Bending Rigidity of Fluctuating

Membranes

K.R. Mecke

Bergische Universitat Wuppertal, Fachbereich Physik, D-42097 Wuppertal

Federal Republic of Germany

24 August 1994

Abstract. A lattice model of random surfaces is studied including configurations

with arbitrary topologies, overhangs and bubbles. The Hamiltonian of the surface in-

cludes a term proportional to its area and a scale-invariant integral of the squared

mean curvature. We propose a discretization of the curvature which ensures the scale-

invariance of the bending energy on the lattice. Non-perturbative renormalization

groups for the surface tension and the bending rigidity are applied, which are also

valid at high temperatures and scales above the persistence length. We find at van-

ishing surface tensions a closed expression for the scale dependent rigidity including

the usual logarithmic decay at low temperatures. Different scaling behaviours at non-

vanishing tensions occur yielding characteristic length scales, which determine the

structure of homogeneous droplet, lamellar, and microemulsion phases.

PACS: 5.20 Gg, 68.10.-m

1 Introduction

Models of random surfaces can be applied to such diverse problems as mi-

croemulsions, fluid membranes, elementary particle physics and cosmology. In

particular the interfaces or membranes between homogeneous oil and water

domains decorated by amphiphilic molecules are fascinating examples of ran-

dom surfaces which can be studied by experiments. For a review of such self-

assembling amphiphilic systems see, for example, Ref. [1].

In addition to the tension of the interface, its bending energy is needed to

explain many of the properties of an ensemble of fluctuating interfaces or sur-

faces. An interesting feature, for example, is the change of the effective rigidity

1

due to fluctuations yielding a logarithmic dependence on the scale of the in-

terface if the surface tension is small. The interface therefore appears rigid at

scales smaller than a typical persistence length ξP and crumpled at larger scales

due to the loss of orientational coherence at vanishing bending rigidities. This

renormalization of the rigidity has already been computed in a one-loop pertur-

bation theory [2-6] restricting the configurations of the interface to small planar

undulations.

It is therefore interesting to study large amplitudes in general, including the

influence of overhangs and bubbles, which are dominant at high temperatures or

low rigidities. This would allow a consistent treatment of fluctuating interfaces

with changing topologies, which is necessary for the study of microemulsion

phases for example. Also the influence of non-vanishing surface tensions on the

bending rigidity should be studied in greater detail in order to describe struc-

tures appearing in homogeneuous droplet or lamellar phases.

In this paper we propose a lattice model and a non-perturbative renormal-

ization group to study the statistical behaviour of fluctuating interfaces gov-

erned by surface tension and bending energy. This approach includes effects of

changing topologies (bubbles) of the interface and can be applied also for large

surface tensions. We use an approximation of random surfaces by plaquette

configurations on a cubic lattice, where non-perturbative real-space renormal-

ization schemes have been well developed already [7, 8]. However, in using lattice

approximations of surfaces one has to worry about the scale-invariance of the

bending energy, which is not guaranteed in a simple way. In order to ensure this

symmetry, we propose in Section 2 a discretization of the curvature using ad-

ditional plaquette-variables to describe the curvature yielding a scale invariant

expression for the bending energy. In Section 3 we apply two different types of

real-space renormalization groups to obtain the scale-dependence of the bending

rigidity and the phase-diagram of fluctuating interfaces.

2 The model

We want to describe an ensemble of fluctuating interfaces separating homoge-

neous domains of two different media, for example oil and water in the case of

microemulsions. The surface Hamiltonian can be written for continuous fluid

membranes as

H =

∮

[

r +κ

2

(

1

R1+

1

R2

)2]

dO (1)

with the microscopic (bare) surface tension r, the bending rigidity κ, the cur-

vature radii R1 and R2 of the surface and the surface-measure dO [9]. We split

2

the integral of the squared mean curvature into a Gaussian term

κ

∮

1

R1R2dO (2)

and a remaining term

Hκ :=κ

2

∮ (

1

R21

+1

R22

)

dO , (3)

where the curvatures 1/Ri are separated from each other. The first term - pro-

portional to the Euler-characteristic due to the Gauss-Bonnet theorem - will

not be considered in this paper. We neglect also in the expression (1) a term

proportional to the mean curvature H = 12 ( 1

R1+ 1

R2) and to the Gaussian curva-

ture G = 1R1R2

usually included in the elastic free energy of a fluid membrane.

Both terms - proportional to H as well as to G - can be modelled on lattices in

an appropriate way by using integral geometry [10, 11]. This standard method

of discretization of curvatures can not be applied to the integral of the squared

mean curvature used in Eq. (1). Therefore we want to focus here only on the

energy term (3) in order to study a possible discretization of it and the scaling

behavior of the bending rigidity κ. The influence of the Gaussian term G and

the term proportional to the mean curvature H can then be studied as a natural

extension of the model, which will be done in a following paper.

In the next sections we focus entirely on a possible discretization of the term

Hκ as given by Eq. (3) from which we will deduce the scale dependence of the

couplings r and κ by a real-space renormalization group in Section 3.

