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  • Letter
  • Open Access

Non-Hermitian chiral anomalies

Sharareh Sayyad1,2,*, Julia D. Hannukainen3, and Adolfo G. Grushin1

  • 1Université Grenoble Alpes, CNRS, Grenoble INP, Institut Néel, 38000 Grenoble, France
  • 2Max Planck Institute for the Science of Light, Staudtstraße 2, 91058 Erlangen, Germany
  • 3Department of Physics, KTH Royal Institute of Technology, 106 91 Stockholm, Sweden

  • *sharareh.sayyad@mpl.mpg.de
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Phys. Rev. Research 4, L042004 – Published 7 October, 2022

DOI: https://doi.org/10.1103/PhysRevResearch.4.L042004

Abstract

The chiral anomaly underlies a broad number of phenomena, from enhanced electronic transport in topological metals to anomalous currents in the quark-gluon plasma. The discovery of topological states of matter in non-Hermitian systems raises the question of whether there are anomalous conservation laws that remain unaccounted for. To answer this question, we consider both two and four space-time dimensions, presenting a unified formulation to calculate anomalous responses in Hermitianized, anti-Hermitianized, and non-Hermitian systems of massless electrons with complex Fermi velocities coupled to non-Hermitian gauge fields. Our results indicate that the quantum conservation laws of chiral currents of non-Hermitian systems are not related to those in Hermitianized and anti-Hermitianized systems, as would be expected classically, due to different anomalous terms that we derive. We further present some physical consequences of our non-Hermitian anomaly that may have implications for a broad class of emerging experimental systems that realize non-Hermitian Hamiltonians.

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References (59)

