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Scaling behavior at the crumpling transition
Nuclear Physics B342 (1990) 753—763 North-Holland
SCALING BEHAVIOR AT THE CRUMPLING TRANSITION Ray L. RENKEN and John B. KOGUT Loomis Laborato,y of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, IL 61801, USA Received 12 February 1990 (Revised 2 April 1990)
Finite-size scaling is used to study the critical behavior at the crumpling transition in three dimensions. The correlation length critical exponent, v, is estimated to be 0.78(2). The Hausdorff dimension just above the critical coupling is bounded within the range 2
The formulation of dynamically triangulated random surfaces [1,21 as a lattice regularization of the Polyakov string [31has led to progress in the theory of noncritical strings [4]. Exact solutions have been found, including those for dimensions D —2 and D 0 [1,21, and D 1 [4]. In this range of dimensions, the critical exponents obtained agree with the continuum results of Knizhnik, Polyakov and Zamolodchikov (KPZ) [51.However, in D> 1, the dynamically triangulated random surface models with gaussian action appear to be in a physically undesirable crumpled phase. Likewise, for D> 1, the continuum results of KPZ cease to be meaningful. Ambjørn and Durhuus [6,71 have suggested that a model with a non-trivial continuum limit could be constructed by including a term in the action that depends on the extrinsic curvature. Catterall [8] and others [91,working in the “microcanonical” ensemble and in three dimensions, have shown that there is a continuous phase transition at a finite value of the coupling for a particular formulation of the extrinsic curvature term. This discovery provides us with a new critical theory which, if non-trivial, could have physical relevance. From a statistical mechanical point of view, perhaps the most novel feature of the dynamically triangulated random surface models is the functional integration over geometries. In all of the models where an exact solution has been found, the corresponding models with fixed triangulation (i.e. without the functional integration over geometries) have been well understood. In particular, their critical exponents and central charges have been known. This is not the case for Catterall’s transition. Kantor and Nelson [101have studied a model identical to one studied by Catterall except that the geometry is fixed and the gaussian action is replaced with =
=
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R.L. Renken, lB. Kogut
/
Crumpling transition
a “tethering potential” that limits the length of the legs of the triangles. They find that a continuous phase transition separates a crumpled phase from a smooth phase. Their phase diagram is identical to Catterall’s, but presumably the critical behavior is different. Ambjørn et al. have studied a fixed triangulation model with gaussian weight and extrinsic curvature term [11]. Their results suggest that this model also has a second-order phase transition. In this paper, we obtain more detailed information about the critical behavior of this crumpling transition. To be precise, we study the action
S= —~L(X~—X~.)2—AL(1—cos6~1),
(1)
where i and j label sites on a torus with a regular triangularization (illustrated in fig. 1), the sums are over nearest neighbor pairs, the X-coordinates are three-vectors, ~ is the angle in the embedding space between the two triangles sharing the link (i, J), and A is the extrinsic curvature coupling. Scale invariance has been used to eliminate the strength of the gaussian coupling. The functional integration is over all values of the X-coordinates, except that one is held fixed in order to eliminate motion of the surface as a whole. Periodic boundary conditions are used, making the topology that of a torus. A continuum string model with extrinsic curvature was introduced by Kleinert [12] and Polyakov [131several years ago. One interesting result of these investigations was the realization that the model could be studied perturbatively when
Fig. 1. A regular triangulation.
R.L. Renken, lB. Kogut
1/A
/
Crumpling transition
755
a is small. This coupling is asymptotically free with a f3-function, a2 D f3(a)=—-—------2~ 2
(a<<1),
(2)
which indicates that the origin a 0 is an ultraviolet stable and infrared unstable point. The work of Catterall [8] and others suggests that there is another ultraviolet stable fixed point which is non-perturbative. Our paper has the more modest goal of demonstrating that there is an ultraviolet stable fixed point in the fixed connectivity model. We leave a more detailed study of Catterall’s theory to future work. We primarily study two operators that are sensitive to the difference between the crumpled phase and the smooth phase: the curvature part of the action, and the gyration radius. The curvature part of the action, =
Se=(~(1-cosOii))
(3)
is large when A is small and gets smaller as A increases. More importantly, the specific heat (up to constants)
C=A2((S~—KSe)2)
(4)
diverges at a continuous phase transition on an infinite lattice. On a finite lattice its scaling behavior gives the critical exponents. The gyration radius is (X2)
=
N(N- 1) ~ ~((xx)2)
(5)
The sum is over all N nodes of the surface and cr, is the coordination number at the site i. For a regular triangulation, o- 6. The gyration radius is interesting because its scaling behavior gives the Hausdorff dimension, dH. In the large-N limit, =
KX2(N))~N2~.