Figure 1: Edges on a cubic lattice: Two neighbouring plaquettes could join in

three different ways at the common edge. But for interfacial plaquettes only

rectangular edges < ijk > (a) and planar edges < ijkl > (b) are allowed in the

model. Configurations with four interfacial plaquettes at a common edge are

forbidden (c).

3

Figure 2: Discretization of curvatures using Frenet’s equation: The curvature

k(s) of curves in the plane IR2

is given by the derivative of the angle φ(s) with

respect to the arclength s. Discretizing the curve by plaquettes the differential

reduces to a difference of angles φij located at neighbouring interfacial plaquettes

tij = 1 of the lattice.

2.1 Discretization of the interface

In order to discretize Hκ, we introduce first an approximation of continuous

interfaces by plaquettes of a 3-dimensional simple-cubic lattice Λ. The homo-

geneous domains are represented by cubes with edge-length a centered at the

sites i of the lattice. Each plaquette corresponds to a face of such a cube, i.e.,

to a square perpendicular to a bond < ij > joining neighbouring sites (Fig. 1).

We place N Ising spins si ∈ {1,−1} (i = 1, . . . , N) at the vertices of the cu-

bic lattice. Two adjacent and different spins si 6= sj - representing neighbouring

oil and water domains for example - form an interface-plaquette of area a2 per-

pendicular to the bond < ij > between the sites i and j (see Fig. 1). Sometimes

expressions can be simplified using for each unit-cell i the occupation variables

ti = (1 + si)/2 ∈ {0, 1}

indicating whether the cell is filled with water or not. Additionally we introduce

the interface variables

tij =1 − sisj

2= ti + tj − 2titj ∈ {0, 1}

located at the bond < ij > indicating the presence (tij = 1) of an interface if

adjacent cells are occupied differently. Thus tij indicates the membership of the

related lattice plaquette to the interface. Due to the neighbouring occupation

variables ti an interfacial plaquette possesses an inner side (ti = 1) and an outer

side (tj = 0).

4

In Fig. 1 we show three different kinds of edges formed by neighbouring

plaquettes, i.e., three possibilities in which neighbouring plaquettes could join

at a common edge. Two plaquettes can join perpendicularly at a three-site-edge

< ijk > forming a ’rectangular edge’ (see Fig 1 (a)) or parallelly at a four-site-

edge < ijkl > forming a ’planar edge’ (see Fig. 1 (b)). If the plaquettes are

perpendicular they have a common neighboured lattice site, which is denoted

by i in Fig. 1 (a). In Fig. 1 (c) on the right side four plaquettes join at a

common edge. As those configurations allow no determination of the course of

the interface, we will forbid them in this model. Each of the four edges of a

plaquette should have exactly one neighbouring plaquette, and configurations

{ti} of spins such as those illustrated in Fig. 1 (c), where four plaquettes coincide

are not allowed, i.e., ’self-intersections’, are forbidden.

Thus the ensemble {si} of spin configurations restricted by the constraint

of no ’self-intersections’ corresponds to an ensemble of plaquette configurations

{tij} representing closed interfaces and approximating, for example, continuous

membranes between oil and water. So far we have described the usual approach

to discretize interfaces by Peierls-countours on a cubic lattice. This discretization

procedure works well in many cases to describe fluid interfaces. But in order to

describe curvatures of interfaces we have to do an additional step.

Let us consider, for example, a 2-dimensional subspace or sublattice pictured

in Fig. 2, which cuts a planar, closed curve out of a 2-dimensional interface.

The continuous curve on the left side is approximated on the right side by a

sequence of straight lines on the sublattice, i.e., by a sequence of plaquettes on

a 3-dimensional lattice. As a result of this discretization of the curve we loose

information about its slope and therefore about its curvature. In order to restore

this information on the lattice we introduce at each plaquette < ij >, i.e., at

each line segment in Fig. 2, a vector ~tij representing the slope of the continuous

curve. The vector ~tij is assumed to be parallel to the tangent vector ~ts of the

curve pointing in the direction of increasing arclength s.

In 3-dimensional space a 2-dimensional surface has not only one tangent

vector at each surface point but a 2-dimensional tangent plane. Therefore we

introduce an index ν = 1, 2 and distinguish two vectors ~t(ν)ij at each plaquette

of the lattice indicating two perpendicular directions of the tangent plane. The

vectors are lying in the two different 2-dimensional sublattices joining at the

plaquette < ij >. We want to emphasize that within this approach the interface

configuration on the lattice is not only given by plaquettes but additionally by

the directions of the tangent planes represented by the vectors ~t(ν)ij .

5

Figure 3: Typical configurations of interfaces: On the left side a nearly flat

interface is shown with a large number of right-angled edges. Nevertheless the

bending energy is nearly zero due to the small deviations of the tangent vectors.

On the right side, a closed curve with four edges and L plaquettes on each side

is shown. The minimal bending energy of such a configuration scales according

to the reciprocal length 1/L like the bending energy of a continuous circle.

