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We study the many-body localization of the random-field Heisenberg chainusing the nested shift-invert Lanczos method with an iterative linear solver.We use the minimum residual method (MINRES) inside each Lanczos iteration. Thememory consumption of the proposed method is only proportional to the dimensionof the Hilbert space. We also introduce a preconditioner that takes intoaccount the effects of disorder and interaction in the random-field Heisenbergchain. As a probe of many-body localization transition, we propose a unitaryoperator called the twist operator, which has a clear interpretation in thereal space. We discuss its behavior for thermal and localized eigenstates. Wedemonstrate the efficiency of the nested iterative shift-invert diagonalizationmethod with the proposed preconditioner for the many-body localization problemand estimate the transition point of the random-field Heisenberg chain moreprecisely based on the finite-size analysis of the expectation value of thetwist operator.

Nested Iterative Shift-invert Diagonalization for Many-body Localization in the Random-field Heisenberg Chain