A constructive theory of the numerically accessible many-body localized to thermal crossover

The many-body localised (MBL) to thermal crossover observed in exact diagonalisation studies remains poorly understood as the accessible system sizes are too small be an asymptotic scaling regime. We develop a model of short 1D chains which MBL phase is destabilised by formation resonances. reproduces several properties numerically crossover, including apparent correlation length exponent $\nu=1$, exponential growth Thouless time with disorder strength, linear drift critical strength size, scale-free resonances, $1/\omega$ dependence disorder-averaged spectral functions, and sub-thermal entanglement entropy subsystems. In resonances induced local perturbation rare at $L$ smaller than \emph{resonance length} $\lambda$. For $L \gg \sqrt{\lambda}$, typically overlap, this does not describe transition. further controversial numerical observations Refs. [\v{S}untajs et al, 2019] [Sels & Polkovnikov, 2020] claimed inconsistent MBL. thus argue that numerics date consistent thermodynamic limit.