  We examine excitons formed in the bulk of a topological insulator as the
system is tuned via a parameter between topological and trivial insulating
phases, arguing that nontrivial topology has fingerprints in the spectrum of
these excitons. The closely related hydrogen atom problem is well known to have
a degeneracy due to a hidden symmetry, and the changes to the excitonic
spectrum that we find can be understood as a result of breaking of this
underlying symmetry due to the Berry phase. Furthermore, this phase is found to
affect the spectrum in the topological parameter regime much more strongly than
in the trivial regime. We first construct a semiclassical model of the system
to develop qualitative intuition for the effects at play, then move to a more
robust numerical simulation of the full quantum system, working with the
Bernevig-Hughes-Zhang model of a 2D topological insulator.
