  We point out that in the deep band-inverted state, topological insulators are
generically vulnerable against symmetry breaking instability, due to a
divergently large density of states of 1D-like exponent near the chemical
potential. This feature at the band edge is associated with a novel van Hove
singularity resulting from the development of a Mexican-hat band dispersion. We
demonstrate this generic behavior via prototypical 2D and 3D models. This
realization not only explains the existing experimental observations of
additional phases, but also suggests a route to activate additional
functionalities to topological insulators via ordering, particularly for the
long-sought topological superconductivities.
