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Strong self-avoidance and crumpling in random surfaces with extrinsic curvature
Physics Letters B 283 (1992) 55-62 North-Holland
PHYSICS LETTERS B
Strong self-avoidance and crumpling in random surfaces with extrinsic curvature C.F. Baillie Physics Department, University of Colorado, Boulder, CO 80309, USA and D.A. J o h n s t o n Department of Mathematics, Heriot-Watt University, Riccarton, Edinburgh EH14 4AS, Scotland, UK Received 20 February 1992
We introduce strong self-avoidance in simulations of dynamically triangulated and fixed triangulation random surfaces with extrinsic curvature. We compare the results with our previous simulations of weakly self-avoiding surfaces and with molecular dynamics simulations of "real" surfaces in solid-state physics.
It has been clear for some years that the diseases of the bosonic string also manifest themselves in random surface models which represent discretized euclidean string worldsheets. A discretization of the Polyakov string partition function [ 1 ] Z= I DgDXexp(-½
~ d 2 x ~ g g a b O a Xu ObXu) , (1)
[ 3 ] would, however, smooth out the surface and the square of the extrinsic curvature of the surface has precisely this effect. It also has a dimensionless coupling and preserves all the original symmetries of the theory so it is a good candidate for curing the ills of the Polyakov action. The extrinsic curvature may be discretized in two ways,
so=Z
1 2
,
(4)
gives, for a mixed intrinsic area worldsheet dX,.u exp ( - S,)
(2)
where ZT is a sum over different triangulations with the same number of nodes N and the action Sg is just a simple gaussian
s~=½ ~ (x~'-x~'):. (0>
(3)
Analytical work and simulations [ 2 ] have shown that the typical surfaces produced by such an action are very crumpled and that, as a consequence, it is not possible to obtain a sensible continuum theory. Adding "stiffness" to the original Polyakov action
where the j ( i ) are the neighbours of a node i, Oi is the area of the surrounding triangles, and
so= y. (1-,L,-,~,~), Adj
(5)
where the sum is over adjacent triangles and the normals ~ , live on the triangles. We denote the discretization of S, in eq. (4) as the "area" discretization and the discretization of Se in eq. (5) as the "edge" discretization respectively. It appears that the former is afflicted with lattice artifacts but that an action of the form Sg+XSe has a second order phase transition at finite 2 at which the physical string tension is fi-
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nite, thus offering the prospect of defining a continuum theory at this point [4] ~ It is interesting to note that an effective superstring action derived by Wiegmann [ 6 ]
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where the ~¢' are the unit normals to the surface and the e# its tangents, contains an extrinsic curvature term. It might thus seem that in adding a term by hand to a bosonic string action to cure its ills we had arrived at something close to a superstring action which we know is better behaved. Eq. (6) also contains an imaginary W e s s - Z u m i n o term which may have something to do with enforcing self-avoidance for the surfaces [ 7 ], as similar phase factors arise from fermionic degrees of freedom for particle paths [ 8 ]. In ref. [9 ] we explored models with actions Sg+2S~and Sg + 2Sa that included explicit weakself-avoidance and found that their behaviour was largely unaltered: the area action was no better behaved, and the edge action still appeared to retain its second order phase transition. The self-avoidance was enforced by using the spheres and tethers method used in refs. [ 10,1 1 ] in simulations of "real" surfaces in solid-state physics. In this approach, conceptually at least, each node in the surface is surrounded by a hard sphere o f radius ~rand joined to its neighbours by tethers o f some m a x i m u m length l. In actually implementing the algorithm, in order to avoid the time-consuming task of checking for the overlap of hard spheres, one divides coordinate space up into cubic boxes whose side length is the sphere diameter a and makes sure that the boxes are at most singly occupied. I f / < x/3 a it is impossible to thread a sphere, and hence the surface, through three other spheres and the surface is said to be strongly self-avoiding - even distant parts of the surface cannot pass through each other. If, however, l>~ x/3 ~r such intersections are possible, though there is still a local self-avoidance due to the excluded volume of the spheres and we have weak self-avoidance. We chose to simulate weak self-avoidance in ref. [ 9 ]
in order to have the m i n i m u m interference with the "springy" nature o f the gaussian term and keep the action as close as possible to a string action. It is interesting to compare our work, which is essentially motivated by string theory, with simulations intended to represent realistic surfaces such as amphiphilic bilayers or cell membranes. The initial work in this field was done with fixed triangulation ("tethered") surfaces with actions incorporating extrinsic curvature but no self-avoidance and apparently showed a crumpling transition [ 12 ]. However, later work [ I 0 ] which included strong self-avoidance, but still used fixed surfaces, showed no evidence for this and it was suggested in ref. [ 13 ] that this was because the self-avoidance constraint induced an effective rigidity which counteracted crumpling. Perhaps surprisingly some recent simulations [ 11 ] with dynamically triangulated ("fluid") rand o m surfaces (albeit modestly sized, as are ours) that had strong self-avoidance but no explicit extrinsic curvature term claim to have produced crumpled surfaces. This suggests that the effect o f making the surface dynamical is to counteract the induced rigidity coming from the self-avoidance. Our simulations in ref. [ 9 ] cannot be directly compared with this because only weak self-avoidance was enforced. Therefore in this paper we investigate strong self-avoidance, by the simple expedient o f shortening the tether lengths so that l < xfl3 a. If we view our surfaces as a statistical mechanical system rather than an approximation to string theory this is a perfectly natural thing to do and it is always possible that universality will ensure that the change this entails to the gaussian action will still be acceptable in the string case ~2. In what follows we briefly describe the methods used and variables measured and then present the results obtained, before discussing the interesting differences between weak and strong self-avoidance, and comparing the results with those of surfaces simulated in solid-state physics. The methods o f simulation were identical to those used in our earlier work [4] and ref. [9]. We used small meshes of 72 nodes because o f the exploratory
*~ Some recent work has cast doubt on the second order nature of the phase transition, but a theory with a sensible limit would still be possible with a higher order transition, provided the string tension scaled (as it appears to) [ 5 ].
