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 systems1,2,3, in which several quantum phases—such as antiferromagnetic Mott insulators for repulsive interactions4,5,6,7 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 mechanisms9,10.
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The data presented in the figures are available at https://osf.io/u9wj6. More detailed data and information of this study are available from the corresponding author upon request.
The DQMC theory simulations were performed using the QUEST Fortran 90/95 package, version 1.44, from https://code.google.com/archive/p/quest-qmc/.
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This work has been supported by BCGS, the Alexander-von-Humboldt Stiftung, DFG (SFB/TR 185 project B4), Cluster of Excellence Matter and Light for Quantum Computing (ML4Q) EXC 2004/1 - 390534769 and Stiftung der deutschen Wirtschaft.
The authors declare no competing interests.
Peer review information Nature thanks the anonymous reviewer(s) for their contribution to the peer review of this work.
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Gall, M., Wurz, N., Samland, J. et al. Competing magnetic orders in a bilayer Hubbard model with ultracold atoms. Nature 589, 40–43 (2021). https://doi.org/10.1038/s41586-020-03058-x