Field-theory approach to flat polymerized membranes

Abstract We review the field-theoretic renormalization-group (RG) approach toward the critical properties of flat polymerized membranes. We start with a presentation of a flexural effective model that is entirely expressed in terms of a transverse (flexural) field with nonlocal interactions. We then provide a detailed account of the full three-loop computations of the RG functions of this model within the dimensional regularization scheme. The latter allows us to consider the general case of a d -dimensional membrane embedded in a D -dimensional space. Focusing on the critical flat phase of two-dimensional membranes ( d = 2) in three-dimensional space ( D = 3), we analyze the corresponding flow diagram and present the derivation of the anomalous stiffness. The latter controls all the other critical exponents of the theory, such as the roughness exponent and the scaling of the elastic constants. State-of-the-art four-loop results, discussions on the structure of the perturbative series, and comparisons with other approaches are also provided.