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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.

At the historic Shelter Island Conference on the Foundations of Quantum Mechanics in 1947, Willis Lamb reported an unexpected feature in the fine structure of atomic hydrogen: a separation of the 2S(1/2) and 2P(1/2) states(1). The observation of this separation, now known as the Lamb shift, marked an important event in the evolution of modern physics, inspiring others to develop the theory of quantum electrodynamics(2-5). Quantum electrodynamics also describes antimatter, but it has only recently become possible to synthesize and trap atomic antimatter to probe its structure. Mirroring the historical development of quantum atomic physics in the twentieth century, modern measurements on anti-atoms represent a unique approach for testing quantum electrodynamics and the foundational symmetries of the standard model. Here we report measurements of the fine structure in the n = 2 states of antihydrogen, the antimatter counterpart of the hydrogen atom. Using optical excitation of the 1S-2P Lyman-alpha transitions in antihydrogen(6), we determine their frequencies in a magnetic field of 1 tesla to a precision of 16 parts per billion. Assuming the standard Zeeman and hyperfine interactions, we infer the zero-field fine-structure splitting (2P(1/2)-2P(3/2)) in antihydrogen. The resulting value is consistent with the predictions of quantum electrodynamics to a precision of 2 per cent. Using our previously measured value of the 1S-2S transition frequency(6,7), we find that the classic Lamb shift in antihydrogen (2S(1/2)-2P(1/2) splitting at zero field) is consistent with theory at a level of 11 per cent. Our observations represent an important step towards precision measurements of the fine structure and the Lamb shift in the antihydrogen spectrum as tests of the charge-parity-time symmetry(8) and towards the determination of other fundamental quantities, such as the antiproton charge radius(9,10), in this antimatter system.

Precision measurements of the 1S-2P transition in antihydrogen that take into account the standard Zeeman and hyperfine effects confirm the predictions of quantum electrodynamics.