matter physics context, eq. (16)corresponds to the “capillary wave model” [1]; at first order, it describes thetransverse fluctuations of a membrane as free, independent, massless modes,whereas the subleading terms introduce (self-)interactions. In this context, σ can be interpreted as an (asymptotic) interface tension, which does notdepend on the local orientation of the normal to the infinitesimal surfaceelement.A perturbative expansion in powers of ( σL L ) − yields the followingresult for the partition function of the interface with periodic boundaryconditions in both directions [44–46]: Z = λ √ u exp( − σL L ) (cid:12)(cid:12)(cid:12) η ( iu ) /η ( i ) (cid:12)(cid:12)(cid:12) − (cid:20) f ( u ) σL L + O (cid:18) σL L ) (cid:19)(cid:21) , (17)where the parameter λ can be predicted invoking a perturbative argumentfor the φ scalar field theory in three dimensions, τ = iu = iL /L is themodulus of the torus associated with the cross-section of the system, η isDedekind’s function: η ( τ ) = q / ∞ Y n =1 (1 − q n ) , q ≡ exp(2 πiτ ) , (18)and f ( u ) is defined as: f ( u ) = 12 (cid:26)h π uE ( iu ) i − π uE ( iu ) + 34 (cid:27) , (19)where E ( τ ) is the first Eisenstein series: E ( τ ) = 1 − ∞ X n =1 n q n − q n . q ≡ exp(2 πiτ ) , (20)8n particular, for u = L /L = 1 one gets f (1) = 1 / L = L ≡ L takesthe form: F s = σL − ln λ − σL + O (cid:18) σL ) (cid:19) . (21)This is the theoretical expectation which, in the following section, will becompared with our numerical results for F (2) s — see eq. (12).More recently, a different approach to calculate the partition functionwas proposed in ref. [15]: this method is more elegant and powerful withrespect to the perturbative expansion in powers of ( σL L ) − , and it takesadvantage of the standard covariant quantization techniques for the bosonicstring. The power of this method relies in the fact that it allows to resumthe complete loop expansion for the interface partition function at all orders,with a final result for the interface partition function in d dimensions I ( d ) taking the form of a series of Bessel functions: I ( d ) = 2 (cid:16) σ π (cid:17) d − V T √ σ A u ∞ X m =0 m X k =0 c k c m − k (cid:18) E u (cid:19) d − K d − ( σ AE ) , (22)(where A = L L , and V T is the product of the system sizes in the transversedirections) and a consistent, closed-form expression for the spectrum levels: E = E k,m = s πσL (cid:18) m − d − (cid:19) + 4 π σ L (2 k − m ) , (23)that agree with those presented in ref. [41].For the case of an interface with the boundary conditions of a cylinder,an analogous result was derived in ref. [52], while the associated energyspectrum had already been known since the Eighties [53].In eq. (23), for d = 3 and m = k = 0, the argument of the squareroot becomes negative for σL < π/
3. This is known as the tachyonicsingularity in the effective string framework (see [54]) and can be physicallyinterpreted as the signature of a high temperature deconfinement transition:For temperatures higher than T c / √ σ = p /π the string vanishes and quarksare no longer confined. Equivalently, in the dual model, the Ising spinmodel, for L < N t the interface tension vanishes and a transition from aferromagnetic to a paramagnetic phase occurs.This interpretation was discussed for the first time by Olesen in thecase of Polyakov loop correlators [54] and holds essentially unchanged inthe present case, although the Arvis spectrum [53] on which Olesen’s result9as based is very different from the one we have in the present case — seeeq. (23). In fact, the lowest state (the one which drives the phase transition)is the same in both spectra.Notice that one should not expect that this analysis provides an exactresult for the finite temperature transition. Indeed the Svetitsky-Yaffe con-jecture [55] suggests that the finite temperature transition in the Z gaugetheory belongs to the universality class of the 2D Ising spin model. Onthe other hand, the analysis of the string picture gives mean-field criticalexponents (i.e. ν = 1 / Let us define the interface energy as: E s ≡ E a − E p , (24)where E a ( E p ) is the expectation value for the energy of a system withanti-periodic (respectively, periodic) boundary conditions.The internal energy of a system is given by the derivative of the reducedfree energy with respect to β : E ≡ − ∂ ln Z ( β ) ∂β = P { s } exp[ − βH ( { s } )] H ( { s } ) P { s } exp[ − βH ( { s } )] . (25)From eq. (25), by integration from β to β it follows: − ln Z a Z p (cid:12)(cid:12)(cid:12)(cid:12) β = − ln Z a Z p (cid:12)(cid:12)(cid:12)(cid:12) β + Z ββ d ˜ β E s ( ˜ β ) . (26)By adding ln L on both sides of the equation we get: F (1) s ( β ) = F (1) s ( β ) + Z ββ d ˜ β E s ( ˜ β ) . (27)In general it is difficult to determine free energies directly in a singleMonte Carlo simulation. On the other hand, expectation values such as E p and E a can be easily determined.It is rather an old idea to compute free energies (in particular: interfacefree energies) from eq. (27) — see, e.g., the first few references in [56].10or β there are different possible choices: For β < β c the interfacetension vanishes, and with a suitable choice of β < β c (depending on theinterface area) F (1) s ( β ) vanishes to a very good approximation. Alterna-tively, one might start the integration from large values of β ≫ β c , where F (1) s ( β ) can be obtained from the low temperature expansion.Here we follow the strategy discussed in [56]: F (1) s ( β ) is computed usingthe boundary flip algorithm [57] at a β value corresponding to a Z a /Z p ratio of the order of 1 /
