Absence of a crumpling transition in strongly self-avoiding tethered membranes

We report extensive simulations of a model for tethered self-avoiding membranes with bending rigidity. These simulations have been performed at finite and infinite temperature for membranes whose stretched configurations are hexagonal with linear dimensions $L$ in the range 5-19. We have analyzed the shape of the clusters by calculating the eigenvalues ${\lambda }_i$, $i=1-3\mathrm{}$ of the inertia tensor and the structure factor for wave vectors in the direction of the eigenvectors of this matrix. We find that the smallest eigenvalue scales, for all temperatures investigated, as ${\lambda }_1\sim L^{v_1}$ with $v_1\approx 0.65$; the two larger eigenvalues scale as $L$. Thus, in the thermodynamic limit $L\to \infty$, the tethered self-avoiding membrane is flat but rough, in contrast to previous conclusions [Y. Kantor and D. R. Nelson, Phys. Rev. Lett. 58, 2774 (1987); Phys. Rev. A 36, 4020 (1987)], and in contrast to the behavior of membranes without self-avoidance which display a thermodynamic phase transition between a crumpled high-temperature phase and a low-temperature flat phase [Y. Kantor, M. Kardar, and D. R. Nelson, Phys. Rev. Lett. 57, 791 (1986); Phys. Rev. A 35, 3056 (1987)].