Ground state magnetization of the Heisenberg XXZ chain

The Hamiltonian of the Heisenberg XXZ chain (without external field) has the form $$H = -J \sum_{n=1}^{N}\left(S_n^xS_{n+1}^x+ S_n^yS_{n+1}^y + \Delta S_n^zS_{n+1}^z\right).$$ It is known that this model has three phases, for instance, from this book,
1. Ferromagnetic for $$\Delta > 1$$
2. Paramagnetic for $$-1 \leq \Delta \leq 1$$
3. Anti-ferromagnetic for $$\Delta < -1$$

In the ground state, the magnetization per site along the $$z-$$axis is $$\langle\sigma^z\rangle = \pm 1$$ for $$\Delta > 1$$, and zero otherwise.

Is there a formula or expression for the magnetization per site along the $$x-$$axis, $$\langle\sigma^x\rangle$$, as a function of $$\Delta$$? Or is there an order parameter that distinguishes between the paramagnetic and anti-ferromagnetic phases?

As you say, the magnetization $$\langle \sigma^z \rangle$$ is $$\pm 1$$ in the ferromagnetic regime ($$\Delta > 1$$), and is zero otherwise. To distinguish the paramagnetic regime from the antiferromagnetic regime you need to look at the two-point correlator $$\langle \sigma_i \sigma_j \rangle$$. In particular the asymptotic behavior of the correlation function: $$\mathrm{lim}_{(i-j) \rightarrow \infty} (-1)^{( i - j)} \langle \sigma_i \sigma_j \rangle = P_0^2$$ reveals the presence of long-range magnetic order. An expression for $$P_0$$ was obtained by Baxter.