10. For such a choice, F (1) s ( β ) can be accuratelydetermined using a moderate amount of CPU time.In practice, we have performed simulations for a finite number of inversetemperatures β ≤ β i ≤ β to obtain values for E s ( β i ). The integration (27)must then be performed by using some numerical integration scheme. For adetailed discussion of the schemes that we have used see the appendix A.In principle, the numerical integration, along with its (small) systematicerror, could be avoided. One might compute: F ( β + ∆ β ) − F ( β ) = − log [ Z ( β + ∆ β ) /Z ( β )]= − log {h exp[ − ∆ βH ( { s } )] i β } (28)in a Monte Carlo simulation that generates the Boltzmann distribution cor-responding to β . Here ∆ β has to be chosen such that ∆ β p h H i − h H i isof order one, to keep the statistical error under control. Then F ( β ) − F ( β )is computed by a sequence of such ∆ β steps.For an interesting new alternative, using a generalized Jarzynski relation,see ref. [58].Quite a different strategy to compute F s is based on the so-called snakealgorithm [59, 60]: A sequence of boundary conditions is introduced, thatinterpolates between periodic and anti-periodic boundary conditions. Theboundary in the 0-direction is progressively filled with J h xy i = −
1. Inrefs. [3–14] we have used this approach to compute ratios of Polyakov-loopcorrelators.In the present work, however, we do not use the snake algorithm, sincefor boundary conditions close to the anti-periodic boundary conditions thesimulations are very difficult. If the last layer of bonds is not completelyfilled with antiferromagnetic couplings J h xy i = − β , it provides the most efficientway to organize the simulations and to keep the resulting data under control. In order to compute the starting-point free energy F s ( β ), we have used avariant of the boundary flip algorithm [57] — see refs. [13, 61] for a dis-cussion. The update is performed using the single cluster algorithm [62].Typically, we have performed O (10 ) up to O (10 ) measurements. For eachmeasurement, mostly 10 single cluster updates were performed.In order to compute the energy for the systems with periodic and anti-periodic boundary conditions, we have used a demonized local Metropolisalgorithm implemented in multispin coding technique: details can be foundin ref. [63]. This way, n bit systems run in parallel in our implementation( n bit = 32 or n bit = 64, depending on the machine used). Most simulationswere performed with the local algorithm alone.For each measurement, 12 complete update sweeps were performed. For n bit = 32 we performed either 100,000 or 96,000 measurements for each copyof the system and each value of β ; for n bit = 64, 50,000 measurements foreach copy of the system were performed. In order to ensure equilibration, wehave taken 24,000 local update sweeps. Note that this is an overkill for thesmaller lattice sizes and the larger β values. However, due to the enormousnumber of individual simulations, we could not check carefully each of theruns. Therefore we decided to use a common number of thermalizationupdates, that is suitable for all of the parameter choices we considered —including the most difficult cases.For β values close to β , there is a non-negligible probability that morethan 0 (periodic boundary conditions) or 1 (anti-periodic boundary condi-tions) interfaces are formed in the system. Most likely, the local updateis not capable of generating these additional interfaces within the givennumber of update cycles. Therefore, we performed single cluster updatesin addition. In this step, we could not make use of the multispin codingtechnique, therefore the cluster update was performed one-by-one for the n bit systems. One single cluster update is performed per measurement; wemade no attempt to optimize the ratio of cluster and local updates. For adiscussion about a similar combination of algorithms, see ref. [64]. We haveused this method, instead of the local update only, for β = β + m ∆ β with m / The numerical evaluation of integral (27) was done using standard numericalintegration schemes which are summarized in the appendix A.All the schemes that we have considered can be written in the form: F (1) s ( β ) = F (1) s ( β ) + N X j =0 c j ∆ β E s ( β + j ∆ β ) + O ( N − m ) , (29)where ∆ β = ( β − β ) /N and P Nj =0 c j = N . For our final estimates, wehave used schemes with an O ( N − ) integration error. In order to get aquantitative estimate of the integration error, we have compared the re-sult obtained from different schemes; e.g. scheme (A.3) and scheme (A.4).Furthermore, we have performed the numerical integration for the theo-retical predictions of F s ( L , L , σ ), as discussed in section 3, along with σ ( β ) = σ ( β − β c ) × [1 + b ( β − β c ) θ + c ( β − β c )], with coefficients similarto those reported below. We found that the error of the integration is atleast two orders of magnitude smaller than our statistical error, and is henceignored in the further analysis of the data. The statistical error ǫ of F (1) s ( β ) is computed using standard error propaga-tion: ǫ [ F (1) s ( β )] = ǫ [ F (1) s ( β )] + (∆ β ) X j c j ǫ [ E s ( β + j ∆ β )] , (30)where ǫ is the statistical error and the c j coefficient