Bosonic Crystalline Symmetry Protected Topological Phases Beyond the Group Cohomology Proposal

Understand

It is demonstrated by explicit construction that three-dimensional bosonic crystalline symmetry protected topological (cSPT) phases are classified by $H_{\phi}^{5}(G;\mathbb{Z})\oplus H_{\phi}^{1}(G;\mathbb{Z})$ for all 230 space groups $G$, where $H^n_\phi(G;\mathbb{Z})$ denotes the $n$th twisted group cohomology of $G$ with $\mathbb{Z}$ coefficients, and $\phi$ indicates that $g\in G$ acts non-trivially on coefficients by sending them to their inverses if $g$ reverses spacetime orientation and acts trivially otherwise.

• The previously known summand $H_{\phi}^{5}(G;\mathbb{Z})$ corresponds only to crystalline phases built without the $E_8$ state or its multiples on 2-cells of space.
• It is the crystalline analogue of the "group cohomology proposal" for classifying bosonic symmetry protected topological (SPT) phases, which takes the form $H_{\phi}^{d+2}(G;\mathbb{Z})\cong H_{\phi}^{d+1}(G;U(1))$ for finite internal symmetry groups in $d$ spatial dimensions.
• The new summand $H_{\phi}^{1}(G;\mathbb{Z})$ classifies possible configurations of $E_8$ states on 2-cells that can be used to build crystalline phases beyond the group cohomology proposal.