Krylov complexity in many-body localization systems

We study the operator growth and Krylov complexity in many-body localization (MBL) systems. Using the Lanczos algorithm, the operator growth problem can be mapped to a single particle hopping problem on a semi-infinite chain with the hopping amplitudes given by the Lanczos coefficients. We find that the hopping amplitudes grow linearly along the chain with a logarithmic correction in both MBL and ergodicity phases. Moreover, in the MBL phase, the hopping amplitudes have an additional even-odd modulation and some effective randomness. We show numerical evidences which suggest that in MBL, the corresponding single particle hopping problem is localized, resulting in its bounded Krylov complexity of the operators. In addition, we obtain the spectral function and auto-correlation function in MBL from the Lanczos algorithm. By extrapolating and modeling the behavior of the Lanczos coefficients, we discuss the possible features of the spectral function in MBL in the thermodynamic limit.