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 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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Fig. 1: Illustration of the bilayer Hubbard model.
Fig. 2: Detection of intralayer correlations.
Fig. 3: Magnetic correlations between the layers.
Fig. 4: Crossover from the antiferromagnetic Mott insulator to a band insulator of singlets.

Data availability

The data presented in the figures are available at More detailed data and information of this study are available from the corresponding author upon request.

Code availability

The DQMC theory simulations were performed using the QUEST Fortran 90/95 package, version 1.44, from


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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.

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The idea for the experiment was conceived by M.G., N.W., C.F.C. and M.K. Data taking was performed by M.G., N.W. and C.F.C. with contributions by J.S. Data analysis was primarily performed by M.G. and N.W. Numerical simulations were performed by C.F.C. and N.W. The results were discussed and interpreted by all coauthors, and the manuscript was written by M.K. with contributions from all coauthors.

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Correspondence to Michael Köhl.

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The authors declare no competing interests.

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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).

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