2.2 Discretization of the curvature

In order to describe the curvature radius R of discrete surfaces in a local way,

we use the Frenet equation for planar curves parametrized by the arclength s,

which simply relates the curvature

k(s) :=1

R=

∂φ(s)

∂s= lim

ε→0

arccos{~ts · ~ts+ε}ε

(4)

to the derivative of the angle φ(s) between the tangent vector of the curve and

a prescribed space direction (see Fig. 2). Thus the curvature of the curve is

uniquely determined by the function φ(s).

If we discretize curves by plaquettes only, we would loose information about

its slope and therefore about its curvature defined by Eq. (4). However, the

introduction of the vectors ~t(ν)ij allows the definition of angles φ even on the

lattice by

φ(ν)ij = arccos{~nij · ~t(ν)

ij } ∈ [−π

2,π

2] . (5)

These variables describe the angle between the normal vector ~nij of the plaquette

< ij > and the assumed direction ~t(ν)ij of the surface. Note that we use an index

ν = 1, 2 to distinguish between the two perpendicular 2-dimensional sublattices

joining at the plaquette < ij >. Strictly speaking, two angles φ(ν)ij are located

at each plaquette describing the two curvature directions, whereas only one is

depicted in Fig. 2. Approximating the interface by plaquettes of a lattice we

can choose the directions of the principal curvatures of the discrete interface

6

parallel to the axes of the plaquettes. This means that the principal curvatures

are given by the curvatures in these directions.

The discretization of Equation (4) yields immediately a difference equation

for the angles φ(ν)ij . We have to take into account the possible change of the

reference direction ~nij , i.e., the changing definition (5) of the angles φ(ν)ij and

φ(ν)ik located at the interfacial plaquettes forming a rectangular edge < ijk >

(see Fig 1. (a)). Additionally, we have to define the sign of the curvature at the

edge < ijk >, which is choosen to be positive if the common neighbouring site

of the plaquettes is occupied, i.e., is lying on the inner side of the interface (see

Fig. 1). In this way we obtain the discretized curvature

k(ν)ijkl =

φ(ν)ij − φ

(ν)kl

a(6)

located at a four-site-corner < ijkl > of neighbouring planar plaquettes and the

curvature

k(ν)ijk =

φ(ν)ij − φ

(ν)ik

a+

π

2asi (7)

located at the interface edge < ijk > of neighbouring perpendicular plaque-

ttes. Here we assume that the lattice constant a determines a microscopic cell-

curvature k0 = π/a of the interface, which is an upper limit of the bend.

In the following we can neglect the index ν at the curvatures kedge (kedge =

kijk or kijkl) because the sublattice indicated by ν is already determined by the

lattice sites i, j, k, and l. Note that even planar plaquettes shown in Fig 1.(b)

could have a curvature at the common edge < ijkl >.

The term πsi/2a in Eq. (7) accounts for the change of the direction of the

plaquettes, i.e., of the reference direction. The spin si belongs to the cell adjacent

to both plaquettes, i.e., to both broken bonds tij and tik. This term breaks the

symmetry between the two different sides of the interface due to the definition

of positive curvature of a droplet with si = 1 in a matrix with sj = −1. It is

omitted at planar edges.

The direction of the curve, i.e., of the tangent vectors ~t(ν)ij , should not deviate

more than a right angle from the direction of a plaquette. Therefore the angles

φij are constrained to values smaller than π/2 or less. It is important to recognize

that the sum∑

{edges} kedge = 2π of the curvature variables kedge along the

edges of a closed curve is just 2π, because the differences of the angles cancel

each other leaving the net change in the direction of the plaquettes. This result

agrees with the integration of the Frenet equation (4) over a closed curve.

The main result of this Section is the introduction of discrete curvatures of an

interface by differences of angles located at plaquettes. An alternative approach

7

to describe curvatures on a cubic lattice would be the introduction of variables

kedge right at the edges of neighbouring plaquettes describing the curvature

directly instead of in terms of angles φij . Nevertheless, the values of kedge should

be constrained by the inverse lattice constant k0 = π/a - suppressing scales

below a - and by the mentioned integrability condition along closed curves. Then

the variables φij located at plaquettes could be considered as parametrization

of the curvature variables kedge satisfying these two conditions.

However, using the definitions (6) and (7) of discrete curvatures we obtain

the following approximations of continuous integrals

Iσ := a2∑

<ij>

tij ≈∫

dO

Ic := a2∑

<ijk>

kijktij tik + a2∑

<ijkl>

kijkltij tkl

=π

2a∑

(ijk)

sitij tik ≈∫ (

1

R1+

1

R2

)

dO

Iκ := a2∑

<ijk>

k2ijktij tik + a2

∑

<ijkl>

k2ijkltij tkl

≈∫ (

1

R21

+1

R22

)

dO .

(8)

The sums∑

<ij>,∑

<ijk> and∑

<ijkl> run over all bonds < ij >, right-angled

triangles < ijk >, and squares < ijkl > of neighbouring sites of the lattice (see

Fig. 1). The first integral Iσ is just the total area of the surface approximated

by the number of plaquettes. In the second integral the variables φij drop out

and the integral of the mean curvature Ic depends only on the number of signed

interface edges sitij tik in agreement with the results of integral geometry [10, 11].

