A one-to-two-dimensional mapping using a partial Fast Fourier Transform

It will be shown how to map a simple one-dimensional tight binding model with a cosine potential in one dimension exactly to a two-dimensional tight binding model with periodic boundary conditions in the presence of a single flux quantum spread evenly on the torus. The mapping is achieved by a partial sequence of “Fast Fourier Transform” (FFT) steps which if completed would be an exact Fourier transform of the original model. Each step of the FFT recursively maps a tight binding model into two decoupled sublattices of half the lattice length.