Time crystal $ϕ^4$ kinks by curvature coupling as toy model for mechanism of oscillations propelled by mass, like observed for electron and neutrinos

Dirac equation requires $E=mc^2$ energy of resting particle, leading to some $\exp(-iEt/\hbar)$ its evolution - periodic process of $ω=mc^2/\hbar$ frequency, literally propelled by mass of particle, confirmed experimentally e.g. for quantum phase of electron as de Broglie clock/Zitterbewegung (and its angular momentum), or flavor oscillations of neutrinos for 3 masses. Entities having energetically preferred periodic process already in the lowest energy state are recently searched for as time crystals. To understand such mechanism of clock propulsion by mass itself, it would be valuable to recreate something analogous in simple models like wobbling kinks. There is proposed such toy model as 1+1D $(ϕ,ψ)$ Lorentz invariant two-component scalar field theory, extending popular $ϕ^4$ model by second component $ψ$ corresponding to such periodically evolving degree of freedom, which is coupled through powers of curvature $R=\partial_0 ϕ\, \partial_1 ψ-\partial_1 ϕ\,\partial_0 ψ$, as suggested by earlier 3+1D model~\cite{my}. This way kink spatial structure $\partial_x ϕ\neq 0$ brings energetic preference for nonzero $\partial_t ψ$ time derivative, by energy minimization leading to periodic process of $0<ω<\infty$ frequency, as required for time crystals.