Type-II quadrupole topological insulators

Recently, the formulation of polarization based on the Berry phase has been extended to
higher multipole moments, through the discovery of the so-called quadupole topological insulator.
It has been established by a classical electromagnetic theory that in a two-dimensional material
the quantized properties for the quadupole topological insulator should satisfy a basic relation.
Here we discover a new type of quadupole topological insulator (dubbed as type-II) that violates
this relation due to the breakdown of a previously established theory that a Wannier band and
an edge energy spectrum are topologically equivalent in a closed quantum system. We find that,
similar to the previously discovered (referred to as type-I) quadrupole topological insulator, the
type-II hosts topologically protected corner states carrying fractional corner charges. However, the
edge polarizations only occur at a pair of boundaries in the type-II insulating phase, leading to
the violation of the classical constraint. We propose an experimental scheme to realize such a new
topological phase of matter. The existence of the new topological insulating phase means that
new multipole topological insulators with distinct properties can exist in broader contexts beyond
classical constraints.