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Nature Communications

Low-dimensional compact states in 3D moiré lattices

Zixuan Gao, V. V. Konotop, Ruihan Peng, Zhenli Xu, Zhiguo Yang, Fangwei Ye

Published Jul 8, 2025• 6 citations
Publisher

Abstract

Moiré lattices formed by superimposing rotated two-dimensional (2D) periodic sublattices, such as twisted bilayer graphene, can exhibit fascinating properties not observed in their individual constituent layers. Despite extensive research on 2D moiré lattices, the physics of three-dimensional (3D) moiré lattices—formed by superimposing rotated 3D periodic sub-lattices with crystallographic symmetries, which exhibit unique 3D potentials determined by twisting angles—remains largely unexplored. In this work, we demonstrate that depending on the choice of rotation angles, moiré potentials composed of two cubic sub-lattices can exhibit three different phases: they can be fully 3D incommensurate, incommensurate in only one spatial direction (while remaining periodic in the orthogonal plane), or fully periodic. These incommensurate potentials, which can be created for condensates of non-interacting atoms using judiciously combined laser beams, are shown to support compact states of three distinct types: fully localized in space, confined to a line, or confined to a specific plane. Our findings establish a foundation for controlling wave localization in incommensurate 3D moiré systems, with potential applications in cold atom systems, optics, and beyond. 3D moiré lattices can exhibit distinct incommensurate phases depending on twist angles. Here, authors demonstrate phase-controlled emergence of fully localized, line-localized, and plane-localized states, enabling tunable 3D transport in cold atom and optical systems.

