We report the status of a high-statistics Monte Carlo simulation of
non-self-avoiding crystalline surfaces with extrinsic curvature on lattices of
size up to $128^2$ nodes. We impose free boundary conditions. The free energy
is a gaussian spring tethering potential together with a normal-normal bending
energy. Particular emphasis is given to the behavior of the model in the cold
phase where we measure the decay of the normal-normal correlation function.Comment: 9 pages latex (epsf), 4 EPS figures, uuencoded and compressed.
  Contribution to Lattice '9

Abraham

Ambjørn

Aronovitz

Baig

David

G. Thorleifsson

Guitter

Harnish

K. Anagnostopoulos

Kantor

Le Doussal

M. Falcioni

M.J. Bowick

Nelson

Renken

S.M. Catterall

Schmidt

Wheater

English

arXiv

Crossref

The phase diagram of crystalline surfaces

Anagnostopoulos, K.N.

Bowick, M.J.

Catterall, S.M.

Falcioni, M.

Thorleifsson, G.

Copenhagen University Research Information System

We report the status of a high-statistics Monte Carlo simulation of non-self-avoiding crystalline surfaces with extrinsic curvature on lattices of size up to 128^2 nodes. We impose free boundary conditions. The free energy is a gaussian spring tethering potential together with a normal-normal bending energy. Particular emphasis is given to the behavior of the model in the cold phase where we measure the decay of the normal-normal correlation function

Catterall, Simon

Anagnostopoulos, Konstantinos N.

