Crumpling transition and flat phase of polymerized phantom membranes

Abstract

Polymerized phantom membranes are revisited using a nonperturbative renormalization group approach. This allows one to investigate both the crumpling transition and the low-temperature, flat, phase in any internal dimension D and embedding dimension d, and to determine the lower critical dimension. The crumpling phase transition for physical membranes is found to be of second order within our approximation. A weak first-order behavior, as observed in recent Monte Carlo simulations, is however not excluded. 11.10.Hi, 11.15.Tk Membranes form a particularly rich and exciting domain of statistical physics in which the interplay between two-dimensional geometry and thermal fluctuations has led to a lot of unexpected behaviors going from flat to tubular and glassy phases (see [1,2,3, 4] for reviews). Roughly speaking, membranes fall into two groups [4]: fluid membranes, in which the building monomers are free to diffuse. The connectivity is thus not fixed and the membrane displays a vanishing shear modulus. In contrast, in polymerized membranes the monomers are tied together through a potential which leads to a fixed connectivity and to elastic forces. While fluid membranes are always crumpled, polymerized membranes, due to their nontrivial elastic properties, exhibit a phase transition between a crumpled phase at high temperature and a flat phase at low temperature with orientational order between the normals of the membrane [4,5,6,7]. Amazingly, due to the existence of long-range forces mediated by phonons, the correlation functions in the flat phase display a nontrivial infrared scaling behavior [8,9,10]. Accordingly, the lower critical dimension above which an order can develop appears to be smaller than 2 [10], in apparent violation of the Mermin-Wagner theorem.Let us consider the general case of D-dimensional non self-avoiding (phantom) membranes embedded in a d-dimensional space. Early ǫ-expansion [7] performed at one-loop order on the Landau-Ginzburg-Wilson-type model relevant to study the crumpling transition of polymerized membranes has led to predict that just below the upper critical dimension D = 4, the crumpling transition is of second order for d > d cr = 219 while it is of first order for d < d cr . This leaves however open the question of the nature of the transition in the physical (D = 2, d = 3) situation, the case ǫ = 2 being clearly out of reach of such a one-loop order computation. On the numerical side former Monte Carlo (MC) studies (see [11, 12] for reviews) predict a second-order behavior while more recent simulations [13,14] rather favor first-order behaviors. There is however no definite conclusion and no explanation for these versatile results.In parallel to the investigation of the crumpling transition, an effective elastic field theory has been used to probe the flat, low-temperature, phase of membranes [4,5,8,10]. An ǫ-expansion has been performed A flaw of the previous approaches to polymerized membranes is that, due to their perturbative character, they are unable to treat all asp...

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