Fluctuations of Solid Membranes

Statistical Mechanics of Thin Spherical Shells

“…1, where the pressure is 36% of p 0 c ). Similar perturbative divergences in the bending rigidity and Young's modulus of flat membranes of size R 0 (here, the corrections diverge with γ rather than ffiffi ffi γ p [38]) can be handled with integral equation methods [26,29], which sum contributions to all orders in perturbation theory, or alternatively, with the renormalization group [27]. We take the latter approach in the next section.…”

“…and beyond an important thermal length scale l th ∼ κ 0 = ffiffiffiffiffiffiffiffiffiffiffiffiffi k B TY 0 p , the bending rigidity and Young's modulus renormalize with length scale l like flat sheets with κ R ≈ κ 0 ðl=l th Þ η and Y R ≈ Y 0 ðl th =lÞ η u , where η ≈ 0.8 and η u ≈ 0.4 [27]. However, this renormalization is interrupted as one scales out to the shell radius R 0 .…”

“…IV, we also present a simple, intuitive derivation of the scaling relation η u þ 2η ¼ 2, originally derived using Ward identities associated with rotational invariance in Refs. [27,28]. In Sec.…”

“…Unlike long one-dimensional polymers, which perform self-avoiding random walks [24,25], arbitrarily large two-dimensional membranes remain flat at low temperatures because of the strong thermal renormalizations triggered by flexural phonons, which result in strongly scale-dependent enhanced bending rigidities and reduced in-plane elastic constants [26][27][28][29][30][31][32][33][34][35][36][37] (see also books and reviews in Refs. [38][39][40][41]).…”

Scaling behavior of tethered crumpled manifolds with inner dimension close to : Resumming the perturbation theory

“…The reason is that the nonself-avoiding membrane densely fills out the embedding space, such that it always "sees" the interaction. A way to circumvent this problem is to set up the expansion about any point (D < 2, d c (D)) and to extrapolate along an appropriate path in the (D, d)-plane to the physically interesting point (D, d) = (2, 3) [18][19][20][21][22][23][24]. To second order in ε one then finds a radius of gyration exponent of ν ≈ 0.86 [16,17] , which is a strong correction with respect to the only logarithmic dependence in the non-interacting theory, and indicates the existence of a crumpled phase, for which ν ≥ 2/3 follows from the fact that a membrane has a finite volume.…”