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

There are only two one-site matrices, so they cannot span the four-dimensional virtual matrix algebra. For two sites,Every overlap $⟨s_i|j⟩$ is nonzero, and the four outer products $|i⟩⟨s_j|$ are linearly independent. Hence

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Physical ${\sigma }_x$ swaps $A^0$ and $A^1$. Direct multiplication of the four blocked matrices showsThus one may choose

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Apply ${\sigma }_x$ to every odd site and use the first pulling-through identity on every two-site block. Adjacent $V_a^{-1}{V}_a$ factors cancel, while periodicity and cyclicity of the trace cancel the final pair. The MPS is unchanged. The identical argument with $V_b$ applies to every even site. Therefore the state has the two commuting physical symmetrieswhich generate ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$.

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Although the physical symmetry generators commute,Their virtual commutator isThis phase cannot be removed by rephasing $V_a$ and $V_b$. The virtual matrices therefore form a nontrivial projective representation, which is the nontrivial projective virtual symmetry of a matrix product state and proves that the cluster state lies in a nontrivial SPT phase.

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Under symmetry $g$, cyclicity changes the twist matrix byTherefore its charge is the sign in this conjugation. With $V_a=X$ and $V_b=Z$,The four twists realize all four characters of ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$.

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Ignoring the common normalization $1/(2\sqrt{2})$, direct multiplication givesThese four states span the positive eigenspace of $Z\otimes X\otimes Z$. The parent term annihilating that local MPS support is therefore the complementary projector

Translated terms have stabilizers $K_j=Z_{j-1}{X}_j{Z}_{j+1}$. Two such Pauli strings either do not overlap nontrivially or have two X--Z anticommutations, so all $K_j$ commute. Their positive eigenspaces have a common state, making the parent Hamiltonian frustration free. Each alternating global X symmetry flips the two Z factors of every $K_j$ or neither, and therefore commutes with every local term.

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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

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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.

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