Well, the crystal being discussed is a string of interacting ytterbium ions which are triggered to flip spin orientation via laser pulses, which can be accompanied by tuning which sets up a spin-spin interaction with the rest of the string, and optical effects from other lasers are used to set up a disordered MBL or many-body localization effect with the electromagnetic fields they produce.
When you trigger the flip effect the system just oscillates back and forth, it's the dna helix looking pattern on the left here, labeled (a).
When you add the disorder effect it produces a much noiser state oscillation, the second from the left, labeled (b).
When you add the spin-spin interaction along with the other two it stops oscillating and evolves in a smooth fashion over time, the third from the left, labeled (c).
If you increase the amount of spin-spin interaction too high the system tends towards a noisy state, fourth from the left, labeled (d).
This string of ions exhibits symmetry which is preserved unless the interactions are stimulated in a certain fashion, at which point the regular "beat" of the system turns into a steady trend over time which disappears again when the strength of the interactions is increased too far, so it exhibits a spontaneously broken symmetry over time, with the analogue being that a crystal may exhibit symmetry when viewed from one point, but loses that symmetry when the view is translated to a different point, a spontaneously broken symmetry under spatial translations.
I think that's what is going on, I'm a bit tired and it's a rather complicated bit of physics that I'm not as familiar with.
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