We investigate actions for dynamically triangulated random surfaces that consist of a gaussian or
이 논문에서는 Gaussian 액션 함수에 Tight Immersion 액션 함수를 추가하는 방법도 제안한다. 이 액션은 표면의 평탄성과 곡률 모두 고려하며, Fenchel의 정리를 바탕으로 유도되었다. 시뮬레이션 결과, 이 액션은 두 번째 질서의 phase transition을 보여주지 않는다. 그러나 특정 열량에 대한 스펙트럼이 존재하는것은 다른 액션에 비해 유사한 표면을 만드는것이 가능하다는 것을 나타낸다.
한글 요약 끝:
간단하게 요약하자면, 이 논문에서는 다목적 유사 표면의 동역학을 연구하기 위해 다양한 액션 함수를 제안한다. 그 중 하나는 Tight Immersion이라는 이름으로 지명된 모듈러스와 곡률 항만 포함하는 Gaussian 액션 함수이다. 시뮬레이션 결과, 이 액션은 두 번째 질서의 phase transition을 보여주지 않는다. 그러나 특정 열량에 대한 스펙트럼이 존재하는것은 다른 액션에 비해 유사한 표면을 만드는것이 가능하다는 것을 나타낸다.
We investigate actions for dynamically triangulated random surfaces that consist of a gaussian or
arXiv:hep-lat/9303006v1 11 Mar 1993COLO-HEP-310Smooth Random Surfacesfrom Tight Immersions?C.F. BailliePhysics Dept.University of ColoradoBoulder, CO 80309USAD.A.
JohnstonDept. of MathematicsHeriot-Watt UniversityRiccartonEdinburgh, EH14 4ASScotlandOctober 24, 2018AbstractWe investigate actions for dynamically triangulated random surfaces that consist of a gaussian orarea term plus the modulus of the gaussian curvature and compare their behavior with both gaussianplus extrinsic curvature and “Steiner” actions.
Considerable effort has recently been devoted to exploring modifications of the discretized Polyakovpartition function [1] for a random surfaceZ =XTZN−1Yi=1dXµi exp(−Sg),(1)where the sum over triangulations PT means that we have, in effect, a fluid surface. Sg is just a simplegaussian actionSg = 12X
Earlierwork [2] had made it clear that this action (and variations, such as area and edge length actions [3, 4])failed to lead to a sensible continuum theory because the string tension did not scale so, inspired byanalytical work on QCD strings and biological membranes [5], an extrinsic curvature or “stiffness” termwas addedSe =X∆i,∆j(1 −ni · nj),(3)where ni, nj are the normals on neighboring triangles ∆i, ∆j. Simulations of Sg +λSe, called the gaussianplus extrinsic curvature (GPEC) action, seemed to indicate that there was a second order phase transition(the “crumpling transition”) from a small λ crumpled phase to a large λ smooth phase at which one mighthope to define a non-trivial continuum theory [6].
More recent simulations of larger surfaces suggest,however, that the transition is not second order [7, 8] 1 but that the string tension may possibly still bescaling correctly.Simpler spin systems on dynamical triangulations, such as Ising and Potts models [10], provide somereassurance that a non-trivial theory may be lurking at the crumpling transition because they displaythird order transitions and still have a sensible continuum limit.Nonetheless, the rather murky be-havior seen in [7, 8] prompts the question of whether actions with a sharper phase transition can befound. One geometrically appealing suggestion was the Steiner action put forward in [11], and simulatedmicrocanonically in [12]Ssteiner = 12X
It was pointed out in [13] that the grand canonical partition function diverged for Ssteineralone, so we conducted some exploratory simulations of various actions combining edge-length, area orgaussian terms with Ssteiner finding particularly sharp transitions for Area+λSsteiner and Sg +λSsteineractions [14].As this initial work is on the same small triangulations that indicated a second ordertransition for the GPEC action we cannot claim this is strong evidence for a sharp transition with theSteiner actions, without further results from larger lattices [15].In this paper we will explore a further possibility for a random surface action which, like the Steinerterm, is a natural object to consider from a geometrical point of view. For a curve C embedded in threedimensions it was shown by Fenchel that1πZC|κ|ds ≥2(5)where κ is the curvature and the equality holds when C is a plane convex curve [16].
For a surface Mimbedded in three dimensions the Gauss-Bonnet theorem tells us that12πZMKdS = χ(M)(6)where K is now the Gaussian curvature of the surface and χ is the Euler characteristic. As this is atopological invariant it tells us nothing about the configuration of the surface.
To get the equivalent of1 Unlike the GPEC action on rigid lattices, which almost certainly is [9].1
Fenchel’s equ. (5) we take a modulus sign in equ.
