Ising spins on a dynamically triangulated random surface

Volume 213, number 4

PHYSICSLETTERSB

3 November 1988

I S I N G S P I N S ON A DYNAMICALLY TRIANGULATED R A N D O M SURFACE J. JURKIEWICZ a.L, A. KRZYWICKI a, B. PETERSSON b and B. SODERBERG b a Laboratoires de Physique Th~orique 2, B~timent 211, Universit~ de Paris-Sud, F-91405 Orsay, France b FakultiitJ~r Physik, Universittit Bielefeld, D-4800 Bielefeld, Fed. Rep. Germany

Received 24 June 1988

We study a model of random surfacesendowedwith fermionicdegreesof freedom. The critical parameters are estimated using a Monte Carlo simulation method, for the dimensionalitydofthe embeddingspace rangingfrom d=0 to d= 10.

The present paper belongs to a series devoted to a numerical study of dynamically triangulated random surfaces [ 1-3 ]. This time we assume that there are Ising spins living on the vertices of the surface. It is known [4] that Ising spins are in this context equivalent to Majorana fermions. Hence this work is an attempt to extend the previous investigations to surfaces endowed with fermionic degrees of freedom. The importance of studying fermionic surfaces does not need to be advertized, especially in view of apparently intrinsic deceases of purely bosonic surfaces. In most of this work we are dealing with only one species of spins (Maj orana fermions). Towards the end we shall comment briefly on the multi-spin case. A caveat is in order. We assume the surface is embedded in a d-dimensional euclidean space. The "string critical exponents" measured for the purely bosonic model change smoothly from values typical for the Liouville phase to those characteristic for the branched polymer phase when d becomes large enough. As discussed in ref. [ 5 ], it is unclear whether this smooth behaviour is a physical phenomenon or an experimental artifact. There exist theoretical arguments [6,7] pointing towards the existence of a phase transition at d = 1. However, it is not known whether the phase at d > 1 is the branched polymer phase or whether there exists an intermediate scaling t Permanent address: Institute of Physics, Jagellonian University, PL-30059Cracow,Poland. 2 Laboratoireassoci6au CNRS. 0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )

phase, as suggested if the data are taken literally (see also refs. [ 8,9 ] ). As we shall see the present version of the surface with fermions on it looks, in this respect, very similar to the purely bosonic case. Of course, our determination of the critical exponents of the model is fully significant under the assumption that there exists a non-trivial phase at intermediate d. For d = 0 the model can be solved exactly [ 10]. The same critical exponents have been found in the corresponding continuum theory by the authors of ref. [7] (incidentally, this indicates that the summation over triangulations does indeed correctly represent the summation over metrics). The analytic method of ref. [7] fails, however, for d>~ 1, where a "strongly coupled gravity" regime sets in. We wish to study this regime numerically. However, we first simulate on the computer the solvable d = 0 case in order to check and gauge the tools used to extract the critical parameters of the model. Armed with this experience we extend the study to d > 0. We limit ourselves to a micro-canonical simulation. In other words, the statistical averages are taken keeping the number of vertices of the surface (denoted N) constant. We limit to a strict minimum the presentation of the model (cf. ref. [5] for a recent review and for a list of references). The partition function is

× exp(<~> [Xama,, - (Xm --X,~)2] ) •

(1) 511

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Here T refers to a specific triangulation and W ( T ) is the corresponding symmetry factor, qj ( j = 1, ..., N) is the number of the nearest neighbours of the jth vertex of the surface, xj is its embedding space coordinate and a s ( = + 1 ) is the Ising spin living on it. As discussed in ref. [ 5 ], the parameter o~ controls the fluctuations of the geometry. We set a = ½d. Finally, x denotes the coupling of the Ising spins. We assume the surface to have the topology of a sphere. The center of gravity of the surface is kept fixed to avoid a zero mode in the integration over the embedding space. The method used to update the triangulations is the one originally proposed in ref. [ 11 ] and already used by us in ref. [ 1 ]; the links of the lattice are selected at random and "flipped", if the "flip" is accepted by an appropriate Metropolis algorithm. When the number of attempted flips equals the total number of links we say that a "sweep" of the lattice has been completed (although some links have been flipped several times while some others remain, in general, untouched). The spins and the x variables (for d > 0) are updated, using the heat-bath algorithm, once per sweep of the lattice (we have convinced ourselves ~hat this is sufficient). We initially heat the system carrying out 2000-4000 sweeps without measuring anything. Then, the gyration radius, the magnetization and the internal energy of Ising spins are measured once per sweep. After 4096 such production sweeps (one "experiment"), the results of the performed measurements are used to find estimates of observables we are interested in. For a given choice of model parameters we need 16-64 experiments (depending on the accuracy sought) to get reliable estimates of the average values of relevant observables and of the corresponding errors. The error analysis is done using the binning method as explained in refs. [ 1-3 ]. Our goal is the calculation of the critical parameters of the spin system described by the partition function (1) and in addition the calculation of the fractal dimension of the surface at x = Xc for d > 0. The critical parameters are determined from finitesize scaling. This method is commonly considered to be the most reliable one in dealing with problems of the type we face here. We will not discuss here the foundations of the method referring the interested reader to ref. [ 12 ], which contains an example of an application of this method and many references. The 512