(6)
In the crumpled phase the gyration radius goes like ln N, so the Hausdorff dimension is infinite. When A is large, the fluctuations of the surface are small, the Hausdorff dimension is close to two, and the surface is smooth. We calculate expectation values of the gyration radius and the specific heat using the Metropolis algorithm [141.The triangulation is not updated, only the sites are. This makes it easy to vectorize the updates by splitting the lattice into four
756
R.L. Renken, lB. Kogut
/
Crumpling transition
Fig. 2. The four sublattices used to vectorize the Metropolis algorithm. The shaded region indicates which sites cannot be updated in parallel with site 1.
sublattices as shown in fig. 2. The numbers on the sites indicate the sublattices they are assigned to. The shaded region indicates which sites cannot (for example) be updated in parallel with site 1. One test of the calculation can be made by computing the gaussian part of the action,
Sg=~(E(x~_x1)2).
(7)
Scale invariance allows an exact calculation of this for arbitrary A:
D Sg=~(N_1).
(8)
Table 1 gives the results for lattice sizes 8 x 8, 12 x 12, 16 x 16, 24 X 24 and 32 x 32 respectively. The error analysis is done using standard blocking methods. In the case of the specific heat, it is checked that the value obtained by averaging all of the data is equal (within statistics) to the average of the specific heats computed from twenty subsets of the data. Although it was necessary to iterate for millions of sweeps, we did not have the difficulties mentioned in ref. [11] for lattices of more than 400 sites. Finite-size scaling provides a simple method of extracting estimates for the critical exponents from the data. On an L x L lattice, the specific heat per site, CL(A), has a maximum at a value of the extrinsic curvature, Amax(L), which approaches the critical value, A* as L goes to infinity. A scaling hypothesis [15]
R.L. Renken, J.B. Kogut
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Crumpling transition
757
TABLE 1 Expectation values of various operators on 8 x 8, 12 x 12, 16 x 16, 24 x 24 and 32 X 32 lattices as a function of A, the coupling to the extrinsic curvature. Sg is the gaussian part of the 2 denotes the gyration radius (eq. (5)), C is the specific heat (eq. (4)), action (eq. (8)), X and S~is the extrinsic curvature part of the action (eq. (3)). The number of iterations of the Metropolis algorithm is listed in the final column A
S
2 5
C
S~
Sweeps
X 8x8
0.5 0.7 0.8 LO 1.04 1.08 1.12 1.16 1.18 1.2 1.24 1.4 2.0
94.3(3) 94.5(2) 94.5(2) 94.3(3) 94.4(2) 94.1(2) 94.6(3) 94.6(2) 94.5(2) 94.4(2) 94.9(2) 94.5(2) 96(I)
2.07(2) 2.50(1) 2.69(2) 3.03(2) 3.09(1) 3.14(1) 3.20(l) 3.24(l)
0.53(2) 0.93(3) 1.16(4) 1.70(6) 2.05(4) 2.13(4) 2.17(4) 2.10(7) 2.15(8) 2.10(6) 1.92(4) 1.49(4) 0.79(3)
3.25(I) 3.26(1) 3.31(1) 3.36(1) 3.53(4)
124.4(3)
85.6(4) 80.3(4) 76.2(4) 71.6(4) 69.5(3) 67.8(3) 64.4(2) 54.0(l)
2x 2x 3x 2x 1 x 1 x I x I x I x 1 x 1 x 1 x 5x
10~ io~ i0~ lo~ io~ ion i06 io~ i0~ i0~ i0~ i0~ io~
12 x 12 0.8 0.9 0.95 1.0 1.1
214.7(3) 2 14.4(2) 214.3(3) 214.1(3) 214.2(3)
4.66(2) 5.32(1) 5.62(1) 5.87(1) 6.26(1)
1.44(4) 1.71(4) 1.73(5) 1.68(5) 1.54(4)
224.4(6) 192.7(4) 178.2(5) 165.1(4) 143.0(3)
2 2 2 2 2
x x x x x
i0~ to5 io~ io~ iO~’
().60(l) 1.37(2) 1.79(5) 1.71(5) 1.83(4) 1.41(3) 1.32(3) 1.14(2) 0.94(2)
580.6(7) 445.3(7) 372.9(9) 340.1(8) 310(1) 260.3(7) 228.7(5) 116.0(3) 106.6(3)
2 4 4 4 4 2 2 1 5
x x x x x x x X x
i0~ 10” iO~
1.87(5) 2.15(6) 2.18(6) 2.08(7) 1.91(7)
880 (2) 779(1) 740(2) 689 (2) 643 (2)
3 6 5 5 6
x x x x x
10” ion iO” ion 10”
1.68(4) 2.03(6) 2.37(7) 2.5(2) 2.27(7) 1.49(3) 1.10(1)
1708(3) 1527 (3) 1399 (2) 1323 (4) 1247 (3) 1016(2) 755.9(8)
9x 6x 8x 1.4 X 1.1 >< 6x 7x
10”
mx 16 ((.5 (1.7 (1.8 (1.85 ().9 1.0 1.1 1.4 2.0
382.6(3) 382.4(2) 382.6(3) 382.5(2) 382.4(3) 382.4(5) 382.3(5) 382(1) 382(1)
3.36(3) 5.39(3) 7.04(4) 7.84(3) 8.63(4) 9.84(3) 10.46(2) 11.51(4) 12.18(5)
