  The non-interacting band structure of spinless fermions in a two-dimensional
($d=2$) $p$-band honeycomb lattice exhibits two quadratic band touching points
(QBTPs), which lie at the Fermi levels of filling $\nu=1/4$ and its
particle-hole conjugated filling $\nu=3/4$. A weak Hubbard interaction $U$
spontaneously breaks the time-reversal symmetry and removes the QBTP, rendering
the system into a quantum anomalous Hall insulator (QAHI). The first-order
topological nature of QAHI is characterized by a nontrivial Chern number and
supports ($d-1$)-dimensional chiral edge modes. With increasing the interaction
$U$, the system is driven into a Dirac semimetal by breaking the crystal
symmetry through a discontinuous quantum phase transition. The emergent Dirac
points each with Berry flux $\pi$ are generated in pairs, originating from the
$2\pi$ Berry flux of QBTP. A sufficiently large $U$ ultimately drives the
system into a dimerized insulator (DI) by simultaneously annihilating the Dirac
points at the Brillouin zone boundary. The second-order topological nature of
DI is characterized by the quantized polarizations and supports
($d-2$)-dimensional corner states. Our study provides a unique setting for
exploring the topological switch between the first-order and second-order
topological insulators.