is given by the integrationrule.In order to get correct fits for F s at different values of β , we have toevaluate the covariances of F s at different values of β . Let us consider β >β : Due to the fact that F (1) s ( β ) and E s ( ˜ β ) with ˜ β ≤ β are obtained in acommon set of simulations, F (1) s ( β ) and F (1) s ( β ) are statistically correlated.The covariance is defined as:cov( A, B ) := h [ A − h A i ][ B − h B i ] i . (31)13 β β max ∆ β F (1) s ( β )8 0.23607 0.24607 0.0002 4.69471(36)9 0.23407 0.24607 0.0002 4.95168(41)10 0.23007 0.24607 0.0002 4.49995(29)11 0.23007 0.24607 0.0002 4.98701(37)12 0.22907 0.24607 0.0002 5.12142(37)13 0.22807 0.24607 0.0002 5.16150(36)14 0.22707 0.24607 0.0002 5.10299(34)15 0.22667 0.24607 0.0002 5.26780(35)16 0.22607 0.24607 0.0001 5.27054(28)17 0.225802 0.230002 0.00005 5.43001(34)17 0.230002 0.246202 0.000118 0.225302 0.230002 0.00005 5.38722(31)18 0.230002 0.246202 0.000119 0.225002 0.230002 0.00005 5.44569(32)19 0.230002 0.246202 0.000120 0.224902 0.230002 0.00005 5.64529(39)20 0.230002 0.246202 0.000121 0.224702 0.230002 0.00005 5.73892(43)22 0.224502 0.230002 0.00005 5.80747(41)23 0.224302 0.230002 0.00005 5.84791(43)24 0.224002 0.230002 0.00005 5.73910(41)25 0.223602 0.230002 0.00005 5.46928(59)26 0.223602 0.230002 0.00005 5.66167(58)27 0.223602 0.230002 0.00005 5.86440(63)28 0.223502 0.230002 0.00005 5.91713(63)29 0.223402 0.226102 0.00005 5.95322(65)30 0.223402 0.230002 0.00005 6.14879(70)31 0.223302 0.226102 0.00005 6.16267(70)32 0.223152 0.230002 0.00005 6.06153(50)33 0.223052 0.226102 0.00005 6.03701(64)34 0.222952 0.226102 0.00005 5.99466(65)35 0.222902 0.226102 0.00005 6.04115(69)36 0.222852 0.230002 0.00005 6.07957(84)38 0.222752 0.226102 0.00005 6.13242(73)40 0.222652 0.230002 0.00005 6.14808(78)44 0.222552 0.230002 0.00005 6.37175(117)48 0.222452 0.230002 0.00005 6.52109(100)52 0.222352 0.226102 0.00005 6.58323(108)56 0.222252 0.226102 0.00005 6.55408(116)64 0.222152 0.226102 0.00005 6.77107(123)Table 1: Summary of the simulations for the square interfaces. For eachlinear extension L = L = L of the interface, the table gives the startingpoint β of the integration, the maximal inverse temperature β max that hasbeen simulated, and the step-size ∆ β . In the case of L = 17 , , ,
20 wehave two intervals with different ∆ β . The initial value of the integration F (1) s ( β ) has been computed with the boundary flip algorithm. For detailssee the text. 14 β F (1) s ( β )24 0.222402 5.19305(69)28 0.222402 5.53093(93)32 0.222402 5.87624(96)36 0.222402 6.21529(116)40 0.222402 6.54949(141)44 0.222402 6.87239(137)48 0.222402 7.19102(163)Table 2: Summary of runs with asymmetric lattices L = L , in the samenotation as in the previous table. L = 96, L = 64, ∆ β = 0 . β max = 0 . F (1) s ( β ) = F (1) s ( β ) + X j ˜ c j ( β ) E j , (32)where the E j and F (1) s ( β ) are statistically independent. Hence:cov( F (1) s ( β ) , F (1) s ( β )) = var( F (1) s ( β )) + X j ˜ c j ( β )˜ c j ( β )var( E j ) ≈ var( F (1) s ( β )) . (33)The last equality is only approximate, due to the fact that ˜ c j ( β ) = 1 forthe last few j ≤ m . In the limit ∆ β →
0, the approximation becomes exact.In our data, we have checked that the approximation is very good, and it istherefore used in the fits.
First we have analysed the data for the square lattices. In comparisonto our previous work ref. [13], we have results for more than four timeslarger interface areas. This allows us to use the interface tension as a fitparameter, while in ref. [13] we had to take it from ref. [7], where Polyakov-loop correlators were studied.Furthermore, here we have data for a large range of inverse temperatures β , allowing us to address the question of corrections to scaling.15s a starting point, let us first discuss the results for β = 0 . β = 0 . L =8 [67]. In table 3 we have summarized our results for the interface free energy F (2) s at β = 0 . F (1) s , which is the result of our numerical integration, to F (2) s , which is lessdependent on L and is therefore more suitable for the comparison with thetheoretical predictions. For L /L fixed, the difference between F (2) s and F (1) s goes down exponentially as the interface area increases. For our numericalresults at β = 0 . L ≤ F (2) s = σL + c + c σL (34)and F (2) s = σL + c + c σL + c ( σL ) , (35)where σ , c , c and c are the free parameters of the fits. At this stage ofthe analysis we made no attempt to compare with the full NG-predictionwhich can be obtained from eq. (22).Results of fits with the ansatz (34) are given in table 4. Starting from L min = 19, the χ / d.o.f. is smaller than one. The fit result for c is, upto L min = 24, decreasing with increasing L min . For L min = 24 we get c = − . c = − .
25. Sincethe fit result is increasing with L min we might consider c = − . χ / d.o.f. is below one for all L min available. c is now increasing with increasing L min . Unfortunately, no stable estimatefor c is obtained. Higher order corrections seem to play an importantrˆole. The results c = − . L min = 20 might serve as lowerbound for c . Combining the results of the fits with the ans¨atze (34,35) wemight summarize our results as: − . > c > − . σ (0 . . L min = 24. The comparison with results from the ansatz (34)suggests that systematic errors should not be larger than the statistical errorthat is quoted.We have repeated this type of analysis for β = 0 . . . . . . . . .
