Abstract
We consider the Lifshitz topological transitions and the corresponding changes in the galvano-magnetic properties of a metal from the point of view of the general classification of open electron trajectories arising on Fermi surfaces of arbitrary complexity in the presence of magnetic field. The construction of such a classification is the content of the Novikov problem and is based on the division of non-closed electron trajectories into topologically regular and chaotic trajectories. The description of stable topologically regular trajectories gives a basis for a complete classification of non-closed trajectories on arbitrary Fermi surfaces and is connected with special topological structures on these surfaces. Using this description, we describe here the distinctive features of possible changes in the picture of electron trajectories during the Lifshitz transitions, as well as changes in the conductivity behavior in the presence of a strong magnetic field. As it turns out, the use of such an approach makes it possible to describe not only the changes associated with stable electron trajectories, but also the most general changes of the conductivity diagram in strong magnetic fields.
REFERENCES
I. M. Lifshitz, Sov. Phys. JETP 11, 1130 (1960).
I. M. Lifshitz, M. Ya. Azbel, and M. I. Kaganov, Electron Theory of Metals (Nauka, Moscow, 1971; Consultants Bureau, Adam Hilger, New York, 1973).
G. E. Volovik, J. Low Temp. Phys. 43, 47 (2017).
G. E. Volovik, Phys. Usp. 61, 89 (2018).
I. M. Lifshitz, M. Ya. Azbel, and M. I. Kaganov, Sov. Phys. JETP 4, 41 (1957).
I. M. Lifshitz and V. G. Peschansky, Sov. Phys. JETP 8, 875 (1959).
I. M. Lifshitz and V. G. Peschansky, Sov. Phys. JETP 11, 137 (1960).
S. P. Novikov, Russ. Math. Surv. 37, 1 (1982).
A. V. Zorich, Russ. Math. Surv. 39, 287 (1984).
I. A. Dynnikov, Russ. Math. Surv. 47, 172 (1992).
S. P. Tsarev, private commun. (1992–1993).
I. A. Dynnikov, Math. Notes 53, 494 (1993).
A. V. Zorich, in Proceedings of the Conference on Geometric Study of Foliations, Tokyo, November 1993, Ed. by T. Mizutani et al. (World Scientific, Singapore, 1994), p. 479.
I. A. Dynnikov, in Proceedings of the 2nd European Congress of Mathematics ECM2, BuDA, 1996.
I. A. Dynnikov, in Solitons, Geometry, and Topology: On the Crossroad, Ed. by V. M. Buchstaber and S. P. Novikov, Vol. 179 of Am. Math. Soc. Transl., Ser. 2 (AMS, Providence, RI, 1997), p. 45.
I. A. Dynnikov, Russ. Math. Surv. 54, 21 (1999).
S. P. Novikov and A. Ya. Maltsev, JETP Lett. 63, 855 (1996).
A. Ya. Maltsev, J. Exp. Theor. Phys. 85, 934 (1997).
S. P. Novikov and A. Ya. Maltsev, Phys. Usp. 41, 231 (1998).
A. Ya. Maltsev and S. P. Novikov, J. Stat. Phys. 115, 31 (2004).
A. Ya. Maltsev and S. P. Novikov, Proc. Steklov Inst. Math. 302, 279 (2018).
S. P. Novikov, R. De Leo, I. A. Dynnikov, and A. Ya. Maltsev, J. Exp. Theor. Phys. 129, 710 (2019).
A. V. Zorich, Ann. Inst. Fourier 46, 325 (1996).
A. Zorich, in Solitons, Geometry, and Topology: On the Crossroad, Ed. by V. M. Buchstaber and S. P. Novikov, Vol. 179 of Am. Math. Soc. Transl., Ser. 2 (AMS, Providence, RI, 1997), p. 173. https://doi.org/10.1090/trans2/179
A. Zorich, in Pseudoperiodic Topology, Ed. by V. I. Arnold, M. Kontsevich, and A. Zorich, Vol. 197 of Am. Math. Soc. Transl., Ser. 2 (AMS, Providence, RI, 1999), p. 135. https://doi.org/10.1090/trans2/197
R. De Leo, Russ. Math. Surv. 55, 166 (2000).
R. De Leo, Russ. Math. Surv. 58, 1042 (2003).
A. Ya. Maltsev and S. P. Novikov, arXiv: cond-mat/0304471. https://doi.org/10.1023/B:JOSS.0000019835.01125.92
A. Ya. Maltsev and S. P. Novikov, Solid State Phys., Bull. Braz. Math. Soc., New Ser. 34, 171 (2003).
A. Zorich, in Frontiers in Number Theory, Physics and Geometry, Vol. 1: On Random Matrices, Zeta Functions and Dynamical Systems, 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.
R. De Leo and I. A. Dynnikov, Russ. Math. Surv. 62, 990 (2007).
R. De Leo and I. A. Dynnikov, Geom. Dedic. 138, 51 (2009).
I. A. Dynnikov, Proc. Steklov Inst. Math. 263, 65 (2008).
A. Skripchenko, Discrete Contin. Dyn. Sys. 32, 643 (2012).
A. Skripchenko, Ann. Glob. Anal. Geom. 43, 253 (2013).
I. Dynnikov and A. Skripchenko, in Topology, Geometry, Integrable Systems, and Mathematical Physics: Novikov’s Seminar 2012–2014, Vol. 234 of Advances in the Mathematical Sciences, Am. Math. Soc. Transl. Ser. 2, Ed. by V. M. Buchstaber, B. A. Dubrovin, and I. M. Krichever (Am. Math. Soc., Providence, RI, 2014), p. 173; arXiv: 1309.4884.
I. Dynnikov and A. Skripchenko, Trans. Moscow Math. Soc. 76, 287 (2015).
A. Avila, P. Hubert, and A. Skripchenko, Invent. Math. 206, 109 (2016).
A. Avila, P. Hubert, and A. Skripchenko, Bull. Soc. Math. France 144, 539 (2016).
R. D. Leo, in Advanced Mathematical Methods in Biosciences and Applications, Ed. by F. Berezovskaya and B. Toni, Part of STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health (Springer, Cham, 2019), p. 53.
A. Ya. Maltsev and S. P. Novikov, Russ. Math. Surv. 74, 141 (2019).
I. A. Dynnikov, A. Ya. Maltsev, and S. P. Novikov, Russ. Math. Surv. 77 (6), 109 (2022); arXiv: 2306.11257.
A. Ya. Maltsev, J. Exp. Theor. Phys. 124, 805 (2017).
A. Ya. Maltsev, J. Exp. Theor. Phys. 125, 896 (2017).
A. Ya. Maltsev, J. Exp. Theor. Phys. 129, 116 (2019).
C. Kittel, Quantum Theory of Solids (Wiley, New York, 1963).
A. A. Abrikosov, Fundamentals of the Theory of Metals (Elsevier Sci. Technol., Oxford, UK, 1988).
Funding
The research was carried out at the expense of the grant of the Russian Science Foundation no. 21-11-00331, “Geometric methods in the Hamiltonian theory of integrable and almost integrable systems.”
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Maltsev, A.Y. Lifshitz Transitions and Angular Conductivity Diagrams in Metals with Complex Fermi Surfaces. J. Exp. Theor. Phys. 137, 706–724 (2023). https://doi.org/10.1134/S1063776123110079
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DOI: https://doi.org/10.1134/S1063776123110079