Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Letter
  • Published:

Experimental signature of the parity anomaly in a semi-magnetic topological insulator

Nature Physics volume 18, pages 390–394 (2022)Cite this article

Subjects

Abstract

A three-dimensional (3D) topological insulator features a 2D surface state consisting of a single linearly dispersive Dirac cone1,2,3. Under broken time-reversal symmetry, the single Dirac cone is predicted to cause half-integer quantization of Hall conductance, which is a manifestation of the parity anomaly in quantum field theory1,2,3,4,5,6,7,8,9. However, despite various observations of quantization phenomena10,11,12,13,14,15, the half-integer quantization has not been observed because most experiments simultaneously measure a pair of equivalent Dirac cones16 on two opposing surfaces. Here we demonstrate the half-integer quantization of Hall conductance in a synthetic heterostructure termed a semi-magnetic topological insulator, where only one surface state is gapped by magnetic doping and the opposite one is non-magnetic and gapless. We observe half-quantized Faraday and Kerr rotations with terahertz magneto-optical spectroscopy and half-quantized Hall conductance in transport at zero magnetic field. Our results suggest a condensed-matter realization of the parity anomaly4,5,6,7,8,9 and open a way for studying the physics enabled by a single Dirac fermion.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Semi-magnetic topological insulator and the parity anomaly.
Fig. 2: Half-quantized terahertz Faraday and Kerr rotations.
Fig. 3: Half-integer quantization in electrical transport.
Fig. 4: Correspondence between transport and magneto-optics.

Similar content being viewed by others

Data availability

All relevant data within this paper are available from the authors upon reasonable request. Source data are provided with this paper.

References

  1. Fu, L., Kane, C. L. & Mele, E. J. Topological insulators in three dimensions. Phys. Rev. Lett. 98, 106803 (2007).

    Article  ADS  Google Scholar 

  2. Fu, L. & Kane, C. L. Topological insulators with inversion symmetry. Phys. Rev. B 76, 045302 (2007).

    Article  ADS  Google Scholar 

  3. Qi, X.-L., Hughes, T. L. & Zhang, S.-C. Topological field theory of time-reversal invariant insulators. Phys. Rev. B 78, 195424 (2008).

    Article  ADS  Google Scholar 

  4. Niemi, A. J. & Semenoff, G. W. Axial-anomaly-induced fermion fractionization and effective gauge-theory actions in odd-dimensional space-times. Phys. Rev. Lett. 51, 2077–2080 (1983).

    Article  MathSciNet  ADS  Google Scholar 

  5. Redlich, A. N. Gauge noninvariance and parity nonconservation of three-dimensional fermions. Phys. Rev. Lett. 52, 18–21 (1984).

    Article  MathSciNet  ADS  Google Scholar 

  6. Jackiw, R. Fractional charge and zero modes for planer systems in a magnetic field. Phys. Rev. D 29, 2375–2377 (1984).

    Article  MathSciNet  ADS  Google Scholar 

  7. Semenoff, G. Condensed-matter simulation of a three-dimensional anomaly. Phys. Rev. Lett. 53, 2449–2452 (1984).

    Article  ADS  Google Scholar 

  8. Fradkin, E., Dagotto, E. & Boyanovsky, D. Physical realization of the parity anomaly in condensed matter physics. Phys. Rev. Lett. 57, 2967–2970 (1986).

    Article  ADS  Google Scholar 

  9. Haldane, F. D. M. Model for a quantum Hall effect without Landau levels: condensed-matter realization of the ‘parity anomaly’. Phys. Rev. Lett. 61, 2015–2018 (1988).

    Article  ADS  Google Scholar 

  10. Chang, C.-Z. et al. Experimental observation of the quantum anomalous Hall effect in a magnetic topological insulator. Science 340, 167–170 (2013).

    Article  ADS  Google Scholar 

  11. Xu, Y. et al. Observation of topological surface state quantum Hall effect in an intrinsic three-dimensional topological insulator. Nat. Phys. 10, 956–963 (2014).

    Article  Google Scholar 

  12. Yoshimi, R. et al. Quantum Hall states stabilized in semi-magnetic bilayers of topological insulators. Nat. Commun. 6, 8530 (2015).

