  A time-dependent optical lattice with staggered particle current in the
tight-binding regime was considered that can be described by a time-independent
effective lattice model with an artificial staggered magnetic field. The low
energy description of a single-component fermion in this lattice at
half-filling is provided by two copies of ideal two-dimensional massless Dirac
fermions. The Dirac cones are generally anisotropic and can be tuned by the
external staggered flux $\p$. For bosons, the staggered flux modifies the
single-particle spectrum such that in the weak coupling limit, depending on the
flux $\p$, distinct superfluid phases are realized. Their properties are
discussed, the nature of the phase transitions between them is establised, and
Bogoliubov theory is used to determine their excitation spectra. Then the
generalized superfluid-Mott-insulator transition is studied in the presence of
the staggered flux and the complete phase diagram is established. Finally, the
momentum distribution of the distinct superfluid phases is obtained, which
provides a clear experimental signature of each phase in ballistic expansion
experiments.
