= Solution
The <Hamiltonian> has the discrete <symmetry> described in <2/a/solution>, whereas either selected easy-axis <ground state> has a nonzero <order parameter> $\langle S^x\rangle$ and is not invariant under it. This is <spontaneous symmetry breaking>: the equations preserve the symmetry while the selected macroscopic <quantum state> does not.
An infinitesimal longitudinal <magnetic field> selects one orientation. The defining order of limits is
$$
\boxed{m_x=\lim_{h\to0^+}\lim_{L\to\infty}\frac1L\sum_j\langle S_j^x\rangle_h\ne0}
$$
in the ordered zero-temperature phase, with the opposite sign for $h\to0^-$. A symmetric finite-volume <quantum state> can instead have zero <magnetization>. Quantum ordered states are generally dressed by fluctuations rather than exact fully polarized product states, because unequal transverse exchange can create pairs of <spin waves>. Discrete easy-axis symmetry does not imply a gapless <Goldstone mode>; indeed <2/e/solution> finds a nonzero <spin wave> gap. Nor does this argument assert positive-temperature order in a one-dimensional short-range chain.<Statistical Field Theory by David Tong>[Section 1.2.1, spontaneous symmetry breaking and the order of thermodynamic and zero-field limits]{ref}
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