Quasi-periodic moir\'{e} patterns and dimensional localization in three-dimensional quasi-moir\'{e} crystals

Recent advances in spin-dependent optical lattices [Meng et al., Nature \textbf{615}, 231 (2023)] have enabled the experimental implementation of two superimposed three-dimensional lattices, presenting new opportunities to investigate \textit{three-dimensional moir\'{e} physics} in ultracold atomic gases. This work studies the moir\'{e} physics of atoms within a spin-dependent cubic lattice with relative twists along different directions. It is discovered that dimensionality significantly influences the low-energy moir\'{e} physics. From a geometric perspective, this manifests in the observation that moir\'{e} patterns, generated by rotating lattices along different axes, can exhibit either periodic or quasi-periodic behavior--a feature not present in two-dimensional systems. We develop a low-energy effective theory applicable to systems with arbitrary rotation axes and small rotation angles. This theory elucidates the emergence of quasi-periodicity in three dimensions and demonstrates its correlation with the arithmetic properties of the rotation axes. Numerical analyses reveal that these quasi-periodic moir\'{e} potentials can lead to distinctive dimensional localization behaviors of atoms, manifesting as localized wave functions in planar or linear configurations.