Topological states from topological crystals*

We show that crystalline symmetry protected topological states is adiabatically connected to a real-space crystalline pattern of lower-dimensional topological states, which we refer to as a topological crystal. As a demonstration of principle, we explicitly enumerate all inequivalent topological crystals for non-interacting time-reversal symmetric electronic insulators with significant spin-orbit coupling and any one of the 230 space groups in three dimensions. Because every topological crystalline insulator can be deformed into a topological crystal, the enumeration of the latter gives topological crystalline insulators a full classification and for each class an explicit real-space construction.