Monte Carlo simulation in Fourier space

In the context of solving the long-standing problem of computing Landau-Ginzburg free energies including gradient corrections for the ϕ4 model, we recently introduced a new Monte Carlo algorithm for lattice spin systems based exclusively on Fourier amplitudes of the underlying spin configurations [A. Tröster, Phys. Rev. B 76 (2007) 012402]. In the present paper we shall provide additional information on the motivation, main ideas and constructions underlying the algorithm. Also we discuss important details of its construction with emphasis on an analysis of its scaling behavior with system size.