3 November 1988

basic idea is that, in the neighbourhood of a continuous phase transition point, the physics of a finitesize system is controlled by the magnitude of the ratio (linear size) / (correlation length). The (average) linear size of the system L is related to the number of vertices of the lattice as follows: L~N

lID ,

(2)

where D is a fractal dimension, in general different from the naive value D = 2 (see, for example, ref. [ 13 ] ) and which we will not try to determine here. The magnetization is denoted by M: M=~

trj. N

(3)

We first calculate ( M 4) / ( M 2 ) 2. One expects from finite-size scaling that for x close enough to its critical value Xc one has ( M 4) / ( M Z ) 2 = ~ ( X - - x c ) N 1 / v D l ,

(4)

where f i s , a priori, unknown and v is the mass-gap exponent (thus the correlation length ~ Ix-/¢c I - " it should be borne in mind that in this work we focus on the spin system and that the exponents denoted by the same symbols in refs. [1-3,5] are different entities, referring to the geometry of the surface). Hence, the curves describing the variation of ( M 4 ) / ( M 2 ) 2 versus x (at fixed N) should all cross at the same point x = xc. Furthermore, the ratio of slopes of two such curves enables one to find yD. In practice, things are not that simple. Due to the effect of irrelevant operators the curves do not cross exactly at the same point, although the crossing points converge towards xc when larger and larger lattices are used. Also, to get a reliable estimate of x¢ one is advised to use lattices with a size ratio as large as possible. However, at constant statistics, the error in the numerical determination of /
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we have been working with N = 2 0 , 45, 100, 225 and 500. It turns out that with our statistics the lattice with 500 vertices is the largest one that can be used efficiently for the determination of the critical point. The cross-over points closest to the exact value are x(45×500)=0.215+0.002 and x ( 1 0 0 X 5 0 0 ) = 0.218 + 0.003. The difference between these results gives an estimate of the systematic error in the determination of the critical coupling. We give both resuits in table 1. The statistical errors quoted are the conventional errors of least-square fits to the data. From the ratio of slopes of the fits to < M 4 > / ( M 2 > z at the crossing points we find z,D=3.3+0.6 and 3.2+0.3 for the cross-over points 4 5 × 5 0 0 and 100 × 500 respectively. The exact value for t,D at d = 0 is ~D=3. Finite-size scaling leads one to expect at x = xc the following large-N behaviour of the average magnetization < [MI > ~ N - p / ' ° ,

(5)

magnetic susceptibility (6)

Z= xN(M 2) ~ N~/'o '

3 November 1988

internal energy density

(E)=N-t( ~ aflkl~C~+czN(~'-')/~n, (jk >

(7)

and specific heat C m x 2 N ( ( E 2) - ( E ) 2) ~ c 3 +¢4 N a / v D •

(8)

The constants cj are not universal. At d = 0 the analyric results [10] (valid also for the non-degenerate lattice) are: #/~,D=~, 7 / l , D = 23and or~ p D = - ~. We determine the exponents experimentally carrying out runs at N = 100, 200, 500, 750, 1000 and 1500 and keeping x = x(45 × 500) or x( 100 × 500). We give both results in table 1. The difference gives an idea of the systematic error. The results for the "magnetic" exponents, listed in table 1, are in good agreement with the exact values. We find that the asymptotic regime ( 5 ) - (6) is reached only above N = 2 0 0 . The points at N = 100 and 200 are slightly, but systematically off the linear log-log fit to the last four points which produces the good estimates listed in table 1 (a least-squares fit to all the six points yields fl/ u D ~ O . 13 and 7/ z,D,~0.73 ). The measurement of ot is somewhat less successful.

