Consider the 1D quantum XXZ model defined by the Hamiltonian: $$ \begin{aligned} H &= ∑_i S^x_{i}S^x_{i+1} + S^y_{i}S^y_{i+1} + Δ_1 S^z_{i}S^z_{i+1} \\ &= ∑_i \frac{1}{2}(S^+_{i}S^-_{i+1}+\mathrm{H.c.}) + Δ_1 S^z_{i}S^z_{i+1} \end{aligned} $$ Or in terms of spinless fermions from the Jordan-Wigner transformation: $$ H = ∑_i \frac{1}{2}(c^†_{i}c_{i+1}+\mathrm{H.c.}) + Δ_1 \left(n_{i}-\frac12\right)\left(n_{i+1}-\frac12\right) $$ I choose an average filling per site $\bar{n}=1/2$ which means an average magnetization $\bar{m}=0$.
The system is well known for being a Luttinger liquid (LL) (so critical) for $|Δ_1|<1$ and it undergoes a gapless-gapped quantum phase transition of Berezinskii-Kosterlitz-Thouless (BKT) type at $Δ_1=1$ to a z-ordered Néel (antiferromagnetic) state ↑↓↑↓. It occurs at the Luttinger coefficient $K_c=1/2$.
My questions is about the transition occuring at $Δ_1=-1$. Nowhere can I find the nature (the universality class) of this transition to the ferromagnetic state ↑↑↑↑↑↓↓↓↓↓ (also named phase separation because of the $\bar{m}=0$ constraint). Apart from its gapless-gapped nature and that it is a special point for the LL perspective as $u→0$ and $K→∞$, I do not know how to characterize it further. It seems to break the $ \mathbb{Z}_2 $ symmetry while the AFM transition breaks the discrete translational symmetry.