24 and 0 . F (2) s
18 6.00956(34)19 6.40442(38)20 6.81999(45)21 7.25617(50)22 7.71334(51)23 8.19024(56)24 8.68757(58)25 9.20809(77)26 9.74659(78)27 10.30706(84)28 10.88919(87)29 11.48975(92)30 12.11320(98)31 12.7558(10)32 13.4210(11)33 14.1074(12)34 14.8145(13)35 15.5415(13)36 16.2881(15)38 17.8477(16)40 19.4919(17)44 23.0292(22)48 26.9095(24)52 31.1193(28)56 35.6618(32)64 45.7769(40)Table 3: Interface tension F (2) s for square interfaces at β = 0 . L . 17 min σ c c χ / d.o.f.18 0.0105283(7) 2.6589(11) -0.209(4) 1.4019 0.0105273(8) 2.6611(13) -0.219(5) 0.9520 0.0105267(8) 2.6625(15) -0.226(6) 0.8121 0.0105264(9) 2.6635(17) -0.230(7) 0.7922 0.0105262(9) 2.6640(19) -0.234(9) 0.8223 0.0105257(10) 2.6654(22) -0.242(11) 0.7624 0.0105255(11) 2.6661(25) -0.246(13) 0.7825 0.0105262(12) 2.6636(30) -0.229(17) 0.6926 0.0105259(13) 2.6649(34) -0.239(21) 0.70Table 4: Fits with ansatz (34) for square interfaces at β = 0 . c is decreasing with increasing L min , until it starts to fluctuate. In thecase of the fits with ansatz (35) we see that c is increasing with increasing L min for β < . β it decreases. For β < . c = − .
25. For larger values of β , deviations become visible: forinstance, for β = 0 .
24, the result from a fit with ansatz (34) and L min = 12is c = − . c as obtained from fits with the ansatz (34)and √ σL min ≈ β ≤ .
24 we get: c | L min =2 / √ σ = − . − . σ , (36)with χ / d.o.f.= 0 .
55. In figure 1 we show our data along with the result ofthis fit. We conclude that c is affected by corrections to scaling. Howeverthe corrections are, within the numerical precision, proportional to σ ∝ ξ − ,i.e. they vanish much faster than ξ − ω , where ω = 0 . This could be explained bythe fact that the effective interface model only assumes that the symmetriesof the continuous space are restored. The restoration of rotational symmetryis indeed associated with a correction exponent ω ′ ≈ Field theoretical methods give slightly smaller values: ω = 0 . ǫ -expansion and ω = 0 . ω = 0 . φ model on the lattice. σ -0.3-0.28-0.26-0.24-0.22-0.2-0.18 c _2 f o r L _ m i n = σ − / Figure 1: Results for c from fits with the ansatz (34) and L min = 2 / √ σ . L min σ c c c χ / d.o.f.18 0.0105246(11) 2.6701(29) -0.299(22) 0.20(5) 0.7119 0.0105250(12) 2.6689(35) -0.288(29) 0.17(7) 0.7220 0.0105252(14) 2.6682(42) -0.280(38) 0.14(10) 0.7521 0.0105252(15) 2.6678(51) -0.275(50) 0.13(15) 0.7922 0.0105252(15) 2.6679(56) -0.276(60) 0.14(19) 0.84Table 5: Fits with ansatz (35) for square interfaces at β = 0 . C β Figure 2: The constant C as a function of β . The quantity is defined ineq. (37) in the text. The value for β = 0 . c . In the scaling limit, this quantityshould behave like: c ( β ) = C −
12 ln[ σ ( β )] . (37)In fig. 2 we have plotted our results for C as a function of β .Within our numerical precision there is no sign of corrections to scalingwhatsoever for β ≤ . β = 0 . L min ≈ . / √ σ . Asdiscussed above, for the case β = 0 . .2 Global fit of the data for the interface free energy In the neighbourhood of the transition, the interface tension behaves as: σ ( β ) = σ t µ × (1 + at θ + bt + ct θ + dt θ ′ + ... ) , (38)where t = β − β c is the reduced temperature. The most accurate resultfor the inverse critical temperature is β c = 0 . µ of the interface tension is related with the critical exponent ofthe correlation length as µ = 2 ν . The most accurate values given in theliterature are ν = 0 . ν = 0 . ω = 0 . θ = νω = 0 . θ ′ = 1 . θ ≈ θ ′ ≈
1, we take as ansatz for the fits: σ ( β ) = σ t µ × (1 + at θ + bt ) . (39)Fitting our new results for σ ( β ) to this ansatz would require to take intoaccount the cross-correlations among the values of σ ( β ) for different valuesof β . Instead of computing these cross-correlations, we performed fits for F (2) s fitting the L and the β dependence at the same time. The cross-correlations of F (2) s at different values of β can be easily obtained as discussedin subsection 4.3.Based on the results obtained above, we performed a four parameter fitof the data for F s ( L, β ). To this end, we have used the ansatz: F s ( L, β ) = σ ( β ) L + C −
12 ln[ σ ( β )] −
14 1 σ ( β ) L , (40)where the interface tension is given by the ansatz (39), namely, the freeparameters are σ , a , b and C . The critical exponents and the inversecritical temperature are fixed by their best estimates given in the literature,which are quoted above eq. (39).In the fit, we have used results for the interface free energy at thesame values of β as discussed in the previous subsection: β = 0 . . . . . . . . . . β . However, littleinformation would be added this way, since, by construction, the interfacefree energies at close-by values of β are highly correlated.After some experimenting we decided to take our final estimate froma fit with input data characterized by β ≤ . F s − ln L ≥ √ σL ≥
3. In total, 51 data-points satisfythis criterion. The results for the fit parameters are σ = 10 . a = − . b = − . C = 0 . χ /d.o.f. = 0 . L dependence of our result, we have repeated thefit with β ≤ . F s − ln L ≥ √ σL ≥ .
