 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-343/4/e/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

On the square lattice the PEPO intertwines the local operators as

X_{u} X_{v} ⟷ Z_{e = (uv)}, Z_{v} ⟷ A_{v} = \prod_{e ∋ v} X_{e} .

Its image also obeys the automatic zero-flux constraints

B_{p} = \prod_{e \in \partial p} Z_{e} = 1

. Thus the transverse-field Ising Hamiltonian maps to a

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 gives

H_{TC} = - v \sum A_{v} - p \sum B_{p},

the toric code Hamiltonian. Conversely, within a simply connected zero-flux sector one may solve

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.

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