Light-cones and quantum caustics in quenched spin chains*

We discuss the light-cone-like structures in spin chains following a quantum quench, and identify these stuctures as examples of quantum caustics. Specifically, we connect the light-cone to catastrophe theory and show that they arise from singularities in a gradient map due to coalescing saddles in a generating function, which in turn takes the form of an action. The identification of light-cones as catastrophes ensures structural stability and self-similarity scaling relations of both wave- and correlation functions, which asymptotically take the form of well-known diffraction integrals. We present a quench in the transverse-field Ising model as an example, and in addition to the features mentioned above, we demonstrate the existence of a lattice of vortex-antivortex pairs within the light-cone which are sensitive to the presence of the quantum phase transition, and can be connected to the dynamical critical exponent.