1) and L ≤ L max = 44. This means that the range in L is roughly √ σ = 10 . a = − . b = − . C = 0 . χ /d.o.f. = 0 . β -interval of our fit. We included data that satisfythe criteria 0 . ≤ β ≤ . F s − ln L ≥ L ≤ L max = 44. Thereare 55 data points that satisfy these criteria. The results of the fit are σ = 10 . a = − . b = − . C = 0 . χ /d.o.f. = 0 . σ are consistent among the three different fits. Thedifferences, possible within the statistical error, of these results provide anestimate of the systematic error due to finite- L and large- t corrections thatare not taken into account by the ansatz. We arrive at: σ = 10 . β c − . ν − . − . θ − . a = − . − β c − . − ν − . − . θ − . . The number in the brackets gives the systematic error caused by correctionsto the ansatz, as discussed above. In the second line of eqs. (41,42) we givethe dependence of the result on the input parameters β c , ν and θ . Thedependence of C on β c , ν and θ is small enough to be neglected. We take: C = 0 . σ obtained from the global fit with the results obtainedfrom analysing single values of β in the previous subsection. For all valuesof β the results are consistent. 22 σ global fit σ σ at given values of the inverse temperature β . In the second column we give σ as obtained from our global fit. Thestatistical error of σ is properly computed, and the second error quoted isthe systematic one. It is estimated from comparing results from different fitranges. The third column gives σ obtained from fits with the ansatz (34)and L min ≈ . / √ σ . For the three smallest values of β the global fit providesmore accurate results for σ than the fits with the ansatz (34). Using the definitions ˜ t = ( β − β c ) /β c and σ = ˜ σ ˜ t µ × (1 + ... ) we get:˜ σ = 1 . , (44)where the error is dominated by the uncertainty of ν . Note that this resultis perfectly consistent with (and more precise than) the most accurate result˜ σ = 1 . σ is given in table 8 of ref. [66].Also our results for the interface tension at given values of β can becompared with values given in the literature. Here we only give a smallselection of the most recent results. For more see e.g. ref. [75]. In [7], study-ing Polyakov-loop correlators, we find σ = 0 . . β = 0 . . σ = 0 . . . . β = 0 . . .
23 and 0 . σ = 0 . . . . β , taken from our global fit. Our valuesare consistent with those of ref. [36], except for β = 0 . σ = 0 . . . . β =0 . . . . σ = 0 . . . . β = 0 . . β = 0 . . β only small interfaceareas were available and too small areas had been included into the fit. L In figure 3 we plot F (2) s − σL +0 . σ ) as a function of √ σL . The numericalvalues for σ are taken from table 6. Note that these values of σ are obtainedfrom rather large values of √ σL (i.e. are little affected by higher ordercorrections). In addition to the numerical data for square interfaces at β =0 . β = 0 . β for which the finite temperature transition occurs for N t = 16 and N t = 8, where N t denotes the number of lattice spacings in the “inversetemperature” compactified direction) we give the 2-loop prediction and thefull Nambu-Goto result. In the case of the string predictions we have taken C = 0 . √ σL ' . / ( σL ) term for the full NG result is approximately equal to − . β fall on top of the 2-loop and full NG predictions for √ σL ' .
2; notethat the 1-loop approximation predicts F s − σL to be constant. For smaller √ σL , the data rather abruptly depart from the string prediction. Thisindicates that (mainly) not O (1 / ( σL ) ) but rather higher order correctionsare responsible for the deviation. Still down to √ σL ≈ . β fall on top of each other within the error-24 .5 2 2.5 3 3.5 σ1/2 L F _ s - L ^ + . l n ( ) σσ full NGtwo loop β=0.223102β=0.226102 square interfaces Figure 3: Comparison of the 2-loop prediction, the full Nambu-Goto resultwith the data for square interfaces at β = 0 . β = 0 . √ σL , differences become visible, indicating correctionsto scaling. These scales should be compared with the scale of the finitetemperature transition √ σN t ≈ .