    Article  ADS  Google Scholar 

  13. Wu, L. et al. Quantized Faraday and Kerr rotation and axion electrodynamics of a 3D topological insulator. Science 354, 1124–1127 (2016).

    Article  MathSciNet  MATH  ADS  Google Scholar 

  14. Dziom, V. et al. Observation of the universal magnetoelectric effect in a 3D topological insulator. Nat. Commun. 8, 15197 (2017).

    Article  ADS  Google Scholar 

  15. Okada, K. N. et al. Terahertz spectroscopy on Faraday and Kerr rotations in a quantum anomalous Hall state. Nat. Commun. 7, 12245 (2016).

    Article  ADS  Google Scholar 

  16. Nielsen, N. B. & Ninomiya, M. Absence of neutrinos on a lattice: (I). Proof by homotopy theory. Nucl. Phys. B185, 20–40 (1981).

    Article  MathSciNet  ADS  Google Scholar 

  17. Novoselov, K. S. et al. Two-dimensional gas of massless Dirac fermions in graphene. Nature 438, 197–200 (2005).

    Article  ADS  Google Scholar 

  18. Morimoto, T., Hatsugai, Y. & Aoki, H. Optical Hall conductivity in ordinary and graphene quantum Hall systems. Phys. Rev. Lett. 103, 116803 (2009).

    Article  ADS  Google Scholar 

  19. Shimano, R. et al. Quantum Faraday and Kerr rotations in graphene. Nat. Commun. 4, 1841 (2013).

    Article  ADS  Google Scholar 

  20. Thouless, D. J., Kohmoto, M., Nightingale, M. P. & den Nijs, M. Quantized Hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett. 49, 405–408 (1982).

    Article  ADS  Google Scholar 

  21. Jotzu, G. et al. Experimental realization of the topological Haldane model with ultracold fermions. Nature 515, 237–240 (2014).

    Article  ADS  Google Scholar 

  22. Essin, A. M., Moore, J. E. & Vanderbilt, D. Magnetoelectric polarizability and axion electrodynamics in crystalline insulators. Phys. Rev. Lett. 102, 146805 (2009).

    Article  ADS  Google Scholar 

  23. Chu, R.-L., Shi, J. & Shen, S.-Q. Surface edge state and half-quantized Hall conductance in topological insulators. Phys. Rev. B 84, 085312 (2011).

    Article  ADS  Google Scholar 

  24. König, E. J. et al. Half-integer quantum Hall effect of disordered Dirac fermions at a topological insulator surface. Phys. Rev. B 90, 165435 (2014).

    Article  ADS  Google Scholar 

  25. Gu, M. et al. Spectral signatures of the surface anomalous Hall effect in magnetic axion insulators. Nat. Commun. 12, 3524 (2021).

    Article  ADS  Google Scholar 

  26. Chen, R. et al. Using nonlocal surface transport to identify the axion insulator. Phys. Rev. B 103, L241409 (2021).

    Article  ADS  Google Scholar 

  27. Lu, R. et al. Half-magnetic topological insulator with magnetization-induced Dirac gap at a selected surface. Phys. Rev. X 11, 011039 (2021).

    Google Scholar 

  28. Chen, Y. L. et al. Massive Dirac fermion on the surface of a magnetically doped topological insulator. Science 329, 659–662 (2010).

    Article  ADS  Google Scholar 

  29. Maciejko, J., Qi, X.-L., Drew, H. D. & Zhang, S.-C. Topological quantization in units of the fine structure constant. Phys. Rev. Lett. 105, 166803 (2010).

    Article  ADS  Google Scholar 

  30. Tse, W.-K. & MacDonald, A. H. Magneto-optical Faraday and Kerr effects in topological insulator films and in other layered quantized Hall systems. Phys. Rev. B 84, 205327 (2011).

    Article  ADS  Google Scholar 

  31. Hancock, J. N. et al. Surface state charge dynamics of a high-mobility three-dimensional topological insulator. Phys. Rev. Lett. 107, 136803 (2011).

    Article  ADS  Google Scholar 

  32. Aguilar, R. V. et al. Terahertz response and colossal Kerr rotation from the surface states of the topological insulator Bi2Se3. Phys. Rev. Lett. 108, 087403 (2012).

