# Fractional charge bound to a vortex in two dimensional topological crystalline insulators

## TL;DR

This paper links the fractional charge bound to vortices in 2D topological crystalline insulators, like Kekule textured graphene, to bulk topological invariants and explores their relation to 3D axion insulators and symmetry protections.

## Contribution

It establishes a theoretical framework connecting vortex-bound fractional charge in 2D topological crystalline insulators to bulk topology and 3D axion insulator models, including symmetry considerations.

## Key findings

- Fractional charge at vortices relates to bulk topological invariants.
- Effective 3D Hamiltonian describes axion insulator with quantized magnetoelectric polarization.
- Vortex charge is half-quantized under chiral symmetry, otherwise it is not quantized but remains half-quantized with inversion symmetry.

## Abstract

We establish the correspondence between the fractional charge bound to a vortex in a textured lattice and the relevant bulk band topology in two-dimensional (2D) topological crystalline insulators. As a representative example, we consider the Kekule textured graphene whose bulk band topology is characterized by a 2D $\mathbb{Z}_{2}$ topological invariant $\nu_{\rm 2D}$ protected by inversion symmetry. The fractional charge localized at a vortex in the Kekule texture is shown to be related to the change in the bulk topological invariant $\nu_{\rm 2D}$ around the vortex, as in the case of the Su-Schriefer-Heeger model in which the fractional charge localized at a domain wall is related to the change in the bulk charge polarization between degenerate ground states. We show that the effective three-dimensional (3D) Hamiltonian, where the angle $\theta$ around a vortex in Kekule-textured graphene is a third coordinate, describes a 3D axion insulator with a quantized magnetoelectric polarization. The spectral flow during the adiabatic variation of $\theta$ corresponds to the chiral hinge modes of an axion insulator and determines the accumulated charge localized at the vortex, which is half-quantized when chiral symmetry exists. When chiral symmetry is absent, electric charge localized at the vortex is no longer quantized, but the vortex always carries a half-quantized Wannier charge as long as inversion symmetry exists. For the cases when magnetoelectric polarization is quantized due to the presence of symmetry that reverses the space-time orientation, we classify all possible topological crystalline insulators whose vortex defect carries a fractional charge.