Hierarchical zero- and one-dimensional topological states in symmetry-controllable grain boundary.

Structural imperfections can be a promising testbed to engineer the symmetries and topological states of solid-state platforms. Here, we present direct evidence of hierarchical transitions of zero- (0D) and one-dimensional (1D) topological states in symmetry-enforced grain boundaries (GB) in 1T'-MoTe. Using a scanning tunneling microscope tip press-and-pulse procedure, we construct two distinct types of GBs, which are differentiated by the underlying symmorphic and nonsymmorphic symmetries. The GBs with the nonsymmorphic rotation symmetry harbor first-order topological edge states protected by a nonsymmorphic band degeneracy. On the other hand, the edge state of the symmorphic GBs attains a band gap. More interestingly, the gapped edge state realizes a hierarchical topological phase, evidenced by the additional 0D boundary states at the GB ends. We anticipate our experiments will pioneer the material platform for the hierarchical realization of first-order and higher-order topology.