Self-Consistent Theory of Polymerized Membranes

TL;DR

This paper develops a self-consistent theory for polymerized membranes, predicting critical exponents and phase behavior with high accuracy, and providing insights into flat and crumpled membrane phases.

Contribution

It introduces a self-consistent screening approximation that is exact in certain limits, offering new predictions for critical exponents and phase boundaries of polymerized membranes.

Findings

Predicts roughness exponent ζ=0.590 for flat membranes.

Determines size exponent ν=0.732 at the crumpling transition.

Identifies the lower critical dimension D_{lc}={2d}/{d+1}.

Abstract

We study $D$-dimensional polymerized membranes embedded in $d$ dimensions using a self-consistent screening approximation. It is exact for large $d$ to order $1/d$, for any $d$ to order $\epsilon=4-D$ and for $d=D$. For flat physical membranes ($D=2,d=3$) it predicts a roughness exponent $\zeta=0.590$. For phantom membranes at the crumpling transition the size exponent is $\nu=0.732$. It yields identical lower critical dimension for the flat phase and crumpling transition $D_{lc}(d)={2 d \over {d+1}}$ ($D_{lc}={\sqrt{2}}$ for codimension 1). For physical membranes with ${\it random}$ quenched curvature $\zeta=0.775$ in the new $T=0$ flat phase in good agreement with simulations.