论文标题
结晶膜的缩放行为：$ε$ - expansion方法
Scaling behavior of crystalline membranes: an $ε$-expansion approach
论文作者
论文摘要
我们通过重新归一化组（RG）方法和$ε$ - expansion研究了平面中二维（2D）结晶膜的缩放行为。通过维度延续，通过标量复合场与声音介导的相互作用相结合的平面外波动，将问题概括为控制$ε$扩展所必需的非全能维度，这是通过尺寸延续来实现的。有效的理论将被称为高斯曲率相互作用（GCI）模型，相当于弹性$ d $二维流形的理论，在$（d + d_ {c}）$ - 尺寸嵌入空间$ d = 2 $的$（d + d_ {c}）$ d_ $ d_ $ d_ {c} $ d = 2 $中。相反，对于$ d \ neq 2 $，GCI模型不等于弹性膜理论的直接维度延续，它定义了通用内部尺寸$ d $的替代概括。我们以两回路顺序计算明确的RG函数，并确定指定相关函数的长波长缩放为$ε^{2} $中的指数$η$。 GCI模型的自洽筛选近似（SCSA）被证明与O（$ε^{2} $）完全正确。对于$ d_ {c} = 1 $，o（$ε^{2} $）校正被小型数值预取子抑制。结果，尽管$ε= 2 $的大价值，第一和第二订单结果的外推到$ d = 2 $导致数字非常接近，$η= 0.8 $和$η\ simeq 0.795 $。计算出的指数值接近通过非扰动RG，SCSA和数值模拟获得的早期参考结果。这些迹象表明，对GCI模型的扰动分析也可以提供一个有用的框架，即使在$ d = 2 $时，也可以准确地定量预测缩放指数。
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 $D\neq 2$, 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 $D$. We calculate explicitly RG functions at two-loop order and determine the exponent $η$ characterizing the long-wavelength scaling of correlation functions to order $ε^{2}$ in an $ε=(4-D)$-expansion. The self-consistent screening approximation (SCSA) for the GCI model is shown to be exact to O($ε^{2}$). For $d_{c} = 1$, the O($ε^{2}$) correction is suppressed by a small numerical prefactor. As a result, despite the large value of $ε= 2$, extrapolation of the first and second order results to $D = 2$ leads to very close numbers, $η= 0.8$ and $η\simeq 0.795$. The calculated exponent values are close to earlier reference results obtained by non-perturbative RG, the SCSA 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 $D = 2$.