io~ iO~ to~ to~ 10” IO~
24 x 24 ((.75 0.80 0.82 ((.85 ((.88
861.9(6) 861.7(6) 862.9(4) 863.0(5) 862.9(5)
10.21(8) 13.08(8) 14.4(1) 15.9(1) 17.4(1) 32 x 32
((.7 ((.75 ((.78 0.8 0.82 ((.9 1.1
1535.9(6) 1534.4(6) 1534.8(6) 1533.9(4) 1534.4(4) 1535.4(8) 1535(1)
10.9(l) 14.8(2) 18.7(2) 21.2(3) 24.0(2) 32.4(1) 38.61(6)
io~ 10” i0~ 1(1” i06
758
R.L. Renken, lB. Kogut
/
Crumpling transition
implies CL(Amax) —~A(L”’—1) +B,
(9)
w=a/v
(10)
where
and v is the correlation length exponent. Fig. 3 is a plot of the specific heat as a function of A for L 8, 12, 16, 24, and 32. The peaks occur at decreasing values of 2 in eq. (4) are responsible A as L increases. This A-dependence and the factor of A for the decrease in CL(Amax) for small L. For each value of L, the data exhibits a unique maximum as a function of A. At the peak, there are generally several values of A in the data sets that have the same value of the specific heat. Where some choice has to be made for subsequent analysis, we choose a central value as Amax. Fig. 4 is a log—log plot of CL(Amax) versus L. Eq. (9) indicates that for large L the plotted data should approach a straight line with slope w. For L ~ 16, the data does fall on a straight line to within statistical error. A two parameter fit to these three points gives =
w=0.57(8),
(11)
C C,,.
:~.
***~~
4
U
•
* U
C
p C
—
0
0.5
I
I
I
1
1.5
2
I
2.5
Fig. 3. The specific heat as a function of the extrinsic curvature coupling, for L = 8 (marked by an L = 12 (marked by triangles), L = 16 (marked by squares), L = 24 (marked by empty circles) and L = 32 (marked by filled circles).
R.L. Renken, lB. Kogut
5
/
Crumpling transition
‘10
‘
759
20
L Fig. 4. A log—log plot of Cj(Am..) versus L.
with a 60% confidence level for the fit. If the hyperscaling relation holds, dv=2—a, then there are two equations for the two unknowns, a and a=0.44(5),
v=0.78(2).
(12) i-’.
Solving them gives (13),(14)
Note that these error bars do not include unknown systematic finite-L effects. Note also that the data are inconsistent with a first-order transition, which would give a linear increase in the peak of the specific heat as a function of L with the slope equal to a quarter of the square of the jump in the entropy [16]. The data puts an upper bound of 0.05 on the jump of the entropy. The scaling hypothesis also implies A*_Amnix(L) —‘c/L’
(15)
for large L, where t 1/i.’ (however, the constant may vanish leaving terms L —f with t> 1/i.’). This scaling law is much less straightforward to use than eq. (9) because it has three parameters (and we basically have three useful data points) and because the breadth of the peak makes it difficult to determine Amax along with an error bar in an objective fashion. We specified our selection of Amax earlier. We assign one quarter of the estimated width of the peak as the error on these estimates. Fig. 5 is a plot of these estimates of Amax(L) versus 1/Lu”’ with 1) given by eq. (14). Fitting a straight line through the last three points gives a slope =
760
R.L. Renken, J.B. Kogut
/
Crumpling transition
3 ±2 and an intercept 0.76 ±0.03 with a confidence level (given the crude error
estimates) of 80%. Note that a slope of zero is not ruled out. This would mean that c 0 in eq. (15) leaving terms Lt with t> 1/i’ (a scenario that is not uncommon in systems with periodic boundary conditions [15]), and A* would be closer to 0.8. These results are consistent with those of ref. [11] where A* was determined to be in the range from 0.7 to 0.8. Now we turn to the gyration radius, plotted in fig. 6 as a function of A for the five data sets. A log—log plot of the gyration radius versus N for A 0.80 is given in fig. 7. Eq. (6) indicates that the slope of this line should approach a constant, 2/d~. In fact, the slope is actually increasing. If this trend continues, then the slope obtained from the two points with the largest L puts an upper bound on d~: =
=
d 11~2.38(4)
(A=0.80).
