Elasticity, shape fluctuations, and phase transitions in the new tubule phase of anisotropic tethered membranes

We study the shape, elasticity, and fluctuations of the recently predicted [L. Radzihovsky and J. Toner, Phys. Rev. Lett. 75, 4752 (1995)] and subsequently observed (in numerical simulations) [M. Bowick, M. Falcioni, and G. Thorleifsson, Phys. Rev. Lett. 79, 885 (1997); tubule phase of anisotropic membranes, as well as the phase transitions into and out of it. This novel phase lies between the previously predicted flat and crumpled phases, both in temperature and in its physical properties: it is crumpled in one direction, and extended in the other. Its shape and elastic properties are characterized by a radius of gyration exponent $\nu$ and an anisotropy exponent $z$. We derive scaling laws for the radius of gyration $R_G{(L}_{\perp }{,L}_y)$ (i.e., the average thickness) of the tubule about a spontaneously selected straight axis and for the tubule undulations $h_{\mathrm{rms}}{(L}_{\perp }{,L}_y)$ transverse to its average extension. We show that for square membranes (with intrinsic size $L_{\perp }{=L}_y=L)$, $R_G\propto L^{\nu }$, and $h_{\mathrm{rms}}\propto L^{1-{\eta }_{\kappa }z/2\mathrm{}}$, with ${\eta }_{\kappa }$ a bending rigidity anomalous elasticity exponent related to $\nu$ and $z$. For phantom (i.e., non-self-avoiding) membranes, we predict $\nu =\frac{1}{4}$, $z=\frac{1}{2}$, and ${\eta }_{\kappa }=0\mathrm{}$, exactly, in excellent agreement with simulations. For $D=2\mathrm{}$ dimensional membranes embedded in the space of dimension $d<\mathrm{}11\mathrm{}$, self-avoidance greatly swells the tubule and suppresses its wild transverse undulations, changing its shape exponents $\nu$, $z$, and ${\eta }_{\kappa }.$ For a $D$-dimensional membrane embedded in $d>\mathrm{}{d}_{*}$ $\left[d_{*}\mathrm{}\right(D=2\mathrm{})>\frac{7}{2}]$, ${\eta }_{\kappa }=0\mathrm{}$ and $z=(D-1+2\nu )/3\mathrm{}$, while for $d<\mathrm{}{d}_{*},$ ${\eta }_{\kappa }>0\mathrm{}$ and