Ruelle–Pollicott resonances from the truncated propagator: underlying mathematical structure and the operator cascade

In recent years, the truncated propagator with its leading eigenvalues, the Ruelle–Pollicott resonances, has been recognized as a useful tool for studying the nonequilibrium dynamics of chaotic many-body (quantum) systems. In this talk, I will discuss when and why the truncated propagator works and, along the way, clarify the chaos scenario in quantum many-body systems, which is directly relevant to classical truncation-based simulations of quantum dynamics. Specifically, local observables evolve with time into increasingly nonlocal ones (the operator cascade), much as initially smooth phase-space densities evolve into ever finer fractal structures in classical chaos. The fractality of the operator cascade is quantifiable with a fractal dimension connected to the leading decay rate of correlation functions of local observables and is encoded in the diverging structure of the leading eigenvector of the truncated propagator. This diverging eigenvector can be formalized within the rigged Hilbert space framework and interpreted as describing an effective dissipative theory, valid only for local observables, that emerges within purely unitary dynamics. I will support my claims with numerical results for generic quantum systems without conservation laws in 1D (kicked Ising, random brickwall circuits) and 2D (kicked Ising), as well as for 1D models with conservation laws (magnetization-conserving circuits, the Trotter limit of the kicked Ising model), and with exact solutions for dual-unitary circuits and for CMV matrices inspired by generic many-body operator dynamics.
The talk is based on arXiv:2604.16720 (with Marko Žnidarič) and arXiv:2608.28575 (with Friedrich Hübner and Marko Žnidarič).