Thus Ic can be approximated by spin-variables on a lattice without the use of

additional angle-variables.

Nevertheless, in order to approximate the integral of the squared mean cur-

vature Hκ, the angle-variables φij are necessary. If the angles φij are not taken

into account, e.g. by setting φij = 0, the curvature term Iκ reduces to

I0κ =

(π

2

)2 ∑

{<ijk>}

tij tik , (9)

i.e., to the number of right-angled edges tijtik of the configuration, each sup-

porting a curvature 1/a2. Note that we use the lattice constant a explicitly in

8

Figure 4: A finite cell is shown with typical interface configurations. Effective

bending energies veff of the spin-configurations are defined by decimation of

the angle-variables φij inside the finite cell, where the angles at the interface

boundary are identified φij = φkl.

Note that in the low temperature limit x → 0 the effective bending energy

vanishes or reaches the value v/3 depending on the net curvature inside the cell.

Eqs. (8) to emphasize the dimension of the different integrals. Whereas the total

surface area Iσ is proportional to a2 and the mean curvature integral Ic is pro-

portional to a, the integral of the squared mean curvature I0κ does not depend

on the lattice constant anymore.

The expression (9) is often used for the bending energy of membranes on

lattices. But in contrast to the integral of the squared mean curvature Hκ in

the continuum given by Eq. (3), the discrete integral in Eq. (9) is not scale-

invariant. If the size of a configuration is enlarged, for example a globular domain

of spins with si = 1 growing in a sea of spins with sj = −1, the curvature

term I0κ increases. The scale-invariance of the bending energy Hκ is restored

in the derived expression (8) for the squared curvature integral Iκ by taking

into account additional curvature variables kedge with absolut values between

zero and k0 = π/a. This allows scale-invariant configurations depicted in Fig.

3 for example. A straight line - approximated by a sequence of right-angled

plaquettes and by vectors ~tij pointing mainly in the direction of the line - has

a zero bending energy independently of its length. Of course if the vectors are

pointing in the directions of the line segments, which is also a valid interface

configuration, the bending energy would increase proportional to the number

of right-angled edges, i.e., to its length. On the other side a globular domain of

spins with si = 1 surrounded by a sea of spins with sj = −1 can have a bending

energy independently of its linear size L, if the curvature located at the edges

are smoothed by vectors. This will be shown in Section 2.4.

9

2.3 The lattice model

Before we discuss the properties of the curvature term Iκ in Eq. (8) let us define

the partition sum of the model

Z =∑

{si}

∫ +∞

−∞

D[φ]e−βH (10)

by the summation over all plaquette configurations and by the integration over

the additional variables φij . Note that the sum∑

{si}is taken only over spin-

configurations with no self-intersecting interfaces, i.e., with no edges joining

four plaquettes. Using the approximations (8) the Hamiltonian of the interface

is given by

−βH = σ∑

<ij>

tij − v

(

2

π

)2

(11)

∑

<ijk>

(φij − φik +π

2si)

2tij tik +∑

<ijkl>

(φij − φkl)2tijtkl

with the normalized bending rigidity

v :=βκ

2

(π

2

)2

and the negative surface tension

σ = −βra2 .

Note that the size a2 of a plaquette and the square of the microscopic curvature

k0 = π/a cancel each other in the definition of the rigidity v. The integration

measure D[φ] for the angles φij is choosen to be Gaussian

D[φ(1), φ(2)] =∏

ij

dφ(1)ij dφ

(2)ij

m√2π

e−12 m2(φ

(1)ij

+φ(2)ij

) (12)

with a mass m = 2/π to ensure a variance√

< φ2 > = π/2 at vanishing rigidity

v = 0. The angles φij should not be much greater than π/2 to suppress scales

lower than the lattice constant. Of course measures like the step function

D[φ(1), φ(2)] =∏

ij

dφ(1)ij dφ

(2)ij Θ

(

π/2− |φ(1)ij |)

Θ(

π/2 − |φ(2)ij |)

(13)

could also be used, but to perform the integrals the Gaussian measure is more

convenient.

10

Figure 5: Bond-moving-scheme in the Migdal-Kadanoff approximation: A 3×3-

cell with lattice constant L is renormalized to a cell with lattice constant bL

and a scale-factor b = 3. In order to sum over the spins inside the 3 × 3-cell

the couplings are moved to 1-dimensional chains connecting the corners of the

cell. Note that for a interface configuration curved inside the cell only a bending

energy v/b are moved.

The index ν at φ(ν)ij denotes the two different principal curvatures at each

plaquette. As the principal curvatures are assumed to be not coupled in the

Hamiltonian H given in Eq. (11) (see Hκ in Eq. (3)), the total bending energy

of the interface equals the sum of the bending energies along planar interfacial

curves - depicted, for example, in Figs. 2 (b), 3, and 4. Thus considering the

effects of the variables φij on the bending energy, we can neglect the index ν

in the expressions and restrict the discussion to a 2-dimensional sublattice, i.e.,

to a single planar curve on the interface. In the next section we will define an

effective bending rigidity along such 2-dimensional curves.

