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Chaotic Trajectories on Fermi Surfaces and Nontrivial Modes of Behavior of Magnetic Conductivity

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Abstract

We present here a survey of questions related to recently discovered types of non-closed electron trajectories on complex Fermi surfaces, corresponding to chaotic dynamics in the quasi-momentum space. Trajectories of this type have been found and are currently quite well studied theoretically. However, their experimental detection is still to be done. Here we discuss the geometric properties of such trajectories, the likelihood of their occurrence on real Fermi surfaces, and the behavior of the magnetoconductivity in the limit ωBτ → ∞ when they occur. The review includes the latest results of research on chaotic trajectories for dispersion relations of the most general form.

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Notes

  1. Note that in the formulation of the general Novikov’s problem, only the periodicity of the dispersion relation \(\epsilon \)(p) is required.

  2. More precisely, it is true for the contribution to the Hall conductivity from the connected component of the Fermi surface under consideration.

REFERENCES

  1. I. M. Lifshitz, M. Ya. Azbel, and M. I. Kaganov, Sov. Phys. JETP 4, 41 (1957).

    Google Scholar 

  2. I. M. Lifshitz and V. G. Peschansky, Sov. Phys. JETP 8, 875 (1959).

    Google Scholar 

  3. I. M. Lifshitz and V. G. Peschansky, Sov. Phys. JETP 11, 131 (1960).

    Google Scholar 

  4. I. M. Lifshitz, M. Ya. Azbel, and M. I. Kaganov, Electron Theory of Metals (Moscow, Nauka, 1971; Consultants Bureau, New York, 1973).

