In-plane deformation of a triangulated surface model with metric degrees of freedom

# In-plane deformation of a triangulated surface model with metric degrees of freedom

Hiroki Mizuno and Hiroshi Koibuchi
###### Abstract

Using the canonical Monte Carlo simulation technique, we study a Regge calculus model on triangulated spherical surfaces. The discrete model is statistical mechanically defined with the variables , and , which denote the surface position in , the metric on a two-dimensional surface and the surface density of , respectively. The metric is defined only by using the deficit angle of the triangles in . This is in sharp contrast to the conventional Regge calculus model, where depends only on the edge length of the triangles. We find that the discrete model in this paper undergoes a phase transition between the smooth spherical phase at and the crumpled phase at , where is the bending rigidity. The transition is of first-order and identified with the one observed in the conventional model without the variables and . This implies that the shape transformation transition is not influenced by the metric degrees of freedom. It is also found that the model undergoes a continuous transition of in-plane deformation. This continuous transition is reflected in almost discontinuous changes of the surface area of and that of , where the surface area of is conjugate to the density variable .

Keywords: Triangulated surface model; Regge calculus model; Monte Carlo; First-order transition; In-plane transition

PACS Nos.: 11.25.-w, 64.60.-i, 68.60.-p, 87.10.-e, 87.15.ak

## 1 Introduction

A surface model for membranes is defined by a mapping from a two-dimensional surface to . Not only the mapping but also the metric of is assumed as dynamical variables of the model in its statistical mechanical study . However, the variable is always assumed to be the Euclidean metric or the induced metric of the mapping in the numerical studies that have been performed so far . In those studies, the variable is fixed in the partition function, and as a consequence a role of in the phase structure of the model remains unclear.

The metric of can be discretized as a variable by using the edge length of triangles in the Regge calculus approach to the triangulated surface model. The model is defined by the Hamiltonian , where and are the Gaussian bond potential and the bending energy, and is the bending rigidity . In this approach, one can see whether or not influences the discontinuous transition between the smooth phase at and the collapsed phase at . This transition and related phenomena have long been studied numerically as well as analytically by many groups .

Such a Regge calculus model was recently studied in Ref. ?, where is embedded in and the metric is slightly extended from the conventional Regge metric by incorporating a degree of freedom for a deficit angle of the triangles in . One of the non-trivial results reported in Ref. ? is that the deficit angle varies almost discontinuously at the transition point, although the discontinuity is very small compared to the deficit angle itself.

However, it is still unclear whether or not such a discontinuity in the internal geometric quantities can also be seen in the case where the edge length of triangles is fixed while the deficit angle is varied. Moreover, it is also nontrivial whether the fluctuating metric influences the phase structure of the model with a constant metric.

In this paper, we study a surface model with a metric variable, which depends only on the deficit angle of triangles. Since the surface is embedded in , the numerical simulation is more time consuming than that of the model in Ref. ?. However, the metric is simplified by fixing the edge length of the triangle in such that , and hence the numerical simulations are greatly simplified compared to those for the model in Ref. ?.

We should note that the surface area of varies if the parameter in is varied while is fixed. However, the parameter can always be fixed to because of the scale transformation . Indeed, changes from to while the partition function remains unchanged; it changes up to a multiplicative constant, under this scale transformation. This implies that can always be fixed to .

## 2 Discrete surface model

A discrete Hamiltonian and the partition function are introduced in this section. These discrete quantities are closely related to those in Ref. ?, where the two-dimensional surface is embedded in by a mapping , while in this paper is embedded in . Fig. 1: (a) The smooth triangle Δ in M, (b) the image X(Δ) of Δ by a mapping X from M to the external space R3, (c) a local coordinate domain D in R2 and the regular triangle Ψ(Δ), and (d) a unit normal vector n0 of X(Δ) and those ni(i=1,2,3) of the nearest neighbor triangles.

A two-dimensional surface is smoothly triangulated, and a triangle is shown in Fig. 1(a), where denotes the edge length, and denotes the internal angle of . The triangle in is mapped into by as shown in Fig. 1(b), where the triangle is a linear one. The edge lengths and the internal angles of are denoted by and , respectively.

Let denote a coordinate mapping from to a domain in , then we have a linear triangle in , which is shown in Fig. 1(c). The edge length of is assumed to be identical to the one in , while the internal angle of differs from of , where . In the conventional Regge calculus model, there is no difference between in Fig. 1(a) and in Fig. 1(c) because .

As mentioned in the Introduction, one basic assumption is that

 L1=L2=L3=1 (1)

for all . This implies that . Another assumption is expressed as

 Φ1=Φ2=Φ3=Φ, (2)

where depends on , and is not always identical to as in the model of Ref. ?. We should note that ”smoothly triangulated” does not always mean , where is an internal angle meeting at the vertex of in .

