Universal properties of the many-body Lanczos algorithm at finite size

TL;DR

This paper investigates the universal finite-size properties of the Lanczos algorithm in many-body quantum systems, revealing scaling behaviors of Lanczos coefficients linked to autocorrelation functions and hydrodynamics.

Contribution

It introduces a conjecture on the scaling of Lanczos coefficients in large systems and supports it with numerical evidence across various models.

Findings

Lanczos coefficient ratios scale with system size according to autocorrelation tail properties

Universal scaling laws depend on hydrodynamic behavior of local operators

Numerical results confirm the conjectured scaling across different models

Abstract

We study the universal properties of the Lanczos algorithm applied to finite-size many-body quantum systems. Focusing on autocorrelation functions of local operators and on their infinite-time behaviour at finite size, we conjecture that in the large $n$ limit, the ratios between consecutive Lanczos coefficients should have specific scalings with the size of the lattice that we make precise and that depend on the hydrodynamic tail of the autocorrelation function. The scaling associated with strong or approximate zero-modes is also discussed. We support our conjecture with a numerical study of different models.