We study the scaling behavior of two-dimensional (2D) crystalline membranes in the flat phase by a renormalization group (RG) method and an ε-expansion. Generalization of the problem to non-integer dimensions, necessary to control the ε-expansion, is achieved by dimensional continuation of a well-known effective theory describing out-of-plane fluctuations coupled to phonon-mediated interactions via a scalar composite field, equivalent for small deformations to the local Gaussian curvature. The effective theory, which will be referred to as Gaussian curvature interaction (GCI) model, is equivalent to theories of elastic D-dimensional manifolds fluctuating in a $(D+d_c)$-dimensional embedding space in the physical case $D=2$ for arbitrary $d_c$. For $D\ne 2$, instead, the GCI model is not equivalent to a direct dimensional continuation of elastic membrane theory and it defines an alternative generalization to generic internal dimension D. After decoupling interactions through a Hubbard-Stratonovich transformation, we study the GCI model by perturbative field-theoretic RG within the framework of an expansion in $\epsilon =(4-D)$. We calculate explicitly RG functions at two-loop order and determine the exponent η characterizing the long-wavelength scaling of correlation functions to order ${\epsilon }^2$. The value of η at this order is shown to be insensitive to Feynman diagrams involving vertex corrections. As a consequence, the self-consistent screening approximation for the GCI model is shown to be exact to O(${\epsilon }^2$). In the physical case of a single out-of-plane displacement field, $d_c=1$, the O(${\epsilon }^2$) correction is suppressed by a small numerical prefactor. As a result, despite the large value of $\epsilon =2$, extrapolation of the first and second order results to $D=2$ leads to very close numbers, $\eta =0.8$ and $\eta \simeq 0.795$. The calculated exponent values are close to earlier reference results obtained by non-perturbative renormalization group, the self-consistent screening approximation and numerical simulations. These indications suggest that a perturbative analysis of the GCI model could provide an useful framework for accurate quantitative predictions of the scaling exponent even at $D=2$.

我们通过重归一化组（RG）方法和ε-膨胀研究了二维（2D）晶体膜在平坦相中的结垢行为。将问题推广到控制ε扩展所必需的非整数维是通过众所周知的有效理论的维连续来实现的，该理论描述了平面外涨落，该涨落通过标量复合场耦合到声子介导的相互作用，等效对于局部高斯曲率的微小变形。有效的理论，将被称为高斯曲率相互作用（GCI）模型，等效于弹性D维流形在流体中波动的理论。$（d+d_C）$物理情况下的三维嵌入空间 $d=2$ 对于任意 $d_C$。对于$d\ne 2$，相反，GCI模型不等同于弹性膜理论的直接维数延续，它定义了通用内部维数D的替代概括。在通过Hubbard-Stratonovich变换将相互作用解耦后，我们通过扰动场理论RG在扩展的框架内研究了GCI模型。$\epsilon =（4-d）$。我们以两环阶计算显式RG函数，并确定表征相关函数到阶的长波长缩放的指数η${\epsilon }^2$。已显示此顺序的η值对涉及顶点校正的费曼图不敏感。结果，GCI模型的自洽筛选近似被证明对O（${\epsilon }^2$）。在单个平面外位移场的物理情况下，$d_C=\mathrm{1个}$，${\epsilon }^2$）校正可以通过较小的数值前置因子来抑制。结果，尽管$\epsilon =2$，将一阶和二阶结果外推到 $d=2$ 导致非常接近的数字， $\eta =0.8$ 和 $\eta \simeq 0.795$。计算出的指数值接近于非扰动重归一化组，自洽筛选近似和数值模拟所获得的早期参考结果。这些迹象表明，对GCI模型的扰动分析甚至可以为标度指数的精确定量预测提供有用的框架。$d=2$。