Since is not always given by , the deficit angle can be defined such that

 φ=3Φ−π, (3)

or equivallently

 Φ=Φ0(1+φπ)=13(π+φ). (4)

We should note that is not always a linear triangle, because the edges of are curved and the deficit angle is not always zero. On the contrary, the triangle is linear, because the deficit angle of is fixed to be zero.

The surface model is defined by the discrete metric

 gab=(1FF1),F=cosΦ,|F|<1, (5)

where represents the deficit angle . This is the induced metric of the coordinate mapping . The inequality in Eq. (2) implies that is positive definite. The area of in is given by

 AΔ=12√1−F2. (6)

The area varies according to the variation of although the edge length is fixed. Since varies as a function of , we call the model as a Regge calculus model even though is fixed.

The Hamiltonian is defined by

 S(X,g)=S1+bS2, (7) S1=16∑ΔS1(Δ)/AΔ,S2=16∑ΔS2(Δ)/AΔ,

with

 S1(Δ)=ℓ21+ℓ22+ℓ23−FΔ(ℓ1ℓ2cosϕ3+ℓ2ℓ3cosϕ1+ℓ3ℓ1cosϕ2) (8)

and

 S2(Δ)=3−n0⋅n1−n0⋅n2−n0⋅n3 −FΔ[(n1−n0)⋅(n2−n0)+(n2−n0)⋅(n3−n0)+(n3−n0)⋅(n1−n0)], (9)

where the symbol is an internal angle of as mentioned above. in Eqs. (8) and (2) denotes that the value of depends on the triangle . The unit normal vectors are shown in Fig. 1(d).

The partition function is defined by

 Z(b)=∫Dg∫DXexp[−S(X,g)]. (10)

The integration in is given by the multiple -dimensional ones such that

 ∫DX=∫′N∏i=1dXi, (11)

where denotes that the center of mass of the surface is fixed to the origin of . The symbol in is defined by

 ∫Dg=∫NT∏i=1dFi∫NT∏i=1dρiexp(−λFSF−λρASρA−λρSρ), |Fi|<1,ρi>0. (12)

The integration is included in although is a variable independent of the surface geometry. The symbol in Eq. (2) is the total number of triangles. In the exponential factor in Eq. (2), is the interaction term for the variable given by

 SF=∑ij|Fi−Fj|, (13)

where represents all nearest neighbor triangles and . In the presence of the interaction , the metric can be smoothened as a function on the surface . Thus, it is physically natural to include as a measure term. We should note that the interaction in Eq. (13) can also be expressed by , however we use the expression for numerical simplicity.

and in Eq. (2) are given by

 SρA=∑ΔρΔAΔ,Sρ=∑ij|ρi−ρj|, (14)

where (or ) is the conjugate variable to the area (or ) and can be called the ”surface density”. We should note that the surface density is not an external field but introduced as a variable field on the surface to see the in-plane surface deformation.

The density is interconnected to the geometric variables and only through the interaction . The term is introduced to define an interaction between the fileds . Since the variable couples to in , then becomes nonzero finite (or well-defined). We should note also that can vary without the density field because depends on which varies on the surface . However, the interaction is expected to influence the in-plane deformation of .

By including the measure terms of the exponential factor of Eq. (2) in the Hamiltonian, we have the effective Hamiltonian such that

 S(X,F,ρ)=S1+bS2+λFSF+λρASρA+λρSρ. (15)

Note also that the conventional Hamiltonians and are restored up to irrelevant multiplicative factors if . In this case, we have and , and therefore we find that both and are independent of the surface and can be neglected.

## 3 Continuous surface model

The surface model of Helfrich and Polyakov is defined by a mapping from a two-dimensional surface to , described by . The surface is assumed to be of sphere topology in this paper just like in Ref. ?. The symbol denotes a local coordinate of . The image corresponds to a real physical membrane, however, the self-avoiding property is not assumed in , and hence the mapping is not always injective.

The Hamiltonian of the model is given by a linear combination of the Gaussian bond potential and the extrinsic curvature energy such that

 S=S1+bS2, S1=∫√gd2xgab∂aXμ∂bXμ, (16) S2=12∫√gd2xgab∂anμ∂bnμ,

where is the bending rigidity . The matrix in and is the metric on , is the determinant of , and is its inverse. The symbol in is a unit normal vector of the surface. The continuous Hamiltonian is invariant under the conformal transformation such as for any positive function , and it is also invariant under the reparametrization such as , which changes both of the variables and . These symmetries allows us to use a constant metric , however, in the case of constant metric the in-plane deformation of is prohibited, and as a consequence the area of surface remains constant as long as the surface tension coefficient in is fixed. No information of the in-plane transformation in membranes is obtained from such a model with .