    Article  ADS  Google Scholar 

  33. Zhang, S. et al. Anomalous quantization trajectory and parity anomaly in Co cluster decorated BiSbTeSe2 nanodevices. Nat. Commun. 8, 977 (2017).

    Article  ADS  Google Scholar 

  34. Chong, S. K. et al. Topological insulator-based van der Waals heterostructures for effective control of massless and massive Dirac fermions. Nano Lett. 18, 8047–8053 (2018).

    Article  ADS  Google Scholar 

  35. Gluschke, J. G. et al. Impact of invasive metal probes on Hall measurements in semiconductor nanostructures. Nanoscale 12, 20317–20325 (2020).

    Article  Google Scholar 

  36. Huckestein, B. Scaling theory of the integer quantum Hall effect. Rev. Mod. Phys. 67, 357–396 (1995).

    Article  ADS  Google Scholar 

  37. Nomura, K. & Nagaosa, N. Surface-quantized anomalous Hall current and the magnetoelectric effect in magnetically disordered topological insulators. Phys. Rev. Lett. 106, 166802 (2011).

    Article  ADS  Google Scholar 

  38. Qi, X.-L., Li, R., Zang, J. & Zhang, S.-C. Inducing a magnetic monopole with topological surface states. Science 323, 1184–1187 (2009).

    Article  MathSciNet  MATH  ADS  Google Scholar 

  39. Lian, B. et al. Topological quantum computation based on chiral Majorana fermions. Proc. Natl Acad. Sci. USA 115, 10938–10942 (2018).

    Article  MathSciNet  MATH  ADS  Google Scholar 

  40. He, Q. L. et al. Chiral Majorana fermion modes in a quantum anomalous Hall insulator-superconductor structure. Science 357, 294–299 (2017).

    Article  MathSciNet  MATH  ADS  Google Scholar 

  41. Kayyalha, M. et al. Absence of evidence for chiral Majorana modes in quantum anomalous Hall-superconductor devices. Science 367, 64–67 (2020).

    Article  ADS  Google Scholar 

  42. He, J. J., Liang, T., Tanaka, Y. & Nagaosa, N. Platform of chiral Majorana edge modes and its quantum transport phenomena. Commun. Phys. 2, 149 (2019).

    Article  Google Scholar 

  43. Alyabyeva, L. N., Zhukova, E. S., Belkin & Gorshunov, B. P. Dielectric properties of semi-insulating Fe-doped InP in the terahertz spectral region. Sci. Rep. 7, 7360 (2017).

    Article  ADS  Google Scholar 

  44. Büttiker, M. Absence of backscattering in the quantum Hall effect in multiprobe conductors. Phys. Rev. B 38, 9375–9389 (1988).

    Google Scholar 

  45. Seeger, K. Semiconductor Physics: An Introduction (Springer, 2004).

Download references

Acknowledgements

We thank J. G. Checkelsky for enlightening discussions, and K. N. Okada, S. Iguchi, M. Ogino, Y. Hayashi, H. Shishikura, D. Murata and Y. D. Kato for support of the terahertz measurements. This research project was partly supported by the JSPS/MEXT Grant-in-Aid for Scientific Research (nos. 15H05853, 15H05867, 17J03179, 18H04229 and 18H01155) and JST CREST (nos. JPMJCR16F1 and JPMJCR1874).

Author information

Author notes

  1. M. Mogi

    Present address: Department of Physics, Massachusetts Institute of Technology, Cambridge, MA, USA

Authors and Affiliations

  1. Department of Applied Physics and Quantum Phase Electronics Center (QPEC), University of Tokyo, Bunkyo-ku, Tokyo, Japan

    M. Mogi, Y. Okamura, K. Yasuda, T. Morimoto, N. Nagaosa, M. Kawasaki, Y. Takahashi & Y. Tokura

  2. RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama, Japan

    M. Kawamura, R. Yoshimi, K. S. Takahashi, N. Nagaosa, M. Kawasaki, Y. Takahashi & Y. Tokura

  3. Institute for Materials Research, Tohoku University, Sendai, Miyagi, Japan

    A. Tsukazaki

  4. Tokyo College, University of Tokyo, Bunkyo-ku, Tokyo, Japan

    Y. Tokura

  5. Department of Physics, Massachusetts Institute of Technology, Cambridge, MA, USA

    K. Yasuda

Authors
  1. M. Mogi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Y. Okamura
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. Kawamura
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. R. Yoshimi
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. K. Yasuda
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. A. Tsukazaki
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. K. S. Takahashi
    View author publications

