Abstract
The high complexity of many-body quantum dynamics means that essentially all approaches either exploit special structure or are approximate in nature. One such approach--the memory function formalism--involves a carefully chosen split into fast and slow modes. An approximate model for the fast modes can then be used to solve for Green's functions G(z) of the slow modes. Using a formulation in operator Krylov space known as the recursion method, we prove the emergence of a universal random matrix description of the fast mode dynamics. This is captured by the level-n Green's function G_n (z), which we show approaches universal scaling forms in the fast limit n→∞. Notably, this emergent universality can occur in both chaotic and non-chaotic systems, provided their spectral functions are smooth. This universality of G_n (z) is precisely analogous to the universality of eigenvalue correlations in random matrix theory (RMT), even though there is no explicit randomness present in the Hamiltonian. At finite z we show that G_n (z) approaches the Wigner semicircle law, while if G(z) is the Green's function of certain hydrodynamical variables, we show that at low frequencies G_n (z) is instead governed by the Bessel universality class from RMT. As an application of this universality, we give a numerical method--the spectral bootstrap--for approximating spectral functions from Lanczos coefficients. Our proof involves a map to a Riemann-Hilbert problem which we solve using a steepest-descent-type method, rigorously controlled in the n→∞ limit. We are led via steepest-descent to a Coulomb gas optimization problem, and we discuss how a recent conjecture--the `Operator Growth Hypothesis--is linked to a confinement transition in this Coulomb gas. These results elevate the recursion method to a theoretically principled framework with universal content.