(6) and find 212πZM|K|dS ≥4 −χ(M)(7)with the equality holding when the surface is imbedded as a convex surface in three dimensional space.This term discretizes toStight =Xi|2π −Xj(i)φij|(8)where the outer sum is over all the vertices of the triangulation and the inner sum is round the neighborsj of a node i. φij is the angle subtended by the jth triangle at the ith vertex – see Fig.1. The behaviorof Stight was measured for the GPEC action in [8] (rather than being included in the action), where itwas found to correlate closely with the extrinsic curvature - dropping sharply in value at the crumplingtransition.
This raises the hope that including Stight in the dynamics may be sufficient to give a crumplingtransition without the assistance of an extrinsic curvature term.We will examine the phase structure of both S1 = Sg + λStightS1 = 12X
We included a local factor in the measure for compatibility with our earlier simulationswhich can be exponentiated to giveSm = d2Xilog(qi),(11)where qi is the number of neighbors of point i, and d = 3 dimensions. We thus simulated S1,2 + Sm.
Wemeasured < Sm > and the mean maximum number of neighbors < max(qi) > to get some idea of thebehavior of the intrinsic geometry. The extrinsic geometry was observed by measuring < Stight > andits associated specific heatC = λ2N< S2tight > −< Stight >2(12)as well as the gyration radius X2, a measure of the mean size of the surface as seen in the embeddingspace,X2 =19N(N −1)XijXµi −Xµj2 qiqj.
(13)For S1 the expectation value of Sg can be shown to be d(N −1)/2 by exactly the same argument that isused to give this result for the GPEC action, because Stight shares the scale invariance of the extrinsiccurvature term in the GPEC action. This serves as a useful check of equilibration in this case.
A furtheruseful check is provided by removing the modulus sign in equ. (8) which should then sum to give 4π forevery configuration by the Gauss-Bonnet theorem.The simulation used a Monte Carlo procedure which we have described in some detail elsewhere [17].It first goes through the mesh moving the X’s, carrying out a Metropolis accept/reject at each step, andthen goes through the mesh again carrying out the “flip” moves on the links, again applying a Metropolisaccept/reject at each stage.
The entire procedure constitutes a sweep. Due to the correlated nature ofthe data, a measurement was taken every tenth sweep and binning techniques were used to analyze theerrors.
We carried out 10K thermalization sweeps followed by 30K measurement sweeps for each datapoint. The acceptance for the X move was monitored and the size of the shift was adjusted to maintainan acceptance of around 50 percent.
The acceptance for the flip move was also measured, but in this2As a particular case of the theorem: Let Mn be a compact oriented C∞manifold immersed in En+N, such that thetotal absolute curvature equals 2. Then Mn belongs to a linear subvariety of dimension n + 1, and is imbedded as a convexhypersurface in En+1.
The converse is also true [16].2
case there is nothing to adjust, so as for GPEC actions this dropped with increasing λ (but was stillappreciable even for quite large λ).If we look at the numerical results for S1 (Sg + λStight) in Table 1 firstλsweepsSgSmStightCtightX2max(qi)0.50030K106.62(0.03) 122.79(0.00) 109.94(0.03) 0.46( 0.00) 2.40(0.01) 12.07(0.00)1.00030K106.33(0.01) 123.60(0.00) 66.40(0.00) 0.77( 0.00) 2.54(0.01) 10.92(0.00)1.25030K106.34(0.05) 123.76(0.00) 54.96(0.01) 0.84( 0.00) 2.53(0.01) 10.70(0.00)1.50030K106.45(0.03) 123.83(0.00) 46.70(0.02) 0.86( 0.00) 2.80(0.04) 10.59(0.00)1.75030K106.58(0.10) 123.89(0.00) 40.77(0.02) 0.91( 0.00) 2.77(0.05) 10.52(0.00)2.00030K107.25(0.21) 123.91(0.00) 36.10(0.05) 0.93( 0.00) 2.94(0.12) 10.49(0.00)2.25030K106.28(0.02) 123.92(0.00) 32.11(0.01) 0.90( 0.00) 2.16(0.00) 10.48(0.00)2.50030K106.34(0.07) 123.91(0.00) 29.42(0.02) 0.90( 0.00) 2.17(0.01) 10.48(0.00)3.00030K106.52(0.08) 123.90(0.00) 25.18(0.00) 0.82( 0.00) 2.14(0.01) 10.49(0.00)3.50030K106.48(0.10) 123.88(0.00) 22.53(0.01) 0.78( 0.00) 2.15(0.01) 10.51(0.00)4.00030K106.02(0.05) 123.84(0.00) 20.56(0.01) 0.73( 0.00) 2.12(0.02) 10.53(0.00)4.50030K106.08(0.37) 123.80(0.00) 19.15(0.00) 0.68( 0.00) 2.20(0.07) 10.57(0.01)Table 1Results for S1, N = 72we can see that Stight does indeed drop offwith increasing λ just like the extrinsic curvature. The behaviorof Sm and max(qi) is also reminiscent of the GPEC action, with the internal geometry becoming moreregular with increasing λ.
