  Topological insulator is an exciting discovery because of its robustness
against disorder and interactions. Recently, higher-order topological
insulators have been attracting increasing attention, because they host 1D
topologically-protected hinge states in 3D or 0D corner states in 2D. A
significantly critical issue is whether the higher-order topological insulators
also survive interactions, but it is still unexplored. We study the effects of
weak Coulomb interaction on 3D second-order topological insulators, with the
help of a renormalization group calculation. We find that the higher-order
topological insulators are always unstable, suffering from two types of
topological phase transitions. One is from higher-order topological insulator
to first-order topological insulator, the other is to normal insulator. The
first type is accompanied by emergent time-reversal and inversion symmetries
and has a dynamical critical exponent $z=1$. The second type does not have the
emergent symmetries and has non-universal dynamical critical exponents $z<1$.
Our results may inspire more inspections on the stability of higher-order
topological states of matter and related novel quantum criticalities.
