TY - JOUR

T1 - Masses in graphenelike two-dimensional electronic systems

T2 - Topological defects in order parameters and their fractional exchange statistics

AU - Ryu, Shinsei

AU - Mudry, Christopher

AU - Hou, Chang Yu

AU - Chamon, Claudio

PY - 2009/11/18

Y1 - 2009/11/18

N2 - We classify all possible 36 gap-opening instabilities in graphenelike 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 sixth competing mass, the one added by Haldane to obtain the quantum Hall effect in graphene without magnetic fields, which breaks time-reversal symmetry and competes with all other masses in any of the quintuplets. Topological defects in these five-dimensional order parameters can generically bind excitations with fractionalized quantum numbers. The problem simplifies greatly if we consider spin-rotation invariant systems without superconductivity. In such simplified systems, the possible masses are only 4 and correspond to the Kekulé dimerization pattern, the staggered chemical potential, and the Haldane mass. Vortices in the Kekulé pattern are topological defects that have Abelian fractional statistics in the presence of the Haldane term. We calculate the statistical angle by integrating out the massive fermions and constructing the effective field theory for the system. Finally, we discuss how one can have generically non-Landau-Ginzburg-type transitions with direct transitions between phases characterized by distinct order parameters.

AB - We classify all possible 36 gap-opening instabilities in graphenelike 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 sixth competing mass, the one added by Haldane to obtain the quantum Hall effect in graphene without magnetic fields, which breaks time-reversal symmetry and competes with all other masses in any of the quintuplets. Topological defects in these five-dimensional order parameters can generically bind excitations with fractionalized quantum numbers. The problem simplifies greatly if we consider spin-rotation invariant systems without superconductivity. In such simplified systems, the possible masses are only 4 and correspond to the Kekulé dimerization pattern, the staggered chemical potential, and the Haldane mass. Vortices in the Kekulé pattern are topological defects that have Abelian fractional statistics in the presence of the Haldane term. We calculate the statistical angle by integrating out the massive fermions and constructing the effective field theory for the system. Finally, we discuss how one can have generically non-Landau-Ginzburg-type transitions with direct transitions between phases characterized by distinct order parameters.

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U2 - 10.1103/PhysRevB.80.205319

DO - 10.1103/PhysRevB.80.205319

M3 - Article

AN - SCOPUS:77950853233

SN - 1098-0121

VL - 80

JO - Physical Review B - Condensed Matter and Materials Physics

JF - Physical Review B - Condensed Matter and Materials Physics

IS - 20

M1 - 205319

ER -