  We study the many-body localization of the random-field Heisenberg chain
using the nested shift-invert Lanczos method with an iterative linear solver.
We use the minimum residual method (MINRES) inside each Lanczos iteration. The
memory consumption of the proposed method is only proportional to the dimension
of the Hilbert space. We also introduce a preconditioner that takes into
account the effects of disorder and interaction in the random-field Heisenberg
chain. As a probe of many-body localization transition, we propose a unitary
operator called the twist operator, which has a clear interpretation in the
real space. We discuss its behavior for thermal and localized eigenstates. We
demonstrate the efficiency of the nested iterative shift-invert diagonalization
method with the proposed preconditioner for the many-body localization problem
and estimate the transition point of the random-field Heisenberg chain more
precisely based on the finite-size analysis of the expectation value of the
twist operator.
