Fourier Monte Carlo renormalization-group approach to crystalline membranes

The computation of the critical exponent $\eta$ characterizing the universal elastic behavior of crystalline membranes in the flat phase continues to represent challenges to theorists as well as computer simulators that manifest themselves in a considerable spread of numerical results for $\eta$ published in the literature. We present additional insight into this problem that results from combining Wilson's momentum shell renormalization-group method with the power of modern computer simulations based on the Fourier Monte Carlo algorithm. After discussing the ideas and difficulties underlying this combined scheme, we present a calculation of the renormalization-group flow of the effective two-dimensional Young modulus for momentum shells of different thickness. Extrapolation to infinite shell thickness allows us to produce results in reasonable agreement with those obtained by functional renormalization group or by Fourier Monte Carlo simulations in combination with finite-size scaling. Moreover, our method allows us to obtain a decent estimate for the value of the Wegner exponent $\omega$ that determines the leading correction to scaling, which in turn allows us to refine our numerical estimate for $\eta$ previously obtained from precise finite-size scaling data.

Figure 1

Raw simulation data for systems of size $L=240$ with outer cutoff $l=24$, inner cutoff $l^{\prime }=10$, and 2D Young modulus $K=9.16$. Enumerating the displayed lines from bottom to top, the index $j=1,\cdots ,10$ labels the underlying tracer configurations of type (24) with arms (42) parametrized by the wave numbers $k_j$ as defined in Eq. (41).

Figure 2

Illustration of our numerical procedure for parameters $L=240$, $l=24$, and $l^{\prime }=10$. Left column plots: fits of data for $\tilde{\kappa }\left(K\right)$ and $\tilde{K}\left(K\right)/K$ to the ansatz (49). Each data point shown is derived from fits of the functions $a\left(k\right),b\left(k\right)$ based on up to $l^{\prime }$ different simulation data of the type shown in Fig. 1. Note the excellent compliance of the numerical data with the necessary conditions ${lim}_{K\to 0}\tilde{\kappa }\left(K\right)={lim}_{K\to 0}\tilde{K}\left(K\right)/K=1$. Right column, upper plot: illustration of the numerical solution of the equations $\eta \left(K\right)\equiv {\eta }_K\left(K\right)$ as defined in Eqs. (35) and (36). Right column, lower plot: determination of the slope of function $K^{\prime }\left(K\right)-K^{*}$ at $K=K^{*}$. The direction of the RG flow is schematically indicated. In this particular example, the intersection point determined by Eq. (37) is located at $K^{*}=9.80127$, and we derive exponents $\eta =0.86507$ and $\omega =1.54456$, respectively.

Figure 3

Numerical results obtained for systems of sizes $L=120,240,360$ with outer cutoff parameters $l=12,24,36$, respectively. Upper plot: $b$-dependent location $K^{*}\left(b\right)$ of a fixed-point value of parameter $K$. Middle plot: $b$-dependent value $\eta \left(b\right)$ of exponent $\eta$. For comparison, the black dotted horizontal line indicates the functional RG result $\eta =0.85$, while the horizontal gray bar refers to the FSS estimate of Ref. [18]. Lower plot: $b$-dependent value $\omega \left(b\right)$ of exponent $\omega$.

Figure 4

Results of least-squares fits of the ansatz (51) to the data for $\langle {\left(\Delta f\right)}^2\rangle$ obtained in Ref. [18] for various choices $0.6\le \omega \le 2.0$. Data not within the range $4/3\le \omega \le 3/2$ are plotted in gray as a guide to the eye. Top left panel: fit results for $\eta$. Top right panel: $\omega$ dependence of goodness-of-fit parameter $Q$ and reduced ${\chi }^2$ parameter of the fits. Bottom right panel: $\omega$ dependence of fit parameter $\delta$. Note the discontinuous change of sign around $\omega \approx 1.22$. Remaining panels: $\omega$ dependence of other fit parameters $\alpha$, $\beta$, and $\gamma$.

Figure 5

Illustration of the fit procedure outlined in Eqs. (A1, A2, A3, A4, A5). Main plot: original data set $F_{20}$ produced from the function (A1) and final fit with Eq. (A3) [but only shown in the interval $(0,0.6)$]. Inset: normalized weights obtained for all 20 data points of the data set $F_{20}$ (note the logarithmic scale).