Spontaneous breaking of continuous scale invariance and Efimovian-like dynamics in a driven-dissipative harmonic oscillator

Abstract

The Efimov effect manifests discrete scale invariance through a geometric series of three-body bound states. Analogous discrete scale symmetry has been observed in the Efimovian expansion of scale-invariant strongly interacting Fermi gases. However, it is unclear whether strong correlation is necessary. Here we investigate the spontaneous breaking of continuous scale invariance in a simple driven-dissipative harmonic oscillator whose driving and dissipation parameters scale as $1/t$. By transforming to logarithmic time, we show that the dynamics is governed by a non-Hermitian superoperator whose spectral decomposition reveals the emergence of a pair of eigenmodes with opposite real parts and equal imaginary parts. This signals the spontaneous breaking of continuous scale invariance and gives rise to Efimovian-like oscillations: log-periodic decay oscillations obeying discrete scaling laws. The onset of oscillations coincides with a PT-symmetry breaking transition in logarithmic time. The Efimovian-like oscillations are robust against weak nonlinear interactions and perturbations. Our work implies that the spontaneous breaking of continuous temporal scale symmetry and the emergence of Efimovian-like dynamics can even be observed in a simple driven-dissipative system without many-body interactions.