The Crumpling Transition of Dynamically Triangulated Random Surfaces

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The Crumpling Transition of Dynamically Triangulated Random Surfaces

Christian Münkel, Dieter Heermann

To cite this version:

Christian Münkel, Dieter Heermann. The Crumpling Transition of Dynamically Triangulated Random Surfaces. Journal de Physique I, EDP Sciences, 1993, 3 (6), pp.1359-1370. �10.1051/jp1:1993184�.

�jpa-00246799�

Oassification Physics Abstracts

05.90 64.60A 64.60F 68.10 87.20C 12.90

The Crumpling Transition of Dynamically Wiangulated Random Surfaces

Christian Mfinkel and Dieter W. Heermann

(~) ^{Institut fur} theoretische Physik, Universitit Heidelberg, Philosophenweg 19, 6900 Heidelberg, Germany

(~) Interdisziplinires Zentrum fur wissenschaftliches Rechnen der Universitit Heidelberg, 6900

Heidelberg, Germany

(Received

2 February J993, accepted 9 February

1993)

Abstract. We present the crumpling transition in three-dimensional Euclidian _space of dynamically triangulated random surfaces with edge extrinsic curvature and fixed topology of

a sphere _as well _as simulations of _a dynamically triangulated torus. We used longer _runs than previous simulations and give _new and _more accurate estimates of critical exponents. Our data indicate _a cusp singularity in the specific heat. The transition temperature, _as well _as the exponents are topology dependent.

1. Introduction.

In this _{paper we are} concerned with the statistical mechanics of surfaces. A

possible approach

in _space dimensions D _> I is to discretize the surface

using

_a

triangulation [1-4].

For this

we need to

specify

the Hamiltonian 7i. The surface S is

replaced by

_a

simplical triangulation T, specified by

the number of nodes N, of links and

triangles,

and the X-coordinate field

by

the coordinates X of the nodes, The metrical fluctuations of the manTold _are modeled

by summing

over

triangulations

induced

by link-flips [5-7].

^The Hamiltonian is _now

choosen,

such

that the

partition

function is not dominated

by configurations

with

spikes.

In order to surpress these

spikes

_one adds _a term with extrinsic curvature. As _a function of the extrinsic curvature the model shows _a transition

(crumpling transition)

at finite

rigidity

of the surface

[8-18].

In _a

previous

_{paper, we} discussed the extrinsic and intrinsic

geometrical properties

of

dynamically triangulated

random surfaces

(for example

the

Hausdorff-dimension, spectral

and

spreading dimension) above,

below and at the transition

[32].

The

partition

function of the above model _can be written _as

ZN _"

jd~Xo j ^~~ _{fl d~X;e~'~ (1)}

where the translational mode is

integrated

out. The Hamiltonian 7i is defined _a8

N

~ ₌

p, (

(xi xj)2

₊ >

£ (i

_na,

nb>)

_"

~

~°~ " ~~~

ii,)) ~

,

.>~J

~ fi

7i~ _~

The Gaussian part ^{of the} Hamiltonian

zig

^is a sum over the

positions

X in

embedding

Euclidian space of all nearest

neighbour nodes,

I-e- all links of the

triangulation.

We shall _use

fl

₌ I

because of the

rescaling

invariance.

7ie is _an

edge

extrinsic curvature term

[19-26]. £~

~ denotes _a summation _over all

adjacent triangles

which share _an

edge

and _na, na, ^is

the"scilar _product

_{of the}

vectors normal to _a

triangle.

The third part of the Hamiltonian 7im is the discretization of the square ^root of the metric.

a; denotes the number of nearest

neighbours

of node I. _a

depends

_on the _measure. We used

a =

D/2

for the simulations in D ₌ 3 dimensional

embedding

_space.

Here _we

investigate

such _a model

numerically (Monte

Carlo

simulations), applying

finite-size

scaling

ideas. These ideas have _proven successful in other contexts for the

analysis

of

phase

transitions. We expect ^{that the above model shows} a

phase

transition _{at some} critical value of the

coupling

I. There _are several

questions

which _we want to address: What is the order of the transition? If the transition is of second

order,

what _are the

exponents?

Are the exponents

topology dependent?

Is the transition temperature

topology dependent?

