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 signatures of the mixed axial–gravitational anomaly in the Weyl semimetal NbP

Abstract

The conservation laws, such as those of charge, energy and momentum, have a central role in physics. In some special cases, classical conservation laws are broken at the quantum level by quantum fluctuations, in which case the theory is said to have quantum anomalies1. One of the most prominent examples is the chiral anomaly2,3, which involves massless chiral fermions. These particles have their spin, or internal angular momentum, aligned either parallel or antiparallel with their linear momentum, labelled as left and right chirality, respectively. In three spatial dimensions, the chiral anomaly is the breakdown (as a result of externally applied parallel electric and magnetic fields4) of the classical conservation law that dictates that the number of massless fermions of each chirality are separately conserved. The current that measures the difference between left- and right-handed particles is called the axial current and is not conserved at the quantum level. In addition, an underlying curved space-time provides a distinct contribution to a chiral imbalance, an effect known as the mixed axial–gravitational anomaly1, but this anomaly has yet to be confirmed experimentally. However, the presence of a mixed gauge–gravitational anomaly has recently been tied to thermoelectrical transport in a magnetic field5,6, even in flat space-time, suggesting that such types of mixed anomaly could be experimentally probed in condensed matter systems known as Weyl semimetals7. Here, using a temperature gradient, we observe experimentally a positive magneto-thermoelectric conductance in the Weyl semimetal niobium phosphide (NbP) for collinear temperature gradients and magnetic fields that vanishes in the ultra-quantum limit, when only a single Landau level is occupied. This observation is consistent with the presence of a mixed axial–gravitational anomaly, providing clear evidence for a theoretical concept that has so far eluded experimental detection.

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

Access options

Rent or buy this article

Prices vary by article type

from$1.95

to$39.95

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

Figure 1: Positive magneto-conductance G(B) and magneto-thermoelectric conductance GT(B) in the Weyl semimetal NbP.
Figure 2: Chiral anomaly in NbP.
Figure 3: Evidence of the mixed axial–gravitational anomaly in NbP.
Figure 4: Longitudinal thermopower.

Similar content being viewed by others

References

  1. Bertlmann, R. A. Anomalies in Quantum Field Theory (Oxford Univ. Press, 2000)

  2. Adler, S. L. Axial-vector vertex in spinor electrodynamics. Phys. Rev. 177, 2426–2438 (1969)

    Article  ADS  Google Scholar 

  3. Bell, J. S. & Jackiw, R. A PCAC puzzle: π0 → γγ in the σ-model. Nuovo Cimento A 60, 47–61 (1969)

    Article  ADS  CAS  Google Scholar 

  4. Nielsen, H. B. & Ninomiya, M. The Adler–Bell–Jackiw anomaly and Weyl fermions in a crystal. Phys. Lett. B 130, 389–396 (1983)

    Article  ADS  MathSciNet  Google Scholar 

  5. Landsteiner, K., Megías, E. & Pena-Benitez, F. Gravitational anomaly and transport phenomena. Phys. Rev. Lett. 107, 021601 (2011)

    Article  ADS  Google Scholar 

  6. Lucas, A., Davison, R. A. & Sachdev, S. Hydrodynamic theory of thermoelectric transport and negative magnetoresistance in Weyl semimetals. Proc. Natl Acad. Sci. USA 113, 9463–9468 (2016)

    Article  ADS  CAS  Google Scholar 

  7. Xu, S.-Y. et al. Discovery of a Weyl Fermion semimetal and topological Fermi arcs. Science 349, 613–617 (2015)

    Article  ADS  CAS  Google Scholar 

  8. Xu, S.-Y. et al. Discovery of a Weyl fermion state with Fermi arcs in niobium arsenide. Nat. Phys. 11, 748–754 (2015)

    Article  CAS  Google Scholar 

  9. Huang, S.-M. et al. A Weyl fermion semimetal with surface Fermi arcs in the transition metal monopnictide TaAs class. Nat. Commun. 6, 7373 (2015)

    Article  ADS  CAS  Google Scholar 

  10. Nielsen, H. B. & Ninomiya, M. Absence of neutrinos on a lattice: (I). Proof by homotopy theory. Nucl. Phys. B 185, 20–40 (1981)

    Article  ADS  MathSciNet  Google Scholar 

  11. Son, D. T. & Spivak, B. Z. Chiral anomaly and classical negative magnetoresistance of Weyl metals. Phys. Rev. B 88, 104412 (2013)

    Article  ADS  Google Scholar 

  12. Xiong, J. et al. Evidence for the chiral anomaly in the Dirac semimetal Na3Bi. Science 350, 413–416 (2015)

    Article  ADS  MathSciNet  CAS  Google Scholar 

  13. Huang, X. et al. Observation of the chiral-anomaly-induced negative magnetoresistance in 3D Weyl semimetal TaAs. Phys. Rev. X 5, 031023 (2015)

