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

Use the $X$ eigenbasis $|s_i{⟩}_X$, $s_i\in {\mathbb{Z}}_2$, and put a dual qubit on every edge with $t_i=s_i+s_{i+1}(\mathrm{mod}2)$. The Kramers--Wannier intertwiner isAn explicit matrix product operator tensor isContracting neighboring virtual indices ${\beta }_i={\alpha }_{i+1}$ around the ring is the graphical MPO: each tensor copies its input bit to the left virtual leg and outputs the XOR of its two virtual legs.

Directly from the domain-wall definition,Therefore $DH(\lambda )=\tilde{H}(\lambda )D$, whereAfter exchanging $X$ and $Z$ and rescaling, this is the same Ising family at reciprocal coupling, so corresponding symmetry sectors have the same spectrum and $H(\lambda )$ is dual to $|\lambda |H(1/\lambda )$ up to the elementary sign conventions. In particular the Ising spectrum is self-dual at $|\lambda |=1$.