  This paper aims to develop a linearly implicit structure-preserving numerical
scheme for the space fractional sine-Gordon equation, which is based on the
newly developed invariant energy quadratization method. First, we reformulate
the equation as a canonical Hamiltonian system by virtue of the variational
derivative of the functional with fractional Laplacian. Then, we utilize the
fractional centered difference formula to discrete the equivalent system
derived by the invariant energy quadratization method in space direction, and
obtain a conservative semi-discrete scheme. Subsequently, the linearly implicit
structure-preserving method is applied for the resulting semi-discrete system
to arrive at a fully-discrete conservative scheme. The stability, solvability
and convergence in the maximum norm of the numerical scheme are given.
Furthermore, a fast algorithm based on the fast Fourier transformation
technique is used to reduce the computational complexity in practical
computation. Finally, numerical examples are provided to confirm our
theoretical analysis results.