2. Simulation Method and Autocorrelations.

First let _us

study

the effect of the extrinsic curvature _on the autc-correlation time of several

observables,

I-e-, _{we are} ^interested in the

dynamics

of _a random surface induced

by

_a Monte Carlo _process. Such _a

study

is of _{course a}

prerequisite

for _a detailed numerical

study

of the

apparent ^{transition which such surfaces}

undergo

_{as a} function of _a

bending rigidity.

From _our results which _we present ^below we must conclude that extensive computer resources must be

applied

because of the _very

long

relaxation times which _even for moderate surface sizes _can

reach several tenthousand _sweeps.

We want to look _at the closed

dynamically triangulated

random surfaces without self- avoidance. As models for such surfaces _we take the first two

topologically

closed surfaces:

The

sphere

and the _torus

(cf. Fig. I).

The torus is

specified by identifying edges

of the _pa-

rameter space P whereas the

sphere,

because of its genus needs _a

slightly

_more

complicated

set-up

procedure.

What _{we are} interested in is to calculate the autc-correlation function of _an observable O _as

a function of time

~°~~'~~

^~

~~~~i~~< I$(~~

~ _~~~

and its

dependence

_on the number of nodes N of the surface. From the autc-correlation function _{we are} then able to extract the autc-correlation time _To

(N).

This is done

calculating

the

integrated

autc-correlation

function,

I-e-,

+OJ tint,O ^"

j j

d~ _PO (~i

N) (4)

-OJ

We also

compared

the

integrated

autc-correlation time with the

exponential

correlation time and the statistical

inefficiency,

^{but found} no

disagreement

within the _errors. If the

successively

Fig-I-

Shown _are two examples of configurations of dynamically triangulated random surfaces.

The left picture shows _a sphere and the right part _a torus.

generated configurations

_can be considered _as

non-interacting

beads _as is done in the Rouse

theory [27,

28] we will find for

example

for the radius of

gyration

rR» c~ N

(5)

syr

and in

general

_we expect

To cc N~

(6)

To calculate observables such _as the

gaussian

part

zig

^of the Hamiltonian _or the radius of

gyration Rgyr,

we use the standard Monte Carlo

algorithm [29-31].

^Such an

algorithm

induces _a stochastic

dynamics

and all _our results

apply only

to such _an induced

dynamics.

One Monte Carlo _sweep is

completed

when each

triangulation point

_was

given

the chance for _a

displacement

from its

previous position

and the

edges

of the

triangulation

_were

given

the chance to re-connected _or

flip

to an

orthogonal position (cf. Fig.

2. The

flip operation implements

the

sum over all

possible triangulation).

This

corresponds

to _one new

configuration.

Fig.2.

This figure demonstrates the elementary moves which _are made during _one update of _a

triangulated random surfaces.

Table I summarizes _our results for the exponent _a for the

sphere.

The exponents where extracted from the data of system sizes from 36 _up to 288. At this

point

_we should mention that the

expected

transition is at

lc

_cS I-S- Out data _was collected above and below the

transition.

Above and below the transition the radius of

gyration

shows the Rouse behaviour. The Gaussian part ^{of the} Hamiltonian shows

clearly

a

dependence

on the

bending rigidity.

Above the transition

point

the autc-correlation exponent

changes

to _one and below it is almost _zero.

The _same holds for the

edge

part ^{of the} Hamiltonian.

Table I. Table

of

^{the results}

for

^the exponent a on the auto-correlation times

of dynamically triangulated

random

sphere.

I is the _measure

of

the

bending rigidity. Skiff

and

flips give

^the acceptance

probability for

_a

displacement of

a node and

flip for

an

edge change.

r rate

0.00 0.2+0.3 0.2+0.2 1.1+0.2 0.51 0.76

1.75 1.0 + 0.1 0.8 + 0.2 1.1 + 0.2 0.47

+

Almost all of the observations for the

spherical

case carry over to the _case of the torus.

3. Results of the Simulations.

Finite size

scaling

_assumes, that there is

only

_one relevant linear

length scale,

which is _com-

pared

to the correlation

length.

To

apply

finite size

scaling

to the

crumpling

transition of

dynamically triangulated

random surfaces

(DTRS)

_{one must} therefore _{assume a}

single length

scale determined

by

