A statistical mechanism for operator growth

TL;DR

The paper proves a lower-bound form of the universal operator growth hypothesis for the chaotic Ising chain and its higher-dimensional analogs by recasting operator growth into a sign-problem-free path-sum over Pauli strings in the Liouvillian graph. By showing positivity of the Liouvillian for a broad class of Hamiltonians, the authors convert quantum interference into countable paths, enabling rigorous lower bounds on the even moments $\\mu_{2n}$ and hence on the high-frequency tail $\\Phi(\\omega)$. They demonstrate that, in 1D and 2D Ising-like models and in disordered MBL chains, the high-frequency decay is at least as slow as $e^{-|\\omega|/\\omega_0}$ with computable bounds on $\\omega_0$ (up to model-dependent constants), and that MBL does not qualitatively alter this tail. The work provides a robust, combinatorial approach to operator growth, complementary to spectral statistics, and opens avenues for Monte Carlo evaluations of the path sums in broader settings. Overall, it strengthens the link between local Hamiltonian structure, operator growth, and universal bounds on spectral tails, with implications for thermalization, quantum chaos diagnostics, and prethermalization under periodic driving.

Abstract

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 $d \ge 2$ 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 decay 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.