Fig. 2: Models and physical consequences of projective symmetry algebras.

a The canonical model for P2. The dashed line marks the unit cell. The four classes of translation-related rotation centers are colored in red, green, purple and blue, respectively. Each rotation center is the center of a plaquette shadowed with the same color of the rotation center. The four colors correspond to the four α-invariants in Table 1. b The model of P2 that realizes the projective symmetry algebra (PSA) with α₁ = α₂ = 1 and α₃ = α₄ = − 1. The dashed line marks the unit cell, which contains four sites. The bonds with red color have a negative hopping amplitude, which makes the shaded plaquettes having a π flux. a and b Are lattice vectors. Here, we added a dimerization pattern in hopping to open spectral gaps, as in (d). c Due to the PSA in (b), high-symmetry momenta are shifted and the Zak phase θ_b over a G_b-periodic path must be nontrivial. Here, k_a,b are the wave-vector components for the lattice vectors (a and b) in (b). d Spectrum of the model in (b) on the slab geometry with the b dimension confined. The spectrum is parametrized by k_a. The in-gap edge states are colored in red, which arise from the nontrivial Zak phase. To construct a canonical model for P3m1, we first build a one-layer lattice model in (e) to accommodate the σ invariant. Then, we double it into a bilayer model in (f) to further accommodate the β invariant. In (f), black and white colors mean the two sites are inequivalent, e.g., they may have different on-site energies. (g) Band structure of a P3m1 model in (f), which exhibits an eightfold nodal point at Γ. Note that along Γ-M (Γ-K), each band is twofold (fourfold) degenerate. h Dispersion in the vicinity of the eightfold degenerate nodal point.