 Solution

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 $|\lambda |=1$.