  We consider quantum rings realized in materials where the dynamics of charge
carriers mimics that of two-dimensional (2D) Dirac electrons. A general
theoretical description of the ring-subband structure is developed that applies
to a range of currently available 2D systems, including graphene,
transition-metal dichalcogenides, and narrow-gap semiconductor quantum wells.
We employ the scattering-matrix approach to calculate the electronic
two-terminal conductance through the ring and investigate how it is affected by
Dirac-electron interference. The interplay of pseudo-spin chirality and
hard-wall confinement is found to distinctly affect the geometric phase that is
experimentally accessible in mesoscopic-conductance measurements. We derive an
effective Hamiltonian for the azimuthal motion of charge carriers in the ring
that yields deeper insight into the physical origin of the observed transport
effects, including the unique behavior exhibited by the lowest ring subband in
the normal and topological (i.e., band-inverted) regimes. Our work provides a
unified approach to characterizing confined Dirac electrons, which can be used
to explore the design of valley- and spintronic devices based on quantum
interference and the confinement-tunable geometric phase.
