Emergent Random Matrix Universality in Quantum Operator Dynamics

The high complexity of many-body quantum dynamics means that essentially all analytical or numerical 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, and the success of this approach depends on the accuracy of the fast space approximation. 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 nonchaotic systems, provided their spectral functions are sufficiently smooth. This universality of G n ( z ) turns out to be precisely analogous to the universality of eigenvalue correlations in random matrix theory (RMT), even though there is present in the Hamiltonian. Concretely, 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 new numerical method, the , for approximating spectral functions, including hydrodynamic transport data, from a finite number of 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. Via the steepest-descent procedure, we are led to a related Coulomb gas optimization problem, and we discuss how a recent conjecture—the “Operator Growth Hypothesis”—implies that chaotic operator dynamics can generically be identified with the critical point of a confinement transition in this Coulomb gas. These results elevate the recursion method from a useful numerical technique to a theoretically principled framework with universal content.