81 (for a discussion of this number seesection 6 below; the effective string model gives √ σN t = p π/ ≈ . In this subsection we compare our results for rectangular interfaces withthe effective string predictions discussed in sect. 3. Note that the effec-tive string results have quite a non-trivial dependence on the aspect ratio u = L /L . Therefore this comparison is rather a stringent test of the theo-retical predictions. We have simulated lattices with the linear sizes L = 96and L = 64. In the remaining direction, the linear size assumes the values L = 24 , , , ,
40 or 48. We use the values of σ and C obtained in the25 L L F (1) nums ( β ) F (2) nums ( β ) F ths ( β ), full F ths ( β ), 1-l. F ths ( β ), 2-l.24 64 96 6.8887(20) 6.8855(20) 6.7495 6.9974 6.834728 64 96 7.6935(21) 7.6929(21) 7.6537 7.7875 7.682132 64 96 8.4626(20) 8.4625(20) 8.4518 8.5380 8.463236 64 96 9.1996(21) 9.2012 9.2632 9.206240 64 96 9.9227(23) 9.9231 9.9713 9.925344 64 96 10.6203(23) 10.6278 10.6674 10.628848 64 96 11.3138(25) 11.3209 11.3548 11.3213Table 7: Free energy of rectangular interfaces, at β = 0 . F (2) nums ( β ) is reported only when different with respect to F (1) nums ( β ). Thesixth, seventh and eighth column display, respectively, the all-loop (full),one-loop (1-l.) and two-loop (2-l.) prediction of the Nambu-Goto stringmodel. L L L F (1) nums ( β ) F (2) nums ( β ) F ths ( β ), full F ths ( β ), 1-l. F ths ( β ), 2-l.24 64 96 7.9988(22) 7.9985(22) 7.9397 8.1043 7.980228 64 96 9.0175(23) 9.0066 9.1016 9.021232 64 96 9.9965(22) 9.9962 10.0594 10.002336 64 96 10.9407(23) 10.9455 10.9917 10.948240 64 96 11.8682(25) 11.8707 11.9070 11.871944 64 96 12.7728(25) 12.7803 12.8102 12.780848 64 96 13.6734(27) 13.6791 13.7049 13.6793Table 8: Same as in table 7, but at β = 0 . F (2) s at β = 0 . . . . N t = 16, 14, 12 and 8. Note that only for β = 0 . L = 24the difference between F (2) s and F (1) s is slightly larger than the statisticalerror. Our numerical results, along with the various string predictions aresummarized in tables 7 to 10, and shown in the plots 4 and 5.Since we have fixed values of L , L and L , our data do not allow for acheck of scaling corrections. Instead, we assume that they are of similar sizeas in the case of the square interfaces and therefore are small for the valuesof β that we consider here. 26 L L F (2) nums ( β ) F ths ( β ), full F ths ( β ), 1-l. F ths ( β ), 2-l.24 64 96 9.7374(23) 9.7111 9.8216 9.730228 64 96 11.0702(24) 11.0653 11.1318 11.072532 64 96 12.3602(24) 12.3572 12.4024 12.360336 64 96 13.6150(25) 13.6141 13.6476 13.615540 64 96 14.8520(27) 14.8492 14.8757 14.849944 64 96 16.0688(27) 16.0699 16.0918 16.070148 64 96 17.2816(29) 17.2804 17.2993 17.2804Table 9: Same as in table 7, but at β = 0 . L L L F (2) nums ( β ) F ths ( β ), full F ths ( β ), 1-l. F ths ( β ), 2-l.24 64 96 18.4131(26) 18.4121 18.4555 18.415228 64 96 21.2414(27) 21.2450 21.2724 21.246332 64 96 24.0310(27) 24.0306 24.0497 24.031236 64 96 26.7859(28) 26.7873 26.8017 26.787540 64 96 29.5271(30) 29.5250 29.5365 29.525144 64 96 32.2449(31) 32.2498 32.2594 32.249848 64 96 34.9623(33) 34.9653 34.9736 34.9653Table 10: Same as in table 7, but at β = 0 . √ σL ≈ .
8, i.e. they can not be discriminated, given the accuracy of our MonteCarlo results for the Ising model.In the range √ σL ' .
8, the Monte Carlo data perfectly agree with thefull NG result and the two-loop approximation. Up to our largest values of √ σL , there is a clear discrimination against the 1-loop approximation.For √ σL < . √ σL close to √ σ/T c , the numerical data follow closely the 2-loopprediction.In our opinion, this behaviour at very small distances is a mere coinci-dence, related to the fact that the 2-loop approximation gives (by chance)the same critical exponent as the 2D Ising universality class and also thevalue for T c / √ σ is very close to the one of the Z gauge theory. In this section we compute universal amplitude ratios of the interface tensionand the correlation length. Generically, these amplitude ratios have theform: R = lim t → σ ( t ) ξ ( t ) , (45)where t is the reduced temperature. In particular, we shall consider thesecond moment correlation length in the high temperature and the low tem-perature phase and the exponential correlation length (i.e. the inverse ofthe lightest mass) in the low temperature phase. The second moment correlation length is defined by: ξ = µ dχ , (46)where: µ = X x x h s s x i (47)28 .2 1.4 1.6 1.8 2 2.2 2.4 2.6 σ1/2 L1 σ F - L L s Ising data1-loop2-loopfull NG
L =96, L =640 2 β=0.223102 σ1/2 L1 σ F - L L s Ising data1-loop2-loopfull NG
L =96, L =640 2 β=0.223452
Figure 4: In the two figures we give F (2) s − σL L as a function of √ σL for β = 0 . β = 0 . L = 96 and L = 64. Notethat in the case of the Monte Carlo results the statistical error is smallerthan the symbol (circle). The 1-loop, 2-loop and full NG predictions aregiven as solid black, red and blue lines, respectively.29 .5 2 2.5 3 3.5 σ1/2 L1 σ F - L L s Ising data1-loop2-loopfull NG
L =96, L =640 2 β=0.223952 σ1/2 L1 σ F - L L s Ising data1-loop2-loopfull NG
L =96, L =640 2 β=0.226102
Figure 5: Same as in the previous figure, but for β = 0 . β =0 . β = 0 . χ is the magnetic susceptibility: χ = X x h s s x i . (48)The coefficients of the high temperature series for µ and χ up to O ( β ) arereported in ref. [76]. Using these, we computed the coefficients for ξ . Quitea straightforward method is the so-called matching method (see e.g. [77]):We start from a power-law ansatz like: ξ = f ( β c − β ) − ν (49)[1 + a ( β c − β ) θ + b ( β c − β ) + c ( β c − β ) θ + d ( β c − β ) θ + ... ] , where we fix β c = 0 . ν = 0 . θ = 0 . γ = 1 . β . The remaining free parameters of the ansatz (49) can thenbe determined by matching the coefficients with those of the exact result forthe HT-series for ξ . As our final result for the amplitude, we obtain: f = 0 . × [1 + 1024 ( β c − . −