    You can also search for this author in PubMed Google Scholar

  8. T. Morimoto
    View author publications

    You can also search for this author in PubMed Google Scholar

  9. N. Nagaosa
    View author publications

    You can also search for this author in PubMed Google Scholar

  10. M. Kawasaki
    View author publications

    You can also search for this author in PubMed Google Scholar

  11. Y. Takahashi
    View author publications

    You can also search for this author in PubMed Google Scholar

  12. Y. Tokura
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Y. Tokura conceived and supervised the project. M.M., R.Y. and K.Y. fabricated the samples with help from A.T., K.S.T. and M. Kawasaki. M.M., Y.O. and Y. Takahashi performed the terahertz spectroscopy measurements and analysed the data. M.M. and M. Kawamura performed the transport measurements and analysed the data. T.M. and N.N. contributed to the theoretical discussions. M.M., M. Kawamura, T.M., N.N. and Y. Tokura wrote the manuscript, with input from all the other authors.

Corresponding authors

Correspondence to M. Mogi or Y. Tokura.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature Physics thanks Liuyan Zhao and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Additional information

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

Extended data

Extended Data Fig. 1 Quantized Faraday and Kerr rotations in a QAH state.

a,b, θF and ηF (a) and θK and ηK (b) spectra at T = 1 K for the QAH insulator film, which is partly the same as Fig. 2c and d in the main text, with a slight variation of external magnetic fields (μ0H = 0, 0.01, and 1 T). c, Measured fine-structure constant αmeas which is calculated from (a) and (b) by using a relation of \(\alpha _{{{{\mathrm{meas}}}}} = ({{{\mathrm{tan}}}}\theta _{{{\mathrm{F}}}}{{{\mathrm{tan}}}}\theta _{{{\mathrm{K}}}} - {{{\mathrm{tan}}}}^2\theta _{{{\mathrm{F}}}})/({{{\mathrm{tan}}}}\theta _{{{\mathrm{K}}}} - 2{{{\mathrm{tan}}}}\theta _{{{\mathrm{F}}}})\) (refs. 29,30). d, e, θF, ηF (d) and θK, ηK (e) spectra at μ0H = 0 T and at various temperatures (T = 1, 1.6, 4.2, 15.6, 32.6, and 56.7 K). f, T dependence of θF and θK taken at ħω = 2 meV, suggesting that the Curie temperature is about 50 K and that the integer quantization subsists possibly up to 4.2 K. The inset is the magnified view of f. The error bars in a-f represent the standard error of the mean.

Extended Data Fig. 2 Optical microscope image of a typical Hall bar device used in the transport measurements.

The black broken lines indicate the shape of the TI film below the gate electrode, formed into the Hall bar structure.

Extended Data Fig. 3 Kerr rotation in the semi-magnetic TI under magnetic fields.

a, Representative complex Kerr rotation spectra for the semi-magnetic TI film used for Fig. 4a in the main text. The open circles at ħω = 0 meV indicate the values anticipated from the measured dc conductivity values. b, Background Kerr spectra of the InP substrate without any TI films at 7 T. The inset shows the Faraday rotation spectra at 7 T, where an observable polarization rotation occurs at ħω > 2 meV, possibly due to the magnetic resonance of magnetic impurities involved in InP substrates. The error bars in a and b represent the standard error of the mean.

Supplementary information

Supplementary Information

Supplementary Sections I–XII and Figs. 1–14.

Source data

Source Data Fig. 2

Source data for Fig. 2.

Source Data Fig. 3

Source data for Fig. 3.

Source Data Fig. 4

Source data for Fig. 4.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mogi, M., Okamura, Y., Kawamura, M. et al. Experimental signature of the parity anomaly in a semi-magnetic topological insulator. Nat. Phys. 18, 390–394 (2022). https://doi.org/10.1038/s41567-021-01490-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41567-021-01490-y

This article is cited by

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing