Operator growth in many-body systems of higher spins

Abstract: We study operator growth in many-body systems with on-site spins larger than $1/2$, considering both non-integrable and integrable regimes. Specifically, we compute Lanczos coefficients in the one- and two-dimensional Ising models for spin values $S=1/2$, $1$, and $3/2$, and observe asymptotically linear growth $b_n \sim n$. On the integrable side, we investigate the Potts model and find square-root growth $b_n \sim \sqrt{n}$. Both results are consistent with the predictions of the Universal Operator Growth Hypothesis. To analyze operator dynamics in this setting, we employ a generalized operator basis constructed from tensor products of shift and clock operators, extending the concept of Pauli strings to higher local dimensions. We further report that the recently introduced formalism of equivalence classes of Pauli strings can be naturally extended to this setting. This formalism enables the study of simulable Heisenberg dynamics by identifying dynamically isolated operator subspaces of moderate dimensionality. As an example, we introduce the Kitaev-Potts model with spin-$1$, where the identification of such a subspace allows for exact time evolution at a computational cost lower than that of exact diagonalization.