17 ( ν − . − . θ − . . Combining the results of eqs. (41,50), we get: R + = 0 . β c − . . ν − . − . θ − . . Inserting the errors of the input parameters, we arrive at: R + = 0 . R + = 0 . R + = 0 . σ ξ exp σξ exp ξ σξ ∗ ∗ β → β c ∗ the results of ref. [81] and ref. [82] are averaged.The numbers for the second moment correlation length are all taken fromref. [80]. In the last row we give the extrapolation to the scaling limit. Fordetails see the text. Here we use the Monte Carlo results for the second moment correlationlength ξ obtained in ref. [80] and the exponential correlation length fromrefs. [81,82] to compute the combination σ ( t ) ξ ( t ) at finite values of t . Withour present results for the interface tension, the error of the combinationis completely dominated by the error of the correlation length. We fit thecombination with the ansatz: σξ = R − + c σ ω/ , (53)using ω = 0 . β ≤ . ω is rather small and can be neglectedcompared with the statistical error of R − . In the case of the exponentialcorrelation length our final estimate is obtained from data with β ≤ . ω can be ignored. Thevalues of the interface tension, the correlation length and the amplitudecombination R − as well as the final results of our fits are summarized intable 11.For comparison, table 12 reports previous estimates obtained for theIsing model in refs. [56, 66, 74, 81, 83].Also field-theoretical predictions for the universal constant R are givenin the literature: 32ear authors(s) Ref. R − ∗ et al. [81] 0.1056(19)1997 Hasenbusch and Pinn [56] 0.1040(8) ∗ Table 12: Comparison of a number of estimates for R − taken from theliterature. The estimate of Zinn and Fisher is based on data of [66]. Agostini et al. used the true instead of the second moment correlation length. Theresult marked with a star refer to the second moment correlation length,while the others refer to the exponential one. • From the ǫ -expansion: Br´ezin and Feng [84] computed R from the ǫ -expansion. Their result reads:14 πR = 2 π ǫ " − ǫ π ) − γ − π √ ! + O ( ǫ ) , (54)with γ = 0 . ... . Their numerical evaluation for ǫ = 1 gives resultsin the range from R ≈ .
051 up to R ≈ . / • From perturbation theory in 3D fixed: The study of interfaces usingperturbation theory in three dimensions was pioneered by M¨unster [85].The starting point of this calculation is the classical solution (i.e. theconfiguration with minimal action) of a system with fixed boundaryconditions in 3-direction. At one boundary, the field is fixed to thenegative minimum, and at the other, to the positive minimum of thepotential. Then, fluctuations around this classical solution are stud-ied. In the more recent paper [86], this analysis was extended to twoloops. Their result is: R = 2 u ∗ R " σ l u ∗ R π + (cid:18) σ l u ∗ R π (cid:19) + O ( u ∗ R ) , (55)with: σ l = 14 (cid:18) (cid:19) − − . Note that they use the second moment definition of the correlation length. σ l = − . . (57)Using Pad´e and Pad´e-Borel analysis, with u ∗ R = 14 . R = 0 . u ∗ R . At this point, one should also note the principle problemsrelated with the definition of u ∗ R , as discussed in ref. [82].Our value σξ exp = 0 . m / √ σ = 3 . Z gauge theory might be compared with 4 . . . . SU (2), SU (3), SU (4) and SU (5)gauge theories in 2+1 dimensions, respectively [87]. Note that there areclear differences between the results for the different gauge groups, indicat-ing that the 0 ++ glueball state probes short distances, which can not bedescribed by an effective string. This is in contrast to higher exited glueballstates, where much less dependence on the gauge group is found [81, 87].In a similar way, we have analysed σ/T c , using the results of ref. [67]for T c . We arrive at σ/T c = 0 . T c / √ σ = 1 . /T c ) ≥ T c . Our new result can be compared with ourprevious estimate T c / √ σ = 1 . /T c = 12) and T c / √ σ = 1 . p /π = 0 . ... obtainedfrom the effective string picture [54].Finally we might compare with SU ( N ) gauge theories in 2+1 dimensions:In the literature one finds T c / √ σ = 1 . . N = 2 , T c / √ σ = 0.892(3), 0.879(3), 0.877(3) for N = 4 , , N = 4 , , N t = 3 and no continuum extrapolation is performed.Similar to the case of m / √ σ we observe a clear dependence on thegauge group. Nevertheless, it is quite remarkable that the simple stringpicture gives the correct value with less than 30% deviation. In this paper, we have presented the results of an accurate numerical studyof the interface free energy in the three-dimensional Ising model. The moti-vation for this study was twofold: 34
To investigate the dynamics of a fluctuating interface with periodicboundary conditions only, avoiding non-trivial effects of Dirichlet bound-ary conditions. • To obtain high precision estimates of the interface/string tension ina large range of the inverse temperature, allowing us to compute thescaling limit of several universal amplitude ratios with high precision.To this end, we have determined the interface free energy by numericalintegration of the interface energy over the inverse temperature β . Theinterface energy is given by the difference of the internal energies in a systemwith periodic and a system with antiperiodic boundary conditions in onedirection. The interface free energy at the starting point of the integrationwas measured using the boundary-flip algorithm. For our simulations wehave used efficient combinations of cluster and multispin coded Metropolisupdates. This approach has allowed us to strongly improve the precisionof the numerical results, as compared to analogous studies presented in theliterature. In particular, the large range of β -values that we have studiedgives us a good control over corrections to scaling (or, in the language oflattice gauge theory: finite a effects). It turns out that, in the regimenot too close to the finite temperature transition, the interface free energy F s ( √ σL , √ σL ) approaches its continuum limit quite fast, characterizedby a correction exponent ω ′ ≈
2. This observation might be explained bythe fact that the effective interface model only assumes the restoration ofthe symmetries of the continuous space-time, which indeed comes with anexponent ω ′ ≈ √ σL ≈ .
