Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 343 / 4 / e / Solution

On the square lattice the PEPO intertwines the local operators asIts image also obeys the automatic zero-flux constraints $B_p={\prod }_{e\in \partial p}{\tilde{Z}}_e=1$. Thus the transverse-field Ising Hamiltonian maps to a ${\mathbb{Z}}_2$ lattice-gauge Hamiltonian generated by edge fields and stars in the zero-flux sector.

At the commuting-projector fixed point, promoting the image constraints to energetic terms givesthe toric code Hamiltonian. Conversely, within a simply connected zero-flux sector one may solve ${\tilde{Z}}_{uv}=X_u{X}_v$ and recover vertex Ising variables. On a torus, the remaining noncontractible loop eigenvalues correspond to the different Ising boundary twists, which accounts for the toric code's topological ground-state sectors.