Competing magnetic orders in a bilayer Hubbard model with ultracold atoms.

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, in which several quantum phases-such as antiferromagnetic Mott insulators for repulsive interactions and charge-density waves for attractive interactions-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.