~2 This certainly seems to be the case for simulations of a random surface without extrinsic curvature in one dimension, where gaussian and cut-off propagators (as one would get with a short tether) give similar results [ 14].
Seff=fd2x{( ~¢~,1g ~ab OaX/l ObXp) +~r~[(egObe~')2+(Vafi~)z]}+W,(A,),
56
(6)
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nature of the work and carried out a sufficient number of m e a s u r e m e n t sweeps to get reasonable statistics. The meshes had spherical topology and were e m b e d d e d in three dimensions; one o f the nodes was pinned down to remove the translational zero mode. We used a standard Monte Carlo scheme with Metropolis accept/reject, the above restrictions o f boxes and tethers being i m p l e m e n t e d as part o f this. 10K sweeps were allowed for the meshes to equilibrate in each case, then measurements were m a d e over the next 200K or 300K sweeps, doing one measurement per 10 sweeps. A sweep consisted o f going through the mesh and performing both X moves and flip moves on the d y n a m i c a l meshes, and only X moves on the fixed meshes. In order to m a i n t a i n the traditional a p p r o x i m a t e l y 50% acceptance rate for X moves the Monte Carlo high-size has to be scaled with the m a x i m u m tether length, which in turn scales as the hard sphere d i a m e t e r ~r. We found in practice that this hit-size is typically 0.25cr. As in ref. [9], we used boxes o f sizes (i.e. side lengths, which correspond to the hard sphere d i a m e t e r s ) 0.093, 0.16, 0.25 and 0.5. For these box sizes with d y n a m i c a l meshes we have 30K, 30K, 30K and 20K measurements respectively at each value of 2; and for fixed meshes we have 20K each. We have run at our usual values o f 2 : 0 for no extrinsic curvature, 0.1, 0.2, ..., 1.5 for area curvature and 0.25, 0.5, 0.6, ..., 1.5, 1.75, 2.0, 2.25, 2.5, 3.0 for edge. As in ref. [9 ] we have used an iterative algorithm to generate the 72-node starting mesh configuration from an icosahedron (which has 12 nodes) by repeatedly adding nodes in the m i d d l e of its faces. F o r weak self-avoidance the only constraint is that there is only one node per box so if the box in which we wish to place the new node is already occupied then we simply keep moving it by a small a m o u n t in a rand o m direction until we find an e m p t y box. This scheme works surprisingly well, requiring no more than three such moves per node to build the 72-node mesh. However, for strong self-avoidance we have the additional constraint that the tether lengths must be less than ,,/3 times the box size (sphere d i a m e t e r ) . Therefore when doing the r a n d o m moves to find an e m p t y box there is a danger that we will not find one within the range o f moves allowed by the m a x i m u m tether length. In fact we have found this to be the case for meshes with more than a p p r o x i m a t e l y 100 nodes.
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Fortunately, for 72 nodes this scheme works well enough to generate the initial mesh (although it requires as much as 1000 moves per n o d e ) . In order to simulate strongly self-avoiding r a n d o m surfaces with more than 100 nodes we will have to start from some triangulation of a sphere other than the simple icosahedron. As we are planning to do this in the near future, we are currently developing an algorithm which triangulates a sphere with an arbitrary n u m b e r o f nodes all roughly the same distance from their neighbours, automatically satisfying both the box and tether constraints. The constraint placed on the X variables by the selfavoidance means that the scaling argument which gives
( Sg) =½d(N-1)
(7)
for the expectation value o f the Polyakov part o f the action, for a mesh with N nodes e m b e d d e d in d dimensions, is no longer valid so this numerical value could not be used as a rigorous check on equilibration as in previous simulations. The change in size o f the surface as 2 increases could, however, be followed in the usual m a n n e r by measuring the discretized expression for gyration radius X2 1 X2= 9N(N-1)
~
(X~-Xy)2q;qJ'
(8)
where q; is the n u m b e r o f neighbours of a node i and the sum is over all pairs o f X's. We also measured the eigenvalues o f the m o m e n t o f inertia tensor 1
~ (Xi-XJ. ) ( X , - X f ) ,
(9)
whose sum isj ust X2 ~3. In a smooth phasefor a planar configuration one would expect to find two similar eigenvalues and one much smaller, and in the crumpled phase one would expect all three to be small and o f equal magnitude. F o r the roughly spherical configuration we expect to find for large 2 in our simulations we should see a p p r o x i m a t e l y equal values for the eigenvalues. The order o f the crumpling transition, if one still ~3 We have been a little careless in dropping the covariantizing factors of q; in eq. (9), but this has very little effect on the numerical results. 57
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exists, was determined by measuring the specific heat as in previous simulations
4 J u n e 1992 I
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Ca,e=
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~
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•
(10)
where the subscripts denote the use o f the area and edge extrinsic curvatures respectively. We recall that without self-avoidance Ca displayed a cusp which did not grow with surface size, suggesting a third or higher order transition, and Ce displayed a peak which grew with surface size, suggesting a second order transition. These results held for both fixed and dynamically triangulated surfaces. With weak self-avoidance we found that the crumpling transition persists essentially unchanged for the area curvature for both fixed and dynamically triangulated surfaces, whereas for the edge curvature the crumpling transition remains the same for the fixed surfaces but is shifted to lower 2 and the peak in the specific heat slightly suppressed for the dynamical surfaces. With strong selfavoidance we shall see below that things are dramatically different. In addition to measuring the usual properties of the Monte Carlo algorithm, namely the acceptance rates for the X moves and the flip moves, we measured some new properties