No abstract

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