Switching Spinless and Spinful Topological Phases with Projective $PT$ Symmetry

A fundamental dichotomous classification for all physical systems is according to whether they are spinless or spinful. This is especially crucial for the study of symmetry-protected topological phases, as the two classes have distinct symmetry algebra. As a prominent example, the spacetime inversion symmetry $PT$ satisfies $(PT{)}^2=\pm 1$ for spinless/spinful systems, and each class features unique topological phases. Here, we reveal a possibility to switch the two fundamental classes via ${\mathbb{Z}}_2$ projective representations. For $PT$ symmetry, this occurs when $P$ inverses the gauge transformation needed to recover the original ${\mathbb{Z}}_2$ gauge connections under $P$. As a result, we can achieve topological phases originally unique for spinful systems in a spinless system, and vice versa. We explicitly demonstrate the claimed mechanism with several concrete models, such as Kramers degenerate bands and Kramers Majorana boundary modes in spinless systems, and real topological phases in spinful systems. Possible experimental realization of these models is discussed. Our work breaks a fundamental limitation on topological phases and opens an unprecedented possibility to realize intriguing topological phases in previously impossible systems.

Figure 1

(a) Schematic figure of the 1D spinless chain. Each unit cell (indicated by the green cubes) contain eight sites. The hopping amplitudes along $x$ are marked in the figure. The red bonds have a negative hopping amplitude $-t_z$. (b) Calculated bulk band structure. Each band is twofold degenerate. (c) Spectrum of a finite chain with a length of 30 unit cells. The four zero modes form two Majorana Kramers pairs. Each pair is localized at one end of the chain.

Figure 2

(a) Schematic figure for the 2D model, consisting of two layers of square lattices. The unit cell consists of four sites as marked by the green square. The red colored bonds have a negative hopping amplitude $-t_z$. (b) Bulk band structure of the model (13). There are eight twofold real Dirac points at zero energy, protected by ${\nu }_{1\mathrm{D}}=1$. Here, we take parameters $t_x=t_y=t_z=1$ and $\lambda =0.5$.

Figure 3

(a) Schematic figure for the 3D generalized Kane-Mele model. Each 2D honeycomb layer is a 2D Kane-Mele model. The red colored bonds have negative hopping amplitudes $-J_1$ or $-J_2$. The unit cell contains four sites, as indicated by the shaded square in the right panel. (b) The bulk band structure contains four real nodal loops. Each loop is characterized by a twofold topological charge $({\nu }_{1\mathrm{D}},{\nu }_{2\mathrm{D}})=(1,1)$, as indicated by the inset. (c) Calculated spectrum of the system with a tubelike geometry extended along $z$ and with a diamond-shaped cross section (see inset). The hinge Fermi arc states can be clearly visualized. Here, we take parameters $t_1=1$, $t_2=0.08$, $J_1=0.3$, $J_2=0.48$, $m_1=0.4$, and $m_2=0.3$.