Quantum Matter Seminar

Spectral transitions in OTOCs: an alternative approach to scrambling in random quantum circuits

Out-of-time-order correlators (OTOCs) are a central tool for characterizing operator spreading and scrambling in quantum systems. While the spatio-temporal dynamics of a standard OTOC are well understood in random quantum circuits, higher-order correlators become increasingly challenging to study both numerically and via effective statistical mechanics models. Recently, an alternative perspective has emerged wherein the decay of OTOCs at all orders is encoded in a transition in the spectrum of operator products. Building upon these recent results, we develop a coherent picture of the spectral transition of OTOCs in random quantum systems. We begin by considering Brownian unitary evolution, where we show that the OTOC spectral density satisfies a conservation equation related via Hilbert transform to the inviscid Burgers' equation. This approach is complemented by an effective random matrix model which allows for an exact solution of the spectral density dynamics and critical properties of the transition. In the presence of a global symmetry, we show that the equations governing the spectral density are universal for Gaussian ensembles. Moreover, we find that restriction to a fixed charge sector can yield source terms which induce distinct steady-state behavior. Finally, we show close agreement between the random matrix model results and numerical simulation of random quantum circuits, commenting on the role of symmetries and locality in OTOC dynamics.