The crumpling transition in the presence of quantum gravity

Nuclear Physics B354 (1991) 328-338 North-Holland

THE CRUMPLING TRANSITION IN THE PRESENCE OF QUANTUM GRAVITY Ray L . RENKEN and John B . KOGUT Lootnis Laboratory of Physic.%, University of 111inots at Urbana-Champaign, 1110 West Green Street, Urbana, IL 61801, USA Received I October 1990 (Revised 12 November 1990) Critical exponents of the crumpling transition on a dynamically triangulated random surface embedded in three dimensions are calculated. The results are that a = 0.24(5) and vd = 1 .76(5).

Polyakov's formulation of the bosonic string [1], T S= _ 2

f d2t7Vgabd"X~`dnxw M

(1)

involves two independent fields: the X-coordinates of the surface in the embedding space, and the metric on the world-sheet, h. In the euclidean path integral used to define the quantization of this theory, both the coordinates in the embedding space and the metric of the world-sheet are integrated over . A discrete formulation of this theory [2,3] replaces the worldsheet manifold, M, with a triangulation, T, specified by N points and an adjacency matrix, S,l(T ), which is one if points i and j are nearest neighbor and zero otherwise. This adjacency matrix is reminiscent of the world-sheet metric in the sense that it tells which points are nearby and which are not. In fact, in the discrete formulation, the functional integration over worldsheet metrics is replaced with a sum over triangulations. Polyakov's action is replaced with

In aiâ ûimensions where there is an exact solution to both the continuum and discrete models, they yield identical results [4]. This is the main justification for viewing the dynamically triangulated random surfaces as a discretization of the Polyakov string. 0550-3213/91/5()3 .50?) 1991 - Elsevier Science Publishers B.V . (North-Holland)

R.L. Reitken . J.B. Koguf / Crumpling transition

329

One of the most basic properties of the lattice string that can be determined is its Hausdorff dimension . The gyration radius 1 ~X=>=N(N-1)

Q, o,

36

((X,-X,

)2)

(3)

(where o,,, is the coordination number of site i) is related to the Hausdorff dimension via ( X =i -N =/`1"

(4)

at large N. In a positive number of dimensions, the model described above actually has a gyration radius with the behavior [5J (X = >-InN If this equation is forced into the form of eq . (4), the result is that the Hausdorff dimension is very large. Physically, this means that quantum fluctuations have modified the classical surface into a collapsed spiky object . This theory is interesting in its own right, but this paper will focus on another theory which has a finite value of the Hausdorff dimension . A simple way to discourage the surface from collapsing [6,7) is to add a term to the action that makes the surface pay a penalty for the production of spikes . Such a term is S,= -A Y~ (1 - cosOj, CU)

(6)

where A is the coupling, the sum is over nearest neighbors, and 6,J is the angle between the two triangles sharing the link between the two neighbors i and j (see fig. 1). A three-dimensional embedding space has been assumed, since that is the space which will be used throughout this paper. Eq. (6) is essentially an extrinsic curvature term . At large enough values of A, the surface is forced to become smooth and the Hausdorff dimension is near two. There are two distinct phases : a crumpled phase at small A and a smooth phase for large A. There is no exact

