Resonant level model from a Krylov perspective: Lanczos coefficients in a quadratic model

TL;DR

This paper analytically investigates the Lanczos coefficients in a quadratic resonant level model, revealing their dependence on coupling structures and challenging their use as indicators of system integrability or chaos.

Contribution

It provides explicit formulas for Lanczos coefficients in a quadratic impurity model and shows their growth patterns are highly sensitive to coupling choices, questioning their physical significance.

Findings

Lanczos coefficients can be approximately constant, exactly constant, square root-like, or linear depending on coupling.

Different coupling structures lead to similar exponential decay of autocorrelation functions.

Lanczos coefficients are unreliable indicators of system integrability or chaos.

Abstract

We study the Lanczos coefficients in a quadratic model given by an impurity interacting with a multi-mode field of fermions, also known as resonant level model. We analytically derive closed expressions for the Lanczos coefficients of Majorana fermion operators of the impurity for different structures of the coupling to the hybridization band at zero temperature. While the model remains quadratic, we find that the growth of the Lanczos coefficients structurally depends strongly on the chosen coupling. Concretely, we find $(i)$ approximately constant, $(ii)$ exactly constant, $(iii)$ square root-like as well as $(iv)$ linear growth in the same model. We further argue that in fact through suitably chosen couplings, essentially arbitrary Lanczos coefficients can be obtained in this model. These altogether evince the inadequacy of the Lanczos coefficients as a reliable criterion for classifying the integrability or chaoticity of the systems. Eventually, in the wide-band limit, we find exponential decay of autocorrelation functions in all the settings $(i)-(iv)$, which demonstrates the different structures of the Lanczos coefficients not being indicative of different physical behavior.