Table 1 The critical parameters of the model. The notation is conventional. The upper (lower) figure corresponds to 45 × 500 ( 100 × 500). For d = 0 we also list the exact values. Fractal dimensions found in ref. [ 1 ] for purely bosonic surfaces are listed in parentheses.

0

3

5

10

re

0.215 ± 0.002 0.218 ± 0.003 0.216

0.231 ± 0.004 0.236 ± 0.007

0.292 + 0.029 0. 327 ± 0.016

0.334±0.004

pD

3.3 3.2 3

4.5 5.0

8.5 2.2

1.9

fl/vD

0.17 +0.01 0.16 +0.03 ± 6

0.18 ±0.03 0.12 ±0.04

0.02 ±0.01 0.14 ±0.06

0.13 ±0.10

),/pD

0.68 +0.02 0.71 +0.04

0.67 ±0.05 0.76 +0.05

0.98 +0.05 0.80 +0.08

0.77 ±0.14

±0.6 +0.3

+0.3 +0.7

±2.2 _+1.0

±0.3

t dv

7.3 8.3 (8.3

±0.2 ±0.4 ±0.1)

6.4 6.2 (6.4

±0.1 ±0.1 ±0.1)

4.4

±0.1

(4.9

±0.1)

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From the specific heat we get a/vD=-0.15_+0.13 and - 0.26 _+0.16 for 45 X 500 and 100 X 500 respectively, which is compatible with the theoretical value - ~ , but not very precise. From the internal energy we find ( a - 1 ) / ~ , D = - l . 0 0 _ + 0 . 1 1 and - 1 . 0 5 + 0.21, which is only marginally compatible with the theoretical value - ~. The point is that both C and E are dominated by the constant coming from the nonsingular part o f the free energy, while the N-dependent term is a rapidly decreasing correction due to the negative sign o f a. Our fits are exceedingly sensitive to data taken at N = 100 and 200, where other corrections are operating. The data at large N, which in this case are particularly relevant, are unfortunately measured with an error comparable to the magnitude o f the non-trivial term in (7) or (8). Anticipating slightly, let us mention that all our errors become larger for d > 0, so that the determination o f a / ~ D becomes completely unreliable. All we can say is that ct < 0. For (ix - 1 ) / ~D we consistently get estimates close to - 1. We have applied the procedure, which at d = 0 turned out to be quite successful to data taken at d = 3, 5 and 10. For d > 0 we have also measured the fractal dimension of the surface at x = xc. Already for d = 5 and especially for d = 10 we have encountered a serious problem. The value of ( M 4 ) / ( M 2 ) 2 at the crossover points 45 × 500 and 100 X 500 becomes close to 1, which is the asymptotic value at large K, and the distance between the two crossing curves, after the somewhat uncertain cross-over, appears to be smaller than our experimental errors. This is illustrated in fig. 1, where we compare the favourable case d = 0 ( d = 3 looks similar) to the much less favourable case d = 10. A least squares fit to the data yields a quite precise estimate ofxc also at d = 10, but the error estimate is likely to be optimistic. Furthermore, presumably as a result of fluctuations, x(45 × 500) and x ( 100 × 500) almost coincide for d = 10. Actually, we cannot exclude the possibility that at large d the observed crossover is merely a finite-lattice artifact, which would mean that the spin system does not undergo any phase transition at all. Notice, that the value of dE close to 4 at d = 10 is an indication that the surface tends to degenerate into a branched polymer, like in the purely bosonic case. The results of the analysis are given in table 1. On the whole the variation with d o f the critical expo514

3 November1988 2,2 °

I~

2,0"

N=IO0 ~

N=500

1,8" 1,8" 1,4" 1,2-

1,0

.

,

.

,

.

,

.

,

.

,

.

,

.

,

0,18 0,19 0,20 0,21 0,22 0,25 0,24

(a) 2 , 2

"

/'22'0"~1=100 ~.

