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) similar to 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.