Emergent Dimensionality in Diluted Power-Law Quantum Walks
We investigate single-excitation quantum walks across the one-to-two-dimensional crossover in diluted power-law hopping models. Using arrays of $\ell$ coupled chains of length $L$, we identify the finite-size localization crossover from the longitudinal spreading of an initially localized excitation. The crossover filling is well described by an exponential approach to its two-dimensional value with increasing $\ell$, defining a characteristic transverse scale $\ell_e$, and effectively two-dimensional behavior emerges already for $\ell\ll L$. An independent spectral analysis shows that $\ell_e$ is set by the transverse extent of the eigenstates participating in the dynamics, providing a microscopic interpretation of the dimensional crossover. Our results reveal an intrinsic eigenstate scale governing emergent dimensionality in diluted systems with power-law hopping, a regime directly relevant to recently developed arrays of polar molecules, Rydberg atoms, and other quantum simulators with long-range interactions.