Topological protection conjecture for edge modes in perturbed honeycomb structures

Let ${\mathscr{P}}_\delta[\zeta]:H^2[\zeta]\to L^2[\zeta]$ be the family of edge operators, with Dirac points $(\xi_\star^A,E_\star)$ and $(\xi_\star^B,E_\star)$ satisfying $\operatorname{(H1)}$--$\operatorname{(H4)}$. A family $B_\delta(\zeta)$ is admissible if, for sufficiently small $\delta>0$, each $B_\delta(\zeta):H^2[\zeta]\to L^2[\zeta]$ is compact, depends continuously on $\zeta$, is $2\pi$-periodic, and the perturbed operator has no eigenvalues in the essential spectral gap containing $E_\star$. Topological protection conjecture. Assume that $\operatorname{(H1)}$--$\operatorname{(H4)}$ hold for both Dirac points $(\xi_\star^A,E_\star)$ and $(\xi_\star^B,E_\star)$. There exists $\delta_0>0$ such that for every $\delta\in(0,\delta_0)$, there is an admissible family $B_\delta(\zeta)$ for which ${\mathscr{P}}_\delta[\zeta]+B_\delta(\zeta)$ has no eigenvalues in the essential spectral gap containing $E_\star$. This asserts that the edge index vanishes and the edge modes are not topologically protected. The statement concerns robustness under periodic compact perturbations; its resolution is not supplied here.