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

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