the number of nodes N and the internal dimension d of the surface

L _cc

N"~ (7)

This internal dimension d also

depends

_on the external

properties

of the surface and 1

[32].

So let _us first look at the

specific

heat. If the transition is of second order _we would have

C(I, L)

₌

L°'"i~ ((I lc)L~"] (8)

where

t~

_is

a

scaling function,

which

depends

_on how _one

implements

the surface. At the critical I the

scaling

function is

regular

and the

scaling hypothese

leads to

Cj$~

_c~

^AN°'"~

+

(9)

for the

scaling

of the

peak

in the

specific

heat. If _{we assume a} first order transition then Cj$~~

diverges

_as

^L~

because of the b-distribution _of

Ccc (T) [33, 35].

An evaluation of the

specific

heat C of DTRS

(neglecting

the metric contribution

7im) gives

the

following expression

Call " +

~ (< 7il

_{> < 7ie}

>~) (lo)

The first part is related to the Gaussian Halniltonian

zig

and the second to the

specific

heat C of the

edge

extrinsic curvature 7ie. The

specific

heat for the

edge

extrinsic _curvature is shown

in

figure

3 for the two

topologies

considered. The

interpolation

_was done

by

the method of

Ferrenberg

and Swendsen [36, 37]

using histograms

of 7ie

6 6

36 _-

~~

64 - s -

64 _--

5 108 _--

100 _-- 144 _--

144 -

288 - 4

196 _-

4

432 -

256 -

400 -

~ 3

2 2

0 0

O-S I I-S 2 2,5 3 O-S I I-S 2 2.5 3

~ ~

Fig.3.

Specific heat c

(edge

extrinsic curvature

part)

of the sphere

(left part)

and of the _torus

(right part),

6

5.5

~

# 5

~- ~-

~iZ _4.5

#

E

4 torus --

sphere --

3,5

3

50 100 150 200 300 400500

N

FigA.

Maximum of the specific heat cfl~~ of the torus

(o)

and the sphere

(D).

Using

the data of the

specfic

heat obtained

by applying

the

extrapolation method,

_{we can}

get

_{a very} accurate estimate of the

positions

of the maxima and of the

heights. Figure

4 shows

CJ/~~

^{of the} torus and the

sphere.

A

change

to L~ behaviour is _very

unlikely

and for that

reason the data

strongly

suggest a continuous

phase

^{transition in}

agreement

^with

previous

work [8, 14-17,

38, 39].

From the data shown in

figure

⁴ _we can obtain the

following

_upper

boundaries of critical exponents

Sphere: ~

_{s 0.00 + 0.04 Torus:}

~

_{s 0.06 + 0.02}

₍₁₁₎

Using

this

simple method,

_we cannot

distinguish

_a

diverging specific

heat with _very small but

positiv

_{a, a}

logarithmic divergence

_{a, =} 0 and _{a power} law _{cusp a, <}

0,

where _a, denotes the exponent of the

singular

part of the

specific

heat. We will _use the abbreviation _a instead of _a,.

Following

Fisher [40], ^{these three} cases may be

distinguished

with _a fit of the form

C(Al)

₌ A

(Al~~ I)

+ B

,

Al =

[lf _{1[ (12)}

Figure

5 shows such fits for _a

= 0.I

(power law),

_{a =} -0.01

(near logarithmic)

and _a

=

-1.2,

-2.0

(power

law

cusp).

The fits

clearly

favour _a power law _cusp of the

specific

heat with

a

large negative

^value of _a,

although

_{we were} not able _to estimate the exponent a

precisely.

6 6

5.5 36 - 5.5 36 _{- .--=1}

~ 64 _- 64 4"I

~ IDS -

~ lot

-

4.5 144 _{- 4.5} 144 - _,

~ 28K _-

~ 28K _-

432 _- 432

3 5 3 5

3 "

3

2.5 2.5

~ ~

l.5 1.5

l i

o.5 o.5

lo 5 o 5 lo lo 5 o 5 lo

x x

6 6

5.5 36 - 5.5 36

64 64

log -

5 lo~ -

4.5 144 - 4.5 144 -

~ 288 _- 28K -

432 _- ⁴ 432 -

3.5 3.5

3 ^" 3

25 2.5

2 2

1.5 l.5

l j

o.5 05

08 -0.6 -DA -0.2 0 0.2 0A 0.6 0.8 0.6 DA 02 0 o.2 DA 0.6

x x

Fig-S-

Plot of the specific heat c against _z

=

(al~° -1)

with _a

= 0.1

(upper left),

-0.01

(upper right),

-1.2

(lower left)

and _a a

= -2.0

(lower right).

Our results _are in contrast to the estimates of Renken and

Kogut

[39]

a/vd

₌

0.14(3)

for the

sphere. They

assumed

scaling

above N ₌ 72, ^but the _more accurate data in

Figure

4 shows that this

assumption

is not true. The

decreasing slope

in

Figure

4 is also present ⁱⁿ their

Figure

5 [39]

although

this is

slightly

masked

by larger

_error barrs and less data

points.

Catteral et all. [38]

reported

_an estimate of

a/vd

=

0.185(50),

but

they

used _a different discretization based

on

#~ graphs

with _a

flip

acceptance rate of

only O(I$l)

at

lc

_cS I-S and

larger

correlation times of the

edge

extrinsic curvature than in _our discretization.

Another

possibility

to determine the order of the transition is the cumulant VN of the

edge

extrinsic curvature

VN _:= 1-

