Hamiltonian dynamics of a sum of interacting random matrices

Abstract

In ergodic quantum systems, physical observables have a nonrelaxing component if they ``overlap'' with a conserved quantity. In interacting microscopic models, how to isolate the nonrelaxing component is unclear. We compute exact dynamical correlators governed by a Hamiltonian composed of two large interacting random matrices $H=A+B$. We analytically obtain the late-time value of $\ensuremath{\langle}A(t)A(0)\ensuremath{\rangle}$; this quantifies the nonrelaxing part of the observable $A$. The relaxation to this value is governed by a power law determined by the spectrum of the Hamiltonian $H$, independent of the observable $A$. For Gaussian matrices, we further compute out-of-time-ordered correlators and find that the existence of a nonrelaxing part of $A$ leads to modifications of the late-time values and exponents. Our results follow from exact resummation of a diagrammatic expansion and hyperoperator techniques.