  Whether or not the thermodynamic entropy is equal to the entanglement entropy
of an eigenstate, is of fundamental interest, and is closely related to the
`Eigenstate thermalization hypothesis (ETH)'. However, this has never been
exploited as a diagnostic tool in many-body localized systems. In this work, we
perform this diagnostic test on a clean interacting system (subjected to a
static electric field) that exhibits three distinct phases: integrable,
non-integrable ergodic and non-integrable many-body-localized (MBL). We find
that in the non-integrable ergodic phase, the equivalence between the
thermodynamic entropy and the entanglement entropy of individual eigenstates,
holds. In sharp contrast, in the integrable and non-integrable MBL phases, the
entanglement entropy shows large eigenstate-to-eigenstate fluctuations, and
differs from the thermodynamic entropy. Thus the non-integrable MBL phase
violates ETH similar to an integrable system; however, a key difference is that
the magnitude of the entanglement entropy in the MBL phase is significantly
smaller than in the integrable phase, where the entanglement entropy is of the
same order of magnitude as in the non-integrable phase, but with a lot of
eigenstate-to-eigenstate fluctuations. Quench dynamics from an initial CDW
state independently supports the validity of the ETH in the ergodic phase and
its violation in the MBL phase.
