Krylov complexity of many-body localization: Operator localization in Krylov basis

TL;DR

This paper investigates operator growth and complexity in many-body localization systems using Krylov basis and Lanczos algorithm, revealing localization properties and differences from ergodic systems.

Contribution

It introduces a novel analysis of operator growth in MBL systems via Krylov basis, highlighting the scaling of Lanczos coefficients and the localized nature of the emergent hopping problem.

Findings

Lanczos coefficients in MBL scale as n/ln(n) with even-odd effects

The emergent single-particle hopping problem is localized in MBL

Krylov complexity grows linearly in the phenomenological MBL model

Abstract

We study the operator growth problem and its complexity in the many-body localization (MBL) system from the Lanczos algorithm perspective. Using the Krylov basis, the operator growth problem can be viewed as a single-particle hopping problem on a semi-infinite chain with the hopping amplitudes given by the Lanczos coefficients. We find that, in the MBL systems, the Lanczos coefficients scale as $\sim n/\ln(n)$ asymptotically, same as in the ergodic systems, but with an additional even-odd alteration and an effective randomness. We use a simple linear extrapolation scheme as an attempt to extrapolate the Lanczos coefficients to the thermodynamic limit. With the original and extrapolated Lanczos coefficients, we study the properties of the emergent single-particle hopping problem via its spectral function, integrals of motion, Krylov complexity, wavefunction profile and return probability. Our numerical results of the above quantities suggest that the emergent single-particle hopping problem in the MBL system is localized when initialized on the first site. We also study the operator growth in the MBL phenomenological model, whose Lanczos coefficients also have an even-odd alteration, but approach constants asymptotically. The Krylov complexity grows linearly in time in this case.