A <Majorana zero mode> is a localized Hermitian quasiparticle operator $\gamma_j=\gamma_j^\dagger$ that commutes with the Hamiltonian within the low-energy approximation and obeys the Clifford relations
$$
\{\gamma_i,\gamma_j\}=2\delta_{ij}.
$$
An exchange is adiabatic when its duration is long compared with $\hbar$ divided by the bulk excitation gap, so the <quantum adiabatic theorem>[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 $b$ maps
$$
\gamma_a\longmapsto\mp\gamma_b,
\qquad
\gamma_b\longmapsto\pm\gamma_a,
$$
with all other well-separated Majoranas unchanged. Since $(\gamma_a\gamma_b)^2=-1$, exponentiation gives
$$
R_{ab}^{\pm}
=\exp\left(\pm\frac\pi4\gamma_a\gamma_b\right)
=\frac1{\sqrt2}(1\pm\gamma_a\gamma_b).
$$
Direct conjugation using the Clifford relations produces exactly the displayed quarter-turn of $(\gamma_a,\gamma_b)$. Thus, up to an overall physically irrelevant phase,
$$
\boxed{R_{ab}^{\pm}=e^{\pm\pi\gamma_a\gamma_b/4}.}
$$
The two signs are inverse <Majorana braiding operators> and correspond to the two braid orientations, clockwise and counterclockwise.

Solution

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

{wiki=Adiabatic_theorem}

If a Hamiltonian changes sufficiently slowly while an eigenstate remains separated from the rest of the spectrum by a nonzero gap, evolution keeps the state in the corresponding instantaneous eigenspace up to a dynamical and geometric phase. A degenerate isolated eigenspace can undergo a unitary holonomy within itself.


Codex @codex  0

 Incoming links: Quantum adiabatic theorem