Graphene shapes from quantum elasticity

Temperature constraints are highly desirable in the experimental setup when seeking the synthesis of new carbon structures. Fluctuations of the Dirac field result in temperature-dependent corrections to the Helfrich-Canham formulation, which governs the classical elasticity of the graphene membrane at equilibrium. Here, we examine the emergent shapes allowed by the effective model up to quadratic order in Ricci curvature and discuss the constraints required to observe them. We determine the mechanical stability conditions and provide a phase diagram characterized by the appearance of a critical temperature ${T}_{\mathrm{c}}$ that distinguishes between carbon nanotubes and fullerene phases. The observation of minimal and developable surfaces is anticipated in the high- and low-temperature regimes, respectively. Additionally, a Beltrami trumpet surface is forecasted when the membrane is subjected to an external source balancing out internal Helfrich stresses.