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 R₆L^{ₓ} with L the intrinsic size of the membrane. This phase is more likely to persist down to d0ex{0ex}=0ex{0ex}3 than the crumpled phase. In Flory theory, the universal exponent ₓ0ex{0ex}=0ex{0ex}3/4, which we conjecture is an exact result. We study the elasticity and fluctuations of the tubule state, and the transitions into it.