Bulk-boundary correspondence and edge modes

Consider an interface along y and a mass varying across x from negative to positive. At zero tangential momentum, the Dirac equation becomes a first-order spatial equation. Multiplying by the x Pauli matrix shows that a definite y-Pauli eigenvector can decay on both sides. The positive y eigenvector works for this mass orientation, while the negative one grows. Normalizability, not an arbitrary decision about a spin direction, selects the surviving state.

For a sharp sign wall with positive mass magnitude m0, the amplitude falls as exp(−|x|/ξ), with ξ equal to hbar v divided by m0. Restoring tangential momentum gives a linear dispersion whose sign follows the selected y-Pauli eigenvalue. Reversing the mass wall reverses that eigenvalue and propagation direction. This low-energy calculation supplies one interface branch; an ultraviolet lattice completion is needed to assign absolute integer Chern numbers to the individual Dirac regions.

For fixed axes and the positive-i Berry convention, define the left bulk at negative x and the right bulk at positive x. The signed number of positive-y minus negative-y interface channels equals the right-minus-left Chern difference in this convention. Additional counterpropagating pairs can appear through boundary reconstruction. Such pairs do not alter the net index and can generally couple away, unlike the unmatched chiral imbalance.

The ordinary band correspondence assumes a local charge-conserving Hamiltonian, insulating bulk gaps, and an interface separated from other boundaries on the localization scale. A boundary potential may shift dispersions or add trivial localized levels while preserving the required net crossing of a bulk gap. Observing a peak near a surface at one energy is consequently not enough to identify a protected channel; its connection across the bulk gap and response to permitted perturbations matter.

Opposite edges of a narrow strip provide opposite propagation directions and can hybridize through their exponentially small wavefunction overlap. A resulting finite-size gap does not contradict the semi-infinite bulk-boundary statement. Similarly, disorder can obscure momentum-resolved dispersions while a chiral transport channel persists; a mobility-gap formulation is then more appropriate than a clean plot of energy against edge momentum. Lead coupling and inelastic scattering introduce additional transport scales.

Symmetry-protected helical boundaries use a different obstruction, and breaking their protecting symmetry can gap them. Intrinsically topologically ordered systems may have gapped boundaries with nontrivial anyon boundary conditions, while some three-dimensional interacting surfaces can acquire their own topological order. The present net-chirality equation is therefore a precise band-interface statement. It should not be promoted into an unrestricted claim that all uses of the word topology guarantee a metallic surface.