The discrete Hamiltonians in Eqs. (2)–(2) are obtained from the continuous one in Eq. (3) as follows. On the triangulated surfaces shown in Figs. 1(a)–1(c), the partial derivatives in in Eq. (3) can be replaced by , , where denotes the position of the vertex such that , . The derivatives in in Eq. (3) can also be replaced by , , where are shown in Fig. 1(d).

We make and to be symmetric under the permutation of the indices of and such that , , . By including those terms which are cyclic under the permutation, and by multiplying a factor , we have the discrete and in Eqs. (2)–(2).

This symmetrization is necessary, because if it were not for the symmetrization, the Hamiltonian becomes dependent on the choice of local coordinates on the triangles. Thus, the symmetrization , , of and is considered to be a lattice analogue of the reparametrization invariance of the continuous Hamiltonian in Eq. (3). We should note that the component of remains unchanged under the symmetrization. The reason of this is because the internal angle is assumed to be independent of the vertices just as shown in Eq. (2).

## 4 Monte Carlo technique

The Metropolis Monte Carlo simulation technique is employed to update the variables , and . The update of these variables is accepted with the probability , where .

The vertex position is updated such that with a random vector in a small sphere. The function is updated such that , where is a random number, under the constraint . The density field is updated by , where . The constraint is imposed on this update. The acceptance rate for is approximately , and those for and are and respectively.

One Monte Carlo sweep (MCS) consists of sequential updates of , sequential updates of , and sequential updates of . The total number of MCS performed after sufficiently large number of thermalization MCS is at the transition region on the and surfaces, and relatively small number of MCS is performed at non-transition region and on smaller surfaces.

## 5 Simulation results

To see the dependence of physical quantities on the bending rigidity , we fix the parameters , and to the following values:

 λF=0,λρA=1,λρ=1(case1), λF=2,λρA=2,λρ=0(case2). (17)

In case 1, the interaction term is neglected while the density term and the interaction term are included in the Hamiltonian. In case 2, both terms and are included while is neglected. Fig. 2: The mean square size X2 vs. b in (a) case 1 and (b) case 2, the variance CX2 vs. b in (c) case 1 and (d) case 2, and log-log plots of the peak CmaxX2 vs. N in (e) case 1 and (f) case 2. The solid lines in (a), (b), (c) and (d) are drawn by the multi-histogram re-weighting technique. The error bars on the symbols denote the standard errors obtained by the binning analysis.

We firstly show in Figs. 2(a) and 2(b) the mean square size

 X2=1N∑i(Xi−¯X)2,¯X=1N∑iXi (18)

vs. . The transition region in case 1 is slightly smaller than that in case 2, however, the behavior of against the variation in case 1 is almost identical to that in case 2. The error bars on is the standard errors obtained by the binning analysis, and the solid lines connecting the data symbols are drawn by the multi-histogram re-weighting technique . The large errors shown at the transition region in Figs. 2(a) and 2(b) imply that discontinuously changes.

Figures 2(c) and 2(d) show the variance of defined by

 CX2=1N⟨(X2−⟨X2⟩)2⟩. (19)

We see that the peaks increase with increasing . To see the order of the transition, we show vs. in Figs. 2(e) and 2(f) in log-log scales. The straight lines are drawn by a least squares fitting, and we have

 CmaxX2∼Nα, α=1.19±0.04(case1), (20) α=1.32±0.06(case2).

The results in Eq. (20) indicate that is slightly larger than in both cases. From the finite-size scaling theory , () implies that the transition is of first (second) order. Therefore, the results in Eq. (20) imply that the model undergoes a discontinuous transition between the smooth spherical and collapsed phases just like the model without the metric degree of freedom . Fig. 3: The bending energy S2/NB vs. b in (a) case 1 and (b) case 2, the specific heat CS2 vs. b in (c) case 1 and (d) case 2, and log-log plots of the peak CmaxS2 vs. N in (e) case 1 and (f) case 2. The largest three data are used for the least squares fitting in both (e) and (f).

Large errors seen in also imply a discontinuous change in at the transition region (Figs. 3(a), 3(b)). The specific heat is defined by

 CS2=b2N⟨(S2−⟨S2⟩)2⟩. (21)

We see an expected peak in each and find that increases with increasing at the transition point where has its peak. The vs. are plotted in Figs. 3(e) and 3(f) in log-log scales. The straight lines are drawn by fitting the largest three data, and we have

 CmaxS2∼Nσ, σ=0.92±0.09(case1), (22) σ=1.19±0.08(case2).

