Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 342 / 3 / d / Solution

The two new boundaries surround opposite sides of the same bulk. Their induced orientations are opposite, so their chiral edge modes counterpropagate and $\mathrm{sgn}{v}_1=-\mathrm{sgn}{v}_2$.

For $v_1=v$ and $v_2=-v$, define the massIt approaches $+t$ on the left and $-t$ on the right. The zero-energy first-order equations have one normalizable real solution, whose envelope can be chosen asafter choosing the constant Majorana spinor with the appropriate relative sign. The corresponding operatoris self-adjoint and commutes with the Hamiltonian, so it is a Majorana zero mode at a mass domain wall.

Outside the core it decays onIf the phase varies approximately linearly across the core, then $m(x)\simeq -\pi tx/R_c$ near zero and the central envelope is Gaussian with widthThus the spatial extent is of order $\max (\xi ,{\ell }_{\mathrm{core}})$, up to constants depending on the detailed vortex profile. When $|v_1|\ne |v_2|$, continuously changing their magnitudes to equality never changes their opposite signs or closes the asymptotic mass gap. The mass domain wall therefore retains its odd, particle-hole-protected zero mode; only its two component amplitudes and localization lengths change.