1,6 1,4

f

/"

f

N=500

~

1,2"

1,0 0,8

0,20

N

0,25

0,30

0,35

(b)

Fig. 1. Estimates of the critical coupling of Ising spins from the cross-over phenomenon for d=0 (a) and for d= 10 (h). nents appears to be weak. It seems that 7/vD tends to increase while fl/vD tends to decrease. The behaviour o f vD is somewhat uncertain, because of our large systematic error at d = 5 and the surprisingly low estimate at d = 10. In table 1 we also give the fractal dimension and compare it with our old results for the purely bosonic model. As can be seen from the table, the introduction o f the Ising spins leaves the fractal dimension almost unchanged. Finally, we would like to mention that we have carried out some exploratory runs with d spins/site. One can argue that the fermionic integration yields a determinant which to some extent compensates the (inverse) determinant coming from the integration over x variables. If this were true, introducing many spins per site would be equivalent to an effective decrease o f d and the fractal dimension o f the surface would increase. This is not what is observed. At d = l0 and with 10 spins/site we get dE = 4.7 _ 0.1, a result very close to the one listed in table 1. We have not yet achieved a full understanding o f this issue, which is under study, and we can only propose a conjecture:

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the expected c o m p e n s a t i o n is n o t occurring, b e c a u s e the f e r m i o n field, for a n y fixed t r i a n g u l a t i o n , is m a s sive while the b o s o n field x is massless. M o r e precisely, the e i g e n v a l u e s c o r r e s p o n d i n g to the least localized f e r m i o n m o d e s are n o t small e n o u g h to insure a n a p p r o x i m a t e s u p e r s y m m e t r y , w h i c h w o u l d p r e v e n t the surface f r o m degenerating into a b r a n c h e d polymer. T h i s w o r k was s u p p o r t e d b y the D e u t s c h e Fors c h u n g s g e m e i n s c h a f t , D F G , u n d e r research g r a n t Pe 3 4 0 / 1 - 2 . We a c k n o w l e d g e the s u p p o r t o f the Boc h u m C o m p u t e r C e n t e r for u s i n g t h e i r C Y B E R 205, a n d the H L R Z at the K e r n f o r s c h u n g s a n l a g e Jiilich for the use o f t h e i r C r a y - X M P . T h e S i e m e n s V P 2 0 0 o f the C N R S c o m p u t i n g center C I R C E in Orsay has also b e e n extensively u s e d a n d we wish to t h a n k J.M. T e u l e r for assistance. O n e o f us (J.J.) acknowledges the s u p p o r t f r o m the projects C P B P 01.03 a n d 01.09.

References [ 1 ] J. Jurkiewicz, A. Krzywicki and B. Petersson, Phys. Lett. B 168 (1986) 273.

3 November 1988

[ 2 ] J. Jurkiewicz, A. Krzywicki and B. Petersson, Phys. Lett. B 177 (1986) 89. [ 3 ] F. David, J. Jurkiewicz, A. Krzywicki and B. Petersson, Nucl. Phys. B 290 (1987) 218. [ 4 ] M.A. Bershadsky and A.A. Migdal, Phys. Lettl B 174 (1986) 393. [ 5 ] A. Krzywicki, Proc. Intern. Symp. on Field theory on the lattice, Nucl. Phys. B (Proc. Suppl. ) 4 (1988) 64, and references therein; see also J. Ambjorn, Proc. Intern. Symp. on Field theory on the lattice, Nucl. Phys. B (Proc. Suppl. ) 4 ( 1988 ) 71; I.K. Kostov, Proc. Intern. Symp. on Field theory on the lattice, Nucl. Phys. B (Proc. Suppl.) 4 (1988) 98. [6] J. Ambjorn and B. Durhuus, Phys. Lett. B 188 (1987) 253. [7] V.G. Knizhnik, A.M. Polyakov and A.B. Zamolodchikov, Fractal structure of 2D quantum gravity, preprint. [ 8 ] J. Ambjorn, B. Durhuus and J. Fr6hlich, Nucl. Phys. B 275 (1986) 161; J. Ambjorn, Ph. de Forcrand, F. Koukiou and D. Petritis, Phys. Lett. B 197 (1987) 548. [9] D. Boulatov, V.A. Kazakov, I.K. Kostov and A.A. Migdal, Nucl. Phys. B 275 (1986) 641. [ 10 ] D.V. Boulatov and V.A. Kazakov, Phys. Lett. B 186 ( 1987 ) 379. [ 11 ] V.A. Kazakov, I.K. Kostov and A.B. Migdal, Phys. Lett. B 157 (1985) 295. [12]M.N. Barber, R.B. Pearson, D. Toussaint and J.L. Richardson, Phys. Rev. B 32 ( 1985 ) 1720. [ 13 ] A. Billoire and F. David, Nucl. Phys. B 275 (1986) 617.

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