  An implicit variable-step BDF2 scheme is established for solving the space
fractional Cahn-Hilliard equation, involving the fractional Laplacian, derived
from a gradient flow in the negative order Sobolev space $H^{-\alpha}$,
$\alpha\in(0,1)$. The Fourier pseudo-spectral method is applied for the spatial
approximation. The proposed scheme inherits the energy dissipation law in the
form of the modified discrete energy under the sufficient restriction of the
time-step ratios. The convergence of the fully discrete scheme is rigorously
provided utilizing the newly proved discrete embedding type convolution
inequality dealing with the fractional Laplacian. Besides, the mass
conservation and the unique solvability are also theoretically guaranteed.
Numerical experiments are carried out to show the accuracy and the energy
dissipation both for various interface widths. In particular, the
multiple-time-scale evolution of the solution is captured by an adaptive
time-stepping strategy in the short-to-long time simulation.