7 Fig. 1 . An illustration of the angle. B, appearing in eq . (6).

330

R.L . Renken, J.B. Kogut / Crumpling transition

solution of this model, but refs . [6,7] and the KPZ picture suggest that these two phases are separated by a continuous phase transition . This phase transition is very interesting because it potentially defines a new continuum theory with a finite Hausdorff dimension . In this paper we apply finite-size scaling to the transition identified in refs . (6,7] in order to obtain more detailed information about it. The Polyakov string is a theory of free bosons in the presence of two-dimensional quantum gravity . Quantum gravity can be turned off by fixing the world-sheet metric. rather than summing over it . In the discrete formulation this corresponds to fixing the triangulation instead of summing over it. In order to get a flat background metric, a regular triangulation should be chosen . Knizhnik, Polyakov and Zamolodchikov (KPZ) [8,9] have found that the scaling dimensions of a theory in the presence of quantum gravity can be related to the scaling dimensions of the underlying theory where quantum gravity is turned off. In other words, quantum gravity "dresses" the operators of the underlying theory. The only parameter needed to specify the relationship between the two sets of scaling dimensions is the central charge of the underlying theory . Unfortunately, the KPZ work is only applicable when the central charge is less than one. Of course, it is not difficult to understand the Polyakov string when quantum gravity is switched off since the resulting theory is just free bosons. Things are more difficult in the case of the crumpling transition . A number of authors have worked on the crumpling transition in the absence of quantum gravity [10-14]. The picture that has emerged is that this transition is also continuous and probably has a central charge less than one . If this is true, the KPZ work implies that the crumpling transition in the presence of quantum gravity is second order and that a precise study of the scaling dimensions and central charge of the theory without gravity, combined with the KPZ results, would yield the scaling dimensions of the crumpling transition in the presence of quantum gravity . However, work on the theory without gravity has not yet been of a quality to make this route precise . This paper considers the more direct route of studying the full theory itself. It should be mentioned that there is actually one more term that is included in the action . The integration measure over the position field should have the form [s] ®X =

,t-1/2 dD X . U

(7)

The coordination number part of this measure can be absorbed into the action as a term S,=

+ 2 Ein

0-' .

(8)

R. L. Renken, J. B. Kogut / Crumpkng transition

331

Fig. 2. The flip used in updating the triangulation .

Actually, a valid can take on any of a. a = D = 3 The model is

lattice model is obtained with a parameter a, instead of D, which value. The physics of the model does not vary over a broad range is used throughout this paper . simulated using the Metropolis algorithm. The X-coordinates are updated by cycling through each component of each vertex one by one and proposing a new value . The new value is obtained from the old one by adding to it a gaussian random number . This new value is then either accepted or rejected according to the standard algorithm. The acceptance rate for the updates of the X-coordinates is adjusted by varying the width of the gaussian distribution from which the gaussian random numbers are chosen. After updating all of the vertices

once, attention is then turned to the triangulation. It can be proved that any triangulation can be reached from any other triangulation by a sequence of flips of the type illustrated in fig. 2 [5j. These flips are implemented by choosing links at random and proposing the flipped link as the candidate for updating in the Metropolis scheme . This is repeated until the number of links that have been randomly chosen is equal to the number of links in the triangulation . Attention is then returned to the X-coordinates . The existence of two independent fields (in this case the coordinates and the triangulation) complicates the Metropolis scheme a bit . Since the fields are updated in alternation, when one field is being updated it is really being simulated in a background field caused by the other. If both fields are evolving slowly, it is conceivable that equilibrium can be maintained. However, if one field is evolving much more quickly than the other, equilibrium might never be achieved, no matter how long the simulation is run, because the slow field cannot "catch up" with the faster one. It is therefore important to study whether or not the fields are in equilibrium . One parameter available to influence the comparative rate of evolution of the two fields is the acceptance rate of the coordinate updates. The acceptance rate of the link updates cannot be altered because there is no way to propose small flips - a flip is either accepted or it is not . The coordinate sector can be monitored by calculating the expectation value of the gaussian part of the

332

R.L. Renken, J.B. Kogw / Ownpling rransit-, Tnni.e 1 The gyration radius as a function of lattice size and extrinsic curvature coupling

A

N=36

0.0 0.5 1 .0 1 .2 L225 1,25 1,275 1 .3 1 .325 1 .35 1 .375 1 .4 1 .45 1 .5 1 .6 2.0 2.5 3.0

1 .38(1) 1 .72(I) 2.38(3)

(X => N=72 1.71(1) 2.22(3) 3.29(6) 4.14(5)

2.79(1) 2.86(1) 2.89(1) 2.950) 3.0l(1) 3.080) 3.08(1) 3.11(1)

N= 144

N=288

2.11(3) 2.83(6) 4.21(6) 5.67(8)

2.48(3) 3.6(l)

7.50(9)

11 .20

5.6(3)

4 .75(5) 5.12(5) 5.34(5) 5.87(4) 6.12(4) 6.50(5) 6.53(4) 6.53(4)

3.21(I) 3.280) 3.33(3) 3.39(3) 3.38(2)

8.9(I) 9.7(1) 10.70)

