Determination of the Crumpling Fractal Dimension Via k-Space MCRG

TL;DR

This paper uses momentum space MCRG to estimate the fractal dimension at the crumpling transition in triangulated surfaces, providing tentative critical exponent values and insights into the transition's nature.

Contribution

It applies momentum space MCRG to determine critical exponents and fractal dimensions at the crumpling transition, a novel approach in this context.

Findings

Estimated critical exponent η between -1.0 and -1.3

Fractal Hausdorff dimension between 3 and 4

Results suggest a specific universality class for the transition

Abstract

Motivated by the successful application of MCRG in momentum space to $\lambda \phi^4_3$ we determine the critical exponents at the crumpling transition in fixed triangulated surfaces. The results are still tentative, but suggest that $-1.0\ge \eta \ge -1.3$, pointing at a value for the fractal Hausdorff dimension at the crumpling transistion fixed point somewhere between 3 and 4.

Tables

Table 1. Table 1: Dependence on the s 𝑠 s parameter.

s	$⟨A⟩$	$⟨A^2⟩-{⟨A⟩}^2$
${16}^2,3\times {10}^5\mathrm{sweeps},\Delta t=6\times {10}^{-5}$
1	379.6	426.6
2	380.9	429.7
10	380.5	398.4
Exact	382.5	382.5
${32}^2,5\times {10}^5\mathrm{sweeps},\Delta t=8\times {10}^{-5}$
0.1	1542.1	1788.1
0.3	1539.8	1577.6
1	1539.2	1539.9
5	1538.4	1512.8
10	1538.4	1498.3
100	1536.2	1485.4
Exact	1534.5	1534.5

Table 2. Table 2: Comparison between ‘big’ and ‘small’ lattices. 10 6 superscript 10 6 10^{6} sweeps, Δ ​ t = 8 × 10 − 5 Δ 𝑡 8 superscript 10 5 \Delta t=8\times 10^{-5}

per site	big	small
$\lambda {\varphi }_3^4$	$m^2=-3.32$	$\lambda =6.0$
$⟨{\varphi }^2⟩$	0.272	0.267
$⟨{\varphi }^4⟩$	0.161	0.160
$⟨{\varphi }^6⟩$	0.130	0.133
R. S.	$\kappa =0.80$	$s=10$
$⟨A⟩$	1.50	1.50
$⟨A^2⟩-{⟨A⟩}^2$	1.49	0.86
$⟨S_{EC}⟩$	1.35	1.24

Table 3. Table 3:

# of operators	Largest eigenvalue
1st blocking
1	1.56
3	2.09
5	2.11
2nd blocking
1	1.99
3	2.22
5	2.23

Equations

$$S=\sum _{i,j}(x_i-x_j{)}^2+\kappa \sum _{I,J}(1-n_I{n}_J)$$

$$⟨\eta (p,{\tau }_1)\eta (p,{\tau }_2)⟩={\delta }_{{\tau }_1,{\tau }_2}{\delta }_{p+k,0}ϵ(p)$$

$$ϵ(p)=\frac{p^2(p^2+m^2){|}_{max}}{p^2(p^2+m^2)}$$

$$n_I\frac{\partial n_J}{\partial x}\propto (A_J{)}^{-1}$$

$$n_I{n}_J\to n_I{n}_J\exp (-s\Delta t(A_I^{-1}+A_J^{-1}))$$

$$\bar{S}=S+\frac{1}{4}\sum _{i,j}{ϵ}_{i,j}\frac{{\delta }^{2}S}{\delta x_i\delta x_j}+...$$

$$\mathrm{Tr}K=\partial n\partial x$$

$$\mathrm{Tr}{K}^2=\partial n\partial n=S_{EC}$$

$$\mathrm{Tr}{K}^3=({\partial }_{a}n{\partial }_{b}n)({\partial }_{b}n{\partial }_{a}x)$$

$$\mathrm{Tr}{\mathrm{K}}^4=({\partial }_{a}n{\partial }_{b}n)({\partial }_{a}n{\partial }_{b}n)$$

$$\mathrm{Tr}(\nabla K{)}^2=\Delta n\Delta n$$

$$\mathrm{Tr}\nabla K={\partial }_{a}x{\partial }_a{\partial }_{b}n$$

Full text

1 INTRODUCTION

We have to go back to the late eighties[1, 2] to find the first evidence for the existence of a phase transition separating the crumpled and smooth regimes in mathematical models of surfaces defined on a mesh with fixed connectivity. However, the actual values of the critical exponents associated to this phase transition remain largely unknown. Their determination is crucial to identify which type of conformal theory one gets in the vicinity of the critical point.