8, the twostring results can not be discriminated at the level of the accuracy of ourMonte Carlo data. Nevertheless, one might interpret our Monte Carlo dataas a confirmation of the full NG prediction in the sense that e.g. 1 / ( σA ) corrections have indeed a small amplitude.Going close to the finite temperature transition, which occurs at √ σN t =0 . R + = 0 . R + = 0 . m / √ σ of the0 ++ glueball in units of the square root of the string tension and the criticaltemperature of the deconfinement transition T c / √ σ .The comparison with other gauge theories in 2 + 1 dimensions shows avariation of these quantities by roughly 50%, indicating that these quanti-ties can not be described only by an effective string picture, but that alsomicroscopic features of the gauge theory have to be taken into account. Acknowledgements.
M.P. acknowledges support from the Alexander von Humboldt Foundation.36
Integration schemes
One of the simplest methods given in textbooks is the trapezoid rule: Z x N x f ( x )d x = h (cid:20) f + f + f + ... + f N − + 12 f N (cid:21) + O ( N − ) , (A.1)where h = ( x N − x ) /N , f i = f ( x i ) and x i = x + ih . There are rules withfaster convergence as N → ∞ , like the well-known Simpson rule: Z x N x f ( x )d x = h (cid:20) f + 43 f + 23 f + 43 f + 23 f + ... + 43 f N − + 23 f N − + 43 f N − + 13 f N (cid:21) + O ( N − ) . (A.2)However, the disadvantage of the Simpson rule is that f i is not constant inthe middle of the interval: this implies a loss of precision in the final result.Furthermore, N has to be even.A better-suited rule that avoids these problems is given by (see e.g.eq. (4.1.14) in ref. [91]): Z x N x f ( x )d x = h (cid:20) f + 76 f + 2324 f + f + f + ... + f N − + 2324 f N − + 76 f N − + 38 f N (cid:21) + O ( N − ) . (A.3)A similar and maybe slightly better rule (see e.g. eq. (35) in ref. [92]) isgiven by: Z x N x f ( x )d x = h (cid:20) f + 5948 f + 4348 f + 4948 f + f + ... + f N − + 4948 f N − + 4348 f N − + 5948 f N − + 1748 f N (cid:21) + O ( N − ) . (A.4)37 eferences [1] V. Privman, Int. J. Mod. Phys. C (1992) 857[arXiv:cond-mat/9207003].[2] F. J. Wegner, J. Math. Phys. (1971) 2259[3] M. Caselle, M. Hasenbusch and M. Panero, JHEP (2006) 076[arXiv:hep-lat/0510107].[4] M. Panero, JHEP (2005) 066 [arXiv:hep-lat/0503024].[5] M. Caselle, M. Hasenbusch and M. Panero, JHEP (2005) 026[arXiv:hep-lat/0501027].[6] M. Panero, Nucl. Phys. Proc. Suppl. (2005) 665[arXiv:hep-lat/0408002].[7] M. Caselle, M. Hasenbusch and M. Panero, JHEP (2004) 032[arXiv:hep-lat/0403004].[8] M. Caselle, M. Panero and M. Hasenbusch, published in Wako 2003, Color confinement and hadrons in quantum chromodynamics , pag. 460[arXiv:hep-lat/0312005].[9] M. Caselle, M. Hasenbusch and M. Panero, Nucl. Phys. Proc. Suppl. (2004) 593 [arXiv:hep-lat/0309147].[10] M. Caselle, M. Hasenbusch and M. Panero, JHEP (2003) 057[arXiv:hep-lat/0211012].[11] M. Caselle, M. Panero, P. Provero and M. Hasenbusch, Nucl. Phys.Proc. Suppl. (2003) 499 [arXiv:hep-lat/0210023].[12] M. Caselle, M. Panero and P. Provero, JHEP (2002) 061[arXiv:hep-lat/0205008].[13] M. Caselle, M. Hasenbusch and M. Panero, JHEP (2006) 084[arXiv:hep-lat/0601023].[14] M. Bill´o, M. Caselle, M. Hasenbusch and M. Panero, PoS
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