13.1(3) 15.30)

12.49(8) 12 .63(9) 12.7(I)

23 .9(I) 24.7(I) 25A(1)

action . Scale invariance allows this to be computed exactly: ((
X,-X,)2

)

=D(N- 1)

(9)

(D = 3). If the numerical simulation gives the wrong answer, then the coordinates are not keeping up with the triangulations . The only way of checking whether the triangulations are keeping up with the coordinates is to sweep through the triangulations more than once for each sweep through the coordinates and to see if the results are altered. Ref. [61 made systematic checks of this algorithm . In the work reported here, the acceptance rates for the coordinates were kept roughly equal to that for the triangulations . This is smaller than the acceptance rate (50%) recommended by ref. [61 and therefore on the safe side . Still, a study of the systematics on large lattices near the transition has not been made and would be welcome. The results of the simulations are displayed in the tables . The gyration radius is given in table 1. Four lattices sizes were studied (all with the topology of a sphere), N = 36, N _ 72, N = 144 and N = 288. Eq . (4) relates the behavior of the gyration radius to the I-Iausdorff dimension . It is instructive to make a log-log plot of the gyration radius versus the number of points in the triangulation for values of the coupling to the extrinsic curvature on both sides of the transition . Fig. 3 is such a

R.L . Renken. J. B Kogut / Crumpling transmon

333

N Fig. 3. A log-log plot of the gyration radius, (X =), versus the number of points in the triangulation, N. Two values of the coupling are used: A = 0 (triangles) and A = 3 (circles). The tine corresponds to il l = 2 and is just shown for comparison .

plot for A = 0 and A = 3 . At the smaller value of the coupling, the data is fairly horizontal, indicating that the 1-Iausdorff dimension is large and that the surface is crumpled. At the large value of the coupling, the data quickly approaches a d i , = 2 line, indicating that the surface is smooth . Table 2 displays the average value of the largest coordination number of the triangulation, Nn a . This quantity is interesting because it probes the internal geometry of the surface . Nm,, is large when the coupling to the extrinsic curvature is small and small when the coupling to the extrinsic curvature is large. This means that the extrinsic curvature term in the action not only smooths the external geometry of the surface (which it was designed to do), but it also makes the internal geometry of the surface more regular . Nothing more will be said about internal geometry in this paper. Table 3 lists the values of the specific heat . This data is plotted in fig . 4 . (Table 4 gives the number of sweeps used in getting this data, as well as the data in tables 1 and 2 .) The specific heat data is useful because information about the critical exponents of the transition can be extracted from it . The standard scaling hypothesis is [15]

Ci,,_( L) -AL' + B,

(10)

where Cmax(L) is the peak in the specific heat as a function of A for the given value of L, L is the linear dimension of the system, w = al v, and v is the

R.L. Renken, J.B. Kogut / Crtrnrpling rrarcsinon

334

TABLE 2 The average value of the largest coordination number as a function of lattice size and extrinsic curvature coupling

A

N= 36

0.0 0.5 1 .0 1 .2 1 .225 1 .25 1.275 1 .3 1 .325 1 .35 1 .375 1 .4 1 .45 1 .5 1 .6 2.0 2.5 3.0

13 .33(3) 12.82(3) 11 .72(3) 10.89(1) 10.74(1) 10.61(1) 10.47(1) 1(1.33(1) 10.20(1) 10 .07(2) 9.95(2) 9.53(2) 9.30(1) 9.02(2) 8.99(1) 9.01(1)

Nma, N = 72

iJ = 144

N=288

16.07(4) 15.32(4) 13.98(5) 13 .11(7)

18 .49(5) 17 .53(6) 16 .08(3) 15 .(12(9)

20.86(5)

11 .96(4)

13.88(4)

15 .41(2)

11 .61(4)

13.10(5) 12 .53(6) 11 .94(7)

14.76(4) 14.09(4)

10 .64(1) 10 .62(2) 10 .63(2)

11 .28(2) 11 .20(1) 11 .19(1)

19.62(7) 17.8(1)

12 .43(3)

10.77(4) 10.36(5) 991(1) 9.91(1) 290(1)

TABLL 3 The specific heat as a function of lattice size andcoupling to the extrinsic curvature