(16)
We assume that 2 is a lower bound for d11. Although A 0.8 is quite close to A*, close enough that the specific heat scales for L ~ 32, it is presumably larger than A* and may not be close enough to yield the Hausdorff dimension of the critical theory. Eq. (16) is probably just an indication that A 0.8 is in the smooth phase =
=
(d~=2). In order to calculate the Hausdorff dimension reliably at the transition, it will be necessary to know A* much more accurately. A renormalization group calculation might be the best way to do this. Kantor and Nelson [17] have suggested that the
I N C
C.
I
0.00
0.02
I
I
I
004
0.06
0.08
L~’ Fig. 5. Aman versus 1/Lw” for
p
=
0.78.
I
0.10
R.L. Renken, lB. Kogut
/
Crumpling transition
761
p C. .5.
C
a.
C.,
C C. C,,
C
‘a. AC CQ
V p ‘a. 0 C
0 0
C ‘a
~
~
AA A
C
‘
C.
I
0
0.5
1
1.5
2
2.5
N 2) as a function of A. (The error bars are included, but they are hard to see at this scale.)
Fig. 6. (X
C
A V
I
50
‘‘‘‘‘‘I
100
1000
N Fig. 7. A log—log plot of (X2) versus N (including error bars).
762
R.L. Renken, lB. Kogut
_ ____
/
Crumpling transition
_
1(
/
Fig. 8. A possible block-spin scheme for regularly triangulated surfaces.
normals to the triangles are a natural candidate for an order parameter. These normals can be averaged in groups of four as shown in fig. 8. The resulting lattice is isomorphic to the original one and the scale factor is two. The normals can be averaged by adding them as vectors and then dividing by the resulting magnitude. Block averages can be computed for operators built from the angles, Os,, but information about the coordinates, X, is lost. Similar block-spin renormalization group calculations for the three-dimensional Ising model get the critical coupling to five or six decimal places [181. In the absence of a more accurate calculation of A*, statements about the Hausdorff dimension at the crumpling transition are likely to be rather speculative. This work was supported by NSF grants PHY 87-01775 and DMR 89-20538. The numerical calculations were performed using the resources of the National Center for Supercomputer Applications, the Pittsburgh Supercomputer Center, and the Ames Research Center. We thank E. Fradkin, S. Hands, and M. Staudacher for discussions.
References [1] V.A. Kazakov, Phys. Lett. B150 (1985) 282; V.A. Kazakov, 1K. Kostov and A.A. Migdal, Phys. Lett. Bl57 (1985) 295 [2] F. David, Nuci. Phys. B257 (1985) 543; J. Ambjørn, B. Durhuus, and J. Fröhlich, NucI. Phys. B257 (1985) 433 [3] A.M. Polyakov, Phys. Lett. B103 (1981) 207 [4] V.A. Kazakov and A.A. Migdal, NucI. Phys. B311 (1989) 171 [5] V.0. Knizhnik, A.M. Polyakov and A.B. Zamolodchikov, Mod. Phys. Lett. A3 (1988) 819 [6] J. Ambjørn and B. Durhuus, Phys. Lett. B188 (1987) 253
R.L. Renken, lB. Kogut [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
/
Crumpling transition
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J. Ambjørn, B. Durhuus and T. Jonsson, Phys. Rev. Lett. 58 (1987) 2619 S.M. Catterall, Phys. Lett. B220 (1989) 207 C.F. Baillie, D.A. Johnston and R.D. Williams, Caltech preprint Y. Kantor and D.R. Nelson, Phys. Rev. Lett. 58 (1987) 2774 J. Ambjørn, B. Durhuus and T. Jonsson, Niels Bohr Institute preprint number NBI-HE-88-61 H. Kleinert, Phys. Lett. B174 (1986) 335 A.M. Polyakov, NucI. Phys. B268 (1986) 406 N. Metropolis, A.W. Metropolis, M.N. Rosenbluth, A.H. Teller and E. Teller, J. Chem. Phys. 21 (1953) 1087 E. Domany, K.K. Moo, G.V. Chester and ME. Fisher, Phys. Rev. B12 (1975) 5025 C. Dasgupta and B. I. Halperin, Phys. Rev. Lett. 47 (1981) 1556 Y. Kantor and D.R. Nelson, Phys. Rev. A36 (1987) 4020 G.S. Pawley, R.H. Swendsen, D.J. Wallace and K.G. Wilson, Phys. Rev. B29 (1984) 4030
SCALING BEHAVIOR AT THE CRUMPLING TRANSITION Ray L. RENKEN and John B. KOGUT Loomis Laborato,y of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, IL 61801, USA Received 12 February 1990 (Revised 2 April 1990)
Finite-size scaling is used to study the critical behavior at the crumpling transition in three dimensions. The correlation length critical exponent, v, is estimated to be 0.78(2). The Hausdorff dimension just above the critical coupling is bounded within the range 2
The formulation of dynamically triangulated random surfaces [1,21 as a lattice regularization of the Polyakov string [31has led to progress in the theory of noncritical strings [4]. Exact solutions have been found, including those for dimensions D —2 and D 0 [1,21, and D 1 [4]. In this range of dimensions, the critical exponents obtained agree with the continuum results of Knizhnik, Polyakov and Zamolodchikov (KPZ) [51.However, in D> 1, the dynamically triangulated random surface models with gaussian action appear to be in a physically undesirable crumpled phase. Likewise, for D> 1, the continuum results of KPZ cease to be meaningful. Ambjørn and Durhuus [6,71 have suggested that a model with a non-trivial continuum limit could be constructed by including a term in the action that depends on the extrinsic curvature. Catterall [8] and others [91,working in the “microcanonical” ensemble and in three dimensions, have shown that there is a continuous phase transition at a finite value of the coupling for a particular formulation of the extrinsic curvature term. This discovery provides us with a new critical theory which, if non-trivial, could have physical relevance. From a statistical mechanical point of view, perhaps the most novel feature of the dynamically triangulated random surface models is the functional integration over geometries. In all of the models where an exact solution has been found, the corresponding models with fixed triangulation (i.e. without the functional integration over geometries) have been well understood. In particular, their critical exponents and central charges have been known. This is not the case for Catterall’s transition. Kantor and Nelson [101have studied a model identical to one studied by Catterall except that the geometry is fixed and the gaussian action is replaced with =
=
0550-3213/90/$03.50 © 1990
—
=
Elsevier Science Publishers B.V. (North-Holland)
754
R.L. Renken, lB. Kogut
/
Crumpling transition
a “tethering potential” that limits the length of the legs of the triangles. They find that a continuous phase transition separates a crumpled phase from a smooth phase. Their phase diagram is identical to Catterall’s, but presumably the critical behavior is different. Ambjørn et al. have studied a fixed triangulation model with gaussian weight and extrinsic curvature term [11]. Their results suggest that this model also has a second-order phase transition. In this paper, we obtain more detailed information about the critical behavior of this crumpling transition. To be precise, we study the action
S= —~L(X~—X~.)2—AL(1—cos6~1),
(1)
where i and j label sites on a torus with a regular triangularization (illustrated in fig. 1), the sums are over nearest neighbor pairs, the X-coordinates are three-vectors, ~ is the angle in the embedding space between the two triangles sharing the link (i, J), and A is the extrinsic curvature coupling. Scale invariance has been used to eliminate the strength of the gaussian coupling. The functional integration is over all values of the X-coordinates, except that one is held fixed in order to eliminate motion of the surface as a whole. Periodic boundary conditions are used, making the topology that of a torus. A continuum string model with extrinsic curvature was introduced by Kleinert [12] and Polyakov [131several years ago. One interesting result of these investigations was the realization that the model could be studied perturbatively when
Fig. 1. A regular triangulation.
R.L. Renken, lB. Kogut
1/A
/
Crumpling transition
755
a is small. This coupling is asymptotically free with a f3-function, a2 D f3(a)=—-—------2~ 2
(a<<1),
(2)
which indicates that the origin a 0 is an ultraviolet stable and infrared unstable point. The work of Catterall [8] and others suggests that there is another ultraviolet stable fixed point which is non-perturbative. Our paper has the more modest goal of demonstrating that there is an ultraviolet stable fixed point in the fixed connectivity model. We leave a more detailed study of Catterall’s theory to future work. We primarily study two operators that are sensitive to the difference between the crumpled phase and the smooth phase: the curvature part of the action, and the gyration radius. The curvature part of the action, =
Se=(~(1-cosOii))
(3)
is large when A is small and gets smaller as A increases. More importantly, the specific heat (up to constants)
C=A2((S~—KSe)2)
(4)
diverges at a continuous phase transition on an infinite lattice. On a finite lattice its scaling behavior gives the critical exponents. The gyration radius is (X2)
=
N(N- 1) ~ ~((xx)2)
(5)
The sum is over all N nodes of the surface and cr, is the coordination number at the site i. For a regular triangulation, o- 6. The gyration radius is interesting because its scaling behavior gives the Hausdorff dimension, dH. In the large-N limit, =
KX2(N))~N2~.
