  Topology and correlations are fundamental concepts in modern physics, but
their simultaneous occurrence within a single quantum phase is exceptionally
rare. In this study, we present the discovery of such a phase of matter in
Ta2Pd3Te5, a semimetal where the Coulomb interaction between electrons and
holes leads to the spontaneous formation of excitonic bound states below T=100
K. Our spectroscopy unveils the development of an insulating gap stemming from
the condensation of these excitons, thus giving rise to a highly sought-after
correlated quantum phase known as the excitonic insulator. Remarkably, our
scanning tunneling microscopy measurements reveal the presence of gapless
boundary modes in the excitonic insulator state. Their magnetic field response
and our theoretical calculations suggest a topological origin of these modes,
rendering Ta2Pd3Te5 as the first experimentally identified topological
excitonic insulator in a three-dimensional material not masked by any
structural phase transition. Furthermore, our study uncovers a secondary
excitonic instability below T=5 K, which differs from the primary one in having
finite momentum. We observe unprecedented tunability of its wavevector by an
external magnetic field. These findings unlock a frontier in the study of novel
correlated topological phases of matter and their tunability.
