Spin dynamics and dark particle in a weak-coupled quantum Ising ladder with $\mathcal{D}_8^{(1)}$ spectrum

Emergent Ising$_h^2$ integrability is anticipated in a quantum Ising ladder composed of two weakly-coupled critical transverse field Ising chains. The system is remarkable for including eight types of massive relativistic particles, with their scattering matrix and mass spectrum characterized by the $\mathcal{D}_8^{(1)}$ Lie algebra. In this article, by computing the spin dynamical structure factors following analytical form factor approach, we clearly identify dispersive single-particle excitations of (anti-) soliton and breathers as well as their multi-particle continua in the spectra, which is further confirmed by the numerical simulations. We show that the selection rule inherent in the parity and topological charge of the theory, causes a significant result that charge-parity-odd particles, termed as dark particles, cannot be directly excited from the ground state through any local or quasi-local operations. This in turn suggests the long lifetime of the lightest dark particle.