Fermionic atoms in optical lattices have served as a useful model system in which to study and emulate the physics of strongly correlated matter. Driven by the advances of high-resolution microscopy, the current research focus is on two-dimensional systems^1,2,3, in which several quantum phases—such as antiferromagnetic Mott insulators for repulsive interactions^4,5,6,7 and charge-density waves for attractive interactions⁸—have been observed. However, the lattice structure of real materials, such as bilayer graphene, is composed of coupled layers and is therefore not strictly two-dimensional, which must be taken into account in simulations. Here we realize a bilayer Fermi–Hubbard model using ultracold atoms in an optical lattice, and demonstrate that the interlayer coupling controls a crossover between a planar antiferromagnetically ordered Mott insulator and a band insulator of spin-singlets along the bonds between the layers. We probe the competition of the magnetic ordering by measuring spin–spin correlations both within and between the two-dimensional layers. Our work will enable the exploration of further properties of coupled-layer Hubbard models, such as theoretically predicted superconducting pairing mechanisms^9,10.

光学晶格中的费米子原子已成为研究和模拟强相关物质物理学的有用模型系统。在高分辨率显微镜技术进步的推动下，目前的研究重点是二维系统^1,2,3，其中有几个量子相，例如用于排斥相互作用的反铁磁莫特绝缘体^4,5,6,7和电荷-吸引相互作用的密度波⁸——已经观察到。然而，真实材料的晶格结构，如双层石墨烯，是由耦合层组成的，因此不是严格的二维，在模拟中必须考虑到这一点。在这里，我们使用光学晶格中的超冷原子实现了双层 Fermi-Hubbard 模型，并证明层间耦合控制了平面反铁磁有序 Mott 绝缘体和沿层间键的自旋单线态带绝缘体之间的交叉。我们通过测量二维层内和二维层之间的自旋-自旋相关性来探索磁排序的竞争。我们的工作将有助于探索耦合层 Hubbard 模型的进一步特性，例如理论上预测的超导配对机制^9,10。