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

TL;DR

This paper introduces a memory-efficient nested shift-invert Lanczos method with a specialized preconditioner for studying many-body localization in the random-field Heisenberg chain, providing more precise transition point estimates.

Contribution

It develops a novel nested iterative diagonalization technique with a disorder-aware preconditioner for many-body localization analysis.

Findings

Efficient diagonalization method with low memory usage.

Accurate estimation of the localization transition point.

Introduction of the twist operator as a localization probe.

Abstract

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.