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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 anddifferent interesting many-body states were observed. Motivated by theseexperiments, we extend the uniform synthetic magnetic field to a periodic caseand show that a commensurate synthetic magnetic field offers an alternativescheme to realize topological phases in many-body systems of ultra-cold Fermigases in ladder-like optical lattices. Using the exact diagonalization, wenumerically determine the topological band structure, edge states, non-zeroChern numbers, Hofstadter-like-butterfly spectrum, and a complete phase diagramof non-interacting fermionic ladders.

Topological Phases of Fermionic Ladders with Periodic Magnetic Fields