Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 342 / 3 / a / Solution

The continuum Hamiltonian is that of a spinless one-dimensional p-wave topological superconductor. Its topological phase has $\mu >0$, whereas the empty vacuum may be modeled as the trivial phase with $\mu <0$. With superconductor at $x<0$ and vacuum at $x>0$, this is

Near the transition $\mu =0$, momenta are small and the quadratic term may be neglected. In Pauli-matrix notation, the long-wavelength Bogoliubov--de Gennes Hamiltonian isThe zero-energy equation becomesChoose a constant spinor ${\chi }_{+}$ with ${\tau }_y{\chi }_{+}={\chi }_{+}$. Thendecays on both sides because $\mu$ changes from positive to negative. For asymptotically constant $|\mu |$, the Continuum p-wave Majorana interface mode has widthIts characteristic momentum is $|p|\sim |\mu |/\Delta$. The neglected kinetic energy is small compared with $|\mu |$ when $|\mu |\ll m{\Delta }^2$ up to a factor of two, equivalently when $\xi \gg \hslash /(m\Delta )$.

In the Nambu basis, Particle-hole symmetry of a Bogoliubov--de Gennes Hamiltonian lets a zero-energy eigenvector be chosen self-conjugate, $v(x)=u(x{)}^{\ast }$. The corresponding quasiparticle operator isTaking the adjoint and interchanging the two terms gives ${\gamma }^{†}=\gamma$. With the usual normalization it obeys $\{\gamma ,\gamma \}=2$, so it is a Majorana fermion operator localized at the interface.