Description
A many-body Green's function $G(z)$ has a well-known continued fraction representation involving a so-called Krylov operator basis $\{O_{m}\}_{m \geq 0}$. The content of the continued fraction at depth $n$ is denoted by $G_{n}(z)$, and can be thought of as the Green's function for the operator dynamics restricted to the 'fast space' $\{O_{m}\}_{m \geq n}$. In this work we prove that $G_{n}(z)$ can exhibit universality in the $n \to \infty$ limit, approaching different universal scaling forms in different sections of the complex $z$-plane. At finite $z$, we show that $G_{n}(z)$ approaches the Wigner semicircle law from random matrix theory (RMT), the same as the average resolvent in the bulk of the spectrum for the Gaussian Unitary Ensemble with an appropriately rescaled bandwidth. With hydrodynamics in mind, we also study the signatures of power-law decaying autocorrelation functions, showing that at low frequencies $G_{n}(z)$ is instead governed by the Bessel universality class from RMT.
As an application we give a new numerical method, the spectral bootstrap, for approximating spectral functions, including hydrodynamic transport data, from a finite number of Lanczos coefficients. Our proof is complex analytic in nature, involving a map to a Riemann-Hilbert problem which we solve using a steepest-descent-type method, rigorously controlled in the $n\to\infty$ limit. This proof technique requires we make some analyticity and regularity assumptions on the spectral function, and we comment on their possible connections to the eigenstate thermalization hypothesis. We also discuss how a recent conjecture from quantum chaos, the 'Operator Growth Hypothesis', implies that chaotic operator dynamics can generically be identified with the critical point of a confinement transition in a classical Coulomb gas. We then elucidate how this criticality has implications for the computational resources required to reconstruct transport coefficients to a given precision.