2.4 Effective bending rigidity

Performing the Gaussian integrals of the variables φij along a closed curve of

N plaquettes one obtains an effective bending energy

e−βHeffκ (curve) :=

∫

curve

D[φ]e−βHκ (14)

= A−1N · exp{−vI0

κ +κ

2

(π

2

)2~bt∆−1~b}

with the spin interaction~bt∆−1~b = (15)

1

N

N∑

n=1

(

N∑

k=1

(bk − bk−1)[

cos{

k 2πN n}

+ sin{

k 2πN n}]

)2

2 + x − 2 cos{

2πN n}

11

and the normalization

AN =

√

(

1

x(+)

)N

+

(

1

x(−)

)N

− 2

(

1

x

)N

, (16)

x(±) =2

1 + 2/x ±√

1 + 4/x.

In order to simplify the notation, we have defined the abbreviation x = m2/ βκ

and a variable bk = {−1, 0, 1}, b0 = bN , at each edge k indicating if the neigh-

bouring plaquettes are parallel, bk = 0, or right-angled, bk = si = ±1 (see Fig.

1). The sign takes into account the orientation of the curvature, i.e., it is positive

if the common neighbouring site i of the plaquettes is occupied (si = 1).

In addition to the number of edges I0κ in Eq. (9) the effective bending energy

βHeffκ := v I0

κ − κ

2

(π

2

)2~bt∆−1~b + log AN (17)

contains a ’long-ranged’ interaction ~bt∆−1~b of the edges, i.e., of the spins si

along an interfacial curve, which changes the effective rigidity of the interface.

The last term in Eq. (17) can be considered as an entropy contribution of the

angle variables. It vanishes for v = 0, and in the limit v → ∞ it is proportional

to the length of the curve, i.e to an additional area term.

We discuss first the limit βκ >> m2 of large bending rigidity, i.e., x → 0.

The main contributions to the partition sum Z are given by configurations with

smooth interfaces. Instead of performing the integral∫

D[φ] in Eq. (10) we

can take the minimum energy configuration, whose energy scales like 1L for a

closed curve, where L is the number of plaquettes used to approximate the curve

(see Fig. 3). This is a striking result since the bending energy of a cylindrical

configuration of radius R and height h is now proportional to hR in the lattice

as well as in the continuum case. Therefore the energy of a globular domain of

spins with si = 1 surrounded by a sea of spins with sj = −1 is scale invariant.

This scale invariance, easily obtained in the continuum version, requires in an

Ising lattice gas model an interaction mediated by the angle variables φij .

In the limit βκ << m2, i.e., x → ∞, in which the angle variables can

fluctuate strongly and independently, a configuration with φij ≈ 0 contributes

an equal amount to the partition sum as the lowest energy configuration. In

this case the interaction ~bt∆−1~b vanish and the effective bending energy Heffκ

reduces to the number of right-angled edges I0κ (see Eq. (9)). The typical config-

urations are not scale-invariant any more since only the local bending is relevant

and the variables φij do not mediate the global structure.

12

Figure 6: Scale-dependence of the bending-rigidity v ignoring the influence of

surface tension: For scales smaller than the persistence length, L < LP , the

bending rigidity shows a logarithmic decrease v(L) ∼ v(1) − απ/32 log{L},whereas for L > LP an exponential decay occurs. The factor α doesn’t depend

on the initial value v(1) but it depends on the chosen approximation for the

renormalization group. For the Migdal-Kadanoff-approximation (full lines) one

obtains α ≈ 10.1, whereas the finite-cell-renormalization (circles) yields α ≈ 1.4.

3 Renormalization of the bending rigidity

In this Section we discuss two real space renormalization groups for the lattice

Hamiltonian (11). First an instructive but in some points misleading Migdal-

Kadanoff approximation, and secondly a finite-cell renormalization.

The main features of the proposed model are caused by the additional vari-

ables φij introduced in order to approximate the bending energy of interfaces

on lattices in an appropriate scale-invariant way. Therefore let us describe first

the integration procedure for these variables. In Fig. 4 some typical plaquette

interfaces are shown spanning a finite domain or cell of the lattice with edge

length b = 3 in units of the lattice constant a. We want to define an effective

bending rigidity veff (b) for such curves within the cell without assuming a spe-

cific course of the curves outside the cell. For this purpose we identify the angles

φij and φkl at the sides of the cell to simulate closed curves (periodic boundary

condition). Then we integrate over the remaining angles within the cell. Using

Eqs. (15) and (17) we define an effective bending rigidity by

veff = v(

I0κ −~bt∆−1~b

)

. (18)

13

Consider, for example, the right-angled curve depicted in Fig. 4 (a). It shows two

’planar edges’ and one ’rectangular edge’ inside the cell, i.e., it can be described

by b1 = 0, b2 = ±1, and b3 = 0 using the notation of Eq. (15). Thus we obtain

the effective bending rigidity

veff = v1 + x

3 + x

for the curve (a) and in analogy

veff = v1 + 3x

3 + xand veff = v

4x

2 + x

for the curves (b) and (c) shown in Fig. 4.

