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For a one-dimensional periodic spin chain, a Kramers--Wannier intertwiner maps vertex spins in the $X$ basis to edge domain walls. It obeys $D(X_i{X}_{i+1})={\tilde{Z}}_{i+1/2}D$ and $D{Z}_i=({\tilde{X}}_{i-1/2}{\tilde{X}}_{i+1/2})D$ and identifies spin configurations related by the global ${\mathbb{Z}}_2$ symmetry.

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 $|\lambda |=1$.