  We investigate ultra-cold fermions placed into an optical cavity and
subjected to optical lattices which confine the atoms to ladder structures. A
transverse running-wave laser beam induces together with the dynamical cavity
field a two-photon Raman-assisted tunneling process with spatially dependent
phase imprint along the rungs of the ladders. We identify the steady states
which can occur by the feedback mechanism between the cavity field and the
atoms. We find the spontaneous emergence of a finite cavity field amplitude
which leads to an artificial magnetic field felt by the fermionic atoms. These
form a chiral insulating or chiral liquid state carrying a chiral current. We
explore the rich state diagram as a function of the power of the transverse
laser beam, the atomic filling, and the phase imprint during the cavity-induced
tunneling. Both a sudden onset or a slow exponential activation with the
transverse laser power of the self-organized chiral states can occur.
