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...

We study a model of phantom tethered membranes, embedded in three-dimensional space, by extensive Monte Carlo simulations. The membranes have hexagonal lattice structure where each monomer is interacting with six nearest-neighbors (NN). Tethering interaction between NN, as well as curvature penalty between NN triangles are taken into account. This model is new in the sense that NN interactions are taken into account by a truncated Lennard-Jones potential including both repulsive and attractive parts. The main result of our study is that the system undergoes a first-order crumpling transition from low-temperature flat phase to high-temperature crumpled phase, in contrast with early numerical results on models of tethered membranes.

The Ising model on "thin" graphs (standard Feynman diagrams) displays several interesting properties. For ferromagnetic couplings there is a mean field phase transition at the corresponding Bethe lattice transition point. For antiferromagnetic couplings the replica trick gives some evidence for a spin glass phase. In this paper we investigate both the ferromagnetic and antiferromagnetic models with the aid of simulations. We confirm the Bethe lattice values of the critical points for the ferromagnetic model on φ 3 and φ 4 graphs and examine the putative spin glass phase in the antiferromagnetic model by looking at the overlap between replicas in a quenched ensemble of graphs. We also compare the Ising results with those for higher state Potts models and Ising models on "fat" graphs, such as those used in 2D gravity simulations.

The crumpled-to-flat phase transition that occurs in D-dimensional polymerized phantom membranes embedded in a d-dimensional space is investigated nonperturbatively using a field expansion up to order eight in powers of the order parameter. We get the critical dimension dcr(D) that separates a second order region from a first order one everywhere between D = 4 and D = 2. Our approach strongly suggests that the phase transitions that take place in physical membranes are of first order in agreement with most recent numerical simulations.PACS numbers: 87.16. 11.10.Hi, 11.15.Tk Fluctuating or random surfaces are a recurrent concept in physics [1,2]. They occur in soft matter physics or in biology as assemblies of amphiphilic molecules that can form plane or closed structures (vesicles) according to the chemical composition of the membrane itself and its surroundings. Random surfaces also appear in highenergy physics, especially in string theory, as the worldsheet swept out by a string during its spacetime evolution. More recently membranes have received a renewed interest in condensed matter physics where it has been realized that, from the point of view of their mechanical properties, novel materials, like graphene [3], identify with polymerized membranes, providing the first and unique example of genuinely two-dimensional membrane [4,5]. The coexistence of two-dimensional geometry and thermal fluctuations is at the origin of a variety of behaviours depending on the nature of the internal structure of the membrane. Fluid membranes are made of molecules that freely diffuse and re-arrange rapidly when a shear or a stress is performed. This implies that, in absence of an external tension, the dominant energy is the bending energy. It has been shown that, in this case, the height fluctuations are sufficiently strong to prevent the appearance of long-range order; fluid membranes are thus always crumpled [6,7]. Polymerizedor tethered -membranes display a drastically different behaviour. Indeed the existence of an underlying network of linked molecules induces elastic (shearing and stretching) energy contributions that lead to a coupling between height and transverse -phonons -modes. It results from this situation a frustration of the height fluctuations [8] that are strongly reduced at low temperatures giving rise to the appearance of a flat phase with longrange order between the normals [9,10]. The existence of a low-temperature phase accompanied with spontaneous symmetry breaking of rotational invariance is a priori in contradiction with the Mermin-Wagner theorem. However it appears that the effective phonon-mediated interaction between the height fields (more precisely between the Gaussian curvatures) is of long-range kind, allowing to evade the conditions of application of the MerminWagner theorem [9]. Correlatively the low-temperature phase of membranes is characterized by non trivial scaling behaviour in the infrared [11][12][13]:where G hh (q) and G uu (q) are the correlation functions of the out-of-plane and ...