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A fundamental dichotomous classification for all physical systems isaccording to whether they are spinless or spinful. This is especially crucialfor the study of symmetry-protected topological phases, as the two classes havedistinct symmetry algebra. As a prominent example, the spacetime inversionsymmetry $PT$ satisfies $(PT)^2=\pm 1$ for spinless/spinful systems, and eachclass features unique topological phases. Here, we reveal a possibility toswitch the two fundamental classes via $\mathbb{Z}_2$ projectiverepresentations. For $PT$ symmetry, this occurs when $P$ inverses the gaugetransformation needed to recover the original $\mathbb{Z}_2$ gauge connectionsunder $P$. As a result, we can achieve topological phases originally unique forspinful systems in a spinless system, and vice versa. We explicitly demonstratethe claimed mechanism with several concrete models, such as Kramers degeneratebands and Kramers Majorana boundary modes in spinless systems, and realtopological phases in spinful systems. Possible experimental realization ofthese models is discussed. Our work breaks a fundamental limitation ontopological phases and opens an unprecedented possibility to realize intriguingtopological phases in previously impossible systems.

Switching spinless and spinful topological phases with projective $PT$ symmetry