We find that both of the results in Eq. (22) satisfy and are consistent with the result of the conventional model in Ref. ?. This indicates that the model undergoes a first-order transition of surface fluctuations from the finite-size scaling theory . Fig. 4: The variation of S2/NB against MCS in (a) case 1 and (b) case 2, and (c), (d) the normalized distribution (or histogram) h(S2) of S2/NB corresponding to (a), (b), respectively.

In order to see a discontinuity in more clearly, we show the variation of against MCS in Figs. 4(a) and 4(b) at the transition points. The corresponding distribution (or histogram) of are shown in Figs. 4(a) and 4(b). We find a double peak structure in these . This structure depends on and becomes more apparent as increases, although the dependence of on is not shown in the figures. This implies that discontinuously changes at the transition point and indicates that the model undergoes a first-order transition in both cases. We should note that the transition occurs only four times during MCS at the transition point of the surface (Fig. 4(b)). This is the reason for the large errors in data and as mentioned above. Reliable estimate of critical exponents is very difficult at the first-order transition point. Fig. 5: The interaction energy SF/NB vs. b in (a) case 1 and (b) case 2, and log-log plots of CmaxF vs. N in (c) case 1 and (d) case 2. The largest three data in (c) and (d) are fitted to Eq. (23).

The interaction energy vs. is plotted in Figs. 5(a) and 5(b). The peak values of the variance are plotted against in Figs. 5(c) and 5(d) in a log-log scale. The largest three data are found to scale such that

 CmaxF∼Nγ, γ=0.18±0.13(case1), (23) γ=0.13±0.08(case2).

These values satisfy and hence indicate that model undergoes a continuous transition of in-plane deformation . The surface density , which is not plotted in the figures, behaves just like in Figs. 5(a) and 5(b). However, we find no power low scaling behavior in in contrast to in Eq. (23). Fig. 6: The area AΔ vs. b in (a) case 1 and (b) case 2, the area aΔ vs. b in (c) case 1 and (d) case 2, and the variance the variable F vs. b in (e) case 1 and (f) case 2.

The surface area defined by Eq. (2) is an internal geometric variable, and hence it is interesting to see whether or not the phase transitions are reflected in . Figures 6(a) and 6(b) show vs. in case 1 and case 2. We see that the variation of against is quite analogous to that of shown in Fig. 3, although we see that the change of is very small compared to the value itself. The variance defined by , which is not shown as a figure, is also analogous to . However, the ”internal” transition characterized by the fluctuations of is not identified as a phase transition. Indeed, both of the data vs. do not satisfy the scaling relation . Nevertheless, the surface area almost discontinuously changes against the variation of at the transition region (Figs. 6(a),6(b)).

We should note that the variation of the area of the surface in Figs. 6(c) and 6(d) is also small compared to the value of itself. However, varies almost discontinuously just like . This implies that the transition of shape transformation is reflected in the in-plane surface deformation.

The behavior of shown in Fig. 6(e) is identical to that of in Fig. 6(a), while the behavior of vs. in Fig. 6(f) is different from that of in Fig. 6(b). We consider that the interaction described by in Eq. (13) is an origin of this difference. Thus, we find that the in-plane phase structure is sensitive to the measure factor in Eq. (2) such as the factor of . Fig. 7: The Gaussian bond potential S1/N vs. b in (a) case 1 and (b) case 2. The solid lines are drawn as a guide to the eyes.

Finally in this section, we show the Gaussian bond potential vs. in Figs. 7(a) and 7(b). We see that the expected relation is satisfied in both cases. This implies that the simulations are successfully performed. We should emphasize that in both cases is different from either or , and hence and are not always constrained to be a constant. This is in sharp contrast to the case of the conventional model, where the Gaussian bond potential remains constant and corresponds to the surface area.

## 6 Summary and Conclusion

In this paper, using the canonical Monte Carlo simulation technique, we have studied a triangulated surface model, in which the metric is assumed as a dynamical variable. A deficit angle for triangles of triangulated surface is assumed as the metric degrees of freedom. We focus on whether or not the collapsing transition accompanies an in-plane surface deformation. It is also interesting to see whether or not the phase structure of the model is identical to the one of the model with a constant metric such as the Euclidean metric or the induced metric of the mapping .

We find that the model undergoes a first-order transition between the smooth spherical phase at and the collapsed phase at . The transition is almost identical to the one observed in the model with constant metric in Refs. ?, ?. This indicates that the transition of shape transformations is not influenced by the metric variable in Eq. (2). We also find that the phase transition accompanies a continuous in-plane transition, which is confirmed by the finite-size scaling analyses for the peak values of the variance . This in-plane transition is reflected in an almost discontinuous change in both of the internal and external quantities such as the surface density and the external surface area, although the discontinuous changes are very small compared to the values themselves.

Acknowledgment This work is supported in part by a Promotion of Joint Research of Toyohashi University of Technology.

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