Simulation of 2D dynamical triangulation with higher order curvature terms

Physics Letters B 305 (1993) 223-229 North-Holland

PHYSICS LETTERS B

Simulation of 2D dynamical triangulation with higher order curvature terms N T s u d a I and T Yukawa2 National Laboratory for High Energy Physics (KEK), Tsukuba, lbarakl 305, Japan Received 3 March 1993

We present a numerical result of the Monte Carlo simulation of two dimensional random surface generated by dynamical triangulation under influence of higher order curvature terms We measure the fractal dimension and related quantities such as the boundary length distribution for various lattice sizes up to 400 000 triangles The boundary length distribution is found to scale nicely We also find a cross-over transmon between fractal and flat surfaces through varying the strength of higher order terms

1

Introduction

Recently, four dimensional s~mulatlons of q u a n tu m gravity by the dynamical triangulation have appeared [1,2] six years after the pioneering works in two dimensions [3-5] On the other hand analytical studIes of the four dimensional q u a n t u m gravity do not yet enjoy the benefit of recent progress m two & m e n slonal analytic theory after K m z h m k , Polyakov, and Zamolodchikov [6] It reflects that numerical investigations have a smaller gap between different & m e n s~onalltles compared to analyucal studies It is now possible to calculate statistical p r o p e m e s of m a n g u lated space m various dimensions, which have not been available analytically except m two dimensions Furthermore, recent results m four dimensional simulations reveal the existence o f the second order phase transition, which give us hope to obtain the universal continuum hmit in l a m c e theory However, q u a n t u m gravity is one of the most difficult subjects left in theoretical physics m the sense that there are neither experiments nor consistent quantum field theory so far Besides, In numerical simulations of dynamical triangulation the nature of the problem does not allow straightforward use of either vector super-computers or massive parallel comput-

l

2

E-mail address ntsuda@theory kekjp E-mail address yukawa@theory kekjp

ers m contrast to the lattice Q C D s~mulations U n d e r these circumstances it seems that a numerical simulation by itself would not be able to give any definite answer without analytical support Although the importance of four dimensional simulation should not be underestimated, we feel that it would be useful to reconsider two dimensional simulations, in order to get a close comparison between analytical theories and numerical s~mulatlons for narrowing the gap between different dlmenslonallties One of the main topics of numerical simulations in two dimensions is to prove the existence or nonexistence of fractal structure in dynamically triangulated surfaces A typical quantity to characterize the surface ~s the fractal dimensions obtained by counting mangles in a disk covered by a certain steps Such a dimension ~s known to diverge because of branching [7] at least m pure gravity, while for the case with a matter field of central charge c = - 2 the posslblhty of fimte fractal &mension has been argued [8] We shall show here an evidence of fractal by comparing to the theory recently proposed by Kawal et al [9] The second topic is the global geometrical property of surfaces under the influence of higher order curvature terms w~th which fluctuations of local curvature can be controlled We shall show that the numerical slmulat~on reveals two types of surfaces dependlng on the strength of ad&tlonal terms The nature of the surfaces is characterized by a fluctuation of local

0370-2693/93/$ 06 00 Q 1993-Elsevier Science Publishers B V All rights reserved

223

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curvature through which a transition between fractal and flat surfaces is clearly seen Since the higher order curvature term is considered to be irrelevant in two dimensions according to the dimensional analysis, the transition is smooth This paper is organized as follows In section 2, we briefly describe our numerical procedures for the sake of notation In section 3, we discuss the geometrical properties o f surfaces by measuring the loop length distribution, and prove the existence o f fractal structure In section 4, we exhibit the transition between two types of surfaces and g~ve a simple InterpretaUon of those surfaces The last section (section 5) is reserved for discussions

13 May 1993

v2

t3

/~

-../

tl

\I

vl

v3

© 2. Numerical method

~2

In order to fix notations let us describe our numerical procedure briefly We prepare initial configurations consisting N2 triangles with the topology o f the 2-sphere S 2 It is generated by adding vertices sequentially on a surface starting from a tetrahedron in terms of the barycentrlc subdivision o f a randomly chosen triangle When we write the numbers of triangles, hnks, and vertices as N2, Nt, and No respectively, the Euler relation and the manifold condition give the following relationships N2=2N,