(6)
In the crumpled phase the gyration radius goes like ln N, so the Hausdorff dimension is infinite. When A is large, the fluctuations of the surface are small, the Hausdorff dimension is close to two, and the surface is smooth. We calculate expectation values of the gyration radius and the specific heat using the Metropolis algorithm [141.The triangulation is not updated, only the sites are. This makes it easy to vectorize the updates by splitting the lattice into four
756
R.L. Renken, lB. Kogut
/
Crumpling transition
Fig. 2. The four sublattices used to vectorize the Metropolis algorithm. The shaded region indicates which sites cannot be updated in parallel with site 1.
sublattices as shown in fig. 2. The numbers on the sites indicate the sublattices they are assigned to. The shaded region indicates which sites cannot (for example) be updated in parallel with site 1. One test of the calculation can be made by computing the gaussian part of the action,
Sg=~(E(x~_x1)2).
(7)
Scale invariance allows an exact calculation of this for arbitrary A:
D Sg=~(N_1).
(8)
Table 1 gives the results for lattice sizes 8 x 8, 12 x 12, 16 x 16, 24 X 24 and 32 x 32 respectively. The error analysis is done using standard blocking methods. In the case of the specific heat, it is checked that the value obtained by averaging all of the data is equal (within statistics) to the average of the specific heats computed from twenty subsets of the data. Although it was necessary to iterate for millions of sweeps, we did not have the difficulties mentioned in ref. [11] for lattices of more than 400 sites. Finite-size scaling provides a simple method of extracting estimates for the critical exponents from the data. On an L x L lattice, the specific heat per site, CL(A), has a maximum at a value of the extrinsic curvature, Amax(L), which approaches the critical value, A* as L goes to infinity. A scaling hypothesis [15]
R.L. Renken, J.B. Kogut
/
Crumpling transition
757
TABLE 1 Expectation values of various operators on 8 x 8, 12 x 12, 16 x 16, 24 x 24 and 32 X 32 lattices as a function of A, the coupling to the extrinsic curvature. Sg is the gaussian part of the 2 denotes the gyration radius (eq. (5)), C is the specific heat (eq. (4)), action (eq. (8)), X and S~is the extrinsic curvature part of the action (eq. (3)). The number of iterations of the Metropolis algorithm is listed in the final column A
S
2 5
C
S~
Sweeps
X 8x8
0.5 0.7 0.8 LO 1.04 1.08 1.12 1.16 1.18 1.2 1.24 1.4 2.0
94.3(3) 94.5(2) 94.5(2) 94.3(3) 94.4(2) 94.1(2) 94.6(3) 94.6(2) 94.5(2) 94.4(2) 94.9(2) 94.5(2) 96(I)
2.07(2) 2.50(1) 2.69(2) 3.03(2) 3.09(1) 3.14(1) 3.20(l) 3.24(l)
0.53(2) 0.93(3) 1.16(4) 1.70(6) 2.05(4) 2.13(4) 2.17(4) 2.10(7) 2.15(8) 2.10(6) 1.92(4) 1.49(4) 0.79(3)
3.25(I) 3.26(1) 3.31(1) 3.36(1) 3.53(4)
124.4(3)
85.6(4) 80.3(4) 76.2(4) 71.6(4) 69.5(3) 67.8(3) 64.4(2) 54.0(l)
