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

TL;DR

This paper classifies all possible mass gaps in graphene-like 2D systems, explores topological defects with fractional statistics, and discusses unconventional phase transitions beyond Landau-Ginzburg theory.

Contribution

It provides a comprehensive classification of 36 mass instabilities and analyzes topological defects with fractional exchange statistics in graphene-like systems.

Findings

Identified 36 mass gaps and their coexistence properties.

Demonstrated topological defects can host fractionalized excitations.

Calculated fractional statistics of vortices in Kekulé patterns.

Abstract

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. 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 Kekul\'e dimerization pattern, the staggered chemical potential, and the Haldane mass. Vortices in the Kekul\'e 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.