  In this work, we propose a model with Lorentz symmetry violation which
describes the electronic low energy limit of the AA-bilayer graphene (BLG)
system. The AA-type bilayer is known to preserve the linear dispersion relation
of the graphene layer in the low energy limit. The theoretical model shows that
in the BLG system, a time-like vector can be associated with the layer
separation and contributes to the energy eigenstates. Based on these
properties, we can describe in a $(2+1)$-dimensional space-time the fermionic
quasi-particles that emerge in the low-energy limit with the introduction of a
Lorentz-violating parameter, in analogy with the $(3 + 1)$-dimensional Standard
Model Extension (SME). Moreover, we study the consequences of the coupling of
these fermionic quasi-particles with the electromagnetic field, and we show via
effective action that the low-energy photon acquires a massive spectrum.
Finally, using the hydrodynamic approach in the collisionless limit, one finds
that the LSV generates a new kind of anomalous thermal current to the vortexes
of the system via coupling of the LSV vector.