particular to the self-avoidance constraints: for the X moves which are rejected we measured the fraction forbidden by the tether restriction ( l < x ~ or) and the fraction forbidden by the box being occupied; and for the X moves accepted we measured the acceptance rate for moves into the same box and the acceptance rate for moves into a different box. We firstly look at the results for the fixed triangulation random surfaces. In fig. 1 we plot X2 against for the various box sizes used, along with results (labeled as " N O B O X " ) from simulations with no selfavoidance, on the fixed surfaces for the edge curvature (the corresponding graph for the area curvature is very similar). Firstly, we see that (except for the box size of 0.5 which may be too large, as was the case for weak self-avoidance in ref. [ 9 ] ) the surface with strong self-avoidance is smaller than that without, for all 2. This is because the m a x i m u m tether length constraint has forced all the points to be much closer to their neighbours. Secondly there does not appear to be any dramatic increase in X2 as 2 increases, the surface merely expands slowly. This lack of a crumpling 58
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transition for a fixed surface with strong self-avoidance is confirmed by the specific heat for the edge curvature shown in fig. 2 - there is no peak (the corresponding graph for the area specific heat is similar). This result is dramatically different from that
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for a fixed surface with weak self-avoidance where the crumpling transition remains almost unchanged. This is consistent with the simulations for p l a n a r fixed triangulation strongly self-avoiding surfaces [ 10] where no crumpling transition is observed and thus extends the result to spherical surfaces. N o w we look at the results for dynamically triangulated r a n d o m surfaces. Fig. 3 shows X2 against 2 for the various box sizes used on the dynamical surfaces for the edge curvature (the corresponding graph for the area curvature is very similar). Again we see that, for the three smaller box sizes, the surface is smaller with strong self-avoidance. However, we now see something rather striking: )(2 decreases suddenly as it is increased, signaling the presence of a transition between a m o r e extended state at small it and a more compact state at large it. This is the reverse o f the behaviour without strong self-avoidance where we see a sharp increase in X2 at the crumpling transition between the small it crumpled state and the large it smooth state. The existence o f a phase transition at a r o u n d it= 1.0 is also clear from the plot o f the edge specific heat in fig. 4. As there is no discontinuity in the energy at this point we know that this transition is not first order, but to determine whether it is sec-
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Fig. 4. The specific heat (~ versus it on a dynamical mesh. ond or higher order requires simulations o f larger surfaces to examine the scaling o f the peak in the specific heat. F o r the area curvature it is much harder to confirm the existence of a phase transition from its specific heat plot - the cusp seen with no self-avoidance and with weak self-avoidance disappears leaving only a shoulder in the curve. This rounding is almost certainly a result of finite-size effects so we must defer further c o m m e n t s until larger surfaces are simulated. We can also see in figs. 1 and 3 that X2 seems to increase as square o f the box size at a given 2. In fact it increases exactly as the square o f the box size (even for box size of 0.5 which was too large for weak self-avoidance but may be acceptable for strong selfa v o i d a n c e ) , as is shown in fig. 5 for values o f 2 in the crumpled phase (0.25), near the phase transition (1.0) and in the smooth phase (3.0). Thus the box size sets the length scale of the simulations. In order to discover why the surface is becoming larger at low 2, contrary to our naive expectations, we examine the eigenvalues of the m o m e n t of inertia tensor in fig. 6. We see that at large it all three eigenvalues are a p p r o x i m a t e l y the same, which implies the surface is roughly spherical. However, as we decrease 2 one eigenvalue becomes much larger while the other two become only slightly smaller. Thus the overall size 59
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(BOX SIZE) z Fig. 5. The gyration radius X2 versus the square of the box size for ).=0.25, 1.0, 3.0 on both fixed and dynamical meshes. The lines are drawn through the origin and the ).= 1.0 data points. o f the surface (X2 is basically the sum o f the eigenvalues, but see footnote 3) increases and it becomes long in one direction and thinner in the other two. If we render the surface in three dimensions, fig. 7, we see that it is indeed extended in one direction and appears to have m o r e than one branch. It is t e m p t i n g to call such a configuration a b r a n c h e d polymer, but observations o f a surface with only 72 nodes are obviously not conclusive. Simulations o f larger surfaces would allow us to e x a m i n e the low 2 phase more closely by both the extraction o f the external Hausd o r f f d i m e n s i o n dH (4 for a b r a n c h e d p o l y m e r ) and direct observation o f the surface configurations. It is i m p o r t a n t to do this because our numerical results for the case o f no extrinsic curvature ( 2 = 0 ) agree with those o f ref. [ 11 ] (in particular, our ratios o f the second a n d third largest eigenvalues to the largest, averaged over all four box sizes, are 0.34 ( 2 ) and 0.19 ( 1 ) respectively; these values from ref. [ 11 ], for a 6 l - n o d e surface, are correspondingly 0.38 ( 14 ) and 0.16 ( 5 ) while our interpretation o f t h e m disagrees substantially. In ref. [11] the authors assume that their p l a n a r r a n d o m surface is in a c r u m p l e d phase (which has oo H a u s d o r f f d i m e n s i o n ) rather than the 60
Fig. 6. Eigenvalues of the moment of inertia tensor versus 2 for the edge curvature on a dynamical mesh. The three eigenvalues for each box size are plotted with the same symbol. The y-axis is logarithmic in order to separate the plots for clarity.
