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# A monolayer transition-metal dichalcogenide as a topological excitonic insulator

## Abstract

Monolayer transition-metal dichalcogenides in the T′ phase could enable the realization of the quantum spin Hall effect1 at room temperature, because they exhibit a prominent spin–orbit gap between inverted bands in the bulk2,3. Here we show that the binding energy of electron–hole pairs excited through this gap is larger than the gap itself in the paradigmatic case of monolayer T′ MoS2, which we investigate from first principles using many-body perturbation theory4. This paradoxical result hints at the instability of the T′ phase in the presence of spontaneous generation of excitons, and we predict that it will give rise to a reconstructed ‘excitonic insulator’ ground state5,6,7. Importantly, we show that in this monolayer system, topological and excitonic order cooperatively enhance the bulk gap by breaking the crystal inversion symmetry, in contrast to the case of bilayers8,9,10,11,12,13,14,15,16 where the frustration between the two orders is relieved by breaking time reversal symmetry13,15,16. The excitonic topological insulator is distinct from the bare topological phase because it lifts the band spin degeneracy, which results in circular dichroism. A moderate biaxial strain applied to the system leads to two additional excitonic phases, different in their topological character but both ferroelectric17,18 as an effect of electron–electron interaction.

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## Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## Code availability

Many-body perturbation theory calculations were performed by means of the codes Yambo (http://www.yambo-code.org/) and Quantum ESPRESSO (http://www.quantum-espresso.org), which are both open source software. Results for the two-band model were obtained through custom Fortran codes that are available from M.R. upon reasonable request.

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## Acknowledgements

M.P. thanks G. Cicero for illuminating discussions in the early stages of this project and acknowledges Tor Vergata University for financial support through the mission sustainability project 2DUTOPI. This work was supported in part by the MaX (‘MAterials design at the eXascale’, www.max-centre.eu) European Centre of Excellence funded by the European Union H2020-INFRAEDI-2018-1 programme, grant number 824143, project 453. This work was also supported by MIUR-PRIN2017 number 2017BZPKSZ ‘Excitonic insulator in two-dimensional long-range interacting systems (EXC-INS)’. We acknowledge PRACE for awarding us access to the Marconi system based in Italy at CINECA.

## Author information

Authors

### Contributions

D.V. and M.P. developed the first-principles many-body perturbation theory calculations and analysis; M.R. developed the self-consistent mean-field model and wrote the paper; and all authors initiated this project, contributed to the analysis of data and critically discussed the paper.

### Corresponding author

Correspondence to Massimo Rontani.

## Ethics declarations

### Competing Interests

The authors declare that they have no competing financial interests.

Peer review information Nature Nanotechnology thanks Onur Erten and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Extended data

### Extended Data Fig. 1 Screened Coulomb interaction.

Comparison between first-principles (black dots) and model (empty red circles) screened Coulomb interaction, W(q), shown as a function of qx, with qy = 0. The plotted data include form factors due to the wave function overlap between Bloch states: the long-wavelength value (q = 0) is extracted from Bloch states located at the Λ point.

### Extended Data Fig. 2 Exciton wavefunction within the effective-mass approximation.

Contour plot of the exciton wavefunction square modulus in k space within the electron–hole centre-of-mass frame. This is the lowest-energy exciton responsible for the instability that drives the transition into the QSHX phase. The plot compares well with its first-principles counterpart (Fig. 2b).

### Extended Data Fig. 3 Excitonic hybridization and band structure of the trivial ferroelectric excitonic insulator.

a,b, Contour plot in k space of the real (panel a) and imaginary (panel b) part of the self-consistent excitonic hybridization, ΔX(k). c,d, Energy bands along directions kx = 0 (panel c) and ky = Λ (panel d). Black and red curves label opposite spin projections. Since the two-fold screw-axis symmetry along y is broken, Kramers degeneracy is lifted throughout the Brillouin zone. Here T = 0 K and δ = –100 meV.

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Varsano, D., Palummo, M., Molinari, E. et al. A monolayer transition-metal dichalcogenide as a topological excitonic insulator. Nat. Nanotechnol. 15, 367–372 (2020). https://doi.org/10.1038/s41565-020-0650-4