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Selection rules are often considered a hallmark of symmetry. When a symmetryis broken, e.g., by an external perturbation, the system exhibits selectionrule deviations which are often analyzed by perturbation theory. Here, weemploy symmetry-breaking degrees of freedom as synthetic dimensions, todemonstrate that symmetry-broken systems systematically exhibit a new class ofsymmetries and selection rules. These selection rules determine the scaling ofa system's observables (to all orders in the strength of the symmetry-breakingperturbation) as it transitions from symmetric to symmetry-broken. Wespecifically analyze periodically driven (Floquet) systems subject to twodriving fields, where the first field imposes a spatio-temporal symmetry, andthe second field breaks it, imposing a symmetry in synthetic dimensions. Wetabulate the resulting synthetic symmetries for (2+1)D Floquet group symmetriesand derive the corresponding selection rules for high harmonic generation (HHG)and above-threshold ionization (ATI). Finally, we observe experimentally HHGselection rules imposed by symmetries in synthetic dimensions. The new class ofsymmetries & selection rules extends the scope of existing symmetry breakingspectroscopy techniques, opening new routes for ultrafast spectroscopy ofphonon-polarization, spin-orbit coupling, and more.

Selection rules in symmetry-broken systems by symmetries in synthetic dimensions