Quantum chaos as delocalization in Krylov space

We analyze local operator growth in nonintegrable quantum many-body systems. A recently introduced universal operator growth hypothesis proposes that the maximal growth of Lanczos coefficients in the continued fraction expansion of the Green's function reflects chaos of the underlying system. We first show that the continued fraction expansion, and the recursion method in general, should be understood in the context of a completely integrable classical dynamics in Krylov space. In particular, the time-correlation function of a physical observable analytically continued to imaginary time is a tau-function of integrable Toda hierarchy. We use this relation to generalize the universal operator growth hypothesis to include arbitrarily ordered correlation functions. We then proceed to analyze the singularity of the time-correlation function, which is an equivalent sign of chaos to the maximal growth of Lanczos coefficients, and we show that it is due to delocalization in Krylov space. We illustrate the general relation between chaos and delocalization using an explicit example of the Sachdev-Ye-Kietaev model.