  A second-order topological insulator in $d$ dimensions is an insulator which
has no $d-1$ dimensional topological boundary states but has $d-2$ dimensional
topological boundary states. It is an extended notion of the conventional
topological insulator. Higher-order topological insulators have been
investigated in square and cubic lattices. In this paper, we generalize them to
breathing Kagome and pyrochlore lattices. First, we construct a second-order
topological insulator on the breathing Kagome lattice. Three topological
boundary states emerge at the corner of the triangle, realizing a 1/3
fractional charge at each corner. Second, we construct a third-order
topological insulator on the breathing pyrochlore lattice. Four topological
boundary states emerge at the corners of the tetrahedron with a 1/4 fractional
charge at each corner. These higher-order topological insulators are
characterized by the quantized polarization, which constitutes the bulk
topological index. Finally, we study a second-order topological semimetal by
stacking the breathing Kagome lattice.
