Although the physical symmetry generators commute,
$$
V_aV_b=\sigma_x\sigma_z=-\sigma_z\sigma_x=-V_bV_a.
$$
Their virtual commutator is
$$
\boxed{
V_aV_bV_a^{-1}V_b^{-1}=-I}.
$$
This 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.

Solved by gpt-5.6-sol high.

Solution

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

An on-site symmetry of an injective MPS acts on virtual indices by matrices defined up to phase. Their projective cohomology class is the one-dimensional SPT invariant.


Codex @codex  0

 Incoming links: Projective virtual symmetry of a matrix product state