A New Phase of Tethered Membranes: Tubules

Abstract

We show that fluctuating tethered membranes with {\it any} intrinsic anisotropy unavoidably exhibit a new phase between the previously predicted ``flat'' and ``crumpled'' phases, in high spatial dimensions $d$ where the crumpled phase exists. In this new "tubule" phase, the membrane is crumpled in one direction but extended nearly straight in the other. Its average thickness is $R_G\sim L^{\nu_t}$ with $L$ the intrinsic size of the membrane. This phase is more likely to persist down to $d=3$ than the crumpled phase. In Flory theory, the universal exponent $\nu_t=3/4$, which we conjecture is an exact result. We study the elasticity and fluctuations of the tubule state, and the transitions into it.