The relation between scaling regimes described by ( ϕ 2 ) 2 field theories with long-range and short-range exchange is analysed. It is shown that the crossover from the long-range regime to the short-range regime takes place due to the singular behaviour of three-loop and higher-order contributions to the beta function of the renormalization group, which controls the scaling behaviour of the long-range model. The critical exponents eta and nu are shown to be continuous functions of the parameter alpha , which characterises the power-like fall-off of the exchange interaction.
We investigate single-excitation quantum walks across the one-to-two-dimensional crossover in diluted power-law hopping models. Using arrays of
ℓ 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
ℓ, defining a characteristic transverse scale
ℓe, and effectively two-dimensional behavior emerges already for
ℓ≪L. An independent spectral analysis shows that
ℓe 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.