Fig. 1: Tuning cycles of 1D and 2D insulators with 0D boundary states.

a Terminology for the bulk and boundary of 2D and 3D systems. b A y-directed 1D SSH chain with quantized polarization, enforced by either 3D inversion \( {\mathcal{I}} \) or 1D mirror symmetry along the chain (e.g., the operation M_y, which takes y → −y)^16,17. c In a 2D crystal with \( {\mathcal{I}} \) symmetry, k_x can be treated as a parameter that periodically tunes between a y-directed SSH chain (double black lines in c and d) with zero polarization (k_x = 0) and another with e∕2 polarization (k_x = π), yielding a Chern insulator^17,40 with chiral edge modes (blue lines). d In a 2D crystal with M_y symmetry instead of \( {\mathcal{I}} \), the Hamiltonian at each value of k_x is equivalent to that of a y-directed SSH chain with a quantized polarization of 0 or e∕2; because the polarization cannot change continuously, a periodic tuning cycle indexed by k_x between SSH polarizations 0 and e∕2 must pass through a pair of gapless points. This yields a 2D band-inverted semimetal with topological polarization modes (red lines) analogous to those in zigzag-terminated graphene^1,44. e A \( {{\mathbb{Z}}}_{2} \) quantized quadrupole insulator (QI)²³ invariant under wallpaper group p4m. f A C_4z-broken, mirror-preserving pumping cycle of a QI (double black lines in f and g) is equivalent to a 3D 2nd-order Chern insulator^24,25,26,27 with chiral hinge modes (blue lines), whereas g a p4m-preserving cycle is equivalent to a 3D Dirac semimetal with higher-order Fermi arcs (HOFAs) on its 1D hinges (red lines).