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In 2D Chern insulators (2D CI), the topology of the bulk states is capturedby a topological invariant, the Chern number. The scalar bulk-boundarycorrespondence (sBBC) relates the change in Chern number across an interfacewith the number of 1D chiral edge modes at the interface. However, 3D Cherninsulators (3D CI) can be characterized by a Chern vector C = (Cx, Cy, Cz) anda more general vector bulk-boundary correspondence (vBBC) is needed tocorrectly predict the propagation of the surface modes. In this work thepossible interfaces between 3D photonic CIs are explored, focusing on possiblechanges in Chern vector orientation. To formulate a 3D vBBC, a link is derivedbetween the Chern vector discontinuity across an interface and the winding ofthe surface equifrequency loops on the boundary. Lastly, it is demonstrated howto correctly predict the number and the propagation direction of topologicalphotonic surface modes in 3D CIs.

Vectorial Bulk-Boundary Correspondence for 3D Photonic Chern Insulators