  We consider disordered many-body systems with periodic time-dependent
Hamiltonians in one spatial dimension. By studying the properties of the
Floquet eigenstates, we identify two distinct phases: (i) a many-body localized
(MBL) phase, in which almost all eigenstates have area-law entanglement
entropy, and the eigenstate thermalization hypothesis (ETH) is violated, and
(ii) a delocalized phase, in which eigenstates have volume-law entanglement and
obey the ETH. MBL phase exhibits logarithmic in time growth of entanglement
entropy for initial product states, which distinguishes it from the delocalized
phase. We propose an effective model of the MBL phase in terms of an extensive
number of emergent local integrals of motion (LIOM), which naturally explains
the spectral and dynamical properties of this phase. Numerical data, obtained
by exact diagonalization and time-evolving block decimation methods, suggests a
direct transition between the two phases. Our results show that many-body
localization is not destroyed by sufficiently weak periodic driving.
