  By applying the linearly implicit conservative difference scheme proposed in
[D.-L. Wang, A.-G. Xiao, W. Yang. J. Comput. Phys. 2014;272:670-681], the
system of repulsive space fractional coupled nonlinear Schr\"odinger equations
leads to a sequence of linear systems with complex symmetric and
Toeplitz-plus-diagonal structure. In this paper, we propose the diagonal and
normal with Toeplitz-block splitting iteration method to solve the above linear
systems. The new iteration method is proved to converge unconditionally, and
the optimal iteration parameter is deducted. Naturally, this new iteration
method leads to a diagonal and normal with circulant-block preconditioner which
can be executed efficiently by fast algorithms. In theory, we provide sharp
bounds for the eigenvalues of the discrete fractional Laplacian and its
circulant approximation, and further analysis indicates that the spectral
distribution of the preconditioned system matrix is tight. Numerical
experiments show that the new preconditioner can significantly improve the
computational efficiency of the Krylov subspace iteration methods. Moreover,
the corresponding preconditioned GMRES method shows space mesh size independent
and almost fractional order parameter insensitive convergence behaviors.
