A New Phase of Tethered Membranes: Tubules

We show that fluctuating tethered membranes with 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 $d=3\mathrm{}$ than the crumpled phase. In Flory theory, the universal exponent ${\nu }_t=3/4\mathrm{}$, which we conjecture is an exact result. We study the elasticity and fluctuations of the tubule state, and the transitions into it.