Quasi-periodic moir\'e patterns and dimensional localization in three-dimensional quasi-moir\'e crystals
Abstract
Recent advances in spin-dependent optical lattices [Meng et al., Nature 615, 231 (2023)] have enabled the experimental implementation of two superimposed three-dimensional lattices, presenting new opportunities to investigate 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.
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