Fig. 7. The 72-node dynamical mesh at 2 = 0 rendered in three dimensions.
branched p o l y m e r phase we see for spherical r a n d o m surfaces. One could try to attribute the difference to the fact that the surface has a b o u n d a r y in ref. [ 11 ],
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however they find that the radius o f gyration X2 o f this surface scales as its size N to the power of 0.8. As X 2 ~ N 2/a~ this implies that dH=2.5, which is certainly greater than 2 (the H a u s d o r f f d i m e n s i o n of a flat two-dimensional surface) but is closer to 4 than ~ . Thus we believe that the low 2 phase is indeed that o f a b r a n c h e d p o l y m e r and intend to do larger simulations to confirm it. This makes sense since with strong self-avoidance, even for fluid surfaces, it does not seem possible for the surfaces to collapse to the infinitely-dense c r u m p l e d phase, rather it is more reasonable for t h e m to become tree-like b r a n c h e d polymers. We end with a few c o m m e n t s on the acceptance rates particular to the self-avoidance constraints. Firstly these rates are essentially the same for fixed and d y n a m i c a l surfaces. They are also the same for the smallest three box sizes but differ by about 10% for the largest, again suggesting that the 0.50 box may be too big. F o r the X moves which are rejected, both the fraction forbidden by the tether restriction and the fraction forbidden by the box being occupied decrease (from 0.26 to 0.23 for the former and from 0.18 to 0.13 for the latter) as 2 increases. This is to be expected since at large 2 the surface is spherical and the c o o r d i n a t i o n numbers o f the points are less, so there is on average more r o o m for each point to move around. F o r small 2 the d y n a m i c a l surface is extended in one direction so some points have a lot o f room to move a r o u n d in but most o f them do not. By the same reasoning we expect and find that for the X moves accepted, both the acceptance rate for moves into the same box and the acceptance rate for moves into a different box increase as )~ increases. For the former the fraction changes from 0,50 to 0.54 and for the latter from 0.06 to 0.10. These changes are all fairly small so roughly speaking we can say that for all 2 half the moves are to the same box, a tenth are to a different box and the rest (just over a t h i r d ) are rejected (almost twice as often due to the tether restriction than to box occupancy). To s u m m a r i z e the results of this p a p e r briefly: we have e x a m i n e d the limit o f strong self-avoidance (tether length < x / ' 3 a ) in r a n d o m surface models with an explicit extrinsic curvature term and found that the crumpling transition is suppressed completely on fixed triangulations; on dynamical triangulations a transition still exists but the low 2 phase
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is b r a n c h e d p o l y m e r like rather than being crumpled. The large 2 phase, however, remains smooth. In our earlier work [ 9] we looked at the limit o f weak selfavoidance (tether length >/x/3 ~r ~4) and found that this d i d not a p p e a r to fundamentally alter the crumpling transition. It would be interesting to examine whether our results for both weak and strong selfavoidance were confirmed by simulations o f larger surfaces, which would also enable us to investigate the nature o f the transitions observed m o r e closely and confirm that in the low 2 phase the surface does become a b r a n c h e d polymer. It would also be worth investigating the intermediate range of tether lengths between ~ (actually 2! ) and x / 3 a. F r o m our results so far we expect to see the weak b e h a v i o u r observed in ref. [9] until the m a x i m u m tether length is reduced close to x / 3 cr at which point there should be a transition to the strong b e h a v i o u r observed in this paper. The question is how dramatic is this transition. This work was supported in part by N A T O collaborative research grant CRG910091. C.F.B. is supported by DOE under contract DE-AC02-86ER40253 and by A F O S R G r a n t AFOSR-89-0422. The computations were performed on the TC2000 Butterfly and the Sequent Symmetry at Argonne N a t i o n a l Laboratory, the G P 1 0 0 0 Butterfly at Michigan State University, and the Myrias SPS-2 at the University of Colorado. We would like to thank R.D. Williams for help in developing initial versions o f the d y n a m ical mesh code. ~4 In ref. [9 ] we actually had an upper limit of 2, which is significantly larger than ,/3 a, on the tether length to suppress interactions between distant points on the surface.
References [ 1] A.M. Polyakov, Phys. Lett. B 103 ( 1981 ) 207. [2] J. Ambjorn, B. Durhuus and J. Frohlich, Nucl. Phys. B 257 (1985) 433; D. Boulatov, V. Kazakov, 1. Kostov and A. Migdal, Nucl. Phys. B 275 (1986) 641; A. Bi[loire and F. David, NucL Phys. B 275 (1986) 617; J. Jurkiewicz, A. Krzywicki and B. Petersson, Phys. Lett. B 168 (1986) 273, 61
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[3] A. Polyakov, Nucl. Phys. B 268 (1986) 406: H. Kleinert, Phys. Lett. B 174 (1986) 335; W. Helfrich, J. Phys. 46 (1986) 1263. [4] S. Catterall, Phys. Lett. B 220 (1989) 207; B 243 (1990) 121: C.F. Baillie, D.A. Johnston and R.D. Williams, Nucl. Phys. B335 (1990) 469; R. Renken and J. Kogut, Nucl. Phys. B 354 ( 1991 ) 328; C.F. Baillie, R.D. Williams, S.M, Catterall and D.A. Johnston, Nucl. Phys. B 348 ( 1991 ) 543; J. Ambjorn, J. Jurkiewicz, S. Varsted, A. Irback and B. Petersson, Critical properties of the dynamical random surface with extrinsic curvature, Niels Bohr Institute preprint NBI-HE-91-14. [5] J. Ambjorn, J. Jurkiewicz, S. Varsted, A. Irback and B. Petersson, presented at Lattice 91, Nucl. Phys. B (Proc. Suppl. ), to appear. [6] B.P. Wiegmann, Nucl. Phys. B 323 (1989) 311. [7]A.R. Kavalov and A.G. Sedrakyan, Nucl. Phys. B 285 (1987) 264.
