A statistical mechanism for operator growth

It was recently conjectured that in generic quantum many-body systems, the spectral density of local operators has the slowest high-frequency decay as permitted by locality. We show that the infinite-temperature version of this 'universal operator growth hypothesis' holds for the quantum Ising spin model in d > 2 dimensions, and for the chaotic Ising chain (with longitudinal and transverse fields) in one dimension. Moreover, the disordered chaotic Ising chain that exhibits many-body localization can have the same high-frequency spectral density asymptotics as thermalizing models. Our argument is statistical in nature, and is based on the observation that the moments of the spectral density can be written as a sign-problem-free sum over paths of Pauli string operators.

Automated summary

In this study, it is shown that the universal operator growth hypothesis holds for the quantum Ising spin model and chaotic Ising chain, with the disordered chaotic Ising chain exhibiting similar high-frequency spectral density asymptotics as thermalizing models. The argument is statistical in nature and relies on the observation that moments of the spectral density can be expressed as a sign-problem-free sum over paths of Pauli string operators.