  Coupling a many-body localized system to a thermal bath breaks local
conservation laws and washes out signatures of localization. When the bath is
non-thermal or when the system is also weakly driven, local conserved
quantities acquire a highly non-thermal stationary value. We demonstrate how
this property can be used to study the many-body localization phase transition
in weakly open systems. Here, the strength of the coupling to the non-thermal
baths plays a similar role as a finite temperature in a $T=0$ quantum phase
transition. By tuning this parameter, we can detect key features of the MBL
transition: the divergence of the dynamical exponent due to Griffiths effects
in one dimension and the critical disorder strength. We apply these ideas to
study the MBL critical point numerically. The possibility to observe critical
signatures of the MBL transition in an open system allows for new numerical
approaches that overcome the limitations of exact diagonalization studies. Here
we propose a scalable numerical scheme to study the MBL critical point using
matrix-product operator solution to the Lindblad equation.