NI = 3 N ,

N0=N+2,

with an integer parameter N These initial configurations are then thermahzed by flipping sequentially the c o m m o n link between a pair of neighboring triangles which is randomly chosen For each flip move two types of acceptance check are imposed, namely, (l) geometry check We accept only those configurations contalnmg no singular sub-dmgrams such as tadpoles and self-energies when the triangulated surface is represented in the dual space O0 Metropohs check We employ the usual Metropolis algorithm for thermallzatlon with higher order curvature terms in the action For the higher order curvature terms we employ the scalar curvature square action, tip [ v/gR 2 d2x, J 224

( 1)

Fig 1 A unit cell o f triangulation stored The o corresponds

to a triangle, and the • corresponds to a vertex A triangle (t) as formed by three vertices (v l, re, v3), and it has three neighboring trmngles (tl, t2, t3) which can be expressed as fl ~ , (q, - 6)2/q, in the triangulated space where q, is the coordination number of t h e / t h vertex and fl is the dimensionless strength parameter while the physical parameter fly has the dimension of the volume After sufficient thermahzatIon flip moves of typically 100 to 500 sweeps depending on the lattice size and the value of fl, we store 500 configurations at every 200 sweeps as members o f the ensemble for statistical averaging In practice, configurations are stored for each triangle (t), the three vertices (Vl, ~2, v3 ) and the three neighboring triangles (tl, t2, t3), as in fig 1, and the coordination number qv for each vertex (v)

3. Fractal properties of the surface The curvature of the surface generated by the dynamical triangulation seems to fluctuate rather extremely according to the coordination number distribution which has a Polssonian character When we employ an action containing only the cosmological term and the Einstein term, all the possible triangulations contribute equally for a fixed number o f triangles In this case no scale parameter is involved, and we expect the surface to acquire scahng property in

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the large volume hm~t, or m other words fractal For representing such a surface quantataUvely at ~s customary to define the fractal dxmensaon m terms of the n u m b e r of vertaces or trmngles (1 e vertices m dual space) covered within r steps, V ( r ) , m trmngulated surface or ats dual surface as d(r)

In V (r) In r

-

(2)

These two definxtxons of fractal d~mens~on are almost ~dent~cal when we rescale the n u m b e r of steps by a factor of 2, i e the ratxo Nz/No Since statlstacal propert~es of the surface are expected to be the same e~ther m the real space or the dual space, we shall use our data calculated m the dual space because of h~gher statxstlcs It has been known that the fractal d~mens~on tends to dwerge as the n u m b e r of steps r mcreases m the numerical s~mulat~on [7] Th~s divergence ~s considered to be the consequence of branchmg In fact, when we measure the n u m b e r of boundaries M (°) (r) m a d~sk covered by r steps, ~t shows a tendency of d~vergence These numerical results gave us some doubts on the s~mple fractal p~cture of the random surface expected from the c o n t i n u u m theory [10] However, ~t ~s so natural to expect fractal m such a surface, and we think what we need is rather to find the right quantRy to show it than to g~ve i t up In searching for good scahng varmbles we have measured the boundary length distribution and found a race scahng property Eventually we were reformed that Kawal et al [9] have just obtained an analytic formula for the boundary length which IS exact m the c o n t i n u u m hm~t According to them the n u m b e r of boundaries with a length l counted m a disk with a radius d is given by the function f(l,d)

-

12

10 3

10 2

[. . . . D=I3 I . . . . . D= I4 ~ Theory

o~x~ \ ~

101

qn

10 °

,1

101

10 2 0

1

10

%

Fig 2 The umversal function fD 2 averaged over 50 configurahons of 400K triangles wRh the theoretical curve

the term with x -5/2 It gives divergences in the calculaUons of the n u m b e r of boundaries M (°) (d), and the total length of boundaries M (t) (d) This is conSlstent w~th the d~verglng fractal d~mens~on m numerical s~mulaUons However, the lattice calculation inevitably introduces a cutoff at small x values, and the divergence can be controlled Using eq (3) the cutoff (lo) dependence of the nth m o m e n t M (") is calculated as M (") (10, d) =

3

( d ) 2"

x { r ( n - 3,Xo) + ½ r ( n - ½,xo)

1

7v/'~ d2

X (X -5/2 "k- ½X-3/2

13 May 1993

"b ~ X

+ ~ r ( n + 3,xo)}, 1/2) e - x ,

(4)