  1. S. L. Adler, Axial-vector vertex in spinor electrodynamics, Phys. Rev. 177, 2426 (1969).
  2. J. S. Bell and R. Jackiw, A PCAC puzzle: π0→γγ in the σ-model, Nuovo Cimento A 60, 47 (1969).
  3. D. E. Kharzeev, L. D. McLerran, and H. J. Warringa, The effects of topological charge change in heavy ion collisions: “Event by event P and CP violation”, Nucl. Phys. A 803, 227 (2008).
  4. K. Fukushima, D. E. Kharzeev, and H. J. Warringa, Chiral magnetic effect, Phys. Rev. D 78, 074033 (2008).
  5. M. Kaminski, C. F. Uhlemann, M. Bleicher, and J. Schaffner-Bielich, Anomalous hydrodynamics kicks neutron stars, Phys. Lett. B 760, 170 (2016).
  6. S. Basilakos, N. E. Mavromatos, and J. Solà Peracaula, Gravitational and chiral anomalies in the running vacuum universe and matter-antimatter asymmetry, Phys. Rev. D 101, 045001 (2020).
  7. H. B. Nielsen and M. Ninomiya, The Adler-Bell-Jackiw anomaly and Weyl fermions in a crystal, Phys. Lett. B 130, 389 (1983).
  8. J. Fröhlich, Chiral anomaly, topological field theory, and novel states of matter, Rev. Math. Phys. 30, 1840007 (2018).
  9. A. A. Burkov, Chiral anomaly and transport in Weyl metals, J. Phys.: Condens. Matter 27, 113201 (2015).
  10. K. Landsteiner, Notes on anomaly induced transport, Acta Phys. Pol. B 47, 2617 (2016).
  11. S. L. Adler, Anomalies to all orders, in 50 Years of Yang-Mills Theory, edited by G. ‘t Hooft (World Scientific, Singapore, 2005), pp. 187–228.
  12. R. A. Bertlmann, Anomalies in Quantum Field Theory (Oxford University Press, Oxford, UK, 1996).
  13. J. Xiong, S. K. Kushwaha, T. Liang, J. W. Krizan, M. Hirschberger, W. Wang, R. J. Cava, and N. P. Ong, Evidence for the chiral anomaly in the Dirac semimetal Na3Bi, Science 350, 413 (2015).
  14. X. Huang, L. Zhao, Y. Long, P. Wang, D. Chen, Z. Yang, H. Liang, M. Xue, H. Weng, Z. Fang, X. Dai, and G. Chen, Observation of the Chiral-Anomaly-Induced Negative Magnetoresistance in 3D Weyl Semimetal TaAs, Phys. Rev. X 5, 031023 (2015).
  15. R. D. dos Reis, M. O. Ajeesh, N. Kumar, F. Arnold, C. Shekhar, M. Naumann, M. Schmidt, M. Nicklas, and E. Hassinger, On the search for the chiral anomaly in Weyl semimetals: the negative longitudinal magnetoresistance, New J. Phys. 18, 085006 (2016).
  16. G. E. Marsh, The chiral anomaly, Dirac and Weyl semimetals, and force-free magnetic fields, Can. J. Phys. 95, 711 (2017).
  17. S. Liang, J. Lin, S. Kushwaha, J. Xing, N. Ni, R. J. Cava, and N. P. Ong, Experimental Tests of the Chiral Anomaly Magnetoresistance in the Dirac-Weyl Semimetals Na3Bi and GdPtBi, Phys. Rev. X 8, 031002 (2018).
  18. X. Yuan, C. Zhang, Y. Zhang, Z. Yan, T. Lyu, M. Zhang, Z. Li, C. Song, M. Zhao, P. Leng, M. Ozerov, X. Chen, N. Wang, Y. Shi, H. Yan, and F. Xiu, The discovery of dynamic chiral anomaly in a Weyl semimetal NbAs, Nat. Commun. 11, 1259 (2020).
  19. R. Ilan, A. G. Grushin, and D. I. Pikulin, Pseudo-electromagnetic fields in 3D topological semimetals, Nat. Rev. Phys. 2, 29 (2020).
  20. N. P. Armitage, E. J. Mele, and A. Vishwanath, Weyl and Dirac semimetals in three-dimensional solids, Rev. Mod. Phys. 90, 015001 (2018).
  21. D. Bernard and A. LeClair, A classification of non-Hermitian random matrices, in Statistical Field Theories, edited by A. Cappelli and G. Mussardo (Springer, Dordrecht, 2002), pp. 207–214.
  22. W. Chen, Ş. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, Exceptional points enhance sensing in an optical microcavity, Nature (London) 548, 192 (2017).
  23. S. Lieu, Topological symmetry classes for non-Hermitian models and connections to the bosonic Bogoliubov–de Gennes equation, Phys. Rev. B 98, 115135 (2018).
  24. J. Carlström and E. J. Bergholtz, Exceptional links and twisted Fermi ribbons in non-Hermitian systems, Phys. Rev. A 98, 042114 (2018).
  25. E. J. Bergholtz and J. C. Budich, Non-Hermitian Weyl physics in topological insulator ferromagnet junctions, Phys. Rev. Res. 1, 012003(R) (2019).
  26. U. Magnea, Random matrices beyond the Cartan classification, J. Phys. A: Math. Theor. 41, 045203 (2008).
  27. G. Mussardo, Statistical Field Theory: An Introduction to Exactly Solved Models in Statistical Physics (Oxford University Press, New York, 2010).
  28. Y. Xu, S.-T. Wang, and L.-M. Duan, Weyl Exceptional Rings in a Three-Dimensional Dissipative Cold Atomic Gas, Phys. Rev. Lett. 118, 045701 (2017).
  29. H. Zhou and J. Y. Lee, Periodic table for topological bands with non-Hermitian symmetries, Phys. Rev. B 99, 235112 (2019).
  30. K. Kawabata, K. Shiozaki, M. Ueda, and M. Sato, Symmetry and Topology in Non-Hermitian Physics, Phys. Rev. X 9, 041015 (2019).
  31. W. Xi, W.-Z. Yi, Y.-S. Wu, and W.-Q. Chen, A 1D lattice realization of chiral fermions with a non-Hermitian Hamiltonian, arXiv:2010.09375.
