Masses in graphene-like two-dimensional electronic systems: topological defects in order parameters and their fractional exchange statistics

Understand

We classify all possible 36 gap-opening instabilities in graphene-like structures in two dimensions, i.e., masses of Dirac Hamiltonian when the spin, valley, and superconducting channels are included.

• These 36 order parameters break up into 56 possible quintuplets of masses that add in quadrature, and hence do not compete and thus can coexist.
• There is additionally a 6th competing mass, the one added by Haldane to obtain the quantum Hall effect in graphene without magnetic fields, that breaks time-reversal symmetry and competes with all other masses in any of the quintuplets.
• Topological defects in these 5-dimensional order parameters can generically bind excitations with fractionalized quantum numbers.