The Fast Mode-Fourier-Transform and the bands of the Bethe chain

A many-body Fourier transformation with logarithmic scaling in the number of necessary two-site gates is implemented. The protocol is applied to study the Bethe chain as a prototype of a translationally invariant system. The resulting band diagram features a flat pattern highly correlated to interaction mechanisms. The introduced protocol can be applied to a wide spectrum of scenarios and can offer new possibilities of simulation and analysis.