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


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.

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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.

Data availability

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


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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

Authors and Affiliations



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.

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Nature Physics thanks Liuyan Zhao 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 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.

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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.

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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).

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