  5. M. I. Kaganov and V. G. Peschansky, Phys. Rep. 372, 445 (2002). https://doi.org/10.1016/S0370-1573(02)00275-2

    Article  ADS  Google Scholar 

  6. C. Kittel, Quantum Theory of Solids (Wiley, Chichester, 1963).

    MATH  Google Scholar 

  7. J. M. Ziman, Principles of the Theory of Solids (Cambridge Univ. Press, Cambridge, 1972).

    Book  Google Scholar 

  8. A. A. Abrikosov, Fundamentals of the Theory of Metals (Elsevier Sci. Technol., Oxford, UK, 1988).

    Google Scholar 

  9. Yu. A. Dreizin and A. M. Dykhne, J. Exp. Theor. Phys. 36, 127 (1973).

    ADS  Google Scholar 

  10. Yu. A. Dreizin and A. M. Dykhne, J. Exp. Theor. Phys. 57, 1024 (1983).

    ADS  Google Scholar 

  11. S. P. Novikov, Russ. Math. Surv. 37 (5), 1 (1982). https://doi.org/10.1070/RM1982v037n05ABEH004020

    Article  Google Scholar 

  12. A. V. Zorich, Russ. Math. Surv. 39, 287 (1984). https://doi.org/10.1070/RM1984v039n05ABEH004091

    Article  Google Scholar 

  13. S. P. Tsarev, Private Commun. (1992–1993).

  14. I. A. Dynnikov, Russ. Math. Surv. 47, 172 (1992). https://doi.org/10.1070/RM1992v047n03ABEH000901

    Article  Google Scholar 

  15. I. A. Dynnikov, Math. Notes 53, 495 (1993). https://doi.org/10.1007/BF01208544

    Article  MathSciNet  Google Scholar 

  16. I. A. Dynnikov, in Proceedings of the ECM2, BuDA, 1996.

  17. I. A. Dynnikov, Am. Math. Soc. Transl., Ser. 2 179, 45 (1997). https://doi.org/10.1090/trans2/179

    Article  Google Scholar 

  18. I. A. Dynnikov, Russ. Math. Surv. 54, 21 (1999). https://doi.org/10.1070/RM1999v054n01ABEH000116

    Article  Google Scholar 

  19. S. P. Novikov and A. Y. Maltsev, JETP Lett. 63, 855 (1996). https://doi.org/10.1134/1.567102

    Article  ADS  Google Scholar 

  20. S. P. Novikov and A. Y. Maltsev, Phys. Usp. 41, 231 (1998). https://doi.org/10.1070/PU1998v041n03ABEH000373

    Article  ADS  Google Scholar 

  21. A. Ya. Maltsev, J. Exp. Theor. Phys. 124, 805 (2017). https://doi.org/10.1134/S1063776117040148

    Article  ADS  Google Scholar 

  22. A. Ya. Maltsev, J. Exp. Theor. Phys. 125, 896 (2017). https://doi.org/10.1134/S106377611711005X

    Article  ADS  Google Scholar 

  23. A. V. Zorich, in Proceedings of the Conference on Geometric Study of Foliations, Tokyo, November 1994, Ed. by T. Mizutani et al. (World Scientific, Singapore, 1994), p. 479. https://doi.org/10.1142/9789814533836

  24. A. V. Zorich, Ann. Inst. Fourier 46, 325 (1996). https://doi.org/10.5802/aif.1517

    Article  Google Scholar 

  25. A. Zorich, in Solitons, Geometry, and Topology: On the Crossroad, Ed. by V. M. Buchstaber and S. P. Novikov, Transl. AMS, Ser. 2 179, 173 (1997). https://doi.org/10.1090/trans2/179

  26. A. Ya. Maltsev, J. Exp. Theor. Phys. 85, 934 (1997). https://doi.org/10.1134/1.558398

    Article  ADS  Google Scholar 

  27. A. Zorich, in Pseudoperiodic Topology, Ed. by V. I. Arnold, M. Kontsevich, and A. Zorich, Transl. AMS, Ser. 2 197, 135 (1999). https://doi.org/10.1090/trans2/197

  28. R. de Leo, Russ. Math. Surv. 55, 166 (2000). https://doi.org/10.1070/RM2000v055n01ABEH000252

    Article  Google Scholar 

  29. R. de Leo, Russ. Math. Surv. 58, 1042 (2003). https://doi.org/10.1070/RM2003v058n05ABEH000669

    Article  Google Scholar 

  30. A. Ya. Maltsev and S. P. Novikov, arXiv: cond-mat/0304471. https://doi.org/10.1023/B:JOSS.0000019835.01125.92

  31. A. Ya. Maltsev and S. P. Novikov, Bull. Brazil. Mat. Soc., New Ser. 34, 171 (2003). https://doi.org/10.1007/s00574-003-0007-2

    Article  Google Scholar 

  32. A. Ya. Maltsev and S. P. Novikov, J. Stat. Phys. 115, 31 (2004). https://doi.org/10.1023/B:JOSS.0000019835.01125.92

    Article  ADS  Google Scholar 

  33. A. Zorich, in Frontiers in Number Theory, Physics and Geometry, Vol. 1: On Random Matrices, Zeta Functions and Dynamical Systems, Proceedings of the Ecole de physique des Houches, France, March 9–21, 2003, Ed. by P. Cartier, B. Julia, P. Moussa, and P. Vanhove (Springer, Berlin, 2006), p. 439.

  34. R. de Leo and I. A. Dynnikov, Russ. Math. Surv. 62, 990 (2007). https://doi.org/10.1070/RM2007v062n05ABEH004461

    Article  Google Scholar 

  35. R. de Leo and I. A. Dynnikov, Geom. Dedic. 138, 51 (2009). https://doi.org/10.1007/s10711-008-9298-1

    Article  Google Scholar 

  36. I. A. Dynnikov, Proc. Steklov Inst. Math. 263, 65 (2008). https://doi.org/10.1134/S0081543808040068

    Article  MathSciNet  Google Scholar 

  37. A. Skripchenko, Discrete Contin. Dyn. Sys. 32, 643 (2012). https://doi.org/10.3934/dcds.2012.32.643

    Article  MathSciNet  Google Scholar 

  38. A. Skripchenko, Ann. Glob. Anal. Geom. 43, 253 (2013). https://doi.org/10.1007/s10455-012-9344-y

    Article  MathSciNet  Google Scholar 

  39. I. Dynnikov and A. Skripchenko, in Topology, Geometry, Integrable Systems, and Mathematical Physics: Proceedings of the Novikov’s Seminar 2012–2014, Advances in the Mathematical Sciences, Am. Math. Soc. Transl. Ser. 2 234, 173 (2014); arXiv: 1309.4884. https://doi.org/10.1090/trans2/234/09