~~~~ )

,

(13) ie~)~g

which behaves

quite differently

at temperature ^driven first- and second-order transitions

[33,

34]:

1, and 2.order: VN)ruin

'~I" T#Tcfixed (14)

2,order: VN)ruin

'~1"

^~

T ₌

Tc(N) (15)

1-order: VN)mjn

'~I"

^~

^~~~

^~

^~~ _f

_T

=

Tc(N) (16)

3(E(

₊

Et

E+

and E- _are the

energies

of the system above and below the transition. For _{a very} weak first-order transition

(E+

_cS

E-)

_we also have VN)~n;n *

2/3.

We

computed

VN defined

by equation (13) using again

the method of

Ferrenberg

and Swend-

sen

[36, 37].

The

resulting figures

show the

predicted single peak

minima with

wings following equation (14).

The finite size

dependence

of the minima of VN)~n;n ^{shown in}

Figure

6

distinctly

favours the

asymptotic

behaviour in

equation (15)

and therefore _a continuous

phase

transition

or a very weak first order transition.

0.67

0.65

0.64 _a

)

~

~

0.63

-.

~ ~

0.62 0.61 0.6

0.59

0 0.005 0.01 0.015 0.02 0.025 0.03

IN

Fig.6.

Finite size dependence of the minimum VN )nfin of the reduced cumulant of the edge extrinsic curvature.

(o)

denotes data of the sphere,

(D)

those of the torus and the _arrow the large N limit

2/3

of _a continuous transition.

In

general,

finite size

scaling predicts

also _a shift Al

=

If _-16f'

of the effective transition

'temperature' If proportional

to

N~~(= L~~)

for _a first- and

proportional

to

N~Q"~(=

L~Q")

for _a second-order transition.

Unfortunatly I~°

of

dynanfically triangulated

random surfaces is not known. For that _{reason, we} have to _{use a} twc-parameter fit with unknowns

IT

and vd. But _{we can}

improve

the

reliability

of this

fit,

because _we know _more.

First,

the fit

has to be _a

straight

line

(neglecting

corrections to

scaling)

^and we have

limN-cc Al(N)

₌ 0.

Second,

the shift

Al(N)

is different for the

specific

heat and the cumulant in

general.

Therefore the

slope

of the fitted lines will be different in

general,

but _we stiff have

fimN-co Al(N)

₌ 0 for both observables.

