Emergent random matrix universality in quantum operator dynamics

• URL: http://arxiv.org/abs/2504.18311v3
• Date: Mon, 06 Oct 2025 10:56:17 GMT
• Title: Emergent random matrix universality in quantum operator dynamics
• Authors: Oliver Lunt, Thomas Kriecherbauer, Kenneth T-R McLaughlin, Curt von Keyserlingk,
• Abstract summary: We show the emergence of a universal random matrix description of the fast mode dynamics.<n>This emergent universality can occur in both chaotic and non-chaotic systems.<n>We also show how a recent conjecture--the Operator Growth Hypothesis--is linked to a confinement transition in this Coulomb gas.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $n\to\infty$. 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 $n\to\infty$ 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.

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