Ruelle-Pollicott resonances from truncated operator dynamics in diffusive qubit circuits

Correlation functions of local observables characterize the dynamical properties of physical systems. For strongly chaotic systems without conservation laws, the correlation functions are expected to decay exponentially for any physical observable, with the slowest decay rate being the Ruelle-Pollicott resonance. While Ruelle-Pollicott resonances have previously been studied mainly for classical or semiclassical systems, interest has recently shifted to many-body quantum systems. In this work, we numerically calculate the Ruelle-Pollicott resonances of qubit circuits, which correspond to the leading eigenvalues of the quasi-momentum-resolved truncated propagator of extensive observables. While this method is known to predict the relaxation time of systems without conservation laws, we show that it also gives access to phenomenological transport constants in systems with a U(1) conserved quantity. Due to the non-exponential decay of correlation functions of transport-related observables in such systems, the truncated propagator contains continuums of eigenvalues. For diffusive magnetization-conserving qubit circuits, we show that the leading continuum has a Gaussian quasi-momentum dependence and governs the diffusive transport of magnetization, allowing the numerical extraction of the spin diffusion constant. We expect our conclusions to hold for generic systems with exactly one U(1) conserved quantity.