  The buckling of thin elastic sheets is a classic mechanical instability that
occurs over a wide range of scales. In the extreme limit of atomically thin
membranes like graphene, thermal fluctuations can dramatically modify such
mechanical instabilities. We investigate here the delicate interplay of
boundary conditions, nonlinear mechanics and thermal fluctuations in
controlling buckling of confined thin sheets under isotropic compression. We
identify two inequivalent mechanical ensembles based on the boundaries at
constant strain (isometric) or at constant stress (isotensional)
conditions.Remarkably, in the isometric ensemble, boundary conditions induce a
novel long-ranged nonlinear interaction between the local tilt of the surface
at distant points. This interaction combined with a spontaneously generated
thermal tension leads to a renormalization group description of two distinct
universality classes for thermalized buckling, realizing a mechanical variant
of Fisher-renormalized critical exponents. We formulate a complete scaling
theory of buckling as an unusual phase transition with a size dependent
critical point, and discuss experimental ramifications for the mechanical
manipulation of ultrathin nanomaterials.
