  We study thermal fluctuations of a free-standing bilayer graphene subject to
vanishing external tension. Within a phenomenological theory, the system is
described as a stack of two continuum crystalline membranes, characterized by
finite elastic moduli and a nonzero bending rigidity. A nonlinear
rotationally-invariant model guided by elasticity theory is developed to
describe interlayer interactions. After neglection of in-plane phonon
nonlinearities and anharmonic interactions involving interlayer shear and
compression modes, an effective theory for soft flexural fluctuations of the
bilayer is constructed. The resulting model, neglecting anisotropic
interactions, has the same form of a well-known effective theory for
out-of-plane fluctuations in a single-layer membrane, but with a strongly
wave-vector dependent bare bending rigidity. Focusing on AB-stacked bilayer
graphene, parameters governing interlayer interactions in the theory are
derived by first-principles calculations. Statistical-mechanical properties of
interacting flexural fluctuations are then calculated by a numerical iterative
solution of field-theory integral equations within the self-consistent
screening approximation (SCSA). The bare bending rigidity in the considered
model exhibits a crossover between a long-wavelength regime governed by
in-plane elastic stress and a short wavelength region controlled by monolayer
curvature stiffness. Interactions between flexural fluctuations drive a further
crossover between a harmonic and a strong-coupling regime, characterized by
anomalous scale invariance. The overlap and interplay between these two
crossover behaviors is analyzed at varying temperatures.
