  We study the mechanical buckling of a two dimensional membrane coated with a
thin layer of superfluid. It is seen that a singularity (vortex or anti-vortex
defect) in the phase of the quantum order parameter, distorts the membrane
metric into a negative conical singularity surface, irrespective of the defect
sign. The defect-curvature coupling and the observed instability is in striking
contrast with classical elasticity where, the in-plane strain induced by
positive (negative) disclinations is screened by a corresponding positive
(negative) conical singularity surface. Defining a dimensionless ratio between
superfluid stiffness and membrane bending modulus, we derive conditions under
which the quantum buckling instability occurs. An ansatz for the resulting
shape of the buckled membrane is analytically and numerically confirmed.

Soft Condensed Matter