  An external periodic (Floquet) drive is believed to bring any initial state
to the featureless infinite temperature state in generic nonintegrable isolated
quantum many-body systems in the thermodynamic limit, irrespective of the
driving frequency $\Omega$. However, numerical or analytical evidence either
proving or disproving this hypothesis is very limited and the issue has
remained unsettled. Here, we study the initial state dependence of Floquet
heating in a nonintegrable kicked Ising chain of length up to $L=30$ with an
efficient quantum circuit simulator, showing a possible counterexample: The
ground state of the effective Floquet Hamiltonian is exceptionally robust
against heating, and could stay at finite energy density even after infinitely
many Floquet cycles, if the driving period is shorter than a threshold value.
This sharp energy localization transition/crossover does not happen for generic
excited states. The exceptional robustness of the ground state is interpreted
by (i) its isolation in the energy spectrum and (ii) the fact that those states
with $L$-independent $\hbar\Omega$ energy above the ground state energy of any
generic local Hamiltonian, like the approximate Floquet Hamiltonian, are
atypical and viewed as a collection of noninteracting quasipartiles. Our
finding paves the way for engineering Floquet protocols with finite driving
periods realizing long-lived, or possibly even perpetual, Floquet phases by
initial state design.