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[8] J. Ambjorn, B. Durhuus and T. Jonsson, Nucl. Phys. B 330 (1990) 509. [9] C.F. Baillie and D.A. Johnston, Phys. Lett. B 273 (1991) 380. [ 10] F.F. Abraham, W.E. Rudge and M. Plischke, Phys. Rev. Lett. 62 (1989) 1757; M. Plischke and D. Boal, Phys. Rev. A 38 (1988) 4943; J.-S. Ho and A. Baumgartner, Phys. Rev. Lett. 63 (1989) 1324; D. Boal, E. Levinson, D, Liu and M. Plischke, Phys. Rev. A 40 (1989) 3292. [11 ] J.-S. Ho and A. Baumgartner, Europhys. Lett. 12 (1990) 295. [ 12] Y. Kantor, M. Kardar and D.R. Nelson, Phys. Rev. Lett. 57 (1986) 791; Phys. Rev. A 35 (1987) 3056. [13] F. Abraham and D.R. Nelson, J. Phys. (Paris) 51 (1990) 2653. [ 14 ] Z. Burda, J. Jurkiewicz and L. Karkkainen, Universality of dynamically triangulated random surfaces in onedimensional target space, Bielefeld preprint ( 1992 ).
PHYSICS LETTERS B
Strong self-avoidance and crumpling in random surfaces with extrinsic curvature C.F. Baillie Physics Department, University of Colorado, Boulder, CO 80309, USA and D.A. J o h n s t o n Department of Mathematics, Heriot-Watt University, Riccarton, Edinburgh EH14 4AS, Scotland, UK Received 20 February 1992
We introduce strong self-avoidance in simulations of dynamically triangulated and fixed triangulation random surfaces with extrinsic curvature. We compare the results with our previous simulations of weakly self-avoiding surfaces and with molecular dynamics simulations of "real" surfaces in solid-state physics.
It has been clear for some years that the diseases of the bosonic string also manifest themselves in random surface models which represent discretized euclidean string worldsheets. A discretization of the Polyakov string partition function [ 1 ] Z= I DgDXexp(-½
~ d 2 x ~ g g a b O a Xu ObXu) , (1)
[ 3 ] would, however, smooth out the surface and the square of the extrinsic curvature of the surface has precisely this effect. It also has a dimensionless coupling and preserves all the original symmetries of the theory so it is a good candidate for curing the ills of the Polyakov action. The extrinsic curvature may be discretized in two ways,
so=Z
1 2
,
(4)
gives, for a mixed intrinsic area worldsheet dX,.u exp ( - S,)
(2)
where ZT is a sum over different triangulations with the same number of nodes N and the action Sg is just a simple gaussian
s~=½ ~ (x~'-x~'):. (0>
(3)
Analytical work and simulations [ 2 ] have shown that the typical surfaces produced by such an action are very crumpled and that, as a consequence, it is not possible to obtain a sensible continuum theory. Adding "stiffness" to the original Polyakov action
where the j ( i ) are the neighbours of a node i, Oi is the area of the surrounding triangles, and
so= y. (1-,L,-,~,~), Adj
(5)
where the sum is over adjacent triangles and the normals ~ , live on the triangles. We denote the discretization of S, in eq. (4) as the "area" discretization and the discretization of Se in eq. (5) as the "edge" discretization respectively. It appears that the former is afflicted with lattice artifacts but that an action of the form Sg+XSe has a second order phase transition at finite 2 at which the physical string tension is fi-
0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
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nite, thus offering the prospect of defining a continuum theory at this point [4] ~ It is interesting to note that an effective superstring action derived by Wiegmann [ 6 ]
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where the ~¢' are the unit normals to the surface and the e# its tangents, contains an extrinsic curvature term. It might thus seem that in adding a term by hand to a bosonic string action to cure its ills we had arrived at something close to a superstring action which we know is better behaved. Eq. (6) also contains an imaginary W e s s - Z u m i n o term which may have something to do with enforcing self-avoidance for the surfaces [ 7 ], as similar phase factors arise from fermionic degrees of freedom for particle paths [ 8 ]. In ref. [9 ] we explored models with actions Sg+2S~and Sg + 2Sa that included explicit weakself-avoidance and found that their behaviour was largely unaltered: the area action was no better behaved, and the edge action still appeared to retain its second order phase transition. The self-avoidance was enforced by using the spheres and tethers method used in refs. [ 10,1 1 ] in simulations of "real" surfaces in solid-state physics. In this approach, conceptually at least, each node in the surface is surrounded by a hard sphere o f radius ~rand joined to its neighbours by tethers o f some m a x i m u m length l. In actually implementing the algorithm, in order to avoid the time-consuming task of checking for the overlap of hard spheres, one divides coordinate space up into cubic boxes whose side length is the sphere diameter a and makes sure that the boxes are at most singly occupied. I f / < x/3 a it is impossible to thread a sphere, and hence the surface, through three other spheres and the surface is said to be strongly self-avoiding - even distant parts of the surface cannot pass through each other. If, however, l>~ x/3 ~r such intersections are possible, though there is still a local self-avoidance due to the excluded volume of the spheres and we have weak self-avoidance. We chose to simulate weak self-avoidance in ref. [ 9 ]
in order to have the m i n i m u m interference with the "springy" nature o f the gaussian term and keep the action as close as possible to a string action. It is interesting to compare our work, which is essentially motivated by string theory, with simulations intended to represent realistic surfaces such as amphiphilic bilayers or cell membranes. The initial work in this field was done with fixed triangulation ("tethered") surfaces with actions incorporating extrinsic curvature but no self-avoidance and apparently showed a crumpling transition [ 12 ]. However, later work [ I 0 ] which included strong self-avoidance, but still used fixed surfaces, showed no evidence for this and it was suggested in ref. [ 13 ] that this was because the self-avoidance constraint induced an effective rigidity which counteracted crumpling. Perhaps surprisingly some recent simulations [ 11 ] with dynamically triangulated ("fluid") rand o m surfaces (albeit modestly sized, as are ours) that had strong self-avoidance but no explicit extrinsic curvature term claim to have produced crumpled surfaces. This suggests that the effect o f making the surface dynamical is to counteract the induced rigidity coming from the self-avoidance. Our simulations in ref. [ 9 ] cannot be directly compared with this because only weak self-avoidance was enforced. Therefore in this paper we investigate strong self-avoidance, by the simple expedient o f shortening the tether lengths so that l < xfl3 a. If we view our surfaces as a statistical mechanical system rather than an approximation to string theory this is a perfectly natural thing to do and it is always possible that universality will ensure that the change this entails to the gaussian action will still be acceptable in the string case ~2. In what follows we briefly describe the methods used and variables measured and then present the results obtained, before discussing the interesting differences between weak and strong self-avoidance, and comparing the results with those of surfaces simulated in solid-state physics. The methods o f simulation were identical to those used in our earlier work [4] and ref. [9]. We used small meshes of 72 nodes because o f the exploratory
*~ Some recent work has cast doubt on the second order nature of the phase transition, but a theory with a sensible limit would still be possible with a higher order transition, provided the string tension scaled (as it appears to) [ 5 ].
