<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="script">PT</mml:mi></mml:math>-symmetry breaking and universal spectral statistics in quantum kicked rotors

We investigate the spontaneous parity-time ($\mathcal{PT}$)-symmetry breaking and spectral properties of a $\mathcal{PT}$-symmetric quantum kicked rotor under resonance conditions. At resonance, the QKR reduces to a finite-dimensional system. In the localized regime, we find that increasing the non-Hermitian parameter always induces a transition from a phase where the states exhibit $\mathcal{PT}$ symmetry to one where $\mathcal{PT}$ symmetry is spontaneously broken. In contrast, in the delocalized regime, the existence of such a transition depends on whether the reduced system is $\mathcal{PT}$ symmetric. If the reduced system is not $\mathcal{PT}$ symmetric, $\mathcal{PT}$ symmetry remains in the broken phase regardless of the non-Hermitian parameter. We further analyze the spectral statistics of the system in the delocalized regime. For real energy spectra, the level-spacing distribution transitions from Wigner-Dyson statistics, associated with the Gaussian orthogonal ensemble, to Poisson statistics as the non-Hermitian parameter increases, with the intermediate regime well described by the Brody distribution. For complex spectra, the level-spacing ratios and distributions are governed by time-reversal symmetry. The spectral statistics align with predictions for non-Hermitian random matrix ensembles in classes ${\mathrm{AI}}^{\ifmmode\dagger\else\textdagger\fi{}}$ and $\mathrm{A}$, depending on the presence or absence of time-reversal symmetry. Our results provide insights into the spectral characteristics of non-Hermitian quantum chaotic systems and their connection to $\mathcal{PT}$ symmetry.