We investigate the relation between dilatation and conformal symmetries in the statistical mechanics of flexible crystalline membranes. We analyze, in particular, a well-known model which describes the fluctuations of a continuum elastic medium embedded in a higher-dimensional space. In this theory, the renormalization group flow connects a non-interacting ultraviolet fixed point, where the theory is controlled by linear elasticity, to an interacting infrared fixed point. By studying the structure of correlation functions and of the energy-momentum tensor, we show that, in the infrared, the theory is only scale-invariant: the dilatation symmetry is not enhanced to full conformal invariance. The model is shown to present a non-vanishing virial current which, despite being non-conserved, maintains a scaling dimension exactly equal to $D-1$, even in presence of interactions. We attribute the absence of anomalous dimensions to the symmetries of the model under translations and rotations in the embedding space, which are realized as shifts of phonon fields, and which protect the renormalization of several non-invariant operators. We also note that closure of a symmetry algebra with both shift symmetries and conformal invariance would require, in the hypothesis that phonons transform as primary fields, the presence of new shift symmetries which are not expected to hold on physical grounds. We then consider an alternative model, involving only scalar fields, which describes effective phonon-mediated interactions between local Gaussian curvatures. The model is described in the ultraviolet by two copies of the biharmonic theory, which is conformal, but flows in the infrared to a fixed point which we argue to be only dilatation-invariant.

我们研究了柔性晶体膜统计力学中膨胀与共形对称性之间的关系。我们特别分析了一个众所周知的模型，该模型描述了嵌入高维空间的连续弹性介质的波动。在这个理论中，重整化群流将一个非相互作用的紫外线固定点（该理论受线性弹性控制）连接到一个相互作用的红外固定点。通过研究相关函数和能量-动量张量的结构，我们表明，在红外线中，该理论只是尺度不变的：膨胀对称性没有增强到完全共形不变性。该模型显示出一个不消失的维里电流，尽管它是非守恒的，但它保持一个完全等于的缩放维度$D-1$，即使存在交互。我们将不存在异常维度归因于嵌入空间中平移和旋转下模型的对称性，这些对称性通过声子场的移动来实现，并保护了几个非不变算子的重整化。我们还注意到，在声子变换为主场的假设中，具有位移对称性和保形不变性的对称代数的闭合将需要存在新的位移对称性，这些对称性在物理上不成立。然后我们考虑一个替代模型，只涉及标量场，它描述了局部高斯曲率之间有效的声子介导的相互作用。该模型在紫外线中由双调和理论的两个副本描述，它是共形的，