Can a $\mathbb{Z}_2 \times \mathbb{Z}_2$ symmetry generated by $\prod_{i} \sigma^x_i$ and $\prod_{i} \sigma^z_i$ be broken in a spin-$1/2$ chain?

Question

I am interested in understanding patterns of spontaneous symmetry breaking in spin chains. I want to understand what happens when I have "competing" orders, like symmetry breaking orders that are probed via $\langle \sigma^z_i \sigma^z_{j}\rangle$ and $\langle \sigma^x_i \sigma^x_{j}\rangle$. I call these competing because $\sigma^z_i$ and $\sigma^x_i$ do not commute. I am particularly interested in when I have simultaneous ordering of two competing orders.

In particular, consider a length $L$ ($L$ is even) qubit chain in periodic boundary conditions. Suppose that it explicitly has a $\mathbb{Z}_2 \times \mathbb{Z}_2$ symmetry generated by $\prod_{i=1}^L \sigma^x_i$ and $\prod_{i=1}^L \sigma^z_i$.

What is an example of such a spin-$1/2$ chain that spontaneously breaks both $\prod_{i} \sigma^x_i$ and $\prod_{i} \sigma^z_i$?

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Here are some of the features that I identify with spontaneous symmetry-breaking of all of $\mathbb{Z}_2 \times \mathbb{Z}_2$. I expect there to be a ground-state manifold of four states that are extremely close in energy (splitting decaying like $e^{-cL}$) separated by a gap to the rest of the spectrum. The four states will have eigenvalues $(+1, +1), (-1,+1), (+1, -1),$ and $(-1,-1)$ under the above $\mathbb{Z}_2 \times \mathbb{Z}_2$.

I know that such a symmetry-breaking happens in spin-$1$ chains of a curious flavor, like in $$-\left( \sum_i S^x_i S^x_{i+1} + a (S^y_i e^{i \pi S^z_i}) (e^{i \pi S^x_{i+1}} S^y_{i+1}) + b S^z_i S^z_{i+1} \right)$$ which is identified (after the application of a nonlocal unitary in open boundary conditions) with the string order in regimes of the anisotropic $XYZ$ Heisenberg antiferromagnet.

However, I'm having a hard time finding a spin-1/2 model that shows this sort of symmetry breaking. For spin-$1/2$ particles, $e^{i \pi S_i^z} = i \sigma^z_i$ and so on, and so the above Hamiltonian is no longer $\mathbb{Z}_2 \times \mathbb{Z}_2$ symmetric. I've tried $-\left( \sum_i (S^x_{i} S^x_{i+1} + S^y_{i} S^y_{i+1}) + \epsilon (S^x_{i} S^x_{i+1} S^x_{i+2} S^x_{i+3} + S^y_{i} S^y_{i+1} S^y_{i+2} S^y_{i+3}) \right)$, but that appears to be gapless in the spin-$1/2$ model.

Answer 1

If you do not want to impose translation symmetry, there are trivial (but valid) examples. If the number of sites is a multiple of 4, $$ H = -\sum_{i \text{ even}} (\sigma^x_i\sigma^x_{i+2} + \sigma^z_{i-1}\sigma^z_{i+1}) $$ spontaneously breaks both symmetries and has four ground states $$ (\vert \uparrow \uparrow \uparrow \dots \rangle_{\text{odd}} \pm \vert \downarrow \downarrow \downarrow \dots \rangle_{\text{odd}}) \otimes (\vert \rightarrow \rightarrow \rightarrow \dots \rangle_{\text{even}} \pm \vert \leftarrow \leftarrow \leftarrow \dots \rangle_{\text{even}}) $$ in the four symmetry sectors you mentioned. The case with single-site translation symmetry seems more difficult -- it would be interesting if there is an example there.