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Résumé |
The structure of thermally-fluctuating free-standing membranes has been a subject of investigation for several decades and has
attracted a renewed interest in connection with graphene and atomically-thin two-dimensional materials. A crucial theoretical
prediction is that solid membranes, in absence of applied stress, behave as scale-invariant rough surfaces, with universal scaling
properties controlled by an interacting renormalization-group (RG) fixed point. This presentation will address the relation between
scale and conformal invariance at the corresponding fixed point. By an analysis within the epsilon expansion, it will be shown that the
theory is not symmetric under special conformal transformations, and thus provides an example of a scale-invariant but
nonconformal field theory. The second part of the presentation will analyze the role of quantum mechanical effects on the low-
temperature behavior of free-standing membranes. By a power-counting analysis, it will be shown that the leading behavior can be
described by an effective renormalizable model in which the kinetic energy of in-plane phonons is neglected, alongside other
'irrelevant' terms. The resulting RG equations, combined with arguments of finite-size scaling, will be used to derive general scaling
laws for correlation functions and thermodynamical quantities. The results confirm, in a systematic framework, some predictions of
earlier investigations. Strictly at T=0, the bending rigidity and the elastic moduli are renormalized logarithmically. At small T, the
scaling laws and an explicit analysis of thermal fluctuations corroborate a striking thermodynamical prediction: the thermal
expansion coefficient vanishes for T->0 as a slow, logarithmic function of T. |
