  Considering nonintegrable quantum Ising chains with exponentially decaying
interactions, we present matrix product state results that establish a
connection between low-energy quasiparticle excitations and the kind of
nonanalyticities in the Loschmidt return rate. When domain walls in the
spectrum of the quench Hamiltonian are energetically favored to be bound rather
than freely propagating, anomalous cusps appear in the return rate regardless
of the initial state. In the nearest-neighbor limit, domain walls are always
freely propagating, and anomalous cusps never appear. As a consequence, our
work illustrates that models in the same equilibrium universality class can
still exhibit fundamentally distinct out-of-equilibrium criticality. Our
results are accessible to current ultracold-atom and ion-trap experiments.
