Thresholdless dynamical instability in transverse $\mathcal{PT}$-symmetric scattering systems: The hidden role of bound states in the continuum

Abstract

The stationary scattering properties of transverse parity-time ($\mathcal{PT}$) symmetric systems have been extensively studied, yet their dynamical stability, a prerequisite for any stationary description, remains largely unexplored. Here we uncover a thresholdless dynamical instability in such systems, driven by symmetry-protected bound states in the continuum (BICs). Without gain and loss, the up-down mirror symmetry of the transverse geometry generically protects a BIC, which manifests as the coalescence of an $S$-matrix pole and zero on the real axis of the complex wave-number plane. A Hermitian symmetry-breaking perturbation shifts the poles into the lower half-plane, converting the BIC into a resonance with a Fermi-golden-rule decay width. An anti-Hermitian $\mathcal{PT}$-symmetric perturbation instead reverses the sign of the second-order energy shift, driving the poles into the upper half-plane and producing a time-growing bound state with $\operatorname{Im}E=γ^{2}Γ_{V}/2+O(γ^{3})$, where $γ$ is the gain-loss strength and $Γ_{V}$ is the golden-rule coupling of the BIC to the continuum. The instability therefore sets in at arbitrarily small $γ$ whenever $Γ_{V}>0$. We confirm this mechanism in a three-site side-coupled model that is unstable despite possessing a unitary scattering matrix, and in a four-site rhombic model where transverse and longitudinal gain-loss placements yield vanishing and finite thresholds, respectively. These results establish a microscopic stability criterion and a design principle for stable $\mathcal{PT}$-symmetric scattering devices.

The paper asks whether transverse $\mathcal{PT}$-symmetric scatterers can be dynamically unstable even when their stationary scattering matrix is unitary. It studies mirror-symmetric Hermitian scattering centers coupled to two tight-binding leads, assuming a single nondegenerate, parity-odd bound state in the continuum (BIC). A Feshbach-projection calculation tracks the BIC under a parity-odd perturbation $\eta V$. For real $\eta$, coupling to propagating modes produces the usual second-order radiative decay. For an anti-Hermitian gain–loss perturbation, $\eta=i\gamma$, the second-order shift reverses sign: a spatially localized, growing state appears with $\operatorname{Im}E=\gamma^2\Gamma_V/2+O(\gamma^3)$. Here $\Gamma_V\geq0$ is the on-shell golden-rule coupling calculated from the unperturbed Hermitian system. Thus, whenever $\Gamma_V>0$, the instability threshold is zero, and intensity grows as $e^{\gamma^2\Gamma_Vt}$ to leading order. Exact scattering-pole trajectories and finite-lattice wave-packet simulations illustrate the mechanism. A three-site side-coupled model is unstable despite a unitary real-energy scattering matrix. A four-site rhombus has zero threshold for transverse gain–loss placement, whereas longitudinal placement annihilates the BIC under $V$ and retains a finite threshold associated with other poles. The criterion is conditional on an embedded symmetry-protected state and nonzero continuum coupling; mirror symmetry alone does not ensure an in-band BIC. The paper notes that $\Gamma_V=0$ excludes only second-order instability, while $V|B\rangle=0$ protects the BIC exactly. Weak-gain growth may escape finite observation windows. The treatment is linear: saturation and other nonlinearities can limit growth but are not modeled.