Bowick, Mark

Falcioni, Marco

Syracuse University Research Facility and Collaborative Environment

Syracuse University SURFACE Physics College of Arts and Sciences 9-21-1995 The Phase Diagram of Crystalline Surfaces Simon Catterall Syracuse University Konstantinos N. Anagnostopoulos Niels Bohr Institute Mark Bowick Syracuse University Marco Falcioni Syracuse University G. Thorleifsson Syracuse University Follow this and additional works at: https://surface.syr.edu/phy  Part of the Physics Commons Recommended Citation Catterall, Simon; Anagnostopoulos, Konstantinos N.; Bowick, Mark; Falcioni, Marco; and Thorleifsson, G., "The Phase Diagram of Crystalline Surfaces" (1995). Physics. 490. https://surface.syr.edu/phy/490 This Article is brought to you for free and open access by the College of Arts and Sciences at SURFACE. It has been accepted for inclusion in Physics by an authorized administrator of SURFACE. For more information, please contact surface@syr.edu. arXiv:hep-lat/9509074v1  21 Sep 1995The Phase Diagram of Crystalline SurfacesK.N. Anagnostopoulosa,M.J. Bowickb,S.M. Catterallb,M. Falcionib∗andG. Thorleifssonb†a) Niels Bohr Institute, Blegdamsvej 17,DK-2100 København Ø, Denmark.b) Department of Physics, Syracuse University,Syracuse, NY 13244-1130 U.S.A.February 7, 2008AbstractWe report the status of a high-statistics Monte Carlo simulation ofnon-self-avoiding crystalline surfaces with extrinsic curvature on lat-tices of size up to 1282 nodes. We impose free boundary conditions.The free energy is a gaussian spring tethering potential together witha normal-normal bending energy. Particular emphasis is given to thebehavior of the model in the cold phase where we measure the decayof the normal-normal correlation function.SU-HEP-95-4241-620UFIFT-HEP-95-18∗Poster presented by M. Falcioni at Lattice ’95†bowick, smc, falcioni, thorleif@npac.syr.edu, konstant@nbi.dk11 IntroductionIn recent years there has been a lot of interest in the statistical mechanics ofcrystalline and fluid surfaces [1]. The former is believed to describe physicalpolymerized membranes [2] and the latter may be a regularization of stringtheory.We focus our study on crystalline surfaces with bending rigidity embed-ded in R3. It is conjectured that this model has a second order phase transi-tion driven by the competition between entropy and the bending energy [3].The high temperature phase is characterized by crumpled configurations. Inthe low temperature phase the system is no longer isotropic and the surfacesare roughly flat.One may wonder what stabilizes the flat phase. The theoretical argumentis that the in-plane elastic constants prevent the surface from fluctuatingarbitrarily in the embedding space. This leads to an effective long wavestiffening of the surface [1, 3].The crumpling transition has been studied numerically with simulationsexplicitly incorporating 2-d elastic constants [4, 5, 6]. There have also beensimulations with a simple gaussian spring potential playing the role of thetethering potential [7, 8, 9, 10, 11, 12]. In both cases evidence has beenpresented for a continuous phase transition.In the latter class of models the equilibrium spring length is taken to bezero, and a simple calculation indicates that the microscopic elastic constantsvanish [13]. It is tempting to argue, therefore, that the flat phase of thesemodels is not truly stable, even in the limit of large bending rigidity.Our ultimate aim is to carefully compare the behavior of the appropriateobservables in the cold phase as a function of the equilibrium spring length.2 The ModelConsider a system of particles connected to form a triangular 2–d meshembedded in 3 dimensions. Let each particle be labeled by an internaldiscrete coordinate system x = (x1, x2) denoting its position on the mesh.Its actual position in the embedding space is given by the 3 dimensionalvector r(x1, x2). The action has a tethering potential and a bending energyterm. Our choice is to use simple gaussian springs between the vertices as atethering potential and a normal-normal interaction as the bending energy2term. Therefore the action isS =∑〈xx′〉(lxx′)2 + λ∑〈αβ〉(1− ~nα · ~nβ) . (1)Here the subscripts x, x′ label the vertices and lxx′ is the distance betweenthe vertices x and x′ in the embedding space. The subscripts α, β label thefaces (triangles) of the surface, ~n is the unit normal to the face and λ is thebending rigidity. The sums extend to nearest neighbours. Eq. 1 describesphantom surfaces since it does not include self-avoidance. The surface is arhombus with free boundaries cut out of a triangular lattice. In the case ofa spring of length a, (lxx′) has to be replaced by (lxx′ − a). In this case ourmodel would closely resemble the one of [4] discussed above.We focused our analysis on the following observables: the specific heat,Cv =λ2V(〈S2e 〉 − 〈Se〉2). (2)Here Se is the bending energy term considered above and V is the totalnumber of vertices.The radius of gyration,R2g =13V〈∑ir′i · r′i〉. (3)Here r′i is the position of the node i in the embedding space referred to thecenter of mass. This observable measures the physical extent of the surfaceand its scaling behavior with system size defines the size (Flory) exponentν, via the relation Rg ∝ Lν . The exponent ν is related to the Hausdorffdimension dH via the relation ν = 2/dH .The eigenvalues of the inertia tensor; these eigenvalues give informationon the shape of the surface and how it scales with system size. They areobtained by diagonalizing the anisotropic part of the inertia tensorAab =∑ir′a(i)r′b(i) (4)where a, b refer to the components of the vector r′.The normal-normal correlation function,G(R) = 〈1N∑|α|=R(~nα · ~nO)〉. (5)Here the sum extends to all triangles of the surface which have a geodesicdistance R from the center of the surface O. The angle brackets representthe Monte-Carlo average.30.4 0.6 0.8 1.0 1.2Bending Rigidity0.01.02.03.04.0CvL = 16L = 32L = 46L = 64Figure 1: The specific heat as a function of the bending rigidity.3 Theoretical PredictionsA self-consistent perturbation theory analysis of the continuum model [14]yields predictions for the critical exponents. The exponents of interest arethe size (Flory) exponent ν and the roughness exponent ξ. The roughnessexponent is defined by the scaling of the minimum eigenvalue of the inertiatensor (4),λ ∝ L2ξ. (6)At the critical point the theory predicts ν = ξ = 0.732 while in thecold phase ν = 1 and ξ = 0.59.As far