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 ...

We investigate the flat phase of D-dimensional crystalline membranes embedded in a d-dimensional space and submitted to both metric and curvature quenched disorders using a nonperturbative renormalization group approach. We identify a second-order phase transition controlled by a finite-temperature, finite-disorder fixed point unreachable within the leading order of ε=4-D and 1/d expansions. This critical point divides the flow diagram into two basins of attraction: that associated with the finite-temperature fixed point controlling the long-distance behavior of disorder-free membranes and that associated with the zero-temperature, finite-disorder fixed point. Our work thus strongly suggests the existence of a whole low-temperature glassy phase for quenched disordered crystalline membranes and, possibly, for graphene and graphene-like compounds.

A real space renormalization group technique, based on the hierarchical baby-universe structure of a typical dynamically triangulated manifold, is used to study scaling properties of 2d and 4d lattice quantum gravity. In 4d, the β-function is defined and calculated numerically. An evidence for the existence of an ultraviolet stable fixed point of the theory is presented. May 1995 LPTHE Orsay 95/341 Permanent address: Institute of Physics, Jagellonian University, ul. Reymonta 4, PL-30 059, Kraków, Poland 2 Laboratoire associé au CNRS, URA-D0063 1 1. As is well known, the renormalization group (RG) is a tool providing deep insight into the structure of a quantum field theory. It is certainly worth applying this tool to quantum gravity. This work is devoted to the development and the application of the real space renormalization group technique in the context of euclidean quantum gravity. It is a direct continuation of the work [1].We choose the lattice gravity framework. More precisely we adopt the particularly promising dynamical triangulation approach [2]. The remarkable results obtained within a class of exactly solvable models in two dimensions strongly suggest that the dynamical triangulation recipe is the correct way of discretizing gravity (at least for fixed topology).In conventional statistical mechanics a real space renormalization group transformation has two facets:(a) geometry -cells of the body are "blocked" together.(b) matter fields -"block" fields are defined in terms of the original fields. On a regular lattice it is trivial to perform the step (a) in such a manner that the resulting lattice is identical, modulo rescaling, to the original one.Since the values of critical couplings depend on the lattice type this selfsimilarity feature of the transformation is important. On a random lattice an appropriate definition of (a) requires some thought. In this work we consider pure geometry, without matter fields, and consequently the geometrical aspect of the renormalization group.In ref.[1] a method of "blocking" triangulations that exploits the selfsimilarity feature of random manifolds has been proposed. Without repeating in detail the arguments presented in [1] let us briefly sketch the main idea 3 . The intuitive arguments given below will be replaced progressively by more precise ones later on. We do not wish to give an impression of complexity from the outset.2. In 2d one can show [4] that an infinite randomly triangulated manifold is a self-similar tree obtained by gluing together sub-structures called baby universes (BUs), defined as subuniverses separated from the remaining part of the universe by a narrow neck. We conjecture that a similar picture holds in 4d, at least in the neighborhood of the phase transition point. The results presented in this paper strongly support this conjecture.3 As we have learned from J. Ambjørn during the LATTICE '94 Conference, our ideas partly overlap with those discussed earlier for 2d in ref. [3]. 2We have proposed [1] to define the step (a) as the o...

Zero temperature dynamics of two dimensional triangulations of a torus with curvature energy is described. Numerical simulations strongly suggest that the model get frozen in metastable states, made of topological defects on flat surfaces, that group into clusters of same topological charge. It is conjectured that freezing is related to high temperature structure of baby universes.

-The behaviour of a d-dimensional vectorial N = 3 model at a m-axial Lifshitz critical point is investigated by means of a nonperturbative renormalization group approach that is free of the huge technical difficulties that plague the perturbative approaches and limit their computations to the lowest orders. In particular being systematically improvable, our approach allows us to control the convergence of successive approximations and thus to get reliable physical quantities in d = 3.Introduction. -Lifshitz critical behaviour (LCB) [1] (see also [2][3][4][5]) occurs when a disordered phase encounters both a homogeneous ordered phase and a spatially modulated ordered phase with a modulation wave-vector q mod = 0. In the general case the vector q mod spans a m-dimensional subspace of the d-dimensional space with 0 ≤ m ≤ d. For a N -component order parameter the universal behaviour at criticality is completely determined by the set (m, d, N ). LCB has been proposed to occur in many systems including magnetic models (notably the ANNNI model [6]), liquid crystals, microemulsions, polymer mixtures, ferroelectrics, high-T c superconductors, see [4,5] for reviews. In the domain of magnetic materials there has been a growing activity in the search for LCB behaviour. A clear-cut LCB has been found in manganese phosphide (MnP) [7] and, possibly, in the ternary uranium silicide (UPD 2 Si 2 ) [8]. One can thus expect accurate determinations of the critical quantities from experiments in a near future.