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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?

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As you know from the reference you cite, the XXZ model is solvable using the algebraic Bethe Ansatz. It may be surprising, but although in principle one has an exact solution, actually extracting the behavior of an observable like the magnetization can be surprisingly difficult, as the solution depends on the solution of a large number of algebraic equations.

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.

The paramagnetic regime can also be distinguished from the antiferromagnetic regime by the spectrum of the low-lying energy states. The paramagnetic regime is gapless, while the antiferromagnetic and ferromagnetic regimes are both gapped.

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