Anomalous elasticity in nematic and smectic elastomer tubule

We study anomalous elasticity in the tubule phases of nematic and smectic elastomer membranes, which are flat in one direction and crumpled in another. These phases share the same macroscopic symmetry properties including spontaneously-broken in-plane isotropy and hence belong to the same universality class. Below an upper critical value $D_c =3$ of the membranes' intrinsic dimension D, thermal fluctuations renormalize the elasticity with respect to elastic displacements along the tubule axis so that elastic moduli for compression along the tubule axis and for bending the tubule axis become length-scale dependent. This anomalous elasticity belongs to the same universality class as that of d-dimensional conventional smectics with D taking on the role of d. For physical tubule, D=2, this anomaly is of power-law type and thus might by easier to detect experimentally than the logarithmic anomaly in conventional smectics.