Abstract
The Hubbard model, formulated by physicist John Hubbard in the 1960s1, is a simple theoretical model of interacting quantum particles in a lattice. The model is thought to capture the essential physics of high-temperature superconductors, magnetic insulators and other complex quantum many-body ground states2,3. Although the Hubbard model provides a greatly simplified representation of most real materials, it is nevertheless difficult to solve accurately except in the one-dimensional case2,3. Therefore, the physical realization of the Hubbard model in two or three dimensions, which can act as an analogue quantum simulator (that is, it can mimic the model and simulate its phase diagram and dynamics4,5), has a vital role in solving the strong-correlation puzzle, namely, revealing the physics of a large number of strongly interacting quantum particles. Here we obtain the phase diagram of the two-dimensional triangular-lattice Hubbard model by studying angle-aligned WSe2/WS2 bilayers, which form moiré superlattices6 because of the difference between the lattice constants of the two materials. We probe the charge and magnetic properties of the system by measuring the dependence of its optical response on an out-of-plane magnetic field and on the gate-tuned carrier density. At half-filling of the first hole moiré superlattice band, we observe a Mott insulating state with antiferromagnetic Curie-Weiss behaviour, as expected for a Hubbard model in the strong-interaction regime2,3,7-9. Above half-filling, our experiment suggests a possible quantum phase transition from an antiferromagnetic to a weak ferromagnetic state at filling factors near 0.6. Our results establish a new solid-state platform based on moiré superlattices that can be used to simulate problems in strong-correlation physics that are described by triangular-lattice Hubbard models.
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