    Google Scholar 

  14. Niemann, A. C. et al. Chiral magnetoresistance in the Weyl semimetal NbP. Sci. Rep. 7, 43394 (2017)

    Article  ADS  Google Scholar 

  15. Hirschberger, M. et al. The chiral anomaly and thermopower of Weyl fermions in the half-Heusler GdPtBi. Nat. Mater. 15, 1161–1165 (2016)

    Article  ADS  CAS  Google Scholar 

  16. Li, H. et al. Negative magnetoresistance in Dirac semimetal Cd3As2 . Nat. Commun. 7, 10301 (2016)

    Article  ADS  CAS  Google Scholar 

  17. Arnold, F. et al. Negative magnetoresistance without well-defined chirality in the Weyl semimetal TaP. Nat. Commun. 7, 11615 (2016)

    Article  ADS  CAS  Google Scholar 

  18. Shekhar, C. et al. Observation of chiral magneto-transport in RPtBi topological Heusler compounds. Preprint at https://arxiv.org/abs/1604.01641 (2016)

  19. Alvarez-Gaumé, L. & Witten, E. Gravitational anomalies. Nucl. Phys. B 234, 269–330 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  20. Eguchi, T. & Freund, P. G. O. Quantum gravity and world topology. Phys. Rev. Lett. 37, 1251–1254 (1976)

    Article  ADS  MathSciNet  Google Scholar 

  21. Landsteiner, K., Megías, E. & Peña-Benitez, F. in Strongly Interacting Matter in Magnetic Fields (eds Kharzeev, D. et al.) 433–468 (Springer, 2013)

  22. Kaminski, M., Uhlemann, C. F., Bleicher, M. & Schaffner-Bielich, J. Anomalous hydrodynamics kicks neutron stars. Phys. Lett. B 760, 170–174 (2016)

    Article  ADS  CAS  Google Scholar 

  23. Lundgren, R., Laurell, P. & Fiete, G. A. Thermoelectric properties of Weyl and Dirac semimetals. Phys. Rev. B 90, 165115 (2014)

    Article  ADS  Google Scholar 

  24. Kim, K.-S. Role of axion electrodynamics in a Weyl metal: violation of Wiedemann–Franz law. Phys. Rev. B 90, 121108(R) (2014)

    Article  ADS  Google Scholar 

  25. Sharma, G., Goswami, P. & Tewari, S. Nernst and magnetothermal conductivity in a lattice model of Weyl fermions. Phys. Rev. B 93, 035116 (2016)

    Article  ADS  Google Scholar 

  26. Spivak, B. Z. & Andreev, A. V. Magnetotransport phenomena related to the chiral anomaly in Weyl semimetals. Phys. Rev. B 93, 085107 (2016)

    Article  ADS  Google Scholar 

  27. Burkov, A. A. Chiral anomaly and diffusive magnetotransport in Weyl metals. Phys. Rev. Lett. 113, 247203 (2014)

    Article  ADS  CAS  Google Scholar 

  28. Shekhar, C. et al. Extremely large magnetoresistance and ultrahigh mobility in the topological Weyl semimetal candidate NbP. Nat. Phys. 11, 645–649 (2015)

    Article  CAS  Google Scholar 

  29. Sergelius, P. Berry phase and band structure analysis of the Weyl semimetal NbP. Sci. Rep. 6, 33859 (2016)

    Article  ADS  CAS  Google Scholar 

  30. Liang, T. et al. Evidence for massive bulk Dirac fermions in Pb1−xSnxSe from Nernst and thermopower experiments. Nat. Commun. 4, 2696 (2013)

    Article  ADS  Google Scholar 

  31. Landsteiner, K., Megias, E., Melgar, L. & Pena-Benitez, F. Holographic gravitational anomaly and chiral vortical effect. J. High Energy Phys. 9, 121 (2011)

    Article  ADS  Google Scholar 

  32. Golkar, S. & Sethi, S. Global anomalies and effective field theory. J. High Energy Phys. 5, 105 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  33. Delbourgo, R. & Salam, A. The gravitational correction to PCAC. Phys. Lett. B 40, 381–382 (1972)

    Article  ADS  CAS  Google Scholar 

  34. Landsteiner, K. Notes on anomaly induced transport. Acta Phys. Pol. B 47, 2617–2673 (2016)

    Article  ADS  Google Scholar 

  35. Jensen, K., Loganayagam, R. & Yarom, A. Thermodynamics, gravitational anomalies and cones. J. High Energy Phys. 2, 88 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  36. LeBellac, M. Thermal Field Theory (Cambridge Univ. Press, 2000)

Download references

Acknowledgements

This work was supported by the research grant DFG-RSF (NI616 22/1) ‘Contribution of topological states to the thermoelectric properties of Weyl semimetals’, Severo Ochoa SEV-2012-0249, FPA 2015-65480-P and SFB 1143, by the Helmholtz association through VI-521, and by the DFG (Emmy Noether programme) via grant ME 4844/1. We thank T. Sturm and A. Pöhl, for experimental support. We also acknowledge support by W. Riess, K. Moselund and H. Riel, and thank C. Bollinger for copy-editing.

