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 regime2,3,7,8,9. 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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Data availability
The data that support the plots within this paper, and other findings of this study, are available from the corresponding authors upon reasonable request.
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Acknowledgements
We thank L. Fu for discussions. Research was primarily supported by the US Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES), under award number DE-SC0019481 (optical spectroscopy and theory) and DE-SC0013883 (device fabrication). The transport measurements were supported by ONR N00014-18-1-2368. The growth of WSe2 crystals was supported by the NSF MRSEC programme through Columbia University, Center for Precision Assembly of Superstratic and Superatomic Solids (DMR-1420634), and the growth of the hBN crystals by the Elemental Strategy Initiative of MEXT, Japan and CREST (JPMJCR15F3), Japan Science and Technology Agency. K.F.M. acknowledges support from a David and Lucille Packard Fellowship. A.H.M. acknowledges support from Welch Foundation grant TBF1473.
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Y.T., L.L. and T.L. fabricated the devices; Y.T. and L.L. performed the optical measurements; T.L. performed the transport measurements with the assistance of Y.X. A.H.M. provided theoretical support. S.L. and K.B. grew the bulk TMD crystals, and K.W. and T.T. grew the bulk hBN crystals. K.F.M., J.S. and Y.T. designed the study and performed the analysis. K.F.M., J.S., Y.T. and A.H.M. co-wrote the manuscript. All authors discussed the results and commented on the manuscript.
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Extended data figures and tables
Extended Data Fig. 1 Behaviour of a 0° aligned WSe2/WS2 bilayer.
a, Contour plot of reflection contrast spectrum as a function of gate voltage. As in the data of the 60° aligned sample (Fig. 2), the dashed lines from top to bottom mark the zero, half- and full filling of the first moiré valence band. b, Reflection peak amplitude at the lowest-energy exciton resonance as a function of filling at 1.65 K. The two peaks, denoted by two dashed lines, occur at half- and full filling. c, Filling-dependent g-factor at 1.65 K. The error bars are uncertainties from the linear fitting of the field-dependent Zeeman splitting at small magnetic fields.
Extended Data Fig. 2 Bare-exciton g-factor g0.
a, Exciton Zeeman splitting as a function of magnetic field for four representative fillings at 1.6 K. b, Bare-exciton g-factor (g0) as a function of filling factor. The value is derived from the slope of the field dependence of the exciton Zeeman splitting (a) from 2 to 8 T (solid lines). The error bars are fitting uncertainties. The dashed line (at −4.5) represents the mean value over the entire filling range between 0 and 1.
Extended Data Fig. 3 Curie–Weiss analysis at half-filling for different temperature ranges.
a–d, Temperature dependence of |g0/(g – g0)| ∝ χ−1 (symbols) and the Curie–Weiss law fit (solid lines) for the temperature ranges 1.6–40 K (a), 3–40 K (b), 6.4–40 K (c) and 9.9–40 K (d). Error bars for χ−1 are calculated from uncertainties of the g-factor obtained from a linear fit to the field-dependent Zeeman splitting at a small magnetic field. The extracted Weiss constant is always negative (that is, antiferromagnetic) and has an absolute magnitude of ~1 K. The figure illustrates the reliability of the analysis method.
Extended Data Fig. 4 Zeeman splitting saturation, coupling constant λX and Curie constant C at different fillings.
a, Filling dependence of the Zeeman splitting saturation value evaluated at fields above 1.4 T (symbols, left axis) and the fraction of unpaired holes in a triangular moiré superlattice (red curve, right axis). The latter is proportional to the magnetization saturation value. b, Ratio of the two quantities shown in a, which is proportional to the coupling constant λX, defined as EZ = g0μBμ0(H + λXM). λX shows a weak filling dependence. c, Filling dependence of the inverse of the slope of the Curie–Weiss law fitting curve in Fig. 4b (symbols; error bars are calculated from fitting uncertainties). The inverse of the slope is ~λXC, where C is the Curie constant in the Curie–Weiss law χ = C/(T – θ). Its filling dependence follows that of the magnetization saturation value (that is, the fraction of unpaired holes; red curve, right axis), as expected.
Extended Data Fig. 5 Doping dependence of g-factor at various temperatures.