A

N = 36

0.0 0.5 1 .0 1 .2 1 .225 1 .25

0 0.356(6) 1 .82(5)

1.275 1.3 1.325 1.35 1 .375 1 .4 1 .45 1 .5 1.6 2.0 2.5 3.0

3.37(3) 3.57(3) 3.65(3) 3.72(3) 3.67(3) 3.68(3) 3.57(3) 3.48(5) 2.56(4) 1 .93(3) 1 .04(4) 0.85(2) 0.79(2)

C N=72

N= 144

N= 288

0 0.378(7) 1 .87(4) 3.3(2)

(1 0.373(9) 1 .81(5) 3.2(1)

(1 0.389(9) 1 .88(_5)

4.7(1)

4 .98(9)

5.12(9)

4.53(8)

5.3(1) 4.8(2) 3 .8(1)

5.7(2) 5.18(8)

0.97(3) 0.76(2) 0.70(2)

0.99(2) 0.77(1) 0.70(1)

4.57(8)

3.3(1) 2.14(7) 0.99(2) 0.80(2) (1.73(1)

R .L . Renken, JA Kogut / Crtimpling tranution

335

0

0

U vi -

0

a

O 0

Fib. 4.

The specific heat . C, versus the coupling to the extrinsic curvature on lattices with N=36 (crosses). N= 72 (triangle~). N = 144 (squares) and N = 288 (circles) .

TTUI r 4

The number of sweeps used to generate the data in tables 1-3 A 0.0 0.5 1 .0 1 .2 1 .225 1 .25 1 .275 1.3 1.325 1.35 1.375 1 .4 1 .45 1 .5 1 .6 2.0 2.5 3.0

N=36 6x l0 ; 6x10' 6X10' 1 .6 X 10`' 1 .6 X 1(16 1 .6 x 101, 1 .6X10" 1 .6 x 10`' 1 .6 X 10" 1.6 X 10" 8X 10' 8 X 10 5 8x10 5 6X10 4 6X10' 6 x10'

sweeps

N=72

N= 144

N=288

6x 10' 6x10' 6x10' IXIU`'

8X 1(l' 8XI(l' 8xl0' I x10`'

1x10 5

X 10'

4 X 10'

2x IO`' 2 X lon 2 X I(1°

4X10' 4x 106

8x10' 8xl0' 8X104

2X10 5 2X10' 2X10 5

IX10 ,

1X105

1X1(1`' 1

X l06

1 x 1(1`' I X 10`' )x10`' 6X10' 6X111' 1 .2x10,

2

R.L . Renken, J.B. Kogut / Crumpling transition

336

z

X R

0

Fig . 5 . A log-log plot of the peak of C versus N.

correlation length exponent . The scaling hypothesis is obtained by reasoning that a finite system at criticality cannot have a correlation length larger than its linear dimension. This scaling hypothesis does not immediately apply to the crumpling transition in the presence of quantum gravity because there is no linear dimension L. Only the number of points, N, in the triangulation is known. We must assume that there is some linear scale determined by N, L - N"`t,

(11)

where naively d = 2. This is too naive though because the system has some internal Hausdorff dimension that has not yet been measured (as far as we know) and could differ from two. d is therefore best treated as an unknown internal Hausdorff dimension that acts much like a new critical exponent . The resulting scaling hypothesis is where

Cmax(N) -A'N - +B',

(12)

co' = a/vd .

(13)

Fig. 5 is a log-log plot of C,,,_ versus N. For large N, this plot should approach a straight line with slope tu' . Scaling appears to set in for remarkably small values of N. A fit to the last three points gives w'= 0.14(3)

(14)

R.L. Renken, J.ß. Kogut / Crumpling lransitton

337

Tnui .r 5 A comparison of the critical exponents for the king model and the crumpling transitions in the presence of quantum gravity. The string susceptibility is notyet known for the crumpling transition. Critical exponents for the crumpling transition without quantum gravity are also included Exponent

king + gravity

Crumpling + gravity

Crumpling

- 1 3 a

0.240) 1 .76(5)

0.44(5) 1 .560)

with a confidence level of 60% for the fit. If the hyperscaling relation vd=2-cr

(15)

is assumed, further progress can be made. In every model that has been exactly solved on a dynamically triangulated random surface, the hyperscaling relation is satisfied [4]. For instance [16], the Ising model on a dynamically triangulated random surface has a = - I and vd = 3. Notice that the authors of ref. [16] find themselves in exactly the same situation as here : the correlation length exponent, v, cannot be extracted by itself, but only in combination with the unknown internal Hausdorff dimension d. For the crumpling transition, the hyperscaling relation implies a = 0.24(5),

(16)

vd = 1 .76(5) .

