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The Transverse-field Ising modelcommutes with translations, reflections, and the global spin flipFor $\lambda \gg 1$, the unique product ground state has $Z_i=-1$ at every site; for $\lambda \ll -1$, it has $Z_i=+1$. Both are symmetric paramagnets. For $|\lambda |\ll 1$, the two thermodynamic ground states are ferromagnets $|+{⟩}_X^{\otimes L}$ and $|-{⟩}_X^{\otimes L}$, exchanged by $G$; on a finite ring their even and odd cat combinations have an exponentially small splitting.

The large-field phase has a unique symmetry-preserving ground state, whereas the small-field phase spontaneously breaks the ${\mathbb{Z}}_2$ symmetry. These distinct symmetry realizations cannot be connected while retaining both a nonzero thermodynamic gap and the symmetry, so a phase transition must intervene. Kramers--Wannier duality locates the self-dual transitions at $|\lambda |=1$.

The two-dimensional Kramers--Wannier duality PEPO has one physical input leg at every original vertex and one physical output leg at every edge. A COPY tensor at each vertex sends its binary value to all incident virtual legs. An XOR tensor on each edge compares the two incident virtual bits and writes their sum modulo two to the edge output. Contracting every vertex--edge virtual leg gives the graphical networkrepeated over the graph. In the X-Fourier basis the same network exchanges the COPY and parity constraints. This local tensor description is a projected entangled pair operator.