  The realization of synthetic gauge fields has attracted a lot of attention
recently in relation with periodically driven systems and the Floquet theory.
In ultra-cold atom systems in optical lattices and photonic networks, this
allows to simulate exotic phases of matter such as quantum Hall phases,
anomalous quantum Hall phases and analogs of topological insulators. In this
paper, we apply the Floquet theory to engineer anisotropic Haldane models on
the honeycomb lattice and two-leg ladder systems. We show that these
anisotropic Haldane models still possess a topologically non-trivial band
structure associated with chiral edge modes (without the presence of a net unit
flux in a unit cell), then referring to the quantum anomalous Hall effect.
Focusing on (interacting) boson systems in s-wave bands of the lattice, we show
how to engineer through the Floquet theory, a quantum phase transition between
a uniform superfluid and a BEC (Bose-Einstein Condensate) analog of FFLO
(Fulde-Ferrell-Larkin-Ovchinnikov) states, where bosons condense at non-zero
wave-vectors. We perform a Ginzburg-Landau analysis of the quantum phase
transition on the graphene lattice, and compute observables such as chiral
currents and the momentum distribution. The results are supported by exact
diagonalization calculations and compared with those of the isotropic
situation. The validity of high-frequency expansion in the Floquet theory is
also tested using time-dependent simulations for various parameters of the
model. Last, we show that the anisotropic choice for the effective vector
potential allows a bosonization approach in equivalent ladder (strip)
geometries.