2x 2x 3x 2x 1 x 1 x I x I x I x 1 x 1 x 1 x 5x
10~ io~ i0~ lo~ io~ ion i06 io~ i0~ i0~ i0~ i0~ io~
12 x 12 0.8 0.9 0.95 1.0 1.1
214.7(3) 2 14.4(2) 214.3(3) 214.1(3) 214.2(3)
4.66(2) 5.32(1) 5.62(1) 5.87(1) 6.26(1)
1.44(4) 1.71(4) 1.73(5) 1.68(5) 1.54(4)
224.4(6) 192.7(4) 178.2(5) 165.1(4) 143.0(3)
2 2 2 2 2
x x x x x
i0~ to5 io~ io~ iO~’
().60(l) 1.37(2) 1.79(5) 1.71(5) 1.83(4) 1.41(3) 1.32(3) 1.14(2) 0.94(2)
580.6(7) 445.3(7) 372.9(9) 340.1(8) 310(1) 260.3(7) 228.7(5) 116.0(3) 106.6(3)
2 4 4 4 4 2 2 1 5
x x x x x x x X x
i0~ 10” iO~
1.87(5) 2.15(6) 2.18(6) 2.08(7) 1.91(7)
880 (2) 779(1) 740(2) 689 (2) 643 (2)
3 6 5 5 6
x x x x x
10” ion iO” ion 10”
1.68(4) 2.03(6) 2.37(7) 2.5(2) 2.27(7) 1.49(3) 1.10(1)
1708(3) 1527 (3) 1399 (2) 1323 (4) 1247 (3) 1016(2) 755.9(8)
9x 6x 8x 1.4 X 1.1 >< 6x 7x
10”
mx 16 ((.5 (1.7 (1.8 (1.85 ().9 1.0 1.1 1.4 2.0
382.6(3) 382.4(2) 382.6(3) 382.5(2) 382.4(3) 382.4(5) 382.3(5) 382(1) 382(1)
3.36(3) 5.39(3) 7.04(4) 7.84(3) 8.63(4) 9.84(3) 10.46(2) 11.51(4) 12.18(5)
io~ iO~ to~ to~ 10” IO~
24 x 24 ((.75 0.80 0.82 ((.85 ((.88
861.9(6) 861.7(6) 862.9(4) 863.0(5) 862.9(5)
10.21(8) 13.08(8) 14.4(1) 15.9(1) 17.4(1) 32 x 32
((.7 ((.75 ((.78 0.8 0.82 ((.9 1.1
1535.9(6) 1534.4(6) 1534.8(6) 1533.9(4) 1534.4(4) 1535.4(8) 1535(1)
10.9(l) 14.8(2) 18.7(2) 21.2(3) 24.0(2) 32.4(1) 38.61(6)
io~ 10” i0~ 1(1” i06
758
R.L. Renken, lB. Kogut
/
Crumpling transition
implies CL(Amax) —~A(L”’—1) +B,
(9)
w=a/v
(10)
where
and v is the correlation length exponent. Fig. 3 is a plot of the specific heat as a function of A for L 8, 12, 16, 24, and 32. The peaks occur at decreasing values of 2 in eq. (4) are responsible A as L increases. This A-dependence and the factor of A for the decrease in CL(Amax) for small L. For each value of L, the data exhibits a unique maximum as a function of A. At the peak, there are generally several values of A in the data sets that have the same value of the specific heat. Where some choice has to be made for subsequent analysis, we choose a central value as Amax. Fig. 4 is a log—log plot of CL(Amax) versus L. Eq. (9) indicates that for large L the plotted data should approach a straight line with slope w. For L ~ 16, the data does fall on a straight line to within statistical error. A two parameter fit to these three points gives =
w=0.57(8),
(11)
C C,,.
:~.
***~~
4
U
•
* U
C
p C
—
0
0.5
I
I
I
1
1.5
2
I
2.5
Fig. 3. The specific heat as a function of the extrinsic curvature coupling, for L = 8 (marked by an L = 12 (marked by triangles), L = 16 (marked by squares), L = 24 (marked by empty circles) and L = 32 (marked by filled circles).
R.L. Renken, lB. Kogut
5
/
Crumpling transition
‘10
‘
759
20
L Fig. 4. A log—log plot of Cj(Am..) versus L.
with a 60% confidence level for the fit. If the hyperscaling relation holds, dv=2—a, then there are two equations for the two unknowns, a and a=0.44(5),
v=0.78(2).
(12) i-’.
