Bosonic Crystalline Symmetry Protected Topological Phases beyond the Group Cohomology Proposal

It is demonstrated by explicit construction that the intricate links among short-range-entangled (SRE) states across different dimensions have a vivid embodiment in the realm of symmetry protected topological (SPT) phases with crystalline symmetry. We systematically study three-dimensional bosonic topological phases protected by any space group symmetry G. We prove that these phases are classified by H_φ⁵(G;Z)×H_φ¹(G;Z), where φ indicates g∈G acting on Z as multiplying φ(g)=±1 depending on whether orientation is preserved by g or not. The factor H_φ⁵(G;Z)=H_Borel,φ⁴(G;U(1)), known as the group cohomology proposal for classifying bosonic SPT phases, corresponds to only the phases presented by some SRE 2-skeleton without presence of E₈ state or its multiples (i.e., two-dimensional chiral bosonic phases characterized by quantized thermal Hall effect). The extra factor H_φ¹(G;Z) describes inequivalent E₈ state configurations and be easily read off directly from the international (Hermann-Mauguin) symbol for G. Moreover, our result supports the Generalized Cohomology Hypothesis in the case of crystalline symmetries.