DOI
10.1038/s41467-025-61491-w

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Moiré lattices formed by superimposing rotated two-dimensional (2D) periodic sublattices, such as twisted bilayer graphene, can exhibit fascinating properties not observed in their individual constituent layers. Despite extensive research on 2D moiré lattices, the physics of three-dimensional (3D) moiré lattices—formed by superimposing rotated 3D periodic sub-lattices with crystallographic symmetries, which exhibit unique 3D potentials determined by twisting angles—remains largely unexplored. In this work, we demonstrate that depending on the choice of rotation angles, moiré potentials composed of two cubic sub-lattices can exhibit three different phases: they can be fully 3D incommensurate, incommensurate in only one spatial direction (while remaining periodic in the orthogonal plane), or fully periodic. These incommensurate potentials, which can be created for condensates of non-interacting atoms using judiciously combined laser beams, are shown to support compact states of three distinct types: fully localized in space, confined to a line, or confined to a specific plane. Our findings establish a foundation for controlling wave localization in incommensurate 3D moiré systems, with potential applications in cold atom systems, optics, and beyond. 3D moiré lattices can exhibit distinct incommensurate phases depending on twist angles. Here, authors demonstrate phase-controlled emergence of fully localized, line-localized, and plane-localized states, enabling tunable 3D transport in cold atom and optical systems.
Abstract Recent advances in spin-dependent optical lattices [ Meng et al . 2023 Nature 615 , 231 ] have enabled the experimental implementation of two superimposed three-dimensional lattices, presenting new opportunities to investigate three-dimensional moiré physics in ultracold atomic gases. This work studies the moiré 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 moiré physics. From a geometric perspective, this manifests in the observation that moiré 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 moiré potentials can lead to distinctive dimensional localization behaviors of atoms, manifesting as localized wave functions in planar or linear configurations.
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 moiré physics} in ultracold atomic gases. This work studies the moiré 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 moiré physics. From a geometric perspective, this manifests in the observation that moiré 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 moiré potentials can lead to distinctive dimensional localization behaviors of atoms, manifesting as localized wave functions in planar or linear configurations.
Abstract Observation of strong correlations and superconductivity in twisted-bilayer-graphene have stimulated tremendous interest in fundamental and applied physics. In this system, the superposition of two twisted honeycomb lattices, generating a Moire pattern, is the key to the observed flat electronic bands, slow electron velocity and large density of states. Despite these observations, a full understanding of the emerging superconductivity from the coupled layers and the appearance of a small magic angle remain a hot topic of research. Here, we demonstrate a quantum simulation platform to study superfluids in twisted bilayer lattices based on Bose-Einstein condensates loaded into spin-dependent optical lattices. The lattices are made of two sets of laser beams that independently address atoms in different spin states, which form the synthetic dimension accommodating the two layers. The twisted angle of the two lattices is controlled by the relative angle of the laser beams. We show that atoms in each spin state only feel one set of the lattice and the interlayer coupling can be controlled by microwave coupling between the spin states. Our system allows for flexible control of both the inter- and intralayer couplings. Furthermore we directly observe the spatial Moire pattern and the momentum diffraction, which confirm the presence of atomic superfluid in the bilayer lattices. Our system constitutes a powerful platform to investigate the physics underlying the superconductivity in twisted-bilayer-graphene and to explore other novel quantum phenomena difficult to realize in materials.
Moiré related physics in twisted bilayer two-dimensional (2D) materials has attracted widespread interest in condensed matter physics. Simulation of moiré related physics in cold atom platform is expected to outperform the 2D materials thanks to its advantage of higher tunablility. Here, we demonstrate that, the cold atom platform enables a new mechanism of moiré lattice formation, induced by interlayer interaction with intrinsic "dynamical" character, in contrast to conventional moiré lattice induced by "static" ways such as single-particle interlayer tunneling. Specifically, we consider a twisted bilayer Bose-Hubbard model with vanishing interlayer tunneling, and the bilayer is solely coupled through interlayer interaction that originates from contact interaction of atoms. We find that this system hosts a plethora of novel phases unique to this dynamical lattice, including a variety of Mott insulator (MI) and superfluid (SF) phases either preserving or breaking moiré lattice symmetry, phases with one layer in SF and the other in MI, "interlocked" MI, and self-localized phases at commensurate twist angles, which exhibits the characteristics of Bose glass and quasi-many-body localization in the absence of (quasi)disorder or quasicrystalline lattices. Our prediction can be readily observed in current experimental setup of twisted bilayer optical lattices, opening up new avenues for exploring the rich physics of interaction induced moiré systems in cold atoms.
Moiré superlattices in twisted bilayer graphene and transition-metal dichalcogenides have emerged as a powerful tool for engineering novel band structures and quantum phases of two-dimensional quantum materials. Here we investigate Moiré physics emerging from twisting two independent hexagonal optical lattices of atomic (pseudo-)spin states (instead of bilayers) that exhibit remarkably different physics from twisted bilayer graphene. We employ a momentum-space tight-binding calculation that includes all range real-space tunnelings and show that all twist angles θ≲6° can become magic and support gapped flat bands. Because of the greatly enhanced density of states near the flat bands, the system can be driven to superfluidity by weak attractive interaction. Strikingly, the superfluid phase corresponds to a Larkin-Ovchinnikov state with finite momentum pairing that results from the interplay between flat bands and interspin interactions in the unique single-layer spin-twisted lattice. Our work may pave the way for exploring novel quantum phases and twistronics in cold atomic systems.
The work intends to extend the moiré physics to three dimensions. Three-dimensional moiré patterns can be realized in ultracold atomic gases by coupling two spin states in spin-dependent optical lattices with a relative twist, a structure currently unachievable in solid-state materials. We give the commensurate conditions under which the three-dimensional moiré pattern features a periodic structure termed a three-dimensional moiré crystal. We emphasize a key distinction of three-dimensional moiré physics: In three dimensions, the twist operation generically does not commute with the rotational symmetry of the original lattice, unlike in two dimensions, where these two always commute. Consequently, the moiré crystal can exhibit a crystalline structure that differs from the original underlying lattice. We demonstrate that twisting a simple cubic lattice can generate various crystal structures. This capability of altering crystal structures by twisting offers a broad range of tunability for three-dimensional band structures.