  A quantum phase transition is generally thought to imprint distinctive
characteristics on the nonequilibrium dynamics of a closed quantum system.
Specifically, the Loschmidt echo after a sudden quench to a quantum critical
point $-$ measuring the time dependence of the overlap between initial and
time-evolved states $-$ is expected to exhibit an accelerated relaxation
followed by periodic revivals. We here introduce a new exactly solvable model,
the extended Su-Schrieffer-Heeger model, the Loschmidt echo of which provides a
counterexample. A parallell analysis of the quench dynamics of the three-site
spin-interacting $XY$ model allows us to pinpoint the conditions under which a
periodic Loschmidt revival actually appears
