When dynamical transitions coincide with equilibrium criticality or order

For a quench $H_i \to H_f$, the Loschmidt rate $l(t) = -\lim_{N \to \infty} (1/N) \ln \mid\langle\psi_0\mid \exp(-i H_f t/\hbar) \mid\psi_0\rangle\mid^2$ can be nonanalytic at critical times $t_c$. Determine conditions on $(H_i, H_f)$ under which (a) such $t_c$ exist if and only if the quench crosses an equilibrium critical point and (b) $t_c$ coincide with zeros or sign changes of the order parameter. Answer: a classification valid beyond free-fermion models.

A general criterion proved or verified across interacting nonintegrable models in $d = 1 \text{ and } 2$.