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

Nature volume 547, pages 324327 (20 July 2017) | Download Citation


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.

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  1. 1.

    Anomalies in Quantum Field Theory (Oxford Univ. Press, 2000)

  2. 2.

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

  3. 3.

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

  4. 4.

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

  5. 5.

    , & Gravitational anomaly and transport phenomena. Phys. Rev. Lett. 107, 021601 (2011)

  6. 6.

    , & Hydrodynamic theory of thermoelectric transport and negative magnetoresistance in Weyl semimetals. Proc. Natl Acad. Sci. USA 113, 9463–9468 (2016)

  7. 7.

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

  8. 8.

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

  9. 9.

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

  10. 10.

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

  11. 11.

    & Chiral anomaly and classical negative magnetoresistance of Weyl metals. Phys. Rev. B 88, 104412 (2013)

  12. 12.

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

  13. 13.

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

  14. 14.

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

  15. 15.

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

  16. 16.

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

  17. 17.

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

  18. 18.

    et al. Observation of chiral magneto-transport in RPtBi topological Heusler compounds. Preprint at (2016)

  19. 19.

    & Gravitational anomalies. Nucl. Phys. B 234, 269–330 (1984)

  20. 20.

    & Quantum gravity and world topology. Phys. Rev. Lett. 37, 1251–1254 (1976)

  21. 21.

    , & in Strongly Interacting Matter in Magnetic Fields (eds et al.) 433–468 (Springer, 2013)

  22. 22.

    , , & Anomalous hydrodynamics kicks neutron stars. Phys. Lett. B 760, 170–174 (2016)

  23. 23.

    , & Thermoelectric properties of Weyl and Dirac semimetals. Phys. Rev. B 90, 165115 (2014)

  24. 24.

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

  25. 25.

    , & Nernst and magnetothermal conductivity in a lattice model of Weyl fermions. Phys. Rev. B 93, 035116 (2016)

  26. 26.

    & Magnetotransport phenomena related to the chiral anomaly in Weyl semimetals. Phys. Rev. B 93, 085107 (2016)

  27. 27.

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

  28. 28.

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

  29. 29.

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

  30. 30.

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

  31. 31.

    , , & Holographic gravitational anomaly and chiral vortical effect. J. High Energy Phys. 9, 121 (2011)

  32. 32.

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

  33. 33.

    & The gravitational correction to PCAC. Phys. Lett. B 40, 381–382 (1972)

  34. 34.

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

  35. 35.

    , & Thermodynamics, gravitational anomalies and cones. J. High Energy Phys. 2, 88 (2013)

  36. 36.

    Thermal Field Theory (Cambridge Univ. Press, 2000)

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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


  1. Institute of Nanostructure and Solid State Physics, Universität Hamburg, Jungiusstraße 11, 20355 Hamburg, Germany

    • Johannes Gooth
    • , Anna C. Niemann
    •  & Kornelius Nielsch
  2. IBM Research -Zurich, Säumerstrasse 4, 8803 Rüschlikon, Switzerland

    • Johannes Gooth
    • , Bernd Gotsmann
    •  & Fabian Menges
  3. Leibniz Institute for Solid State and Materials Research Dresden, Institute for Metallic Materials, Helmholtzstraße 20, 01069 Dresden, Germany

    • Anna C. Niemann
    • , Ruben Hühne
    • , Bernd Rellinghaus
    •  & Kornelius Nielsch
  4. Institute for Theoretical Physics, Technical University Dresden, Zellescher Weg 17, 01062 Dresden, Germany

    • Tobias Meng
  5. Department of Physics, University of California, Berkeley, California 94720, USA

    • Adolfo G. Grushin
  6. Instituto de Física Teórica UAM/CSIC, Nicolás Cabrera 13–15, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain

    • Karl Landsteiner
  7. Max Planck Institute for Chemical Physics of Solids, Nöthnitzer Straße 40, 01187 Dresden, Germany

    • Marcus Schmidt
    • , Chandra Shekhar
    • , Vicky Süß
    • , Claudia Felser
    •  & Binghai Yan
  8. Department of Condensed Matter Physics, Weizmann Institute of Science, 7610001 Rehovot, Israel

    • Binghai Yan


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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.

Competing interests

The authors declare no competing financial interests.

Corresponding author

Correspondence to Johannes Gooth.

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