Author information

Authors and Affiliations

Authors

Contributions

J.G. conceived the experiment. M.S., C.S. and V.S. synthesized the single-crystal bulk samples. R.H. characterized the crystal structure. B.R. supervised the micro-ribbon definition and the compositional analysis. A.C.N. fabricated the samples. J.G. carried out the thermoelectric transport measurements with the help of A.C.N. J.G., A.C.N., F.M., B.G., T.M. and A.G.G. analysed the data. B.G., C.F., B.Y. and K.N. supervised the project. A.G.G., T.M. and K.L. provided the theoretical background for the work. All authors contributed to interpreting the data and writing the manuscript.

Corresponding author

Correspondence to Johannes Gooth.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Additional information

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

Extended data figures and tables

Extended Data Figure 1 Material analysis of the NbP micro-ribbon.

a, Sketch of the structure of the NbP crystal. b, SEM image of a NbP micro-ribbon before device processing. The longitudinal direction of the ribbon corresponds to the [100] axis of the crystal. c, Spatial composition of an exemplary NbP micro-ribbon, measured from the top along [100] using SEM-EDX, reveals an average of 53% Nb, 45% P and 2% Ga. x100 is the distance from the end of the sample along [100]. d, XRD spectrum of the NbP at room temperature (Cu Ka radiation). a.u., arbitrary units.

Extended Data Figure 2 Optical micrograph of a measurement device.

The NbP micro-ribbon (green) is placed between two four-probe thermometers (grey), which also serve as electrical probes. The electrically insulated heater line (grey) close to one end of the sample creates a temperature gradient along the length of the sample.

Extended Data Figure 3 Isothermal (T = 0 K) current–voltage (J-V) characteristic of the NbP micro-ribbon at selected temperatures and zero magnetic field (B = 0 T).

The linearity of the curves reveals ohmic electrical contacts.

Extended Data Figure 4 Thermometer calibration.

Resistance R versus base temperature T of the cryostat, measured at isothermal conditions.

Extended Data Figure 5 Temperature gradient T along the sample as a function of the square of the heating voltage VH at different base temperatures, which is proportional to the heating power.

Extended Data Figure 6 Linear response of the thermoelectric current J to the temperature gradient T.

We determine the thermo-conductance GT = J/| T| from the slope of the linear fits. The error bars in Extended Data Fig. 8b are the uncertainties obtained from these fits.

Extended Data Figure 7 Linear response of the thermovoltage Vth to the temperature gradient T.

The thermopower S = −Vth/| T| is determined from linear fits of the data.

Extended Data Figure 8 Zero-field transport.

a, Electrical conductance G at zero magnetic field (B = 0 T) in isothermal conditions ( T = 0 K) as a function of the base temperature. Values of G are obtained from the slope of the linear fits of the data given in Extended Data Fig. 3. b, Thermoelectric conductance GT at zero magnetic field (B = 0 T) and with no electric field imposed (E = 0) as a function of the base temperature. Values of GT are obtained from the slope of the linear fits of the data shown in Extended Data Fig. 6. Error bars, fit uncertainty of the slope. c, Themopower S at zero magnetic field (B = 0 T) as a function of the base temperature. Values of S are obtained from the slope of the linear fits of the data shown in Extended Data Fig. 7. Error bars, fit uncertainty of the slope.

Extended Data Figure 9 Magneto-conductance G(E B) as a function of magnetic field B at selected base temperatures (colour scale).

Extended Data Figure 10 Magneto-thermoelectric conductance GT(T B) as a function of magnetic field B at selected base temperatures (colour scale).

Extended Data Figure 11 Magneto-thermopower S(T B) as a function of magnetic field B at selected base temperatures (colour scale).

Related audio

PowerPoint slides

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gooth, J., Niemann, A., Meng, T. et al. Experimental signatures of the mixed axial–gravitational anomaly in the Weyl semimetal NbP. Nature 547, 324–327 (2017). https://doi.org/10.1038/nature23005

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/nature23005

This article is cited by

Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.

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