Solving them gives (13),(14)
Note that these error bars do not include unknown systematic finite-L effects. Note also that the data are inconsistent with a first-order transition, which would give a linear increase in the peak of the specific heat as a function of L with the slope equal to a quarter of the square of the jump in the entropy [16]. The data puts an upper bound of 0.05 on the jump of the entropy. The scaling hypothesis also implies A*_Amnix(L) —‘c/L’
(15)
for large L, where t 1/i.’ (however, the constant may vanish leaving terms L —f with t> 1/i.’). This scaling law is much less straightforward to use than eq. (9) because it has three parameters (and we basically have three useful data points) and because the breadth of the peak makes it difficult to determine Amax along with an error bar in an objective fashion. We specified our selection of Amax earlier. We assign one quarter of the estimated width of the peak as the error on these estimates. Fig. 5 is a plot of these estimates of Amax(L) versus 1/Lu”’ with 1) given by eq. (14). Fitting a straight line through the last three points gives a slope =
760
R.L. Renken, J.B. Kogut
/
Crumpling transition
3 ±2 and an intercept 0.76 ±0.03 with a confidence level (given the crude error
estimates) of 80%. Note that a slope of zero is not ruled out. This would mean that c 0 in eq. (15) leaving terms Lt with t> 1/i’ (a scenario that is not uncommon in systems with periodic boundary conditions [15]), and A* would be closer to 0.8. These results are consistent with those of ref. [11] where A* was determined to be in the range from 0.7 to 0.8. Now we turn to the gyration radius, plotted in fig. 6 as a function of A for the five data sets. A log—log plot of the gyration radius versus N for A 0.80 is given in fig. 7. Eq. (6) indicates that the slope of this line should approach a constant, 2/d~. In fact, the slope is actually increasing. If this trend continues, then the slope obtained from the two points with the largest L puts an upper bound on d~: =
=
d 11~2.38(4)
(A=0.80).
(16)
We assume that 2 is a lower bound for d11. Although A 0.8 is quite close to A*, close enough that the specific heat scales for L ~ 32, it is presumably larger than A* and may not be close enough to yield the Hausdorff dimension of the critical theory. Eq. (16) is probably just an indication that A 0.8 is in the smooth phase =
=
(d~=2). In order to calculate the Hausdorff dimension reliably at the transition, it will be necessary to know A* much more accurately. A renormalization group calculation might be the best way to do this. Kantor and Nelson [17] have suggested that the
I N C
C.
I
0.00
0.02
I
I
I
004
0.06
0.08
L~’ Fig. 5. Aman versus 1/Lw” for
p
=
0.78.
I
0.10
R.L. Renken, lB. Kogut
/
Crumpling transition
761
p C. .5.
C
a.
C.,
C C. C,,
C
‘a. AC CQ
V p ‘a. 0 C
0 0
C ‘a
~
~
AA A
C
‘
C.
I
0
0.5
1
1.5
2
2.5
N 2) as a function of A. (The error bars are included, but they are hard to see at this scale.)
Fig. 6. (X
C
A V
I
50
‘‘‘‘‘‘I
100
1000
N Fig. 7. A log—log plot of (X2) versus N (including error bars).
762
R.L. Renken, lB. Kogut
_ ____
/
Crumpling transition
_
1(
/
Fig. 8. A possible block-spin scheme for regularly triangulated surfaces.
normals to the triangles are a natural candidate for an order parameter. These normals can be averaged in groups of four as shown in fig. 8. The resulting lattice is isomorphic to the original one and the scale factor is two. The normals can be averaged by adding them as vectors and then dividing by the resulting magnitude. Block averages can be computed for operators built from the angles, Os,, but information about the coordinates, X, is lost. Similar block-spin renormalization group calculations for the three-dimensional Ising model get the critical coupling to five or six decimal places [181. In the absence of a more accurate calculation of A*, statements about the Hausdorff dimension at the crumpling transition are likely to be rather speculative. This work was supported by NSF grants PHY 87-01775 and DMR 89-20538. The numerical calculations were performed using the resources of the National Center for Supercomputer Applications, the Pittsburgh Supercomputer Center, and the Ames Research Center. We thank E. Fradkin, S. Hands, and M. Staudacher for discussions.
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R.L. Renken, lB. Kogut [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
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Crumpling transition
763
J. Ambjørn, B. Durhuus and T. Jonsson, Phys. Rev. Lett. 58 (1987) 2619 S.M. Catterall, Phys. Lett. B220 (1989) 207 C.F. Baillie, D.A. Johnston and R.D. Williams, Caltech preprint Y. Kantor and D.R. Nelson, Phys. Rev. Lett. 58 (1987) 2774 J. Ambjørn, B. Durhuus and T. Jonsson, Niels Bohr Institute preprint number NBI-HE-88-61 H. Kleinert, Phys. Lett. B174 (1986) 335 A.M. Polyakov, NucI. Phys. B268 (1986) 406 N. Metropolis, A.W. Metropolis, M.N. Rosenbluth, A.H. Teller and E. Teller, J. Chem. Phys. 21 (1953) 1087 E. Domany, K.K. Moo, G.V. Chester and ME. Fisher, Phys. Rev. B12 (1975) 5025 C. Dasgupta and B. I. Halperin, Phys. Rev. Lett. 47 (1981) 1556 Y. Kantor and D.R. Nelson, Phys. Rev. A36 (1987) 4020 G.S. Pawley, R.H. Swendsen, D.J. Wallace and K.G. Wilson, Phys. Rev. B29 (1984) 4030