  In recent experiments bosonic [Atala et al., Nat. Phys. 10, 588 (2014), B. K.
Stuhl et al., Science 349, 1514 (2015)] as well as fermionic ladders [M.
Mancini et al., Science 349, 1510 (2015)] with a uniform flux were studied and
different interesting many-body states were observed. Motivated by these
experiments, we extend the uniform synthetic magnetic field to a periodic case
and show that a commensurate synthetic magnetic field offers an alternative
scheme to realize topological phases in many-body systems of ultra-cold Fermi
gases in ladder-like optical lattices. Using the exact diagonalization, we
numerically determine the topological band structure, edge states, non-zero
Chern numbers, Hofstadter-like-butterfly spectrum, and a complete phase diagram
of non-interacting fermionic ladders.
