= Solution
The <Transverse-field Ising model>
$$
H=-\sum_iX_iX_{i+1}+\lambda\sum_iZ_i
$$
commutes with translations, reflections, and the global spin flip
$$
G=\prod_iZ_i.
$$
For $\lambda\gg1$, 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 $|\lambda|\ll1$, the two thermodynamic ground states are ferromagnets $|+\rangle_X^{\otimes L}$ and $|-\rangle_X^{\otimes L}$, 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 $|\lambda|=1$.
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