  Rationally independent free fermions are those where sums of single-particle
energies multiplied by arbitrary rational coefficients vanish only if the
coefficients are all zero. This property guaranties that they have no
degeneracies in the many-body spectrum and gives them relaxation properties
more similar to those of generic systems. Using classic results from number
theory we provide minimal examples of rationally independent free fermion
models for every system size in one dimension. This is accomplished by
considering a free fermion model with a chemical potential, and hopping terms
corresponding to all the divisors of the number of sites, each one with an
incommensurate complex amplitude. We further discuss the many-body spectral
statistics for these models and show that local probes -- like the ratio of
consecutive level spacings -- look very similar to what is expected for the
Poisson statistics. We however demonstrate that free fermion models can never
have Poisson statistics with an analysis of the moments of the spectral form
factor.
