Timeline for 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?
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| Apr 17 at 16:30 | comment | added | user196574 | I agree that LSM is probably at play. Another paper also discusses discrete-symmetry LSM, which I hadn't appreciated. It's tricky because I think LSM doesn't forbid symmetry-breaking and just forbids unique gapped ground states in PBC, but it often points to a big change in the phase diagram. | |
| Apr 17 at 16:16 | comment | added | user196574 | Thanks. I'm tempted that even a weak $\sigma^z_{i} \sigma^z_{i+1}+\sigma^x_{i} \sigma^x_{i+1}$ perturbation might affect the symmetry breaking in the model above. This perturbation is symmetric under the $\mathbb{Z}_2 \times \mathbb{Z}_2$ symmetry in the question but not symmetric under $\prod_{i\, even} \sigma_i^z$ and $\prod_{i\, odd} \sigma_i^x$ which I think is what's broken in this example. This is a little bit of an aside, since a weak perturbation doesn't restore the one-site translation symmetry, but I might ask this as a separate question since now I'm curious. | |
| Apr 17 at 15:08 | comment | added | Nandagopal Manoj | With one-site translation symmetry, there seems to be some LSM physics here, but not sure how illuminating that will be for your question, in particular take a look at section 5.3, theorem 5.2. | |
| Apr 17 at 15:05 | comment | added | Nandagopal Manoj | @user196574 that's right. I think the order should be stable to any symmetric perturbation, but I don't have a proof. I disagree that those are the broken symmetries -- you decide the symmetries first to decide the space of Hamiltonians, and then you check whether a specific Hamiltonian spontaneously breaks any of the chosen symmetries, and whether that is stable to any symmetric perturbation. | |
| Apr 16 at 23:19 | comment | added | user196574 | +1 Thanks for this example. I'm imagining your model as a two-leg square ladder without couplings between the legs. I suppose there should be some choices of weak couplings that would keep the order above, like a weak $\sigma^x_i\sigma^x_{i+2} \sigma^z_{i-1}\sigma^z_{i+1}$, which is a point in the quantum Ashkin-Teller phase diagram. (I think technically the broken symmetries are $\prod_{i\, odd} \sigma_i^x$ and $\prod_{i\, even} \sigma_i^z$, though I need to think on it more :)) If you don't mind, I'm going to explicitly add the constraint of one-site translation invariance to my question. | |
| Apr 16 at 22:44 | history | answered | Nandagopal Manoj | CC BY-SA 4.0 |