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

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

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 ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$.

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.

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 ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$.