$\mathbf{L}=1$ states separated from the ground state by an excitation gap $\sim\hbar^2J$. This picture is believed to be correct as long as the magnitude of the potential energy term is much smaller than the excitation energy for the $\mathbf{L}=1$ state. In the second case, in which the potential energy term dominates (large $S$), we expect that the ground state is a magnetically ordered (Néel state) with $\hat{n}_i=\hat{n}_0$ at all sites $i$, where the excitations are Goldstone modes of the ordered state (spin waves).

It turns out that this naive expectation is valid only in dimensions of $d>1$. In one dimension, the magnetically ordered state is not stable because of quantum fluctuations associated with the Goldstone mode (Mermin-Wigner-Hohenberg Theorem), and the ground state is always quantum disordered [@Mermin66; @Hohenberg67], i.e., a spin liquid state. This result can be shown more rigorously through a renormalization group (RG) analysis of the NL$\sigma$M. We do not go through this analysis in detail in this article; instead, we simply assume that this is the case and examine its consequences. Readers interested in the RG analysis can consult, for example, references [@Polyakov75; @Polyakov87; @Brezin76].

Physically, this result means that after some renormalization, the ground state of 