  A key property of many-body localization, the localization of quantum
particles in systems with both quenched disorder and interactions, is the area
law entanglement of even highly excited eigenstates of many-body localized
Hamiltonians. Matrix Product States (MPS) can be used to efficiently represent
low entanglement (area law) wave functions in one dimension. An important
application of MPS is the widely used Density Matrix Renormalization Group
(DMRG) algorithm for finding ground states of one dimensional Hamiltonians.
Here, we describe two algorithms, the Shift and Invert MPS (SIMPS) and excited
state DMRG which finds highly-excited eigenstates of many-body localized
Hamiltonians. Excited state DMRG uses a modified sweeping procedure to identify
eigenstates whereas SIMPS is a shift-inverse procedure that applies the inverse
of the shifted Hamiltonian to a MPS multiple times to project out the targeted
eigenstate. To demonstrate the power of these methods we verify the breakdown
of the Eigenstate Thermalization Hypothesis (ETH) in the many-body localized
phase of the random field Heisenberg model, show the saturation of entanglement
in the MBL phase and generate local excitations.