Two recent numerical simulations report the following values for the critical exponents $\alpha$ and $\nu$. In [3] a direct measuremt of the correlation length gives $\nu=0.71\pm 0.05$ and $\alpha=0.58\pm 0.1$ (the latter is obtained via scaling relations). A direct measurement of the finite size scaling of the specific heat gives $\nu=0.73\pm 0.06$, in agreement with the previous values. In [4] finite size scaling gives $\omega=\alpha/\nu=1.11\pm 0.1$, that is $\alpha=0.71\pm 0.05$, $\nu=0.64\pm 0.02$. However these values do not agree with a direct fit to the specific heat results on large lattices (up to $128^{2}$), which tend to give much lower values for $\alpha$ [5, 6].

Virtually nothing is known of the other critical exponents $\eta$ and $\gamma$, which cannot be deduced from $\nu$ and $\alpha$ alone using scaling relations. The most interesting one is perhaps $\eta$, the anomalous dimension of the field $x_{i}$ that describes the position of the surface. The fractal dimension $d_{H}$ is related to $\eta$ via the relation $d_{H}=-4/\eta$. Sometimes in random surface theory the field anomalous dimension is introduced through the tangent-tangent two-point function. If we denote the corresponding anomalous dimension by $\bar{\eta}$ it follows that $\eta=\bar{\eta}-2$ and $d_{H}=4/(2-\bar{\eta})$.

In this note we present our preliminary results concerning the above critical exponents obtained via the Monte Carlo renormalization group. The action we shall study is

[TABLE]

with the sums extending over all pair of neighbouring sites ($i,j$) and triangles ($I,J$). $n_{K}$ is the normal vector to the $K$-th triangle. Sites and triangles live on a two dimensional triangulated surface of fixed connectivity embedded in ${\bf R}^{3}$.

2 THE METHOD

Recently we proposed a method of implementing the renormalization group ideas directly in momentum space[7]. It works very well in $\lambda\phi^{4}_{3}$ yielding critical exponents at the Wilson fixed point with a precision comparable, if not better, to any other technique we know of.

Extremely long autocorrelation times difficult the application of local Monte Carlo algorithms to the problem at hand. For a $128^{2}$ system autocorrelation times can be as large as $10^{6}$ sweeps[8].To cure these difficulties a Fourier accelerated Langevin algorithm was proposed in [9] and used in random surfaces in e.g. [5].

Fourier acceleration works best if the Langevin equation is transformed to $k$-space. It is therefore natural to attempt a blocking procedure in momentum space directly. The renormalization group transformation that we use is probably the simplest one; we just discard half the Fourier modes in each direction. The interested reader may wish to consult [7] for a more detailed discussion of the method, here we shall dwell on the details specific to random surfaces.

In Fourier accelerated Langevin algorithms one uses the freedom to select the two-point function for the gaussian stochastic noise that best serves the purpose of updating the different Fourier modes efficiently.

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For a free field theory the choice $\epsilon(p)=(p^{2}+m^{2})^{-1}$, where $m$ is the inverse correlation length of the system, works best. This also seems to be a good choice for $\lambda\phi^{4}_{3}$. It has been suggested[3, 5] that an effective action that describes many features of random surfaces is $S_{eff}=-x\Delta x+\lambda x\Delta^{2}x$ and thus it would seem that

[TABLE]

with $m$ being again the inverse correlation length, would be a sensible choice for $\epsilon(p)$ (actually, of course, the lattice transcription of the above). This expectation is not borne out by actual simulations, however. It turns out to be nearly impossible to get good convergence of the method if one uses the actual inverse correlation length. We have found more convenient to select a relatively large value for the mass, such as $m=1$ which seems to provide a good balance between the need to update efficiently the slow modes and the need to keep their stochastic excursions in check. Most of the results we present below are obtained with this value for $m$.

A potentially more serious drawback of the method is the following. When we update the position of the variable on the $i$-th site we have to compute

[TABLE]

If it just happens that $A_{J}$ is small, the field variable will receive a large kick. Because the resulting configuration is far from the classical trajectory, it will be promptly brought back to order by the Langevin algorithm itself, but such an unphysical large fluctuation spoils the statistical samples. It should be emphasized that this is a problem associated to the finiteness of the Langevin step and not an intrinsic difficulty of our system. One can indeed easily check that reducing the Langevin step reduces these off-equilibrium excursions, but there is a limit to the reduction of the time step as this also increases the autocorrelation times. To alleviate this problem we modify the action in the following form

[TABLE]

Table 1 illustrates the dependence of the results on the parameter $s$ for fixed values of $\Delta t$. Some observables can be calculated exactly and so we know when the method reproduces the right results. We have also analyzed the volume dependence of $s$.

