Self-consistent theory of polymerized membranes

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 \ensuremath{\epsilon}=4-D, and for d=D. For flat physical membranes (D=2, d=3) it predicts a roughness exponent \ensuremath{\zeta}=0.590. For phantom membranes at the crumpling transition the size exponent is \ensuremath{\nu}=0.732. It yields identical lower critical dimension for the flat phase and crumpling transition ${\mathit{D}}_{\mathrm{lc}}$(d)=2d/(d+1) (${\mathit{D}}_{\mathrm{lc}}$= \ensuremath{\surd}2 for codimension 1). For physical membranes with random quenched curvature \ensuremath{\zeta}=0.775 in the new T=0 flat phase in good agremeent with simulations.