  The realization of a genuine phase transition in quantum mechanics requires
that at least one of the Kato's exceptional-point parameters becomes real. A
new family of finite-dimensional and time-parametrized quantum-lattice models
with such a property is proposed and studied. All of them exhibit, at a real
exceptional-point time $t=0$, the Jordan-block spectral degeneracy structure of
some of their observables sampled by Hamiltonian $H(t)$ and site-position
$Q(t)$. The passes through the critical instant $t=0$ are interpreted as
schematic simulations of non-equivalent versions of the Big-Bang-like quantum
catastrophes.