(3)

where the scahng variable x is defined by 4 l i d z The above formula has been derived as a c o n t i n u u m limit of the dynamical triangulation through e --* 0 of d x/~-×number of steps (D), and l ~ e × n u m b e r of triangles of a boundary (L) We plot the values f D 2 at X = 4 L / D 2 for D = 13 and 14 together w~th the theoretical prediction (sohd hne) m fig 2 The remarkable feature of th~s f u n c u o n is the exxstence of

where x0 = 41old z and F ( n , x ) is the incomplete F function From the small x0 behavior o f f (n, x0) M(1) has a d 3 behavior m the large d hmlt, and we expect that the dimension defined by eq (2) will eventually converge to 4 when we perform large s~ze slmulauons Of course d ( r ) = 4 does not necessarily mean the fractal dlmensaon being four since the d3-dependence comes through the short dastance cut-off scale x0 On the other hand, the nth m o m e n t M (") converges for 225

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PHYSICS LETTERS B

10 9 ------

/z

10 8

/

n

/

/

t

10 7 /

10 6

/

/ /

/

/

/

/

/~/////~1

J

//

10 s / 10 4

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n ~> 2, and ~s expected to be proportional to d 2n We show the theoretical and numerical results for n = 0, l, 2, 3 and 4 in figs 3 and 4, respectively Although the overall behavior of both quantities agrees rather well, there remains some discrepancy such as the exponents where our simulation gives a little smaller values for all n There will be finite size effects as well as lattice artifacts and at th~s moment we cannot say much about the reason of the discrepancy At least we have encountered the fractal property of surface which was once almost given up

10 a

10 2

4. Transition between flat and fractal surfaces

10 ~ 10 0

10

d Fig 3 The nth moments of boundary lengths M (n) (theorencal)

I 10 9

•

on=

c

~n=]

[i ' ~

Besides the fractal surface as we have seen in last section, there exists another kind of generic surface, namely the fiat surface In order to study the transition between these two kinds of surfaces we include the scalar curvature square term in the action, which we write as

R(2) =- ~ t

0

(6 z q , ) 2 , q~

(5)

in the triangulated space The effect of this term on the surface can be clearly seen in the coordination n u m b e r distributions for large positive fl as the peak at the coordination n u m b e r being 6 [ 11 ] Let us first take a look at the average value Q2 = (R(Z))/N2 (fig 5) and the fluctuation (fig 6),

.B:2

10 8 10 r

10 6

CR

J 10 ~

{((R(2))

2)

--

((R(2)))2}/N2

(6)

In fig 5 we observe a smooth transition from a weakly fl dependent phase to a fl - l behavior around tic ~ 20 For small values of fl we expect the fractal surface similar to the fl = 0 case because of the universality of the Liouvllle theory, while for large values of fl local flatness of surfaces will show up Indeed, the f l - i dependence of Q2 suggest the equl-parntion law of free gas wRh N2 particles in the ?/d dimensional space, 1 e

10 4 10 a 10 2 10 ~

10 °

10

D Fig 4 The nth moments of boundary lengths M (n) (numerical) 226

=

e-

(R(2)) = ~fl i -1 N2nd

(7)

From the slope of our data n d 1s approximately g~ven as 2 Also the fluctuation CR changes rather suddenly at a similar value of fl As far as the Nz dependence

Volume 305, number 3

10 °

PHYSICS LETTERS B

13 May 1993

~, T

[o

®

o Lb 7

®

102

101 10 ~

® ®

o N2 = 5000 • N2 = 20000

10 0

102 10

.......

100

i'0

.......

1'00

steps

Fig 7 Nb number of boundaries M (°) (r,/Y) and Lb total boundary length M (l) (r,/Y), for ,8 = 300

Fig 5 The fl dependence of Q2

100 O • O

101 rr O

o N2 = 5000

• N2 = 20000

102

o

%

1

10

100

Fig 6 The fl dependence of CR

of Q2 and C~ are concerned, we cannot find any significant difference between N2 = 5000 and 20 000 From these data it may be appropriate to call these two kinds of surfaces the fractal surface for fl below tic, and the flat surface when fl > tic The next q u a n u t y we lake to examine is the n u m b e r of boundaries an the disk covered by r-steps for v a n ous fl, whach we write as M C°)(r, fl) In fig 7 we show M C°) together w~th the total boundary length M (1) for fl = 300 m the same dask The M C°) starts from one, i e a d~sk w~th no holes, and increases as the surface bifurcates into branches Denoting the step n u m b e r where M ~°) exceeds one shghtly as ra, it is regarded to be the mean distance between branches In thts regaon the total boundary length increases linearly as M ~1) ~ 3 2r, indicating a two dlmensmnal fiat dask of radius r It increases as fl increases with fla (a ~ 2 3) The last quanUty we show for an mdacauon of the t r a n s m o n Is the boundary length d~stnbut~on F (1, r, fl) m fig 8 There exist three characteristic regions when/~'s are large the behavaor at small boundary lengths (l = 3, 4) is consadered to be a lattace artifact since they can be removed by a single flap-move and the n u m b e r of these loops should be affected 227