The errors are uncertainties from the linear fitting of the field-dependent Zeeman splitting at small magnetic fields.
Extended Data Fig. 6 Raw data of Fig. 4d.
Exciton Zeeman splitting subtracted by the bare-exciton contribution as a function of magnetic field at a filling factor of 0.61 and various temperatures.
Extended Data Fig. 7 Dependence of the Mott insulating behaviour on an out-of-plane magnetic field.
Two-point resistance as a function of filling factor at 10 K under different magnetic fields. The curves are not displaced. Overall, the resistance increases by 2–3 times from the zero-field case, and most of this increase comes from the change in contact resistance with magnetic field. This observation is consistent with the large onsite Coulomb repulsion energy (U ≈ 15 meV). The Zeeman energy (~4–5 meV) for localized moments (with g-factor of ~8) is small compared to U even at 10 T.
Extended Data Fig. 8 Schematic and optical micrograph of a representative device.
a, Schematic device geometry of a dual-gated WSe2/WS2 bilayer. b, Optical micrograph of a typical device on a SiO2/Si substrate. The red and orange solid lines denote the sample edges of WSe2 and WS2, respectively. The dashed black lines indicate the top and bottom graphene gates. The solid black line denotes the graphene contact electrode. The scale bar is 5 μm.
Extended Data Fig. 9 Determination of crystal orientation by angle-resolved SHG.
Solid squares and circles show polarization-dependent SHG from regions of monolayer WS2 and WSe2, respectively. Dotted curves are fits to the data using D + Acos2[3(ϕ – ϕ0)], where ϕ is the excitation polarization angle, ϕ0 is the initial crystal orientation, D and A are free fitting parameters. The initial crystal orientation ϕ0 is determined to be 20.4° ± 0.1° and 20.0° ± 0.1° for WS2 and WSe2, respectively. The twist angle is determined to be 0.4° ± 0.2°. Empty triangles show the SHG from the overlapping region of the two materials. Because this is much weaker than the SHG from the monolayer-only regions, we conclude that WS2 and WSe2 are nearly 60° aligned, so the second-harmonic dipoles from the two layers are out of phase.
Extended Data Fig. 10
Dependence of moiré lattice constant and moiré density on twist angle in WSe2/WS2 bilayers. aM, moiré lattice constant; n0, moiré density.
Extended Data Fig. 11 Dependence of magnetic response on applied electric field in WSe2/WS2 bilayers.
a, Filling dependence of the exciton g-factor at three different applied electric fields: 0 V nm−1 and ±0.23 V nm−1. b, Magnetic-field dependence of the exciton Zeeman splitting at half-filling and under the same electric fields as in a. c, Temperature dependence of the magnetic susceptibility at half-filling under −0.23 V nm−1. The solid line shows a fit with the Curie–Weiss law. The extracted Weiss constant is −0.8 ± 0.1 K, compared with −0.6 ± 0.2 K under zero electric field. No obvious electric-field dependence of the magnetic response is observed. Error bars in a and c are calculated from the uncertainties of the g-factor obtained from linear fits to the field-dependent Zeeman splitting at small magnetic fields.
Extended Data Fig. 12 Photoluminescence (PL) spectrum of interlayer moiré excitons at different fillings.
The excitation wavelength is 736 nm and the incident power is 1 nW. The curves are vertically displaced for clarity.
Extended Data Fig. 13 Comparison between Zeeman splitting and MCD analysis at half-filling.
a, Magnetic-field dependence of the Zeeman splitting EZ (black filled squares, left axis) and the spectrally integrated MCD (red empty circles, right axis), as defined in Methods. The MCD contrast approaches 0.45 above 1 T. b, Temperature dependence of the g-factor (black filled squares, left axis) and the slope of the MCD contrast in a (red empty circles, right axis) in the low-field limit. The results of the two different analysis methods agree well with each other. Error bars in b are uncertainties from linear fits to the field-dependent data at small magnetic fields in a.
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Tang, Y., Li, L., Li, T. et al. Simulation of Hubbard model physics in WSe2/WS2 moiré superlattices. Nature 579, 353–358 (2020). https://doi.org/10.1038/s41586-020-2085-3
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DOI: https://doi.org/10.1038/s41586-020-2085-3
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