  40. I. Dynnikov and A. Skripchenko, Trans. Mosc. Math. Soc. 76, 287 (2015). https://doi.org/10.1090/mosc/246

    Article  Google Scholar 

  41. A. Avila, P. Hubert, and A. Skripchenko, Invent. Math. 206, 109 (2016). https://doi.org/10.1007/s00222-016-0650-z

    Article  ADS  MathSciNet  Google Scholar 

  42. A. Avila, P. Hubert, and A. Skripchenko, Bull. Soc. Math. France 144, 539 (2016). https://doi.org/10.24033/bsmf.2722

    Article  MathSciNet  Google Scholar 

  43. A. Ya. Maltsev and S. P. Novikov, Proc. Steklov Inst. Math. 302, 279 (2018). https://doi.org/10.1134/S0081543818060147

    Article  Google Scholar 

  44. R. D. Leo, in Advanced Mathematical Methods in Biosciences and Applications, Ed. by F. Berezovskaya and B. Toni, STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics and Health Series (Springer, Cham, 2019), p. 53. https://doi.org/10.1007/978-3-030-15715-9_3

  45. A. Ya. Maltsev and S. P. Novikov, Russ. Math. Surv. 74, 141 (2019). https://doi.org/10.1070/RM9859

    Article  Google Scholar 

  46. A. Ya. Maltsev, J. Exp. Theor. Phys. 127, 1087 (2018). https://doi.org/10.1134/S1063776118100187

    Article  ADS  Google Scholar 

  47. A. Ya. Maltsev, J. Exp. Theor. Phys. 129, 116 (2019). https://doi.org/10.1134/S1063776119050042

    Article  ADS  Google Scholar 

  48. G. E. Zil’berman, Sov. Phys. JETP 5, 208 (1957).

    Google Scholar 

  49. G. E. Zil’berman, Sov. Phys. JETP 6, 299 (1958).

    ADS  Google Scholar 

  50. G. E. Zil’berman, Sov. Phys. JETP 7, 513 (1958).

    Google Scholar 

  51. M. Ia. Azbel, Sov. Phys. JETP 12, 891 (1961).

    MathSciNet  Google Scholar 

  52. A. A. Slutskin, Sov. Phys. JETP 26, 474 (1968).

    ADS  Google Scholar 

  53. A. Alexandradinata and L. Glazman, Phys. Rev. B 97, 144422 (2018). https://doi.org/10.1103/PhysRevB.97.144422

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ACKNOWLEDGMENTS

The article is dedicated to the 90th anniversary of M.Ya. Azbel.

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Authors and Affiliations

  1. Steklov Mathematical Institute, Russian Academy of Sciences, 119991, Moscow, Russia

    I. A. Dynnikov & S. P. Novikov

  2. Landau Institute for Theoretical Physics, Russian Academy of Sciences, 142432, Chernogolovka, Moscow oblast, Russia

    A. Ya. Maltsev & S. P. Novikov

Authors
  1. I. A. Dynnikov
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  2. A. Ya. Maltsev
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  3. S. P. Novikov
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Correspondence to A. Ya. Maltsev.

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Dynnikov, I.A., Maltsev, A.Y. & Novikov, S.P. Chaotic Trajectories on Fermi Surfaces and Nontrivial Modes of Behavior of Magnetic Conductivity. J. Exp. Theor. Phys. 135, 240–254 (2022). https://doi.org/10.1134/S1063776122080106

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