# $Z_2$ symmetry breaking in XXZ model

I have a question about an statement that is said in the paper Entanglement and spontaneous symmetry breaking in quantum spin models (Phys. Rev. A 68, 060301(R), (2013)). It is related to the XXZ model in 1 dimension, given by

$$$$H_\text{xxz} = \sum_{\langle i, j\rangle} -[\sigma^x_i\sigma^x_j + \sigma_i^y\sigma^y_j] + \Delta \sigma^z_i\sigma^z_j.$$$$

In that article it is said that this system presents $$U(1)$$ symmetry (which I understand as the Hamiltonian commutes with $$S_z = \frac{1}{2}\sum^N_i\sigma_i^z$$), and also that ''a global $$\pi$$ rotation about the spin $$x$$ (or $$y$$) axis is also a symmetry ($$Z_2$$).

My question is related to this last symmetry. If I perform a rotation around the $$x$$ axis, for instance, then I should make the change $$\sigma^y_i \rightarrow -\sigma^y_i$$ and $$\sigma^z_i \rightarrow -\sigma^z_i$$ over the previous Hamiltonian (and as the transformation it is global, over each site $$i$$), which leaves it invariant. However, in this article is also said that for values of $$|\Delta| \geq 1$$, this symmetry can be broken in one dimensional systems, which is something that I don't see quite clear, as according to the previous transformation, the change in sign is independent of $$\Delta$$. So I don't know if the transformation that I have described for the $$Z_2$$ is incorrect. Can somebody help me?

One formal way to examine if a symmetry is broken spontaneously is to add a symmetry breaking term and then send it to zero while taking the size of the system to infinity, in the proper order. In your example, it will entail $$H(\eta) = -\sum_{i,j} \left[\sigma^x_i \sigma^x_j + \sigma^y_i \sigma^y_j + \Delta \sigma^z_i \sigma^z_j\right] + \eta \sum_i \sigma^z_i$$ and then examine the magnetization per spin when the proper limits are taken $$m_z = \lim_{\eta\to 0^+}\lim_{N\to\infty}\frac{1}{N}\sum_i\langle \sigma^z_i\rangle$$ this is done in the paper you are referring to, as can be seen in the lower part of Fig.2 for example (I looked at the arXiv version of it)