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

摘要

We classify all possible 36 gap-opening instabilities in graphene-likestructures in two dimensions, i.e., masses of Dirac Hamiltonian when the spin,valley, and superconducting channels are included. These 36 order parametersbreak up into 56 possible quintuplets of masses that add in quadrature, andhence do not compete and thus can coexist. There is additionally a 6thcompeting mass, the one added by Haldane to obtain the quantum Hall effect ingraphene without magnetic fields, that breaks time-reversal symmetry andcompetes with all other masses in any of the quintuplets. Topological defectsin these 5-dimensional order parameters can generically bind excitations withfractionalized quantum numbers. The problem simplifies greatly if we considerspin-rotation invariant systems without superconductivity. In such simplifiedsystems, the possible masses are only 4 and correspond to the Kekul\'edimerization pattern, the staggered chemical potential, and the Haldane mass.Vortices in the Kekul\'e pattern are topological defects that have Abelianfractional statistics in the presence of the Haldane term. We calculate thestatistical angle by integrating out the massive fermions and constructing theeffective field theory for the system. Finally, we discuss how one can havegenerically non-Landau-Ginzburg-type transitions, with direct transitionsbetween phases characterized by distinct order parameters.