Emergent Random Matrix Universality in Quantum Operator Dynamics

Abstract

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” Gn(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 Gn(z) turns out to be precisely analogous to the universality of eigenvalue correlations in random matrix theory (RMT), even though there is no explicit randomness present in the Hamiltonian. Concretely, at finite z we show that Gn(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 Gn(z) is instead governed by the Bessel universality class from RMT. As an application of this universality, 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 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.