We will discuss next the choice of $\Delta t$ and its influence on the autocorrelation time $\tau$. The longest autocorrelation time is always that of the gyration radius, but we present results for a fairly characteristic one, namely that of the extrinsic curvatute $S_{EC}=\sum_{I,J}(1-n_{I}n_{J})$. Figure 1 shows the dependence of $\tau$ on $\Delta t$. (The dependence on $s$ for two values of $m$ is also illustrated — there is virtually none.). As expected $\tau\sim(\Delta t)^{-1}$. In view of what we know about the extremely long autocorrelation times in local Monte Carlo algorithms the present autocorrelation times are truly impressive.

One point that practitioners of the Langevin algorithm always have to bear in mind is that the method is exact only in the $\Delta t\to 0$ limit. Indeed the equilibrium distribution is not the one corresponding to $S$ but rather to

[TABLE]

Even after tunning $s$, some systematic errors remain. Fortunately they can be reduced by working with a small enough value of $\Delta t$. Yet $\Delta t$ has to be large enough so that we get reasonably short autocorrelation times and manage to sample configuration space. After a careful analysis we have settled for $\Delta t=8\times 10^{-5}$ as the optimal value. The renormalization of the parameters in $\bar{S}$ is negligible with this value for $\Delta t$. A detailed analysis will be provided elsewhere. (Alternatively, one could work with a second order Langevin algorithm, where errors are at least of ${\cal O}((\Delta t)^{2})$. We have tried this, but found no real gain.)

2.1 The Fixed-point Action

Now we proceed to discuss the essence of the method. To determine the critical exponent $\nu$ we need to determine (see e.g. [7]) the largest eigenvalue of the matrix $T$ that linearizes the renormalization group transformation in the vicinity of the fixed point associated to the crumpling transition. In practice truncation in the number of operators is required. We have considered a total of nine operators. In continuum notation they are $x\Delta x$, $x\Delta^{2}x$ and $(\partial x\partial x)^{2}$, and those constructed with

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and having dimension $d\leq 2$.

We perform a blocking transformation starting with our bare action (which consists of just two operators), determine $T$ by measuring the appropriate correlators and compute $\nu$. In doing that one needs the field anomalous dimension $\eta$. $\eta$ is determined by demanding that at the fixed point the renormalized and bare actions coincide. It is of course unlikely that the fixed point of this renormalization group transformation lies just in the plane spanned by the two operators of the bare action. However we can assess whether we are close to it by adjusting $\eta$ and seeing how good the agreement between the renormalized and the bare observables is. Table 2 illustrates this point in a $32^{2}\to 16^{2}$ blocking. Similar results are obtained in $64^{2}\to 32^{2}$. For comparison some results for $\lambda\phi^{4}_{3}$ around the Wilson fixed point are also shown. We conclude that we are still at some distance from the fixed point and that a second renormalization is probably necessary. (If we are close enough to the critical surface the flow should drive us closer to the fixed point.)

3 THE RESULTS

It is a very pleasant surprise that the results turn out to be extremely sensitive to the value of $\eta$ used to match the ‘big’ and ‘small’ lattices. Our results from running on a variety of values of $\kappa$ in the vicinity of the crumpling transition with large statistics, both in the crumpled and the smooth phases and on a variety of systems lead us to conclude that $\eta=-1.05\pm 0.05$. from the first blocking transformation.

Our results from the second blocking are still very tentative and we have only results for the $32^{2}\to 16^{2}\to 8^{2}$ blocking, the final lattice being undoubtedly too small. Even so, we get a value $\eta\simeq-1.3$, not too far from the one obtained in the first blocking. From these two results we conclude that $-1.0\geq\eta\geq-1.3$ and $3.1\leq d_{H}\leq 4.0$.

As for $\nu$, Table 3 shows the result of the blocking $32^{2}\to 16^{2}\to 8^{2}$. The second blocking yields very stable results. Should we use these values we would get $\nu=0.82$, but we shall not even quote error bars here as these results are still very preliminary. Yet, they show considerable promise, specially because the computational effort required is relatively small. To mention some figures, to get the above numbers requires about 1.5 hours on a Cray YMP or about 5 hours on a SGI Power Challenge L.