  The interplay between time-reversal symmetry (TRS) and band topology plays a
crucial role in topological states of quantum matter. In
time-reversal-invariant (TRI) systems, the inversion of spin-degenerate bands
with opposite parity leads to nontrivial topological states, such as
topological insulators and Dirac semimetals. When the TRS is broken, the
exchange field induces spin splitting of the bands. The inversion of a pair of
spin-splitting subbands can generate more exotic topological states, such as
quantum anomalous Hall insulators and magnetic Weyl semimetals. So far, such
topological phase transitions driven by the TRS breaking have not been
visualized. In this work, using angle-resolved photoemission spectroscopy, we
have demonstrated that the TRS breaking induces a band inversion of a pair of
spin-splitting subbands at the TRI points of Brillouin zone in EuB$_6$, when a
long-range ferromagnetic order is developed. The dramatic changes in the
electronic structure result in a topological phase transition from a TRI
ordinary insulator state to a TRS-broken topological semimetal (TSM) state.
Remarkably, the magnetic TSM state has an ideal electronic structure, in which
the band crossings are located at the Fermi level without any interference from
other bands. Our findings not only reveal the topological phase transition
driven by the TRS breaking, but also provide an excellent platform to explore
novel physical behavior in the magnetic topological states of quantum matter.