~2 This certainly seems to be the case for simulations of a random surface without extrinsic curvature in one dimension, where gaussian and cut-off propagators (as one would get with a short tether) give similar results [ 14].
Seff=fd2x{( ~¢~,1g ~ab OaX/l ObXp) +~r~[(egObe~')2+(Vafi~)z]}+W,(A,),
56
(6)
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nature of the work and carried out a sufficient number of m e a s u r e m e n t sweeps to get reasonable statistics. The meshes had spherical topology and were e m b e d d e d in three dimensions; one o f the nodes was pinned down to remove the translational zero mode. We used a standard Monte Carlo scheme with Metropolis accept/reject, the above restrictions o f boxes and tethers being i m p l e m e n t e d as part o f this. 10K sweeps were allowed for the meshes to equilibrate in each case, then measurements were m a d e over the next 200K or 300K sweeps, doing one measurement per 10 sweeps. A sweep consisted o f going through the mesh and performing both X moves and flip moves on the d y n a m i c a l meshes, and only X moves on the fixed meshes. In order to m a i n t a i n the traditional a p p r o x i m a t e l y 50% acceptance rate for X moves the Monte Carlo high-size has to be scaled with the m a x i m u m tether length, which in turn scales as the hard sphere d i a m e t e r ~r. We found in practice that this hit-size is typically 0.25cr. As in ref. [9], we used boxes o f sizes (i.e. side lengths, which correspond to the hard sphere d i a m e t e r s ) 0.093, 0.16, 0.25 and 0.5. For these box sizes with d y n a m i c a l meshes we have 30K, 30K, 30K and 20K measurements respectively at each value of 2; and for fixed meshes we have 20K each. We have run at our usual values o f 2 : 0 for no extrinsic curvature, 0.1, 0.2, ..., 1.5 for area curvature and 0.25, 0.5, 0.6, ..., 1.5, 1.75, 2.0, 2.25, 2.5, 3.0 for edge. As in ref. [9 ] we have used an iterative algorithm to generate the 72-node starting mesh configuration from an icosahedron (which has 12 nodes) by repeatedly adding nodes in the m i d d l e of its faces. F o r weak self-avoidance the only constraint is that there is only one node per box so if the box in which we wish to place the new node is already occupied then we simply keep moving it by a small a m o u n t in a rand o m direction until we find an e m p t y box. This scheme works surprisingly well, requiring no more than three such moves per node to build the 72-node mesh. However, for strong self-avoidance we have the additional constraint that the tether lengths must be less than ,,/3 times the box size (sphere d i a m e t e r ) . Therefore when doing the r a n d o m moves to find an e m p t y box there is a danger that we will not find one within the range o f moves allowed by the m a x i m u m tether length. In fact we have found this to be the case for meshes with more than a p p r o x i m a t e l y 100 nodes.
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Fortunately, for 72 nodes this scheme works well enough to generate the initial mesh (although it requires as much as 1000 moves per n o d e ) . In order to simulate strongly self-avoiding r a n d o m surfaces with more than 100 nodes we will have to start from some triangulation of a sphere other than the simple icosahedron. As we are planning to do this in the near future, we are currently developing an algorithm which triangulates a sphere with an arbitrary n u m b e r o f nodes all roughly the same distance from their neighbours, automatically satisfying both the box and tether constraints. The constraint placed on the X variables by the selfavoidance means that the scaling argument which gives
( Sg) =½d(N-1)
(7)
for the expectation value o f the Polyakov part o f the action, for a mesh with N nodes e m b e d d e d in d dimensions, is no longer valid so this numerical value could not be used as a rigorous check on equilibration as in previous simulations. The change in size o f the surface as 2 increases could, however, be followed in the usual m a n n e r by measuring the discretized expression for gyration radius X2 1 X2= 9N(N-1)
~
(X~-Xy)2q;qJ'
(8)
where q; is the n u m b e r o f neighbours of a node i and the sum is over all pairs o f X's. We also measured the eigenvalues o f the m o m e n t o f inertia tensor 1
~ (Xi-XJ. ) ( X , - X f ) ,
(9)
whose sum isj ust X2 ~3. In a smooth phasefor a planar configuration one would expect to find two similar eigenvalues and one much smaller, and in the crumpled phase one would expect all three to be small and o f equal magnitude. F o r the roughly spherical configuration we expect to find for large 2 in our simulations we should see a p p r o x i m a t e l y equal values for the eigenvalues. The order o f the crumpling transition, if one still ~3 We have been a little careless in dropping the covariantizing factors of q; in eq. (9), but this has very little effect on the numerical results. 57
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exists, was determined by measuring the specific heat as in previous simulations
4 J u n e 1992 I
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Ca,e=
