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Study of WSe2/WS2 moire superlattices reveals the phase diagram of the triangular-lattice Hubbard model, including a Mott insulating state at half-filling and a possible magnetic quantum phase transition near 0.6 filling.

The Hubbard model, formulated by physicist John Hubbard in the 1960s(1), 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 states(2,3). 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 case(2,3). 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 dynamics(4,5)), 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 moire superlattices(6) 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 moire superlattice band, we observe a Mott insulating state with antiferromagnetic Curie-Weiss behaviour, as expected for a Hubbard model in the strong-interaction regime(2,3,7-9). 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 moire superlattices that can be used to simulate problems in strong-correlation physics that are described by triangular-lattice Hubbard models.

Since the original work on Bose-Einstein condensation(1,2), the use of quantum degenerate gases of atoms has enabled the quantum emulation of important systems in condensed matter and nuclear physics, as well as the study of many-body states that have no analogue in other fields of physics(3). Ultracold molecules in the micro- and nanokelvin regimes are expected to bring powerful capabilities to quantum emulation(4) and quantum computing(5), owing to their rich internal degrees of freedom compared to atoms, and to facilitate precision measurement and the study of quantum chemistry(6). Quantum gases of ultracold atoms can be created using collision-based cooling schemes such as evaporative cooling, but thermalization and collisional cooling have not yet been realized for ultracold molecules. Other techniques, such as the use of supersonic jets and cryogenic buffer gases, have reached temperatures limited to above 10 millikelvin(7,8). Here we show cooling of NaLi molecules to micro- and nanokelvin temperatures through collisions with ultracold Na atoms, with both molecules and atoms prepared in their stretched hyperfine spin states. We find a lower bound on the ratio of elastic to inelastic molecule-atom collisions that is greater than 50-large enough to support sustained collisional cooling. By employing two stages of evaporation, we increase the phase-space density of the molecules by a factor of 20, achieving temperatures as low as 220 nanokelvin. The favourable collisional properties of the Na-NaLi system could enable the creation of deeply quantum degenerate dipolar molecules and raises the possibility of using stretched spin states in the cooling of other molecules.

NaLi molecules are cooled to micro- and nanokelvin temperatures through collisions with ultracold Na atoms by using molecules and atoms in stretched hyperfine spin states and applying two evaporation stages.