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Fractional Quantum Hall effect (FQHE) is a unique many-body phenomenon, whichwas discovered in a two-dimensional electron system placed in a strongperpendicular magnetic field. It is entirely due to the electron-electroninteractions within a given Landau level. For special filling factors of theLandau level, a many-particle incompressible state with a finite collective gapis formed. Among these states, when the Landau level is half filled, there is aspecial FQHE state that is described by the Pfaffian function and the statesupports charged excitations that obey non-Abelian statistics. Such a$1/2$-FQHE state can be realized only for a special profile of theelectron-electron potential. For example, for conventional electron systems,the $1/2$-FQHE state occurs only in the second Landau level, while in agraphene monolayer, no $1/2$-FQHE state can be found in any Landau level.Another type of low-dimensional system is the bilayer graphene, which consistsof two graphene monolayers coupled through the inter-layer hopping. The systemis quasi-two-dimensional, which makes it possible to tune the inter-electroninteraction potential by applying either the bias voltage or the magnetic fieldthat is applied parallel to the bilayer. It so happens that in the bilayergraphene with AB staking, there is one Landau level per valley where the$1/2$-FQHE state can indeed be present. The properties of that $1/2$-FQHE statehave a nonmonotonic dependence on the applied magnetic field and this can beeven more stable than the one discovered in conventional electron systems.

Interacting Dirac fermions and the rise of Pfaffians in graphene