I am interested in the 'tower-of-states' type continuous spontaneous symmetry breaking (SSB) in quantum systems. For concreteness, assume a $d$ dimensional lattice Hilbert space. That is, suppose a Hamiltonian $H$ is invariant under action of a symmetry group $G$, and that for any finite system size $N$, the ground state of $H$, call it $|\psi_{gs}>$, is unique and therefore $G$-symmetric. By tower of states type SSB, I am referring to examples analogous to the emergence of Neèl order in the 2D (quantum) Heisenberg antiferromagnet, wherein an asymmetric ground state emerges that is not a finite system eigenstate, from linear combinations of many low-lying quasi-degenerate states as $N\rightarrow\infty$.
It is known that a diagnostic for continuous symmetry breaking is the emergence of long-range-order (LRO) in the finite size symmetric state. By long range order, I refer to the following phenomenon: For some local operator localized to lattice site $x$, $O_x$, the correlation function $\langle O_x O_y \rangle \rightarrow constant$ as $|x-y|\rightarrow \infty$. Crucially, this relation holds in the unique symmetric ground state of $H$. It is then stated that the above condition implies spontaneous breaking of the symmetry in the true thermodynamic limit.
My question is, to what extent is this an IFF condition for continuous symmetry breaking? Are there examples such that the symmetric ground state exhibits long range order, yet SSB is prohibited due to strong 'quantum fluctuations'? Conversely, is SSB always accompanied by LRO? What if the symmetric state exhibits quasi-long range order instead? i.e if there is a local operator that decays as a power-law: $\langle O_x O_y\rangle \sim |x-y|^{-\alpha}$ - is it possible that a symmetry is spontaneously broken in this case?
I did come across this related answer: Spontaneous symmetry breaking: proving the equivalence of two definitions
However, the answer primarily seems to discuss the equivalence of SSB (spontaneous acquisition of an order parameter expectation value in the symmetry broken state), and LRO in the symmetry broken state, not the symmetric finite system ground state.
