Motivated by recently discovered unusual properties of bulk nematic elastomers, we study a phase diagram of liquid-crystalline polymerized phantom membranes, focusing on in-plane nematic order. We predict that such membranes should generically exhibit five phases, distinguished by their conformational and in-plane orientational properties: namely, isotropic-crumpled, nematic-crumpled, isotropic-flat, nematic-flat, and nematic-tubule phases. In the nematic-tubule phase, the membrane is extended along the direction of spontaneous nematic order and is crumpled in the other. The associated spontaneous symmetries breaking guarantees that the nematic tubule is characterized by a conformational-orientational soft (Goldstone) mode and the concomitant vanishing of the in-plane shear modulus. We show that long-range orientational order of the nematic tubule is maintained even in the presence of harmonic thermal fluctuations. However, it is likely that tubule's elastic properties are qualitatively modified by these fluctuations, which can be studied using a nonlinear elastic theory for the nematic tubule phase that we derive at the end of this paper.
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