Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 343 / 3 / d

In the X eigenbasis, $M$ is $\mathrm{diag}(I,X)$ on its physical indices. A physical X insertion changes the sign of the minus block and can be pulled through asConjugating both physical legs by Z swaps the two blocks:Hence

Likewise $N=\mathrm{diag}(I,Z)$. Its analogous relations are

Solved by gpt-5.6-sol high.

Pulling the three operators of each cluster stabilizer through the alternating MPO cancels all internal virtual Pauli matrices and givesThus, for $H={\sum }_j(I-K_j)/2$,This is a pair of decoupled ferromagnetic Ising chains, one on each parity sublattice. Its four product ground states independently choose all odd spins up or down and all even spins up or down in the Z basis. In a symmetry-preserving basis they become four cat states.

The dual phase spontaneously breaks the two ${\mathbb{Z}}_2$ spin-flip symmetries. It has ordinary symmetry-breaking order and no nontrivial SPT invariant; the nonlocal MPO has converted the cluster SPT order into symmetry breaking.

Solved by gpt-5.6-sol high.