0.25 0.16

0.14 C -

V _-

02 C _-

V +- 012

O-1 0.15

~~g

~ o-t

~

l'~

~~~

0 .j

i

o ~o.02

0 o.05 o 0.15 o.2 o.25 o.3 o o.05 0. t o.15 o.2

x x

Fig.?.

Best fit of _{y =} Al against _{z =} N~~'"~ _{for the} sphere

(left part)

and the torus

(right part).

(o)

denotes the specific heat c data,

(D)

those of the cumulant V.

Figure

7 shows the best fits for the

sphere

^{and the} torus. The estimates of the parameters

are

lm

= 1.51 + 0.04

,

ud = 3.2 + 0.5

(17)

for the

sphere

and

lm

= 1.47 + 0.02

,

vd = 2.5 + 0.5

(18)

for the torus.

We turn _now to the order parameter itself. Common

practise

is to take

(

₌

R/L (R

is the

typical radius,

L the linear size of the

membrane)

to be _a suitable order parameter

[41-47].

Initially

[47] ^it was also defined

as

Rg(L)

₌

(L (L

_-

oJ),

with the linear size L of the

hexagon

and the radius of

gyration R(c~ £;; IX; X; _[~).

A suitable choice for the order parameter therefore is

(

_:= R~

IN (19)

with

N

~l~

"

j~r(^j~r

i) £'i'i _l'~i -'~I)~ _(~°)

;,j

a; denotes the number of nearest

neighbours,

I-e- the number of links connected to _a node I.

The associated

susceptibility

is

xR» := L~

(1(~) l()~)

⁼

(

(lR~) lR~)~) ⁽²¹⁾

Figure

8 shows the order

parameter (

and

Figure

9 the

susceptibilty

of the

sphere.

The results for the torus are similar.

0.1

~

0.09

~

wP ^°

~

0.08 o° ~$ ⁱ

0.07

~~

°

j~~ ((

.~~

~ +

~

~

~~

0.04

o

~

° _x

432

_*

0.03

~

~

~

0.02

~

~ ~

0.01

0

0 0.5 1.5

2

2.5

3

I

Fig.8.

Order parameter (

=

R~

IN

of the sphere. Errors _are smaller than symbol size.

0.08

DTRS, sphere

~

0.07

~~

0.06

~

~((

^~

0.05

"~

(~

^~*~

0.04

~

°.°3

.j~~J

«

,

o o~

_.jj,

'

'».--q~j",,

~

#a..,_"

__

'~~

.-

-«~'

"".)

i

0

0.7

0.8

0.9

1-I

1.21.3 1.41.5 1.6

1.7

1.8

1.9 2 2.I

I

Fig-g-

Susceptibiity _XR» of the sphere

(Lines

to guide the

eye).

5 4.5

4 _,:.." ~:

3.5

~:.."'

3 _{torus --} _:.""

2 5

sphere

_-- __:.""

~..."'

A 2

~

_.-..'"

ii'

_,:."

~~ 1.5 _{_:.."}

V ,.~..'"

l __:.""

,:a."'

0.5

50 100 150

200

300

400500

N

Fig,10.

Scaling of the radius of gyration squared

(R~(N))

_at the position of the maximum of the specific heat c~~~

(o)

denots the data of the torus,

(D)

data of the sphere.

0.06

0.05 _,..4""

torus -- __:."

~

0.04 sphere

--

_:.""'

0.03 __:.-."~

_,:...'~

0.02

~,:.""

%,....""

o.oi

50 100 150 200 300

400

N

Fig,ll.

Scaring of the susceptibility _X

(Eq. (21), Fig.

9) in the _case of _a torus

(o)

and _a sphere

(°).

We also measured another

possible

order paramter

I'

=

(R'~) IN,

IN

(R'~)

₌

£ _{(X; X)~}

~

i

which exhibits _a small increase _near the

phase

transition and _a slower

decay

of

I'

for 1-0.

The difference is caused

by

_a

change

of the internal geometry near the

phase

transition

[32].

With the data for the

sphere

in

figure

8 and the

corresponding

data of the _{torus one can} estimate the critical exponent

fl/vd

of the order parameter

assuming

_a second order transition.

Here _{we use as} the effective critical temperature the

position

of the

peak

of the

specific

heat.

C~~~

(cf. Fig. 10).

An estimate of

fl/vd using

results in

Torus:

fl/vd

= 0.28 + 0.02

,

Sphere: fl/vd

₌ 0.35 + 0.04

(23)

and with the values of vd in

equation (17)

and

(18)

Torus:

fl

= 0.7 + 0.2

,

Sphere: fl

₌ I.1 + 0.2

(24)

Figure

11 shows the

scaling

of the maxima of the

susceptiblity

_X

(Eq. (21), Fig. 9).

The results _are

Torus:

7/vd

₌ 0.62 + 0.06

,

Sphere: 7/vd

= 0.66 + 0.06

(25)

and, using

the values of vd in

equations (17)

and

(18),

Torus: _{7 =} 1.6 + 0.5

,

Sphere:

_{7 =} 2.1 + 0.5

(26)

With these

large

_error bars and estimated values _a t3 -1.5 the results _are almost

compatible

with the

scaling

relation

a+2fl+7=2 (27)

Acknowiedgements

Partial support

by

the SFB 123 and the BMFT

project

0326657D and 031240284 is

gratefully acknowledged.

References

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Polyakov A., Phys. Lett. 103B

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