  The doubly-connected polygonal geometry of planar graphene rings is found to
bring forth topological configurations for accessing nontrivial relativistic
quantum field (RQF) theory models that carry beyond the constant-mass
Dirac-fermion theory. These include generation of sign-alternating masses,
solitonic excitations, and charge fractionization. The work integrates a RQF
Lagrangian formulation with numerical tight-binding Aharonov-Bohm electronic
spectra and the generalized position-dependent-mass Dirac equation. In contrast
to armchair graphene rings (aGRGs) with pure metallic arms, certain classes of
aGRGs with semiconducting arms, as well as with mixed metallic-semiconducting
ones, are shown to exhibit properties of one-dimensional nontrivial topological
insulators. This further reveals an alternative direction for realizing a
graphene-based nontrivial topological insulator through the manipulation of the
honeycomb lattice geometry, without a spin-orbit contribution.