However, the specific heat C shows only a modest cusp at around λ = 2.00 ascan be seen in Fig.2, which should be contrasted with the larger peaks seen on these small meshes for boththe GPEC and Steiner actions. Similarly X2, plotted in Fig.3, shows no sign of a crumpling transition,with only a small increase in the region of the cusp in C, before it rapidly drops off.
The value of thegaussian term is close to the expected d(N −1)/2, assuring us that the results are equilibrated and boththe metropolis and flip acceptances are reasonable for all the values of λ simulated, so we can be sure thatthe simulation is performing as it should. We also measured the value of the gaussian curvature usingStight with the modulus sign removed and found, as expected (our surfaces have spherical topology), 4πfor every surface generated.
Visual inspection of “snapshots” of the surfaces that arise in the simulationconfirms the absence of a phase transition, with surfaces looking similar for all of the λ values simulated.One of these for λ = 4.0, but which is typical of all the others, is shown in Fig.4 and is obviously notsmooth. Even at the largest λ values simulated the surfaces are still some way from satisfying the lowerbound of 4π on Stight, so it would appear that the disordering effect of Sg overcomes Stight for all λ.The behavior of S2 (Area + λStight) is rather bizarre, as can be seen in Table 2.λsweepsAreaSmStightCtightX2max(qi)0.50030K106.05(0.03) 122.26(0.00) 119.51(0.09) 0.70( 0.01)10.67(0.17)12.66(0.00)1.00030K105.81(0.13) 122.88(0.00) 61.28(0.43) 0.86( 0.01)22.60(3.46)11.91(0.01)1.50030K105.78(0.18) 122.95(0.01) 38.55(0.28) 0.63( 0.01) 113.70(11.84) 11.80(0.02)1.75030K106.11(0.38) 122.93(0.02) 34.01(0.50) 0.69( 0.01) 141.42(42.03) 11.89(0.05)2.00030K105.50(0.10) 122.73(0.03) 31.63(0.14) 0.79( 0.03) 33.69(10.03) 12.42(0.07)2.25030K104.85(0.08) 122.70(0.01) 29.55(0.06) 0.87( 0.03)25.39(3.94)12.57(0.03)3.00030K106.04(0.06) 122.34(0.00) 22.21(0.02) 0.61( 0.00)10.31(0.10)13.60(0.00)3.50030K105.59(0.09) 122.19(0.00) 20.10(0.01) 0.56( 0.00)10.28(0.03)14.08(0.01)4.00030K106.68(0.25) 122.12(0.01) 18.80(0.01) 0.54( 0.00)10.50(0.15)14.16(0.02)4.50030K106.26(0.23) 121.98(0.01) 17.68(0.01) 0.50( 0.00)11.17(0.18)14.67(0.02)Table 2Results for S2, N = 72Again Stight decreases with increasing λ but there is no obvious peak in the specific heat.
max(qi) nowincreases at large λ and there is a huge peak in X2 at around λ = 2.5. For λ > 2.5 the surfaces generatedlook rather similar to those produced by S1, but near λ = 2.5 they are very long, jointed linear structures3
such as that in Fig.5. On its own the area action gives surfaces that are collections of long thin spikesemanating from a central point, and this is also seen for S2 at small λ.
It appears that adding Stightgives (approximately) very long, thin ellipsoids which, while satisfying the convexity property, are notthe generic smooth surfaces that we envisaged. From the evidence of the specific heat there is little signof a phase transition in this region that might be used to define a continuum limit, though it is alwayspossible that a higher order transition may be present.Our conclusions are rather disappointing from the point of view of finding candidate random surfaceactions which might be used similarly to the GPEC action to hunt for a non-trivial continuum stringtheory.
S1 shows little sign of a transition at all, apart from a modest bump in the specific heat, and S2,whilst adding to the bestiary of amusing pathologies that have been observed with various random surfacemodels, also looks unpromising. It may of course be possible to incorporate Stight as an additional termin the GPEC action in order to tune the couplings to see if the approach to the continuum limit could beoptimized.
A further possibility that might be worth pursuing is looking at the effect of self-avoidance,which completely changes the behavior of GPEC actions [18].This work was supported in part by NATO collaborative research grant CRG910091. CFB is supportedby DOE under contract DE-FG02-91ER40672 and by NSF Grand Challenge Applications Group GrantASC-9217394.
The computations were performed on workstations at Heriot-Watt University. We wouldlike to thank R.D.
Williams for help in developing initial versions of the dynamical mesh code.4
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Figure CaptionsFig. 1.
The neighbors and angles used in defining Stight.Fig. 2.
The specific heat C for action S1.Fig. 3.
The gyration radius X2 for action S1.Fig. 4.
A snapshot of a mesh generated by S1 with λ = 4.5.Fig. 5.
A snapshot of a mesh generated by S2 with λ = 2.5.6
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