We study the properties of excited states in one-dimensional many-body localized (MBL) systems using a matrix product state algorithm. First, the method is tested for a large disordered noninteracting system, where for comparison we compute a quasiexact reference solution via a Monte Carlo sampling of the single-particle levels. Thereafter, we present extensive data obtained for large interacting systems of L∼100 sites and large bond dimensions χ∼1700, which allows us to quantitatively analyze the scaling behavior of the entanglement S in the system. The MBL phase is characterized by a logarithmic growth S(L)∼log(L) over a large scale separating the regimes where volume and area laws hold. We check the validity of the eigenstate thermalization hypothesis. Our results are consistent with the existence of a mobility edge.
