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# A tunable bilayer Hubbard model in twisted WSe2

## Abstract

Moiré materials with flat electronic bands provide a highly controllable quantum system for studies of strong-correlation physics and topology. In particular, angle-aligned heterobilayers of semiconducting transition metal dichalcogenides with large band offset realize the single-band Hubbard model. Introduction of a new layer degree of freedom is expected to foster richer interactions, enabling Hund’s physics, interlayer exciton condensation and new superconducting pairing mechanisms to name a few. Here we report competing electronic states in twisted AB-homobilayer WSe2, which realizes a bilayer Hubbard model in the weak interlayer hopping limit for holes. By layer-polarizing holes via a perpendicular electric field, we observe a crossover from an excitonic insulator to a charge-transfer insulator at a hole density of ν = 1 (in units of moiré density), a transition from a paramagnetic to an antiferromagnetic charge-transfer insulator at ν = 2 and evidence for a layer-selective Mott insulator at 1 < ν < 2. The unique coupling of charge and spin to external electric and magnetic fields also manifests a giant magnetoelectric response. Our results establish a new solid-state simulator for the bilayer Hubbard model Hamiltonian.

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## Data availability

Source data are provided with this paper. Additional data that support the findings of this study are available from the corresponding authors upon reasonable request. Source data are provided with this paper.

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## Acknowledgements

We thank L. Fu, Y. Zhang, M. Davydova, A. H. MacDonald, N. Wei and Y. Zeng for fruitful discussions. Y.X. acknowledges support from the National Key R&D Program of China (2021YFA1401300) for data analysis at the Chinese Academy of Sciences. This study was supported by the US Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES), under award number DE-SC0022058 (optical characterization) and DE-SC0019481 (magnetic characterization), the National Science Foundation (NSF) under DMR-2114535 (Rydberg sensing measurement) and the US Office of Naval Research under award number N00014-21-1-2471 (device fabrication). The growth of hBN crystals was supported by the Elemental Strategy Initiative of MEXT, Japan, and CREST (JPMJCR15F3), JST. This work made use of the Cornell NanoScale Facility, an NNCI member supported by NSF grant NNCI-2025233.

## Author information

Authors

### Contributions

Y.X. and K.K. fabricated the devices. Y.X. performed the measurements and analysed the data. K.W. and T.T. grew the bulk hBN crystals. Y.X., K.F.M. and J.S. designed the scientific objectives, oversaw the project and co-wrote the manuscript. All authors discussed the results and commented on the manuscript.

### Corresponding authors

Correspondence to Yang Xu, Kin Fai Mak or Jie Shan.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

## Peer review

### Peer review information

Nature Nanotechnology thanks Angel Rubio and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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## Extended data

### Extended Data Fig. 1 Operating gate-voltage range of the sensor.

a,b, Gate dependence of the reflectance contrast spectrum of the device with the sensor at E = 6 mV/nm (a) and – 55 mV/nm (b). The gate voltage Vtot (= Vtg + Vbg) is proportional to the total charge density in the moiré bilayer and the sensor layer. The red and blue dashed lines mark the limits of gate voltage beyond which the sensor becomes doped, as evidenced by the emergence of the polarons (1.7–1.75 eV) and disappearance of the 2 s state (~ 1.85 eV). c, Operating range of the gate voltage (shaded area). It is primarily determined by the back gate voltage (-2 V, 4 V). The sensor is closer to the back gate. The blue and red symbols are determined from the boundaries in a, b and similar measurements at other electric fields.

### Extended Data Fig. 2 Layer polarization probed by the sensor 2 s exciton.

The reflectance contrast spectrum of the sensor 2 s exciton as a function of electric field at representative doping densities. The case of ν = 1 is shown in Fig. 2a of the main text. The grey dashed lines denote the critical electric fields Ec.

### Extended Data Fig. 3 Layer polarization probed by the moiré exciton.

Reflectance contrast spectrum of the moiré exciton as a function of electric field at representative hole filling factors. The result for ν = 2 is shown in Fig. 2b of the main text. The spectra are measured from the region without the sensor. The white dashed lines mark Ec.

