  Modern theory of electric polarization is formulated by the Berry phase,
which, when quantized, leads to topological phases of matter. Such a
formulation has recently been extended to higher electric 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
quadrupole topological insulator (dubbed type-II) that violates this relation
due to the breakdown of the correspondence that a Wannier band and an edge
energy spectrum close their gaps simultaneously. 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 demonstrate that such new topological phenomena can
appear from quench dynamics in non-equilibrium systems, which can be
experimentally observed in ultracold atomic gases. We also propose an
experimental scheme with electric circuits 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.