Let us consider again the limit βκ >> m2 of large bending rigidity, i.e.,

x → 0. For the curves (a) and (b) the effective bending rigidity is veff = v/3,

i.e., both show a total bend within the cell, whereas for curve (c) veff = 0,

i.e., it is ’planar’. Generally within cells with side-length b the effective bending

rigidity is veff = v/b for bent and veff = 0 for planar curves.

This result is important since it indicates a renormalization of the curvature

radius. Recall that the surface tension σ = −βra2 is a product of the bare

parameter r and the area a2 of plaquettes, whereas the normalized coupling v

results from the bending rigidity κ, the area a2 of plaquettes and the squared

cell curvature k20 = (π/a)2 at an edge (see the definitions for Eq. (11)). More

precisely the rigidity v includes an integral of the squared curvature k20 across

an edge of two right-angled plaquettes multiplied by the length a of the edge.

Thus if the curvature radius 1/k0 of a dominant configuration is renormalized

by a factor b the effective rigidity veff scales as 1/b.

In the opposite limit βκ << m2, i.e., x → ∞, the effective bending rigidity

veff reduces to the number of right-angled edges according to Eq. (9) because

for a typical configuration the curvature remains k0 = π/a of. Thus the renor-

malization of the curvature radius depends on the bare bending rigidity v. If

the bending rigidity is large the curvature radius of dominant configurations

scales according to the size of the cells. But for low rigidities the curve follows

closely the plaquette configuration and the typical curvature radius does remain

constant even when the size of the cell is enlarged.

However, the integration over the angles yields an effective bending rigidity

veff and thus a renormlization of the curvature radius. Now we have to sum

over all plaquette configurations within a cell.

14

Figure 7: Phase-diagram obtained by the Migdal-Kadanoff approximation: Ad-

ditional to the two fixed-points FP1 and FP2 of the 3-dimensional Ising-model

we find a multi-critical fixed-point MFP at T = 0. The disordered phase µ

extends up to T = 0 and separates the homogeneous and lamellar phases. We

suppose that in an extended version of the renormalization group the fixed-point

MFP will move to finite temperatures.

3.1 Migdal-Kadanoff approximation

The Migdal-Kadanoff scheme consists of sucessive bond-moving operations and

spin-summation (dedecoration) of 1-dimensional chains. For a detailed descrip-

tion of the method see Refs. [7, 8]. In Fig. 5 we show a 3 × 3 square cell which

is renormalized by a scale factor b = 3. The increasing lattice constant a′ = La,

i.e., the distance of neighbouring spins, is measured in units of the microscopic

length a. The renormalized quantities are always denoted by primes. The lattice

sites m′ of the renormalized lattice are the sites m on the corners of each cell.

The renormalized spins s′m′ = sm are identified with the unrenormalized ones

located at the same site. Also the angle variables φij located in the middle of a

bond < m′n′ > of the renormalized lattice defines the new variables φ′m′n′ = φij .

In a first step of the Migdal-Kadanoff approximation the nearest-neighbour

couplings σ are moved to 1-dimensional spin-chains connecting the cell-spins s′

at the corners. This bond-moving procedure yields a new coupling

σ = b2σ

between spins in the chains.

For the bond-moving of the bending coupling v, we integrate first over the

variables φij in the limit x → 0 of large bending rigidity βκ compared with m2.

15

This means that an interface spanning the cell has either no effective bending

rigidity (veff = 0 for the ’planar’ interface shown in Fig. 4 (c)) or an effective

bending rigidity veff = v/b (with b = 3 for the ’curved’ interfaces shown in

Fig. 4 (a) and (b)) depending on the net curvature in the cell. The coupling v/b

takes account of the scaling of the curvature radius. If the interface is curved in

the cell, the bending radius scales with a factor b.

Now we localize this bending energy v/b entirely at one edge of the interface

inside the cell. After splitting the energy v/b tijtik in a nearest-neighbour inter-

action −v/2b sisk and a interaction v/2b sisk of diagonal neighbouring spins,

we move them to the sides and the diagonal of the cell (see Fig. 5). The bond

moving along the third cartesian direction (not shown in Fig. 5), results in an

effective coupling

v = bv

b= v .

By this procedure we have taken into account the whole bending energy of the

interface. Nevertheless, we neglect the couplings of the spins inside the cell,

except the couplings between the spins located on the chains shown in Fig. 5.

We define the renormalized couplings v′ and σ′ by the usual dedecoration

transformation

v′ = 2 tanh−1(

tanhb( v2 ))

,

σ′ = 2 tanh−1(

tanhb( σ2 − v

2 ))

+ v′(bL, b) ,(19)

where the renormalized lattice constant bL is enlarged by a factor b. The in-

finitesimal rescaling limit b → 1 + dε yields the differential equations

L∂v(L)

∂L= 2 cosh(

v

2) sinh(

v

2) log | tanh(

v

2)| ,

L∂σ(L)

∂L= 2 cosh(

σ − v

2sinh(

σ − v

2) log | tanh(

σ − v

2)|

+2σ + L∂v(L)

∂L.

