  Recent work has shown that a variety of novel phases of matter arise in
periodically driven Floquet systems. Among these are many-body localized phases
which spontaneously break global symmetries and exhibit novel multiplets of
Floquet eigenstates separated by quantized quasienergies. Here we show that
these properties are stable to all weak local deformations of the underlying
Floquet drives -- including those that explicitly break the defining symmetries
-- and that the models considered until now occupy sub-manifolds within these
larger "absolutely stable" phases. While these absolutely stable phases have no
explicit global symmetries, they spontaneously break Hamiltonian dependent
emergent symmetries, and thus continue to exhibit the novel multiplet
structure. The multiplet structure in turn encodes characteristic oscillations
of the emergent order parameter at multiples of the fundamental period.
Altogether these phases exhibit a form of simultaneous long-range order in
space and time which is new to quantum systems. We describe how this
spatiotemporal order can be detected in experiments involving quenches from a
broad class of initial states.