(17)

The Ising transition and crumpling transition results are gathered together in table 5 for comparison . It is worth noting that there are important properties of the crumpling transition in the presence of quantum gravity that have not yet been measured. Obviously, it would be interesting to have the internal Hausdorff dimension, d, so that v could be determined . Another exponent of great interest, the string susceptibility y,«, is defined by the large N behavior of the partition function ZN -A NNY.~~ - ; ,

(18)

For the Ising model, yur = 4/3. No measurement of this has been made for the crumpling transition . It would also be interesting to repeat the careful calculations of this paper for the crumpling transition of a surface embedded in a higher number of dimensions. Some work [17] suggests that the crumpling transition persists at higher D and

33 8

R.L. Renken, Jß . Kogut / Crumpling transition

continues to be continuous. Presumably the critical exponents vary with D and it would be interesting to know what they are . Work at smaller D, D = 2, indicates that at smaller D the crumpling transition continues to exist (at least in the absence of quantum gravity) but that it is first order [18]. This work was supported by NSF grant PHY 87-01775 . The numerical calculations were performed using the resources of the National Center for Supercomputer Applications, the Pittsburgh Supercomputer Center, and the Ames Research Center. Also, we acknowledge the National Science Foundation support through the Materials Research Laboratory at the University of Illinois, UrbanaChampaign, grant NSF-DMR-20538 . We thank S . Catterall and M . Staudacher for discussions . References [11 A .M. Polyakov, Phys. Lett . 131113 (1981) 207 [2) V .A. Kazakov, Phys . Lett. 13150 (1985) 282; V .A . Kazakov, I .K. Kostov and A .A. Migdal, Phys. Lett . 13157 (1985) 295 [3] F. David, Nucl . Phys. B257 (1985) 543 ; J . Ambjorn, B. Durhuus and J . Fröhlich, Nucl . Phys. B257 (1985) 333 [4] V .A . Kazakov and A .A . Migdal . Nucl . Phys. B31 1 (1989) 171 [5] B .V. Boulatov, V .A . Kazakov, I .K . Kostov and A .A . Migdal, Nuel . Phys. 13275 (1986) 641 [6] S .M . Catterall, Phys . Lett . 13220 (1989) 2(17 [7] C .F . Baillie, D .A . Johnston and R .D . Williams, Nucl. Phys . 13335 (199()) 469 [8] V.G. Knizhnik, A .M . Polyakov and A .B. Zamolodchikov. Mod . Phys. Lett . A3 (1988) 819 [91 J . Distler and H . Kawai, Nucl . Phys . B321 (1989) 509 [10] Y. Kantor and D.R . Nelson, Phys . Rev . Lett. 58 (1987) 2774 [H] J . Ambjorn, B . Durhuus and T. Jonsson, Niels Bohr Institute preprint NBI-IIE-88-61 [121 R . Renken and J . Kogut. Nucl. Phys . B342 (1990) 753 [131 R . Renken and J. Kogut, Nucl. Phys . 13348 (1991) 580 [141 F. David, Nucl . Phys. B (Froc. Suppl .) 17 (1990) 51 [15] E . Domany, K .K. Mon, G .V . Chester and M .E. Fisher, Phys . Rev. 1312 (1975) 51125 [16] D.V . Boulatov and V .A . Kazakov, Phys. Lett . 13186 (1987) 379 [17] C .F . Baillie, R.D. Williams, S.M . Catterall and D.A . Johnston, Phys . Lett. B243 (1990) 358; Nucl . Phys. 13348 (1991) 543 [181 R. Renken and J . Kogut, Nucl . Phys . B.î5() (1991) 554