(20)

We want to emphasize that these renormalization group equations can also be

obtained using the expression I0κ (see Eq. (9)) for the bending energy if one takes

explicitly into account the scaling of the curvature radii. The bending rigidity

v is a product of κ, the area a2 and the cell curvature k20 . This means that the

scaling from v results not only from the renormalization of the bare parameter

κ but also from the scaling of the plaquette size a′ and the curvature radius

1/k′0 of the cell. Therefore the product a′2 k′2

0 is scale-invariant if the curvature

radii scales in accordance with the plaquette size. But this scaling of the cell

curvature drops out in the expression I0κ because π/k′

0 = a is set equal to the

16

microscopic lattice constant. Thus the scaling of the cell curvature k′0 has to be

considered explicitly by setting v = v in the renormalization group equations

(19).

However, in this paper we have defined additional variables to introduce the

curvature radii yielding in a straight-forward manner the correct scaling v = v.

The advantage of this approach, i.e., using the Hamiltonian (11) instead of (9),

is the possibility to break the scaling of the curvature radii at a specified length

scale using an improved renormalization group, which will be discussed below.

The correct scaling of the couplings, i.e., the scale invariance of the bending

energy, can be seen in the low temperature limit T → 0 of the differential

equations (20):

∂v

∂L∼ −1/L ⇒ v(L) ∼ v(1) − log{L} ,

∂σ

∂L∼ [−1 + 2σ]/L ⇒ σ(L) ∼ σ(1) · L2 .

(21)

The surface tension σ scales proportional to the plaquette size L2, whereas the

bending rigidity v remains constant for T → 0.

The first equation in Eq. (20) is independent of σ due to the decoupling of

the spin chains shown in Fig. 5. Therefore, it can be solved exactly, yielding

v(L) = 2 · tanh−1

(

tanh

(

v(1)

2

)L)

, (22)

which is shown in Fig. 6 for two different initial values v(1) (full lines). We

observe a logarithmic decay of the bending rigidity already found in the low

temperature limit given by Eq. (21). But considering the approximation of Eq.

(22) for large scales L,

v(L) →{

v(1) − log(L) if v(1) >> 1

2e−L/L0 if L >> 1 or v(1) << 1, (23)

with L−10 = log | tanh(v(1)/2)|, we find an exponential decay of the bending

rigidity above the persistence length LP := e−v(1), at which the logarithmic

dependence would go to zero. The logarithmic scaling behaviour for large rigidi-

ties v was found previously in a one-loop perturbation theory for continuous

membranes [2-6]. However, the non-perturbative expression (22) for the bend-

ing rigidity is in accordance with the continuum description but allows also a

description above the persistence length LP .

In Fig. 7 we show the phase-diagram obtained by the renormalization group.

We find a homogeneous ordered phase H at large positive surface tensions

17

Figure 8: Influence of the surface tension on scale-dependent bending rigidity v:

The bending rigidity v shows qualitatively different behaviors depending on the

surface tension r. At small values for r the bending rigidity decrease logarithmi-

cally. If r flows to the high-temperature fixed-point at v = r = 0 this decrease

of v remains even at larger scales (uptriangles). But if r increases towards the

homogeneous fixed-point at v = 0, r → ∞, the bending rigidity increases above

a characteristic scale and reaches a constant value (downtriangles). In the op-

posite limit r → −∞, i.e., for flows in the lamellar phase, the rigidity tends to

infinity (squares).

r = −kBTσ/a2 and a lamellar ordered phase L at negative tensions r. The

couplings flow to the low-temperature fixed-points at v = 0, σ → −∞ and at

v = 0, σ → ∞, respectively. In between a disordered phase µ occurs even at low

temperatures in which the couplings flow to the high-temperature fixed-point at

v = σ = 0. This phase behavior is caused by the scale-invariance of the bending

rigidity which allows large fluctuations of the interface at small surface tensions

stabilizing a disordered phase. Additional to the two critical fixed-points FP1

and FP2 of the 3-dimensional Ising-model at vc = 0, σc = 0.28 we find a multi-

critical fixed-point MFP at T = 0. The disordered phase µ extends up to T = 0

and separates the homogeneous and lamellar phase. We suppose that in an ex-

tended version of the renormalization group the fixed-point MFP will move to

finite temperatures yielding a first-order transition between the homogeneous

and the lamellar phase.

18

3.2 Finite-cell approximation

In order to obtain a more detailed scaling behaviour including the effects of the

surface tension σ we apply a finite-cell renormalization to the Hamiltonian (11).

We choose a finite cell of spins with periodic boundary conditions like the

one shown in Fig. 4. The spins are identified in one Cartesian direction, i.e., the

configurations are restricted to surfaces, which are curved only in one direction.

