## Abstract

Quantum spin Hall (QSH) insulators are two-dimensional electronic materials that have a bulk band gap similar to an ordinary insulator but have topologically protected pairs of edge modes of opposite chiralities^{1,2,3,4,5,6}. So far, experimental studies have found only integer QSH insulators with counter-propagating up-spins and down-spins at each edge leading to a quantized conductance *G*_{0} = *e*^{2}/*h* (with *e* and *h* denoting the electron charge and Planck’s constant, respectively)^{7,8,9,10,11,12,13,14}. Here we report transport evidence of a fractional QSH insulator in 2.1° twisted bilayer MoTe_{2}, which supports spin-*S*_{z} conservation and flat spin-contrasting Chern bands^{15,16}. At filling factor *ν* = 3 of the moiré valence bands, each edge contributes a conductance \(\frac{3}{2}{G}_{0}\) with zero anomalous Hall conductivity. The state is probably a time-reversal pair of the even-denominator 3/2-fractional Chern insulators. Furthermore, at *ν* *=* 2, 4 and 6, we observe a single, double and triple QSH insulator with each edge contributing a conductance *G*_{0}, 2*G*_{0} and 3*G*_{0}, respectively. Our results open up the possibility of realizing time-reversal symmetric non-abelian anyons and other unexpected topological phases in highly tunable moiré materials^{17,18,19}.

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## Acknowledgements

We thank C. Jian, A. H. MacDonald, C. Kane, L. Fu, A. Bernevig, N. Regnault, J. Yu, T. Devakul, A. Reddy and E.-A. Kim for their discussions. This work was primarily supported by the US Department of Energy, Office of Science, Basic Energy Sciences, under award no. DE-SC0019481. It was also funded in part by the Air Force Office of Scientific Research under award no. FA9550-20-1-0219, the Cornell University Materials Research Science and Engineering Center DMR-1719875 and the Gordon and Betty Moore Foundation (grant no. GBMF11563; https://doi.org/10.37807/GBMF11563) for device fabrication and thermodynamic measurements. We used the Cornell NanoScale Facility, an NNCI member supported by NSF Grant NNCI-2025233, for sample fabrication. The growth of the hBN crystals was supported by the Elemental Strategy Initiative of MEXT, Japan, and CREST (JPMJCR15F3), JST. K.F.M. acknowledges support from the David and Lucille Packard Fellowship.

## Author information

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### Contributions

K.K., B.S. and Y.Q. fabricated the devices, performed the transport measurements and analysed the data. Y.Z. and Z.X. performed the thermodynamic measurements. K.W. and T.T. grew the bulk hBN crystals. K.K., K.F.M. and J.S. designed the scientific objectives and oversaw the project. All authors discussed the results and commented on the paper.

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## Extended data figures and tables

### Extended Data Fig. 1 Contact resistance characterization.

**a**, Top: schematic of the device contact and channel. The Pt electrodes (grey) are contacted to the tMoTe_{2} channel (blue) through heavily hole-doped tMoTe_{2} contact regions (orange). Middle: schematic valence band alignment of the tMoTe_{2} contact and channel regions (solid red lines). Dashed line: Fermi level; solid black line: spatial variation of the work function. Tunnel barriers are at the Pt junctions, but the heavily-to-lightly doped tMoTe_{2} junctions are transparent. Bottom: equivalent circuit model for two-terminal measurements with *R*_{channel} and *R*_{contact} denoting the channel resistance and the contact resistance at the Pt junction, respectively. **b**, Filling factor dependence of two-terminal resistance *R*_{2t} at different contact gate voltages (*E≈*0, *T=*20 mK and *B*_{⊥}=0.1 T). Inset: measurement configuration. *R*_{2t} at the highest fillings is approximately the sum of resistances in series for two contacts. **c**, Filling factor dependence of four-terminal resistance *R*_{sym} measured with the configuration in the left inset. The resistance drops to about 100 Ω at *ν* = 11, which is nearly temperature independent (right inset). **d**, Temperature dependence of contact resistance (for one contact) calibrated for the two-terminal configuration in the inset.

### Extended Data Fig. 2 Twist angle calibration.

**a**,**b**, Four-terminal resistance *R*_{sym} (**a**) and Hall resistance *R*_{xy} (**b**) versus out-of-plane magnetic field *B*_{⊥} and filling factor *ν* at *E* = 0.06 V/nm and 20 mK. The measurement configuration is shown in the inset of Fig. 1d. Dashed lines denote Landau level *ν*_{LL} = 1, 2 and 3 emerging from *ν* = 2 above *B*_{⊥} ≈ 7-8 T. The dispersing state at low filling factors is the *ν* = 1 Chern insulator. **c**, Line cut of **a**,**b** at *B*_{⊥} = 11 T. The vertical dashed lines mark Landau level *ν*_{LL} = 1, 2 and 3 with nearly quantized *R*_{xy} and *R*_{sym} minimum; the horizontal dashed lines denote the expected quantized value of \({R}_{{xy}}=\frac{h}{{\nu }_{{LL}}{e}^{2}}\). **d**, Twist angle calibrated from Landau level spacing at *B*_{⊥} ranging from 9 T to 11 T. No field dependence is observed. The mean of the twist angle is 2.10 degrees (dashed line) and the uncertainty is about ±0.05 degrees.

### Extended Data Fig. 3 Symmetric device channel.

**a**,**b**, Filling factor dependence of four-terminal nonlocal resistance *R*_{NL} (**a**, *B*_{⊥}=0.1 T) and Hall resistance *R*_{xy} (**b**, *B*_{⊥}=0.2 T) at *E≈*0 and *T* = 20 mK. The inset shows the device channel with contact 1–4. I: current bias; V: voltage probe. The two curves in each panel correspond to swapped source-drain and voltage probe pairs. The nearly identical results demonstrate a highly symmetric two-dimensional channel.