### Extended Data Fig. 4 MCD measurement.

a, Magnetic-field dependence of the MCD spectrum at ν = 2 under zero electric field. b, Linecuts of a at B = 5 T (black line) and -5 T (red line). The MCD is determined as $$\frac{{R_ + (B) - R_ + ( - B)}}{{R_ + (B) + R_ + ( - B)}}$$, where R+(B) is the reflectance contrast of left circular polarized light under magnetic field B.

### Extended Data Fig. 5 MCD at ν = 1.6 under different electric fields.

Magnetic-field dependence of MCD under electric field ranging from 0 to 128 mV/nm. The MCD is obtained by averaging the MCD spectrum over a spectral window of 5 meV centred at the exciton resonance (≈ 1.69 eV). The response is PM at small electric fields. A metamagnetic component emerges above about 8.5 mV/nm, which agrees well with the measured threshold electric field Ec for full layer polarization.

### Extended Data Fig. 6 MCD at different doping densities.

Magnetic-field dependence of MCD for layer-polarized (E = -55 mV/nm) (a) and layer-unpolarized holes (E = 0 mV/nm) (b) at representative filling factors. The MCD is obtained by averaging the MCD spectrum over a spectral window of 5 meV centred at the exciton resonance (≈ 1.69 eV).

### Extended Data Fig. 7 Electric-field effect on the insulating states.

a-f, Gate dependence of the reflectance contrast spectrum of the sensor 2 s exciton at representative electric fields, including 67 mV/nm (a), 43 mV/nm (b), 18 mV/nm (c), 0 mV/nm (d), -18 mV/nm (e), and -43 mV/nm (f). Electrons and holes are introduced into the channel for Vtot above and below about -1 V, respectively. The electric field limits the operating voltage range of the sensor. In addition, notable 2 s exciton energy shift is observed on the hole side, but not on the electron side, when the electric field switches sign. The result indicates strong layer hybridization for the conduction bands in twisted WSe2, the band edge of which is located at the Q point of the Brillouin zone.

### Extended Data Fig. 8 Temperature and magnetic-field effects on the insulating states.

a-f, Top gate dependence of the reflectance contrast spectrum of the sensor 2 s exciton at 1.7 K and under magnetic field of 2 T (a) and 9 T (b), as well as under zero magnetic field at 10 K (c), 36 K (d), 55 K (e) and 80 K (f). The back gate is fixed at 0 V. The corresponding electric field varies for different doping densities. Particularly, the electric field is -43 mV/nm for ν = 1 and -76mV/nm for ν = 2 on the hole side, under both of which the moiré bilayer is fully layer-polarized. The correlated insulating states for holes at ν = 1 and 2 are robust against magnetic field for the entire accessible range (9 T) and against thermal melting up to about 80 K.

### Extended Data Fig. 9 Magnetoelectric (ME) effect at ν = 2.

a,c,d, Normalized MCD (by its value at 9 T) (a) and its derivative with respect to B (c) and to E (d) as a function of the electric and magnetic fields. The four regions (defined by the dashed lines) correspond to: (I) | M | , | P | < 1; (II) | M | < 1, |P | = 1; (III) | M | = 1, |P | < 1; and (IV) | M | , | P | = 1. The boundaries are identified by the derivatives in c and d. The insets in a on the left and right illustrate, respectively, the spin and charge configuration at small magnetic fields and after magnetic saturation. On each side the three insets correspond to P = -1, 0 and 1 from top to bottom. b, Vertical linecuts of a at representative magnetic fields. Strong ME effect is observed for the intermediate magnetic fields, under which the electric field changes the magnetization by over 60%.

### Extended Data Fig. 10 Spatial variation of moiré density.

Top-gate voltage (Vtg) dependence of the reflection contrast spectrum of the sensor 2 s exciton at three sample locations (P1, P2, and P3) that have slightly different moiré densities. Only the position of the ν = 0 state remains unchanged (illustrated by the yellow vertical dashed line) while the positions of the other states (such as the ν = 1,2 states indicated by the black dashed lines) shift from P1 to P3. The same applies to the electron doping case with Vtg > 0. The results demonstrate the slight twist angle variation in the sample.

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Xu, Y., Kang, K., Watanabe, K. et al. A tunable bilayer Hubbard model in twisted WSe2. Nat. Nanotechnol. (2022). https://doi.org/10.1038/s41565-022-01180-7

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• DOI: https://doi.org/10.1038/s41565-022-01180-7