This seems appropriate, as the curvature variables of different space directions

are separated from each other in the Hamiltonian (11). For each interface we

perform the integration over the variables φij , using periodic boundary condi-

tions at the sides of the cell (see Fig. 4), before we sum over all allowed surface

configurations inside the cell. In contrast to the Migdal-Kadanoff approximation

we take into account the full expression (18) for the effective bending rigidity

veff .

Each surface configuration is classified as a ’homogeneous’, ’planar’, or ’curved’

type, as shown for three examples in Fig. 4. The configurations (a) and (b) de-

scribes a net curved interface, thus belonging to the curved class, whereas the

interface (c) is planar. Homogeneous configurations are, for example, interfaces

which remain inside the cell like an isolated bubble.

We define the renormalized couplings σ′ and v′ by

e−βH(s′) =∑

{si}

∫

D′[φ]e−βHP (s′, {si}) , (24)

where the renormalized Hamiltonian is defined for different classes s′ of config-

urations by

βH(s′) =

const. if s′ = homogeneous

const. − σ′ if s′ = planar

const. − σ′ + v′ if s′ = curved

. (25)

Other possibly generated couplings are truncated by this procedure. The main

problem of this kind of renormalization is the weighting factor P (s′, {si}). It is

equal to unity if the spin configuration {si} belongs to the class s′, and zero

otherwise. There is a slight ambiguity in defining which configuration belongs

to which class, but the symmetry of the configuration {si} should be almost the

symmetry of the class s′, and P should obey furthermore the following natural

conditions∑

s′

P (s′, {si}) = 1 ,∑

{si}

P (s′, {si}) = const. , (26)

∑

{si}

tijP (s′, {si}) = const.

19

guaranteeing an invariant partition sum and the existence of an attractive high-

temperature fixed-point at β = 0.

We have considered different choices of P (s′, {si}), i.e., different classifica-

tions of the spin configurations yielding almost similar results.

In Fig. 6 we show the results for the renormalization of the bending rigidity,

where we ignore the influence of the surface tension by setting σ = 0 at every

step. In this way, we can test the influence of different renormalization proce-

dures on the flow of the bending rigidity κ(L) = 8/π2 kBT v(L). We find within

the finite-cell-renormalization the same behaviour as in the Migdal-Kadanoff-

scheme despite a change of the steepness of the logarithmic decay

κ(L) = κ(1) − α

4πkBT log(L) . (27)

The constant α depends on the renormalization group used. In particular, for

the Migdal-Kadanoff scheme we found α = 10.1 for the differential equation

(20), α = 2.9 for the discret version as given by Eq. (19) with b = 3, and

α = 1.4 for the finite-cell-approximation with scaling factor b = 2. These values

are comparable with the field-theoretical results α = 1 (Ref. [2]) and α = 3 (Ref.

[3]). Nevertheless the values of α depend strongly on the choosen approximation

necessary in real-space renormalization groups. The value α = 1.4 seems to be

the most reliable as the finite-cell approximation is the most appropriate one at

low temperatures.

In Fig. 8 we show the influence of the surface tension on the bending rigidity.

Whenever σ flows to the high-temperature fixed-point at zero surface tension

(σ = 0), we observe only a slight deviation of the logarithmic behaviour men-

tioned above. But if it tends to the low temperature fixed-point σ → −∞, i.e.,

towards a homogeneous phase, where interfaces are suppressed by high surface

tensions, the bending rigidity increases above a characteristic length after the

logarithmic decay. The surface tension σ increases proportional to L2 whereas

the bending rigidity κ(L) → κ∞ reaches a constant value. The length at the

turning-point may be identified as the typical size of droplets dispersed in an

homogeneuous phase.

In the opposite limit, where σ flows to the low temperature fixed-point

σ → ∞ forcing a lamellar structure of the membranes the logarithmic decay

of the bending rigidity stops as well, and above the characteristic length-scale

an increase occurs. The length at the turning-point may be identified as the dis-

tance of lamellar sheets, which are the dominant configurations at high tensions

σ. Above this length, the rigidity κ is the bending rigidity of a multilayered

structure and not of a single membrane.

The increase of the effective bending rigidity indicates that this model is ap-

propriate to study fluctuating membranes not only at vanishing tensions but also

20

to describe structures appearing in homogeneuous droplet or lamellar phases.

In particular we find other characteristic lenghts besides the microscopic lattice

size: a persistence length for fluctuating surfaces in the microemulsion phase,

the membrane distance in the lamellar phase and a typical droplet radius in the

homogeneous phase. Extended phase-diagrams and structure functions of this

model will be discussed in a following paper including a detailed analysis of the

finite-cell renormalization.

In this paper we proposed a lattice model and a non-perturbative renor-

malization group for the study of fluctuating interfaces governed by surface

tension and bending rigidity. The discretization and renormalization of curva-

tures within a lattice model may open the possibility to study random surfaces

in a non-perturbative way, including effects of surface tension and changing

topologies of the surfaces.

21

Acknowledgments

It is a pleasure to thank H. Wagner for numerous stimulating discussions and

the encouraging support in Munich. This work was part of my diploma thesis

at the LMU in 1989.

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