Topological insulators owe the protection of their surface state to
the combination of gapped bulk and a discrete symmetry of their
Hamiltonian. Breaking this symmetry removes the protection of the
surface state and allows them to localize. I will introduce an
extension of the topological insulators, "statistical topological
insulators" that survive breaking of the protecting symmetry in a way
that preserves it on average. As a particular example of such a phase
I will show how using the reflection symmetry as the protecting
symmetry allows to generate infinitely many higher dimension
descendants of original topological insulators.