  32. J. Y. Lee, J. Ahn, H. Zhou, and A. Vishwanath, Topological Correspondence between Hermitian and Non-Hermitian Systems: Anomalous Dynamics, Phys. Rev. Lett. 123, 206404 (2019).
  33. K. Kawabata, K. Shiozaki, and S. Ryu, Topological Field Theory of Non-Hermitian Systems, Phys. Rev. Lett. 126, 216405 (2021).
  34. V. M. Martinez Alvarez, J. E. Barrios Vargas, and L. E. F. Foa Torres, Non-Hermitian robust edge states in one dimension: Anomalous localization and eigenspace condensation at exceptional points, Phys. Rev. B 97, 121401(R) (2018).
  35. Y. Xiong, Why does bulk boundary correspondence fail in some non-Hermitian topological models, J. Phys. Commun. 2, 035043 (2018).
  36. F. K. Kunst, E. Edvardsson, J. C. Budich, and E. J. Bergholtz, Biorthogonal Bulk-Boundary Correspondence in Non-Hermitian Systems, Phys. Rev. Lett. 121, 026808 (2018).
  37. V. M. Martinez Alvarez, J. E. Barrios Vargas, M. Berdakin, and L. E. F. Foa Torres, Topological states of non-Hermitian systems, Eur. Phys. J. Spec. Top. 227, 1295 (2018).
  38. S. Yao and Z. Wang, Edge States and Topological Invariants of Non-Hermitian Systems, Phys. Rev. Lett. 121, 086803 (2018).
  39. C. H. Lee and R. Thomale, Anatomy of skin modes and topology in non-Hermitian systems, Phys. Rev. B 99, 201103(R) (2019).
  40. M. N. Chernodub and A. Cortijo, Non-Hermitian chiral magnetic effect in equilibrium, Symmetry 12, 761 (2020).
  41. K. Fujikawa, Path-Integral Measure for Gauge-Invariant Fermion Theories, Phys. Rev. Lett. 42, 1195 (1979).
  42. K. Fujikawa, Path integral for gauge theories with fermions, Phys. Rev. D 21, 2848 (1980).
  43. K. Fujikawa, Energy-momentum tensor in quantum field theory, Phys. Rev. D 23, 2262 (1981).
  44. K. Fujikawa and H. Suzuki, Path Integrals and Quantum Anomalies (Clarendon Press, Oxford, UK, 2004).
  45. See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevResearch.4.L042004 for details on Fujikawa's approach in 1+1 and 3+1 dimensions and further details on the bosonization and one-loop calculation of the 1+1 theory.
  46. We emphasize that the gauge field W is not subject to a gauge principle, but this does not render it less physical. For example, if the Weyl separation is allowed to be a complex quantity, as in a non-Hermitian system, its variations over time and space will lead to the complex electric and magnetic fields.
  47. M. Nakahara, Geometry, Topology and Physics, Graduate Student Series in Physics (Hilger, Bristol, UK, 1990).
  48. To be precise, our employed heat-kernel regularization suppresses states with large eigenvalues in the Euclidean space.
  49. C. K. Kim, Comment on “Bosonization in the chiral Schwinger model”, Phys. Rev. D 38, 3338 (1988).
  50. Aside from these presented topological terms, proportional to ɛμν, in Ã5 and Ã, further nontopological terms, without ɛμν, can also be generated within Fujikawa's method (see SM [45]). It has been pointed out that systems will be anomalous if quantum fluctuations generate topological corrections, irrespective of zero or nonzero nontopological terms [44, 59].
  51. C. Rylands, A. Parhizkar, A. A. Burkov, and V. Galitski, Chiral Anomaly in Interacting Condensed Matter Systems, Phys. Rev. Lett. 126, 185303 (2021).
  52. A. A. Zyuzin, S. Wu, and A. A. Burkov, Weyl semimetal with broken time reversal and inversion symmetries, Phys. Rev. B 85, 165110 (2012).
  53. X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Topological field theory of time-reversal invariant insulators, Phys. Rev. B 78, 195424 (2008).
  54. X.-L. Qi, R. Li, J. Zang, and S.-C. Zhang, Inducing a magnetic monopole with topological surface states, Science 323, 1184 (2009).
  55. A. G. Grushin and A. Cortijo, Tunable Casimir Repulsion with Three-Dimensional Topological Insulators, Phys. Rev. Lett. 106, 020403 (2011).
  56. G. Rosenberg and M. Franz, Witten effect in a crystalline topological insulator, Phys. Rev. B 82, 035105 (2010).
  57. M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory (CRC Press, Boca Raton, FL, 1995).
  58. M. R. Hirsbrunner, T. M. Philip, and M. J. Gilbert, Topology and observables of the non-Hermitian Chern insulator, Phys. Rev. B 100, 081104(R) (2019).
  59. A. Bilal, Lectures on anomalies, arXiv:0802.0634.

Outline

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Phys. Rev. Research 4, L042004– Published 7 October, 2022
  • Received 25 June 2021
  • Revised 14 June 2022
  • Accepted 23 September 2022
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Open access publication funded by the Max Planck Society.

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