Pulling the three operators of each cluster stabilizer through the alternating MPO cancels all internal virtual Pauli matrices and gives
$$
\boxed{
O\,(Z_{j-1}X_jZ_{j+1})
=(Z_{j-1}Z_{j+1})\,O}.
$$
Thus, for $H=\sum_j(I-K_j)/2$,
$$
\boxed{
H'=\frac12\sum_j(I-Z_{j-1}Z_{j+1})},
\qquad
OH=H'O.
$$
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.


Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-343/3/d/ii/solution