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PHYSICS LETTERS B

102

l"

i 10 ~

..... \

I~

\

-" ~--o ,13-- 100 -'1~=2°° ~ ~ = 300

10°

10 ~

102 10 boundary length

100

Fig 8 Boundary length dtstnbutlons for various fl's, at the step D = 50 of 20K mangles strongly by the small size effect There are plateaus f r o m l = 5 to ld ~ 32rd where the branch s i z e l s smaller than the mean distance between branches This suggests that isolated branches bifurcate with equal probability For a branch with the length l > ld the n u m b e r of boundaries decreases with power l a, showing a scahng behavior similar to the fl = 0 case

13 May 1993

(II) Theflat surface For/~ above/~c, the small scale local curvature fluctuaUon is controlled and the surface is locally flat The mean separation &stance between branches gets large and small size loops bifurcate independently Number of small loops is rodependent of the length In the c o n t i n u u m theory the higher order curvature term is written as eq (4) where the physical coupling strength/Jr has the & m e n s m n of volume The correspondmg constant/~ m the latUce formulation ~s damensmnless and expressed as fly ~ a2Nzfl, where a is the lattice constant Therefore, in the continuum and large volume h m n (a2N2 -~ ~ ) with a fixed/?p the couphng strength fl on lattice should be vamshrag, and only the fractal phase will be physical This is of course expected from the &menslonal argument of the higher order curvature term However, th~s term is expected to be important m Mgher d~menslons at d = 4,/? becomes dimensionless, 1 e in this dimension the t r a n s m o n is physical 1fit exists at all We shall continue our mvesUgatlon of R2-gravlty in higher {tlmenslons by the dynamical mangulaUon to check this expectation

Acknowledgement It is a pleasure to acknowledge N lshibashi for many useful &scusslons We are grateful to H Kawal, N Kawamoto, T Mogam~, and Y Watablkl for showlng us their results before pubhcatlon We also thank them for discussions and comments

5. Discussion References Controlling the local curvature by the scalar curvature square term we have been able to study transition between two kinds of surfaces, namely the flat surface and the fractal surface These two surfaces are characterized as follows (I) The fiactal surface For /~ < ]~c ( ~ 20 ) the mean separation between branches is small and bifurcation of branches strongly overlaps The overlapping branches cause large fluctuatmns of local curvature and the surface becomes fractal The n u m b e r of small size loops of the length ! diverges as 1-25 resulting m the &vergence of the fractal & m e n s m n 228

[ l ] J Ambjorn and J Jurklewlcz, Phys Lett B 278 (1992) 42 [2] M E Aglshtem and A A Mlgdal, Mod Phys Lett A 7 (1992) 1039 [3] J Ambjorn, B Durhuus and J Frohhch, Nucl Phys B257 [FSI4] (1985)433, B275 [FS17] (1986) 161 [4] D V Boulatov, V A Kazakov, I K Kostov and A A Mlgdal, Nucl Phys B 275 [FS17] (1986) 641 [5] F David, Nucl Phys B 259 (1985) 45 [6] V G Kmzhmk, A M Polyakov and A B Zamolodchlkov, Mod Phys Lett A 3 (1988) 819 [7] ME Aglshtem and AA Mlgdal, Nucl Phys B 350 (1991) 690

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[8]N Kawamoto, V A Kazakov, Y Saekl and Y Watablkl, Nucl Phys B (Proc Suppl ) 26 (1992) 584, Plays Rev Lett 68 (1992)2113 [9] H Kawal, N Kawamoto, T Mogamx and Y Watablkl, to be pubhshed

13 May 1993

[ 10] H Kawax and M Nlnomlya, Nucl Phys B 336 (1990) 115 [ 11 ] T Yukawa, N Tsuda and A T Sornborger Nuel Phys B (Proc Suppl ) 30 (1993) 791

229