Polymerized Membranes, a Review

TL;DR

This review summarizes recent theoretical advances in understanding polymerized membranes, including their physical properties, phase behavior, renormalization techniques, and effects of disorder, highlighting differences from fluid membranes and polymers.

Contribution

It provides a comprehensive overview of the theoretical framework, renormalization group analysis, and new models for polymerized membranes, including their phase transitions and dynamical properties.

Findings

Prediction of a crumpled phase with fractal dimension ~2.4 in 3D

Distinct tricritical behavior from polymers

Development of a generalized O(N)-model for self-avoiding membranes

Abstract

Membranes are of great technological and biological as well as theoretical interest. Two main classes of membranes can be distinguished: Fluid membranes and polymerized, tethered membranes. Here, we review progress in the theoretical understanding of polymerized membranes, i.e. membranes with a fixed internal connectivity. We start by collecting basic physical properties, clarifying the role of bending rigidity and disorder, theoretically and experimentally as well as numerically. We then give a thorough introduction into the theory of self-avoiding membranes, or more generally non-local field theories with delta-like interactions. Based on a proof of perturbative renormalizability for non-local field-theories, renormalization group calculations can be performed up to 2-loop order, which in 3 dimensions predict a crumpled phase with fractal dimension of about 2.4. The tricritical behavior of membranes is discussed and shown to be quite different from that of polymers. Dynamical properties are studied in the same frame-work. Along the same lines, disorder can be included leading to interesting applications. We also construct a generalization of the O(N)-model, which in the limit N->0 reduces to self-avoiding membranes in analogy with the O(N)-model, which in the limit N->0 reduces to self-avoiding polymers. We also describe the instanton-contribution governing the large-order behavior of perturbation theory. Some technical details are relegated to the appendices.