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Quantum criticality of topological phase transitions in three-dimensional interacting electronic systems

Nature Physics volume 10, pages 774–778 (2014)Cite this article

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Abstract

Topological phase transitions in condensed matter systems accompany emerging singularities of the electronic wavefunction, often manifested by gap-closing points in momentum space. In conventional topological insulators in three dimensions, the low-energy theory near the gap-closing point can be described by relativistic Dirac fermions coupled to the long-range Coulomb interaction; hence, the quantum critical point of topological phase transitions provides a promising platform to test the intriguing predictions of quantum electrodynamics. Here we discover a class of quantum critical phenomena in topological materials for which either the inversion symmetry or time-reversal symmetry can be broken. At the quantum critical point, the emerging low-energy fermions, dubbed the anisotropic Weyl fermions, show both relativistic and Newtonian dynamics simultaneously. The interplay between the anisotropic dispersion and the Coulomb interaction brings about a screening phenomenon distinct from the conventional Thomas–Fermi screening in metals and logarithmic screening in Dirac fermions.

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Figure 1: Generic phase diagram for topological phase transitions in 3D systems lacking either time-reversal symmetry or inversion symmetry.
Figure 2: Feynman diagram for quantum corrections and the resulting renormalization group flow.
Figure 3: Distribution of the screening charge induced by a charged impurity in the non-interacting anisotropic Weyl fermion.

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References

  1. Sachdev, S. Quantum Phase Transitions 2nd edn (Cambridge Univ. Press, 1999).

    MATH  Google Scholar 

  2. Balents, L. Spin liquids in frustrated magnets. Nature 464, 199–208 (2010).

    Article  ADS  Google Scholar 

  3. Xu, C. Unconventional quantum critical points. Int. J. Mod. Phys. B 26, 1230007 (2012).

    Article  ADS  Google Scholar 

  4. Wen, X. G. Topological order: From long-range entangled quantum matter to a unified origin of light and electrons. ISRN Condens. Matter Phys. 2013, 198710 (2013).

    Article  Google Scholar 

  5. Wan, X., Turner, A. M., Vishwanath, A. & Savrasov, S. Y. Topological semimetal and Fermi-arc surface states in the electronic structure of pyrochlore iridates. Phys. Rev. B 83, 205101 (2011).

    Article  ADS  Google Scholar 

  6. Ueda, K. et al. Variation of charge dynamics in the course of metal–insulator transition for pyrochlore type Nd2Ir2O7 . Phys. Rev. Lett. 109, 136402 (2012).

    Article  ADS  Google Scholar 

  7. Balents, L. Viewpoint: Weyl electrons kiss. Physics 4, 36 (2011).

    Article  Google Scholar 

  8. Burkov, A. A. & Balents, L. Weyl semimetal in a topological insulator multilayer. Phys. Rev. Lett. 107, 127205 (2011).

    Article  ADS  Google Scholar 

  9. Hosur, P., Parameswaran, S. A. & Vishwanath, A. Charge transport in Weyl semimetals. Phys. Rev. Lett. 108, 046602 (2012).

    Article  ADS  Google Scholar 

  10. Isobe, H. & Nagaosa, N. Theory of a quantum critical phenomenon in a topological insulator: (3 + 1)-dimensional quantum electrodynamics in solids. Phys. Rev. B 86, 165127 (2012).

    Article  ADS  Google Scholar 

  11. Isobe, H. & Nagaosa, N. Renormalization group study of electromagnetic interaction in multi-Dirac-node systems. Phys. Rev. B 87, 205138 (2013).

    Article  ADS  Google Scholar 

  12. Goswami, P. & Chakravarty, S. Quantum criticality between topological and band insulators in 3 + 1 dimensions. Phys. Rev. Lett. 107, 196803 (2011).

    Article  ADS  Google Scholar 

  13. Luttinger, J. M. Quantum theory of cyclotron resonance in semiconductors: General theory. Phys. Rev. 102, 1030–1041 (1956).

    Article  ADS  Google Scholar 

  14. Murakami, S., Nagaosa, N. & Zhang, S-C. SU(2) non-Abelian holonomy and dissipationless spin current in semiconductors. Phys. Rev. B 69, 235206 (2004).

    Article  ADS  Google Scholar 

  15. Moon, E-G., Xu, C., Kim, Y. B. & Balents, L. Non-Fermi-liquid and topological states with strong spin–orbit coupling. Phys. Rev. Lett. 111, 206401 (2013).

