Particle transmutations in flat-band lattices: bosons to fermions, fermions to composite fermions.

Understanding new quantum states and exotic quasiparticles can often be facilitated by approaching the problem in a different basis. One such example, amongst many, is viewing hard core bosons as fermions coupled to a special gauge-field (called the Chern-Simons gauge field). The non-Maxwellian nature of this gauge field introduces new challenges in treating such a problem. However, progress has been made by treating the problem via various mean-field schemes. The most remarkable one, perhaps, is its application to fractional quantum hall states. In this talk, we will draw our attention to the consequences of one particular scheme that is designed to be applicable for various interesting 2D lattices. We will see how this scheme leads to a particular flux distribution that provides a natural description of the system in the parameter space where a chiral spin-liquid is stabilized for bosons residing in either a Honeycomb lattice or a Kagome lattice. We will view these results in light of other numerical works in the literature. Interestingly, we will also see that when the same scheme is applied to fermions in a lattice, subject to an external magnetic field, we can make interesting predictions related to the composite fermion state: in particular the theory suggests that Graphene with next nearest neighbor hopping at the half-filling condition should be a doped Haldane-Chern insulator of composite fermions.