  The Discrete Truncated Wigner Approximation (DTWA) is a semi-classical phase
space method useful for the exploration of Many-body quantum dynamics. In this
work we investigate Many-Body Localization (MBL) and thermalization using DTWA
and compare its performance to exact numerical solutions. By taking as a
benchmark case a 1D random field Heisenberg spin chain with short range
interactions, and by comparing to numerically exact techniques, we show that
DTWA is able to reproduce dynamical signatures that characterize both the
thermal and the MBL phases. It exhibits the best quantitative agreement at
short times deep in each of the phases and larger mismatches close to the phase
transition. The DTWA captures the logarithmic growth of entanglement in the MBL
phase, even though a pure classical mean-field analysis would lead to no
dynamics at all. Our results suggest the DTWA can become a useful method to
investigate MBL and thermalization in experimentally relevant settings
intractable with exact numerical techniques, such as systems with long range
interactions and/or systems in higher dimensions.