    Article  ADS  Google Scholar 

  16. Yang, B-J. et al. Theory of topological quantum phase transitions in 3D noncentrosymmetric systems. Phys. Rev. Lett. 110, 086402 (2013).

    Article  ADS  Google Scholar 

  17. Murakami, S. & Kuga, S-i. Universal phase diagrams for the quantum spin Hall systems. Phys. Rev. B 78, 165313 (2008).

    Article  ADS  Google Scholar 

  18. Murakami, S. Phase transition between the quantum spin Hall and insulator phases in 3D: Emergence of a topological gapless phase. New J. Phys. 9, 356 (2007).

    Article  ADS  Google Scholar 

  19. Bahramy, M. S., Yang, B-J., Arita, R. & Nagaosa, N. Emergence of non-centrosymmetric topological insulating phase in BiTeI under pressure. Nature Commun. 3, 679 (2012).

    Article  ADS  Google Scholar 

  20. Xi, X. et al. Signatures of a pressure-induced topological quantum phase transition in BiTeI. Phys. Rev. Lett. 111, 155701 (2013).

    Article  ADS  Google Scholar 

  21. Son, D. T. Quantum critical point in graphene approached in the limit of infinitely strong Coulomb interaction. Phys. Rev. B 75, 235423 (2007).

    Article  ADS  Google Scholar 

  22. Abrikosov, A. A. Gapless state of bismuth-type semimetals. J. Low. Temp. Phys. 8, 315–338 (1972).

    Article  ADS  Google Scholar 

  23. Kotov, V. N., Pereira, V. M. & Uchoa, B. Polarization charge distribution in gapped graphene: Perturbation theory and exact diagonalization analysis. Phys. Rev. B 78, 075433 (2008).

    Article  ADS  Google Scholar 

  24. Kotov, V. N., Uchoa, B., Pereira, V. M., Guinea, F. & Castro Neto, A. H. Electron–electron interactions in graphene: Current status and perspectives. Rev. Mod. Phys. 84, 1067–1125 (2012).

    Article  ADS  Google Scholar 

  25. Biswas, R. R., Sachdev, S. & Son, D. T. Coulomb impurity in graphene. Phys. Rev. B 76, 205122 (2007).

    Article  ADS  Google Scholar 

  26. Shankar, R. Renormalization-group approach to interacting fermions. Rev. Mod. Phys. 66, 129–192 (1994).

    Article  ADS  Google Scholar 

Download references

Acknowledgements

We are grateful for support from the Japan Society for the Promotion of Science (JSPS) through the Funding Program for World-Leading Innovative R&D on Science and Technology (FIRST Program), and Grant-in-Aids for Scientific Research (Kiban (S), No. 24224009) from the Ministry of Education, Culture, Sports, Science and Technology (MEXT). B-J.Y. and N.N. greatly appreciate the stimulating discussions with A. Aharony, O. Entin-Wohlman, M. Hermele and M.A. Cazalilla. E-G.M. is grateful for invaluable discussion with L. Balents, M. Metlitski and C. Xu and supported by the MRSEC Program of the National Science Foundation under Award No. DMR 1121053. H.I. is supported by a Grant-in-Aid for JSPS Fellows.

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Authors and Affiliations

  1. RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan

    Bohm-Jung Yang & Naoto Nagaosa

  2. Department of Physics, University of California, Santa Barbara, California 93106, USA

    Eun-Gook Moon

  3. Department of Applied Physics, University of Tokyo, Tokyo 113-8656, Japan

    Hiroki Isobe & Naoto Nagaosa

Authors
  1. Bohm-Jung Yang
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  2. Eun-Gook Moon
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  3. Hiroki Isobe
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  4. Naoto Nagaosa
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Contributions

N.N. conceived the original ideas. B-J.Y., E-G.M. and H.I. performed the calculations. All authors analysed the data and wrote the manuscript.

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Correspondence to Bohm-Jung Yang or Naoto Nagaosa.

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The authors declare no competing financial interests.

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Yang, BJ., Moon, EG., Isobe, H. et al. Quantum criticality of topological phase transitions in three-dimensional interacting electronic systems. Nature Phys 10, 774–778 (2014). https://doi.org/10.1038/nphys3060

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