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Quantum Graphs with Vertices Violating the Time Reversal Symmetry

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Abstract

We discuss quantum graphs with the vertex coupling which violates the time-reversal invariance. For a simple type of this coupling in which the violation is in a sense maximum one we show that it leads to spectral properties determined in the high-energy regime by the graph topology. We illustrate this effect on examples which involve lattice graphs and loop arrays as well as finite graphs associated with Platonic solids. We also show that transport properties of such graphs may differ in the graph bulk and at the edges.

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REFERENCES

  1. N. Nagaosa, J. Sinova, A. Onoda, A. H. MacDonald, and N. P. Ong, “Anomalous Hall effect,” Rev. Mod. Phys. 82, 1539–1593 (2018); arXiv:0904.4154 [cond-mat].

    Article  ADS  Google Scholar 

  2. P. Středa and J. Kučera, “Orbital momentum and topological phase transformation,” Phys. Rev. B 92, 235152 (2015).

    Article  ADS  Google Scholar 

  3. G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs (Am. Math. Soc., Providence, R.I., 2013).

    MATH  Google Scholar 

  4. P. Exner, “Lattice Kronig–Penney models,” Phys. Rev. Lett. 74, 3503–3506 (1995).

    Article  ADS  Google Scholar 

  5. V. Kostrykin and R. Schrader, “Kirchhoff’s rule for quantum wires,” J. Phys. A: Math. Gen. 32, 595–630 (1999); arXiv:math-ph/9806013.

    Article  ADS  MathSciNet  Google Scholar 

  6. P. Exner and M. Tater, “Quantum graphs with vertices of a preferred orientation,” Phys. Lett. A 382, 283–287 (2018); arXiv:1710.02664 [math-ph].

  7. M. Baradaran, P. Exner, and M. Tater, “Ring chains with vertex coupling of a preferred orientation,” Rev. Math. Phys. 33, 2060005 (2021); arXiv:1912.03667 [math.SP].

  8. R. Band and G. Berkolaiko, “Universality of the momentum band density of periodic networks,” Phys. Rev. Lett. 113, 130404 (2013); arXiv:1304.6028 [math-ph].

    Article  Google Scholar 

  9. G. Berkolaiko, Y. Latushkin, and S. Sukhtaiev, “Limits of quantum graph operators with shrinking edges,” Adv. Math. 352, 632–669 (2019); arXiv:1806.00561 [math.SP].

  10. R. Band, G. Berkolaiko, C. H. Joyner, and W. Liu, “Quotients of finite-dimensional operators by symmetry representations,” arXiv:1711.00918 [math-ph] (2017).

  11. P. Exner and J. Lipovský, “Spectral asymptotics of the Laplacian on Platonic solids graphs,” J. Math. Phys. 60, 122101 (2019); arXiv:1906.09091 [math.SP].

  12. P. Exner and J. Lipovský, “Topological bulk-edge effects in quantum graph transport,” Phys. Lett. A 384, 126390 (2020); arXiv:2001.10735 [math-ph].

  13. M. Z. Hasan and C. L. Kane, “Topological insulators,” Phys. Lett. A 82, 3045–3067 (2010); arXiv:1002.3895 [cond-mat.mes-hall].

    Google Scholar 

  14. P. Exner, O. Turek, and M. Tater, “Quantum graphs with vertices of a preferred orientation,” J. Phys. A: Math. Theor. 51, 285301 (2018); arXiv:1804.01414 [math-ph].

  15. P. Exner and O. Turek, “Periodic quantum graphs from the Bethe-Sommerfeld perspective,” J. Phys. A: Math. Theor. 50, 455201 (2017); arXiv:1705.07306 [math-ph].

  16. T. Kottos and U. Smilansky, “Quantum chaos on graphs,” Phys. Rev. Lett. 79, 4794–4797 (1997).

    Article  ADS  Google Scholar 

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Funding

The work reported here was in part supported by the European Union within the project CZ.02.1.01/0.0/0.0/16 019/0000778.

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Correspondence to P. Exner.

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Dedicated to the memory of my friend and colleague Slava Priezzhev

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Exner, P. Quantum Graphs with Vertices Violating the Time Reversal Symmetry. Phys. Part. Nuclei 52, 330–336 (2021). https://doi.org/10.1134/S1063779621020039

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  • DOI: https://doi.org/10.1134/S1063779621020039

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