All plaquette stabilizers commute and square to one. Every term in
$$
H=-\sum_P(S_P^{(X)}+S_P^{(Z)})
$$
is minimized by eigenvalue $+1$, so any ground state satisfies
$$
\boxed{S_P^{(X)}|\psi\rangle
=S_P^{(Z)}|\psi\rangle=|\psi\rangle}
$$
for every plaquette.

The R-edge operator $X_{e_R}$ commutes with every X-type stabilizer. It overlaps the two endpoint R plaquettes in an odd number of vertices and every other plaquette evenly, so it anticommutes only with the two endpoint Z-type stabilizers. It flips those two eigenvalues to $-1$, creating two $r_x$ particles. Multiplying by an adjacent R-edge operator toggles the shared endpoint twice, annihilating that excitation there while creating one at the new endpoint. Repetition forms a <string operator in a topological code> whose two excitations move farther apart.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 342 / 2 / c / Solution

A string operator creates excitations at its endpoints. Multiplying by a successive segment moves an endpoint because the shared intermediate excitation is toggled twice.


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 Incoming links: String operator in a topological code