Symbolic operator growth and recursion method for spin-S Ising and q-states Potts models

Abstract

Operator growth is well understood for systems with a two-dimensional local Hilbert space, and much less so beyond them, where an operator can grow not only spatially but also in depth, inside the local algebra of each site. We compute the moments of infinite-temperature autocorrelation functions for the spin-$S$ Ising model and the $q$-state Potts model exactly and symbolically in the Hamiltonian parameters, in one, two and three dimensions. For the Ising model the moments are symbolic in the spin magnitude as well, so that a single computation covers arbitrary $S$. From them we obtain the Lanczos coefficients, rigorous Taylor bounds on the initial decay of the autocorrelation function, and its intermediate-time behavior via the recursion method. Our results support the Universal Operator Growth Hypothesis beyond local dimension two. Namely, the Lanczos coefficients grow linearly in the non-integrable cases and exhibit a clean square-root growth in the integrable one-dimensional Potts chain. For the spin-$S$ Ising model we show that every moment converges with growing spin to the corresponding moment of the classical spin model. The Lanczos coefficients therefore approach a limiting sequence with $1/S^2$ corrections, so that classicalization occurs at the level of the whole Lanczos sequence rather than of a single observable. This provides a justification for quasiclassical methods in infinite-temperature spin dynamics. All the results are exact, symbolic and obtained directly in the thermodynamic limit, and the computed moments are available in a public repository.