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 symmetries
$$
\boxed{\prod_{j\ {\rm odd}}X_j},
\qquad
\boxed{\prod_{j\ {\rm even}}X_j},
$$
which generate $\mathbb Z_2\times\mathbb Z_2$.

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 / ii / Solution

 Solution

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