Fig. 1: From 2D to 3D magnetic structures.
From: Photonic axion insulator with non-coplanar chiral hinge transport
a 2D QAH effect with insulating bulk but conducting edge, characterized by Chern number (C). Reversing magnetization (M) flips the cyclic direction of edge energy flow. b 3D Chern vector from ferromagnetic stacking, characterized by Chern vector (C). Original 1D edge state expands to a 2D surface. c 3D AXI from antiferromagnetic stacking, characterized by \({{\mathbb{Z}}}_{4}\) topological invariant. The bulk and surface band structures open up a complete bandgap, wherein non-coplanar chiral hinge states exist. The gapless hinge state appears at the hinge formed by two surfaces with opposite signs of surface Dirac mass terms (m).
