The Anderson impurity model from a Krylov perspective: Lanczos coefficients in a quadratic model

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 single impurity Anderson 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 hybridisation 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 $(iv)$ 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.

Figures (4)

• Figure 1: Exemplary Lanczos coefficients and autocorrelation functions for the couplings in Tab. \ref{['tab:Summary']} for different values of $\alpha$. For all cases we set $\mu=1$.
• Figure 2: Dynamics from the rescaled coupling densities given in Eq. (\ref{['eq-rescaled-coupling']}) for different $\alpha$.
• Figure 3: Dynamics of the autocorrelation function $\tilde{\mathcal{C}}(t)$ from the rescaled couplings with $\alpha=20$ depicted for short times.
• Figure 4: Krylov complexity $K(t)$ (in logarithmic scale) for the couplings in Tab. \ref{['tab:Summary']} for different values of $\alpha$. For all cases we set $\mu=1$. The simulation is done using a finite $d \times d$ Liouvillian operator, with $d = 10000$. The dashed lines in (d) indicate the scaling $\sim e^{2\alpha t}$.