### Extended Data Fig. 4 Additional nonlocal two-terminal transport data.

**a**-**d**, Filling factor dependence of two-terminal conductance *G*_{2t} (in units of \(\frac{{e}^{2}}{h}\)) at two electric fields inside the layer-hybridized regime. Inset: measurement configuration. The dashed lines denote the expected quantized values of *G*_{2t} at *ν* = 2, 3, 4 and 6. **e**,**f**, Electric-field dependence of *G*_{2t} (in units of \(\frac{{e}^{2}}{h}\)) at *ν* = 1 (**e**) and *ν* = 6 (**f**) at 20 mK. The horizontal dashed lines denote the expected conductance \(\nu \), \(\frac{3}{4}\nu \), \(\frac{2}{3}\nu \) and \(\frac{1}{2}\nu \) in descending order. The colour of the lines matches the colour of the channels in **a**–**d**. Magnetic field 0.3 T is applied to fully polarize the *ν* = 1 Chern insulator.

### Extended Data Fig. 5 Additional temperature dependence data.

**a**–**c**, Electric-field dependence of two-terminal conductance, *G*_{2t} (in units of \(\frac{{e}^{2}}{h}\)), at varying temperatures for *ν* = 1 (**a**), *ν* = 2 (**b**) and *ν* = 4 (**c**). The horizontal dashed lines denote the quantized value \({G}_{2t}=\frac{\nu {e}^{2}}{h}\). The measurement configuration is shown in the inset of **a**. **d**, Filling factor dependence of four-terminal nonlocal resistance *R*_{NL} at *E* ≈ 0 and varying temperatures. *R*_{NL} at *ν* = 2, 3, 4 and 6 decreases with increasing temperature as bulk conduction becomes more important.

### Extended Data Fig. 6 Twist angle effects.

**a**,**b**, Four-terminal Hall resistance *R*_{xy} (**a**) and *R*_{sym} (**b**) versus out-of-plane electric field and filling factor for 2.1-degree tMoTe_{2} device at *B*_{⊥} = 5 T and 20 mK. The measurement configuration is shown in the inset of **a**. Both *R*_{sym} and *R*_{xy} are small near *ν* = 3 in the layer-hybridized regime even under high magnetic fields. **c**,**d**, Same as **a**,**b** for 3.5-degree tMoTe_{2} device at *B*_{⊥} = 4 T and 1.6 K. A large negative *R*_{xy} is accompanied by a weak *R*_{sym} dip near *ν* = 3 in the layer-hybridized regime (even though at higher temperature and lower magnetic field). The large Hall response suggests a valley-polarized state (rather than a fractional QSH insulator) at *ν* = 3.

### Extended Data Fig. 7 Anisotropic magneto-response.

**a**,**b**, Filling factor dependence of two-terminal resistance *R*_{2t} at varying in-plane (**a**) and out-of-plane (**b**) magnetic fields (*E* = 0.06 V/nm and *T* = 20 mK). Whereas *R*_{2t} is nearly independent of *B*_{⊥}, it increases with *B*_{∥}. The horizontal dashed lines denote the quantized value \({R}_{2t}=\frac{h}{\nu {e}^{2}}\) at *ν* = 2, 3, 4 and 6. **c**, In-plane magnetic field dependence of \(\frac{{R}_{{NL}}({B}_{\parallel }=0T)}{{R}_{{NL}}({B}_{\parallel })}\) at varying temperatures for *ν* = 2, 3, 4 and 6, where *R*_{NL} is the nonlocal four-terminal resistance. At all filling factors the cusp at zero magnetic field is broadened as temperature increases. The anisotropic magneto-response supports that the helical edge states carry Ising spins. Because of spin-*S*_{z} conservation, the edge states are immune to *B*_{⊥}, but are susceptible to gap opening and spin mixing under *B*_{∥}, thus increasing *R*_{2t} and *R*_{NL}. The anisotropic magneto-response also contrasts with the expected bulk-dominant transport, which would show exactly the opposite magnetic field dependence, namely, strong out-of-plane but negligible in-plane magnetic field dependence.

### Extended Data Fig. 8 Incompressibility measurements on 2.2- and 2.7-degree tMoTe_{2}.

**a**,**b**, Electronic incompressibility for 2.2-degree (**a**) and 2.7-degree (**b**) tMoTe_{2} versus vertical electric field *E* and filling factor *ν* at 1.6 K (*B*_{⊥} = 0 T). In addition to the incompressible states at *ν* = 1, 2 and 4, a weak incompressible state is observed at *ν* = 3 in the small electric field, layer-hybridized region. **c**,**d**, The corresponding filling factor dependence of the electronic incompressibility at *E* = 0 V/nm (along the dashed lines in **a** and **b**). The grey-shaded area provides an estimate of the charge gap size at *ν* = 3 (about 0.12 meV and 0.8 meV for 2.2- and 2.7-degree sample, respectively). Note that the true gap size is higher because of the relatively higher base temperature (1.6 K) for the compressibility measurements.

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Kang, K., Shen, B., Qiu, Y. *et al.* Evidence of the fractional quantum spin Hall effect in moiré MoTe_{2}.
*Nature* **628**, 522–526 (2024). https://doi.org/10.1038/s41586-024-07214-5

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DOI: https://doi.org/10.1038/s41586-024-07214-5

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