Emergent time crystals from phase-space noncommutative quantum mechanics

It has been argued that the existence of time crystals requires a spontaneous breakdown of the continuous time translation symmetry so to account for the unexpected non-stationary behavior of quantum observables in the ground state. Our point is that such effects do emerge from position (${\widehat{q}}_i$) and/or momentum (${\widehat{p}}_i$) noncommutativity, i.e., from $[{\widehat{q}}_i,{\widehat{q}}_j]\mathrm{\ne }0$ and/or $[{\widehat{p}}_i,{\widehat{p}}_j]\mathrm{\ne }0$ (for $i\mathrm{\ne }j$). In such a context, a predictive analysis is carried out for the $2$-dim noncommutative quantum harmonic oscillator through a procedure supported by the Weyl-Wigner-Groenewold-Moyal framework. This allows for the understanding of how the phase-space noncommutativity drives the amplitude of periodic oscillations identified as time crystals. A natural extension of our analysis also shows how the spontaneous formation of time quasi-crystals can arise.