as the normal-normal correlation is concerned the only analyticalresult is for λ = 0 (gaussian model). In this case the correlation functionfollows a decay law G(R) ∝ −1/R4.4110100100010 100 1000Rg2 ν = 1.05(4)LFigure 2: The scaling of the radius of gyration squared with system size inthe cold phase(λ = 1.1). The fit gives a value of ν = 1.05(4).4 Numerics and ResultsWe performed Monte-Carlo simulations of systems of sizes 162 to 1282 ver-tices. We used the single hit Metropolis algorithm. The largest lattice wassimulated on a MASPAR MP1 massively parallel processor, while all othersizes were simulated on workstations. We gathered statistics of the order of30–50×106 sweeps per data point for the largest lattices (64 and 128). Ourstatistics are comparable for the smaller lattices.As can be seen in Fig. 1 the specific heat Cv shows a growing peak withsystem size. Presently our statistics are not yet sufficient to allow a reliableestimate of the exponent α which characterizes the growth. Preliminary fitsindicate a value of α ≈ 0.5 consistent with the value obtained in [10, 11]using the same method. Our best estimate for the critical value of thecoupling is around λ ≃ 0.79. Work is currently under way to gather betterstatistics and perform a Ferrenberg-Swendsen type analysis.500.10.20.30.40.50.60.70G(R)5 10 15 20 25 30 35 40rL = 32L = 64λ = 0.8Figure 3: The normal-normal correlation function at λ = 0.8 (around thephase transition).Fig. 2 shows the radius of gyration versus system size at a fixed value ofλ (1.1). The data fits well to a scaling ansatz with ν = 1.05(4), as expectedin the flat phase. In the crumpled phase the data does not fit a power lawbehavior, indicating, as expected [4, 5], a log-like scaling (dH = ∞). Ourestimate of the critical coupling is not precise enough to allow for a fit toRg at the transition.Figures 3 and 4 show the normal-normal correlation around and abovethe phase transition respectively. Fig. 3 demonstrates the effect of finite-size corrections to the value of the critical coupling. For the smaller volumeL = 32 the correlation decays to zero with r, but fits indicate a non-zeroasymptotic value for L = 64. One possible reason is that, due to the volumedependence of the pseudo-critical coupling, the smaller volume is in thecrumpled phase while the larger is in the cold (flat) phase. Note that forlarge r the correlations will always decay to zero because of our choice ofboundary conditions. This is a finite size effect and the data close to the60.30.40.50.60.70.80 10 20G(R)30 40 50L = 32L = 64rλ = 1.1Figure 4: The normal-normal correlation function at λ = 1.1 (in the coldphase).boundary has to be excluded from the fits.Figure 4 shows the correlation in the cold phase. The data for L = 64fits well to a behaviorG(R) ≃1Ra+ b (7)with a = 0.51(1) and b = 0.3(1). Data for higher values of λ and of thesystem size L show a consistent behavior.This is very important—the presence of a non-zero asymptote for thenormal-normal correlation function indicates that the normals remain or-dered on a macroscopic scale.This result supports the existence of a stable flat phase. Our presentfocus is on the precise nature of this phase; in particular we would like toknow if there is a well defined roughness exponent ξ and if so how it dependson the bending rigidity. The existence of a flat phase with a roughness7exponent independent of λ would indicate that the entire flat phase λ > λcis critical and that this model is in the same universality class as [15, 16, 17].Another possibility is that the roughness exponent will depend on λ. Inthis case the flat phase would still be stable but not critical.A possible interpretation of the stability of the flat phase is that themodel discussed here dynamically generates a non-zero length scale whichcan serve to define non-zero renormalized elastic constants. This simplemodel could then be used to study the properties of physical (polymerized)membranes in regimes in which self-avoidance is irrelevant.We are grateful for the use of NPAC computational facilities. Theresearch of M.B. and M.F. was supported by the Department of EnergyU.S.A. under contract No. DE-FG02-85ER40237. M.B. would also like toacknowledge support under NSF grant No. PHY89-04035 from the ITP atSanta Barbara, where some of this work was carried out. S.C. and G.T. weresupported by research funds from Syracuse University. The research ofK.A. was supported by the Department of Energy U.S.A. under contractDE-FG05-86ER-40272. K.A. acknowledges IFT at Gainesville where partof this work was carried out.References[1] “Statistical Mechanics of Membranes and Surfaces”, Vol. 5 of theJerusalem Winter School for Theoretical Physics, D.R. Nelson, T. Pi-ran and S. Weinberg eds., World Scientific, Singapore (1989). F. David,“Introduction to the Statistical Mechanics of Random Surfaces and Mem-branes”, in Vol. 8 of the JerusalemWinter School for Theoretical Physics,D.J. Gross, T. Piran and S. Weinberg eds., World Scientific, Singa-pore (1992). Proceedings to the LXII Session of the Les Houches Sum-mer School on “Fluctuating Geometries in Statistical Mechanics andField Theory”, P. Ginsparg, F. David and C. Zinn-Justin eds., (URLhttp://xxx.lanl.gov/lh94).[2] C.F. Schmidt et al., Science 259 (1993) 952.[3] D.R. Nelson and L. Peliti, J. Phys. France 48 1987 (1085).[4] Y. Kantor and D.R Nelson, Phys. Rev. Lett. 58 (1987) 2774; Phys. Rev.A36 (1987) 4020.8[5] Y. Kantor, M. Kardar and D.R. Nelson, Phys. Rev. Lett. 57 (1986) 791;Phys. Rev. A 35 (1987) 3056.[6] F.F. Abraham and D.R. Nelson, Science 249 (1990) 393.[7] J. Ambjørn, B. Durhuus and T. Jonsson, Nucl. Phys. B 316 (1989) 526.[8] M. Baig, D. Espriu and J. Wheater, Nucl. Phys. B 314 (1989) 587.[9] R. Renken and J. Kogut, Nucl. Phys. B 342 (1990) 753.[10] R.G. Harnish and J. Wheater, Nucl. Phys. B 350 (1991) 861.[11] J. Wheater and T. Stephenson, Phys. Lett. B 302 (1993) 447.[12] M. Baig, D. Espriu and A. Travesset, Nucl. Phys. B 426 (1994) 575.[13] D.R. Nelson, private discussion.[14] P. Le Doussal and L. Radzikovsky, Phys. Rev. Lett. 69 (1992) 1209.[15] J.A. Aronovitz and T.C. Lubensky, Phys. Rev. Lett. 60 (1988) 2634.[16] E. Guitter, F. David, S. Leibler and L. Peliti, Phys. Rev. Lett. 61 (1988)2949; J. Phys. France 50 (1989) 1787.[17] J.A. Aronovitz, L. Golubovic and T.C. Lubensky, J. Phys. France 50(1989) 609.9

The Phase Diagram of Crystalline Surfaces

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