Do dynamical transitions occur only for quenches across critical points
In plain words
Kinks in the return probability were first found for sudden changes that cross an ordinary phase transition. Some models show kinks without such a crossing, and others show a crossing without kinks, and the rule that decides which happens is missing.
Precise statement
For a quench $H_i \to H_f$ with Loschmidt rate $l(t) = -\operatorname{lim}_{N \to \infty} (1/N) \operatorname{ln} \mid\langle\psi_0\mid \operatorname{exp}(-i H_f t/\hbar) \mid\psi_0\rangle\mid^2$, determine the conditions on $(H_i, H_f)$ under which nonanalytic times $t_c$ exist if and only if the quench crosses an equilibrium critical point. Known counterexamples from 2014 on violate both directions in specific models. Answer: a criterion valid for interacting nonintegrable models.
What would settle it
A criterion proved, or verified numerically across interacting nonintegrable models in $d = 1\text{ and }2$.