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

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.

A Majorana zero mode is a localized Hermitian quasiparticle operator ${\gamma }_j={\gamma }_j^{†}$ that commutes with the Hamiltonian within the low-energy approximation and obeys the Clifford relationsAn exchange is adiabatic when its duration is long compared with $\hslash$ divided by the bulk excitation gap, so the adiabatic theorem keeps the state in the ground-state manifold. It should also be short compared with the inverse exponentially small zero-mode splitting if that splitting is not neglected.

Locality and conservation of fermion parity imply that exchanging modes $a$ and $b$ mapswith all other well-separated Majoranas unchanged. Since $({\gamma }_a{\gamma }_b{)}^2=-1$, exponentiation givesDirect conjugation using the Clifford relations produces exactly the displayed quarter-turn of $({\gamma }_a,{\gamma }_b)$. Thus, up to an overall physically irrelevant phase,The two signs are inverse Majorana braiding operators and correspond to the two braid orientations, clockwise and counterclockwise.

For $a\ne b$, the operator $Q_{ab}=i{\gamma }_a{\gamma }_b$ is Hermitian andThereforeFor $f=({\gamma }_a+i{\gamma }_b)/2$, one has $Q_{ab}=2f^{†}f-1$, so these projectors distinguish the two occupations, equivalently the two values of pair fermion parity. This proves thatdescribe a fermion-parity measurement.

Let $P={\Pi }_{+}^{(N0)}$ and suppose $P|\psi ⟩=|\psi ⟩$. The bilinears $i{\gamma }_a{\gamma }_0$ and $i{\gamma }_b{\gamma }_0$ each anticommute with $i{\gamma }_N{\gamma }_0$, whereas their product commutes with it. Expanding the projectors and sandwiching by $P$ therefore givesAfter normalizing the post-measurement state, the sequence consequently implementson the encoded ground space. This is Measurement-only Majorana braiding.

Immediately before measuring $Q_{b0}=i{\gamma }_b{\gamma }_0$, the state has definite parity for a bilinear such as $Q_{a0}$ or the restored reference $Q_{N0}$ that anticommutes with $Q_{b0}$. If $Q|\psi ⟩=s|\psi ⟩$ and $\{Q,Q_{b0}\}=0$, thenHence

If the undesired value of $s_b$ occurs, perform a reference measurement that projects the ancillary pair back toward its previous parity sector, then measure $Q_{b0}$ again. Alternating these anticommuting parity measurements gives a fresh probability $1/2$ of the desired result on each attempt. This Forced Majorana parity measurement has a geometric waiting time, succeeds almost surely, and does not measure the encoded parity $i{\gamma }_a{\gamma }_b$ directly. Apply the same repeat-until-success procedure to each required outcome in ${\Pi }_{+}^{(N0)}{\Pi }_{s_a}^{(a0)}{\Pi }_{s_b}^{(b0)}$; alternatively, keep arbitrary outcomes and track the known inverse-braid or Pauli-frame correction determined by their signs.