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Protected Boundaries and Interfaces of Invertible Matter

Bulk–boundary correspondence constrains the anomalous net content at an interface between two invertible phases. It does not fix every boundary dispersion, velocity, or accidental band. The statement requires locality, a bulk spectral or mobility gap, a declared protecting symmetry, and the difference between the two bulk invariants.

Required background. Chern insulators supplies chiral interface counting; time-reversal topological insulators supplies helical and Dirac boundary parity.

Helpful background. Anomaly inflow supplies the boundary nonconservation and obstruction viewpoint.

A Dirac mass domain wall

Section titled “A Dirac mass domain wall”

Take a continuum two-band Hamiltonian with an interface normal to xx,

H=−ivσx∂x+vkyσy+m(x)σz,m(−∞)m(+∞)<0.H=-iv\sigma_x\partial_x+vk_y\sigma_y+m(x)\sigma_z, \qquad m(-\infty)m(+\infty)<0.

At ky=0k_y=0, multiply Hψ=0H\psi=0 by σx\sigma_x to obtain

∂xψ=−m(x)vσyψ.\partial_x\psi=-\frac{m(x)}{v}\sigma_y\psi.

Exactly one σy\sigma_y eigenvalue produces a wavefunction decaying on both sides, with localization length of order v/∣m∣v/|m|. Projecting vkyσyvk_y\sigma_y onto it gives a single chiral dispersion E(ky)=svkyE(k_y)=s v k_y, where ss is fixed by the mass orientation. Reversing the interface normal reverses the printed chirality. The lattice supplies the ultraviolet completion and makes the difference in Chern numbers integral Hatsugai 1993.

Interface index and allowed reconstruction

Section titled “Interface index and allowed reconstruction”

For class A, the net number of right- minus left-movers equals the appropriately oriented ΔC\Delta C. Smooth potentials can bend velocities and add counterpropagating pairs, but they cannot change the net count while charge conservation and the bulk mobility gap survive. At a time-reversal edge, protection concerns the parity of Kramers pairs; even numbers can gap through symmetry-preserving mixing.

A boundary can become gapped in four distinct ways:

  1. break the protecting symmetry explicitly or spontaneously;
  2. close the adjacent bulk gap and change the phase;
  3. couple to another boundary carrying the opposite anomaly;
  4. with interactions in sufficiently high dimension, develop intrinsic boundary topological order that carries the anomaly.

The fourth mechanism explains why “symmetrically gapped” need not mean “trivial.” Conversely, a finite slab can hybridize its two opposite surfaces and open an exponentially small gap without changing the infinite-system bulk.

Physical tests and failure modes

Section titled “Physical tests and failure modes”

Compare interfaces with known bulk indices, not isolated surface spectra. Confirm that the chemical potential intersects boundary rather than bulk states; vary termination to separate robust net flow from accidental bands; test the protecting symmetry; and examine thickness and system-size scaling. For disorder, a mobility gap and delocalized boundary transport replace a clean band plot. For interacting boundaries, thermal and charge anomalies, symmetry fractionalization, and topological degeneracy may be required Witten 2016, §§ 2–3.

Exercise

Section titled “Exercise”

An interface separates Chern phases with CL=2C_L=2 and CR=−1C_R=-1. How many chiral channels are protected?

Solution

The invariant difference has magnitude ∣ΔC∣=∣CR−CL∣=3|\Delta C|=|C_R-C_L|=3, so three net co-propagating complex-fermion channels are protected. Extra counterpropagating pairs may be present, and the sign of their net direction depends on the chosen interface orientation.

References

Section titled “References”