One-dimensional $\mathcal{P}\mathcal{T}$ -symmetric acoustic heterostructure

The explorations of parity–time ( $\mathcal{P}\mathcal{T}$ )-symmetric acoustics have resided at the frontier in physics, and the pre-existing accessing of exceptional points typically depends on Fabry–Perot resonances of the coupling interlayer sandwiched between balanced gain and loss components. Nevertheless, the concise $\mathcal{P}\mathcal{T}$ -symmetric acoustic heterostructure, eliminating extra interactions caused by the interlayer, has not been researched in depth. Here we derive the generalized unitary relation for one-dimensional (1D) $\mathcal{P}\mathcal{T}$ -symmetric heterostructure of arbitrary complexity, and demonstrate four disparate patterns of anisotropic transmission resonances (ATRs) accompanied by corresponding spontaneous phase transitions. As a special case of ATR, the occasional bidirectional transmission resonance reconsolidates the ATR frequencies that split when waves incident from opposite directions, whose spatial profiles distinguish from a unitary structure. The derived theoretical relation can serve as a predominant signature for the presence of $\mathcal{P}\mathcal{T}$ symmetry and $\mathcal{P}\mathcal{T}$ -symmetry-breaking transition, which may provide substantial support for the development of prototype devices with asymmetric acoustic responses.