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where the subscripts denote the use o f the area and edge extrinsic curvatures respectively. We recall that without self-avoidance Ca displayed a cusp which did not grow with surface size, suggesting a third or higher order transition, and Ce displayed a peak which grew with surface size, suggesting a second order transition. These results held for both fixed and dynamically triangulated surfaces. With weak self-avoidance we found that the crumpling transition persists essentially unchanged for the area curvature for both fixed and dynamically triangulated surfaces, whereas for the edge curvature the crumpling transition remains the same for the fixed surfaces but is shifted to lower 2 and the peak in the specific heat slightly suppressed for the dynamical surfaces. With strong selfavoidance we shall see below that things are dramatically different. In addition to measuring the usual properties of the Monte Carlo algorithm, namely the acceptance rates for the X moves and the flip moves, we measured some new properties particular to the self-avoidance constraints: for the X moves which are rejected we measured the fraction forbidden by the tether restriction ( l < x ~ or) and the fraction forbidden by the box being occupied; and for the X moves accepted we measured the acceptance rate for moves into the same box and the acceptance rate for moves into a different box. We firstly look at the results for the fixed triangulation random surfaces. In fig. 1 we plot X2 against for the various box sizes used, along with results (labeled as " N O B O X " ) from simulations with no selfavoidance, on the fixed surfaces for the edge curvature (the corresponding graph for the area curvature is very similar). Firstly, we see that (except for the box size of 0.5 which may be too large, as was the case for weak self-avoidance in ref. [ 9 ] ) the surface with strong self-avoidance is smaller than that without, for all 2. This is because the m a x i m u m tether length constraint has forced all the points to be much closer to their neighbours. Secondly there does not appear to be any dramatic increase in X2 as 2 increases, the surface merely expands slowly. This lack of a crumpling 58
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transition for a fixed surface with strong self-avoidance is confirmed by the specific heat for the edge curvature shown in fig. 2 - there is no peak (the corresponding graph for the area specific heat is similar). This result is dramatically different from that
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for a fixed surface with weak self-avoidance where the crumpling transition remains almost unchanged. This is consistent with the simulations for p l a n a r fixed triangulation strongly self-avoiding surfaces [ 10] where no crumpling transition is observed and thus extends the result to spherical surfaces. N o w we look at the results for dynamically triangulated r a n d o m surfaces. Fig. 3 shows X2 against 2 for the various box sizes used on the dynamical surfaces for the edge curvature (the corresponding graph for the area curvature is very similar). Again we see that, for the three smaller box sizes, the surface is smaller with strong self-avoidance. However, we now see something rather striking: )(2 decreases suddenly as it is increased, signaling the presence of a transition between a m o r e extended state at small it and a more compact state at large it. This is the reverse o f the behaviour without strong self-avoidance where we see a sharp increase in X2 at the crumpling transition between the small it crumpled state and the large it smooth state. The existence o f a phase transition at a r o u n d it= 1.0 is also clear from the plot o f the edge specific heat in fig. 4. As there is no discontinuity in the energy at this point we know that this transition is not first order, but to determine whether it is sec-
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(BOX SIZE) z Fig. 5. The gyration radius X2 versus the square of the box size for ).=0.25, 1.0, 3.0 on both fixed and dynamical meshes. The lines are drawn through the origin and the ).= 1.0 data points. o f the surface (X2 is basically the sum o f the eigenvalues, but see footnote 3) increases and it becomes long in one direction and thinner in the other two. If we render the surface in three dimensions, fig. 7, we see that it is indeed extended in one direction and appears to have m o r e than one branch. It is t e m p t i n g to call such a configuration a b r a n c h e d polymer, but observations o f a surface with only 72 nodes are obviously not conclusive. Simulations o f larger surfaces would allow us to e x a m i n e the low 2 phase more closely by both the extraction o f the external Hausd o r f f d i m e n s i o n dH (4 for a b r a n c h e d p o l y m e r ) and direct observation o f the surface configurations. It is i m p o r t a n t to do this because our numerical results for the case o f no extrinsic curvature ( 2 = 0 ) agree with those o f ref. [ 11 ] (in particular, our ratios o f the second a n d third largest eigenvalues to the largest, averaged over all four box sizes, are 0.34 ( 2 ) and 0.19 ( 1 ) respectively; these values from ref. [ 11 ], for a 6 l - n o d e surface, are correspondingly 0.38 ( 14 ) and 0.16 ( 5 ) while our interpretation o f t h e m disagrees substantially. In ref. [11] the authors assume that their p l a n a r r a n d o m surface is in a c r u m p l e d phase (which has oo H a u s d o r f f d i m e n s i o n ) rather than the 60
Fig. 6. Eigenvalues of the moment of inertia tensor versus 2 for the edge curvature on a dynamical mesh. The three eigenvalues for each box size are plotted with the same symbol. The y-axis is logarithmic in order to separate the plots for clarity.
Fig. 7. The 72-node dynamical mesh at 2 = 0 rendered in three dimensions.
