We demonstrate that a key elastic parameter of a suspended crystalline membrane—the Poisson ratio (PR) $\nu$—is a nontrivial function of the applied stress $\sigma$ and of the system size $L$, i.e., $\nu ={\nu }_L\left(\sigma \right)$. We consider a generic two-dimensional membrane embedded into space of dimensionality $2+d_c$. (The physical situation corresponds to $d_c=1$.) A particularly important application of our results is to freestanding graphene. We find that at a very low stress, when the membrane exhibits linear response, the PR ${\nu }_L\left(0\right)$ decreases with increasing system size $L$ and saturates for $L\to \infty$ at a value which depends on the boundary conditions and is essentially different from the value $\nu =-1/3$ previously predicted by the membrane theory within a self-consistent scaling analysis. By increasing $\sigma$, one drives a sufficiently large membrane (with the length $L$ much larger than the Ginzburg length) into a nonlinear regime characterized by a universal value of PR that depends solely on $d_c$, in close connection with the critical index $\eta$ controlling the renormalization of bending rigidity. This universal nonlinear PR acquires its minimum value ${\nu }_{\min }=-1$ in the limit $d_c\to \infty$, when $\eta \to 0$. With the further increase of $\sigma$, the PR changes sign and finally saturates at a positive nonuniversal value prescribed by the conventional elasticity theory. We also show that one should distinguish between the absolute and differential PR ($\nu$ and ${\nu }^{\mathrm{diff}}$, respectively). While coinciding in the limits of very low and very high stress, they differ in general: $\nu \ne {\nu }^{\mathrm{diff}}$. In particular, in the nonlinear universal regime, ${\nu }^{\mathrm{diff}}$ takes a universal value which, similarly to the absolute PR, is a function solely of $d_c$ (or, equivalently, of $\eta$) but is different from the universal value of $\nu$. In the limit of infinite dimensionality of the embedding space, $d_c\to \infty$ (i.e., $\eta \to 0$), the universal value of ${\nu }^{\mathrm{diff}}$ tends to $-1/3$, at variance with the limiting value $-1$ of $\nu$. Finally, we briefly discuss generalization of these results to a disordered membrane.

• Received 16 January 2018
• Revised 14 February 2018

DOI:https://doi.org/10.1103/PhysRevB.97.125402