Cascade of singularities in the spin dynamics of a perturbed quantum critical Ising chain

When the quantum critical transverse-field Ising chain is perturbed by a longitudinal field, a quantum integrable model emerges in the scaling limit with massive excitations described by the exceptional E8 Lie algebra. Using the corresponding analytical form factors of the quantum E8 integrable model, we systematically study the spin dynamic structure factor of the perturbed quantum critical Ising chain, where particle channels with total energy up to 5m1 (m1 being the mass of the lightest E8 particle) are exhausted. In addition to the significant single-particle contributions to the dynamic spectrum, each two-particle channel with different masses is found to exhibit an edge singularity at the threshold of the total mass and decays with an inverse square root of energy, which is attributed to the singularity of the two-particle density of states at the threshold. The singularity is absent for particles with equal masses due to a cancellation mechanism involving the structure of the form factors. As a consequence, the dynamic structure factor displays a cascade of bumping peaks in the continuum region with clear singular features which can serve as a solid guidance for the material realization of the quantum E8 model.