branched p o l y m e r phase we see for spherical r a n d o m surfaces. One could try to attribute the difference to the fact that the surface has a b o u n d a r y in ref. [ 11 ],
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however they find that the radius o f gyration X2 o f this surface scales as its size N to the power of 0.8. As X 2 ~ N 2/a~ this implies that dH=2.5, which is certainly greater than 2 (the H a u s d o r f f d i m e n s i o n of a flat two-dimensional surface) but is closer to 4 than ~ . Thus we believe that the low 2 phase is indeed that o f a b r a n c h e d p o l y m e r and intend to do larger simulations to confirm it. This makes sense since with strong self-avoidance, even for fluid surfaces, it does not seem possible for the surfaces to collapse to the infinitely-dense c r u m p l e d phase, rather it is more reasonable for t h e m to become tree-like b r a n c h e d polymers. We end with a few c o m m e n t s on the acceptance rates particular to the self-avoidance constraints. Firstly these rates are essentially the same for fixed and d y n a m i c a l surfaces. They are also the same for the smallest three box sizes but differ by about 10% for the largest, again suggesting that the 0.50 box may be too big. F o r the X moves which are rejected, both the fraction forbidden by the tether restriction and the fraction forbidden by the box being occupied decrease (from 0.26 to 0.23 for the former and from 0.18 to 0.13 for the latter) as 2 increases. This is to be expected since at large 2 the surface is spherical and the c o o r d i n a t i o n numbers o f the points are less, so there is on average more r o o m for each point to move around. F o r small 2 the d y n a m i c a l surface is extended in one direction so some points have a lot o f room to move a r o u n d in but most o f them do not. By the same reasoning we expect and find that for the X moves accepted, both the acceptance rate for moves into the same box and the acceptance rate for moves into a different box increase as )~ increases. For the former the fraction changes from 0,50 to 0.54 and for the latter from 0.06 to 0.10. These changes are all fairly small so roughly speaking we can say that for all 2 half the moves are to the same box, a tenth are to a different box and the rest (just over a t h i r d ) are rejected (almost twice as often due to the tether restriction than to box occupancy). To s u m m a r i z e the results of this p a p e r briefly: we have e x a m i n e d the limit o f strong self-avoidance (tether length < x / ' 3 a ) in r a n d o m surface models with an explicit extrinsic curvature term and found that the crumpling transition is suppressed completely on fixed triangulations; on dynamical triangulations a transition still exists but the low 2 phase
4 June 1992
is b r a n c h e d p o l y m e r like rather than being crumpled. The large 2 phase, however, remains smooth. In our earlier work [ 9] we looked at the limit o f weak selfavoidance (tether length >/x/3 ~r ~4) and found that this d i d not a p p e a r to fundamentally alter the crumpling transition. It would be interesting to examine whether our results for both weak and strong selfavoidance were confirmed by simulations o f larger surfaces, which would also enable us to investigate the nature o f the transitions observed m o r e closely and confirm that in the low 2 phase the surface does become a b r a n c h e d polymer. It would also be worth investigating the intermediate range of tether lengths between ~ (actually 2! ) and x / 3 a. F r o m our results so far we expect to see the weak b e h a v i o u r observed in ref. [9] until the m a x i m u m tether length is reduced close to x / 3 cr at which point there should be a transition to the strong b e h a v i o u r observed in this paper. The question is how dramatic is this transition. This work was supported in part by N A T O collaborative research grant CRG910091. C.F.B. is supported by DOE under contract DE-AC02-86ER40253 and by A F O S R G r a n t AFOSR-89-0422. The computations were performed on the TC2000 Butterfly and the Sequent Symmetry at Argonne N a t i o n a l Laboratory, the G P 1 0 0 0 Butterfly at Michigan State University, and the Myrias SPS-2 at the University of Colorado. We would like to thank R.D. Williams for help in developing initial versions o f the d y n a m ical mesh code. ~4 In ref. [9 ] we actually had an upper limit of 2, which is significantly larger than ,/3 a, on the tether length to suppress interactions between distant points on the surface.
References [ 1] A.M. Polyakov, Phys. Lett. B 103 ( 1981 ) 207. [2] J. Ambjorn, B. Durhuus and J. Frohlich, Nucl. Phys. B 257 (1985) 433; D. Boulatov, V. Kazakov, 1. Kostov and A. Migdal, Nucl. Phys. B 275 (1986) 641; A. Bi[loire and F. David, NucL Phys. B 275 (1986) 617; J. Jurkiewicz, A. Krzywicki and B. Petersson, Phys. Lett. B 168 (1986) 273, 61
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[3] A. Polyakov, Nucl. Phys. B 268 (1986) 406: H. Kleinert, Phys. Lett. B 174 (1986) 335; W. Helfrich, J. Phys. 46 (1986) 1263. [4] S. Catterall, Phys. Lett. B 220 (1989) 207; B 243 (1990) 121: C.F. Baillie, D.A. Johnston and R.D. Williams, Nucl. Phys. B335 (1990) 469; R. Renken and J. Kogut, Nucl. Phys. B 354 ( 1991 ) 328; C.F. Baillie, R.D. Williams, S.M, Catterall and D.A. Johnston, Nucl. Phys. B 348 ( 1991 ) 543; J. Ambjorn, J. Jurkiewicz, S. Varsted, A. Irback and B. Petersson, Critical properties of the dynamical random surface with extrinsic curvature, Niels Bohr Institute preprint NBI-HE-91-14. [5] J. Ambjorn, J. Jurkiewicz, S. Varsted, A. Irback and B. Petersson, presented at Lattice 91, Nucl. Phys. B (Proc. Suppl. ), to appear. [6] B.P. Wiegmann, Nucl. Phys. B 323 (1989) 311. [7]A.R. Kavalov and A.G. Sedrakyan, Nucl. Phys. B 285 (1987) 264.
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[8] J. Ambjorn, B. Durhuus and T. Jonsson, Nucl. Phys. B 330 (1990) 509. [9] C.F. Baillie and D.A. Johnston, Phys. Lett. B 273 (1991) 380. [ 10] F.F. Abraham, W.E. Rudge and M. Plischke, Phys. Rev. Lett. 62 (1989) 1757; M. Plischke and D. Boal, Phys. Rev. A 38 (1988) 4943; J.-S. Ho and A. Baumgartner, Phys. Rev. Lett. 63 (1989) 1324; D. Boal, E. Levinson, D, Liu and M. Plischke, Phys. Rev. A 40 (1989) 3292. [11 ] J.-S. Ho and A. Baumgartner, Europhys. Lett. 12 (1990) 295. [ 12] Y. Kantor, M. Kardar and D.R. Nelson, Phys. Rev. Lett. 57 (1986) 791; Phys. Rev. A 35 (1987) 3056. [13] F. Abraham and D.R. Nelson, J. Phys. (Paris) 51 (1990) 2653. [ 14 ] Z. Burda, J. Jurkiewicz and L. Karkkainen, Universality of dynamically triangulated random surfaces in onedimensional target space, Bielefeld preprint ( 1992 ).