Hamiltonian dynamics of a sum of interacting random matrices

In ergodic quantum systems, physical observables have a non-relaxing component if they "overlap" with a conserved quantity. In interacting microscopic models, how to isolate the non-relaxing 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〈A(t)A(0)〉; this quantifies the non-relaxing 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 (OTOCs) and find that the existence of a non-relaxing 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.