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Characterization of toric systems via transport costs
Angular momentum coupling, Dirac oscillators, and quantum band rearrangements in the presence of momentum reversal symmetries
1. | Department of Applied Mathematics and Physics, Kyoto University, Kyoto 606, Japan |
2. | Department of Physics, Université du Littoral——Côte d'Opale, 59140 Dunkerque, France |
We investigate the elementary rearrangements of energy bands in slow-fast one-parameter families of systems whose fast subsystem possesses a half-integer spin. Beginning with a simple case without any time-reversal symmetries, we analyze and compare increasingly sophisticated model Hamiltonians with these symmetries. The models are inspired by the time-reversal modification of the Berry phase setup which uses a family of quadratic spin-quadrupole Hamiltonians of Mead [Phys. Rev. Lett. 59, 161–164 (1987)] and Avron et al [Commun. Math. Phys. 124(4), 595–627 (1989)]. An explicit correspondence between the typical quantum energy level patterns in the energy band rearrangements of the finite particle systems with compact slow phase space and those of the Dirac oscillator is found in the limit of linearization near the conical degeneracy point of the semi-quantum eigenvalues.
References:
[1] |
V. I. Arnold, Remarks on eigenvalues and eigenvectors of Hermitian matrices, Berry phase, adiabatic connections and quantum Hall effect, Select. Math., 1 (1995), 1–19.
doi: 10.1007/BF01614072. |
[2] |
M. F. Atiyah, V. K. Patodi and I. M. Singer, Spectral asymmetry and Riemannian geometry. I, Math. Proc. Cambridge Phil. Soc., 77 (1975), 49–69.
doi: 10.1017/S0305004100049410. |
[3] |
M. F. Atiyah, V. K. Patodi and I. M. Singer, Spectral asymmetry and Riemannian geometry. Ⅱ, Math. Proc. Cambridge Phil. Soc., 78 (1975), 405–432.
doi: 10.1017/S0305004100051872. |
[4] |
M. F. Atiyah, V. K. Patodi and I. M. Singer, Spectral asymmetry and Riemannian geometry. Ⅲ, Math. Proc. Cambridge Phil. Soc., 79 (1976), 71–99.
doi: 10.1017/S0305004100052105. |
[5] |
J. E. Avron, L. Sadun, J. Segert and B. Simon, Topological invariants in Fermi systems with time-reversal invariance, Phys. Rev. Lett., 61 (1988), 1329–1332.
doi: 10.1103/PhysRevLett.61.1329. |
[6] |
J. E. Avron, L. Sadun, J. Segert and B. Simon, Chern numbers, quaternions, and Berry's phases in Fermi systems, Commun. Math. Phys., 124 (1989), 595–627.
doi: 10.1007/BF01218452. |
[7] |
M. Baake, A brief guide to reversing and extended symmetries of dynamical systems, in Ergodic Theory and Dynamical Systems in Their Interactions with Arithmetics and Combinatorics (eds. S. Ferenczi, J. Kulaga-Przymus and M. Lemanczyk), Lect. Notes Math., 2213, Springer-Verlag, Berlin, Heidelberg, 2018, 35–40. |
[8] |
N. Berglund and B. Gentz, Deterministic slow-fast systems, in Noise-induced Phenomena in Slow-fast Dynamical Systems, Probability and its Applications, Springer Verlag, London, UK, 2006, 17–49. |
[9] |
B. A. Bernevig and T. L. Hughes, Topological Insulators and Topological Superconductors, Princeton Univ. Press, Princeton, NJ, 2013.
doi: 10.1515/9781400846733.![]() ![]() |
[10] |
M. V. Berry, Quantal phase factors accompanying adiabatic changes, Proc. Royal Soc. Lond. A, 392 (1984), 45–57.
doi: 10.1098/rspa.1984.0023. |
[11] |
F. Faure and B. I. Zhilinskií, Topological Chern indices in molecular spectra, Phys. Rev. Lett., 85 (2000), 960–963.
doi: 10.1103/PhysRevLett.85.960. |
[12] |
F. Faure and B. I. Zhilinskií, Topological properties of the Born–Oppenheimer approximation and implications for the exact spectrum, Lett. Math. Phys., 55 (2001), 239–247.
doi: 10.1023/A:1010912815438. |
[13] |
F. Faure and B. I. Zhilinskií, Qualitative features of intra-molecular dynamics. What can be learned from symmetry and topology, Acta Appl. Math., 70 (2002), 265–282.
doi: 10.1023/A:1013986518747. |
[14] |
F. Faure and B. I. Zhilinskií, Topologically coupled energy bands in molecules, Phys. Lett. A, 302 (2002), 985–988.
doi: 10.1016/S0375-9601(02)01171-4. |
[15] |
B. Fedosov, Deformation Quantization and Index Theory, Academie Verlag, Berlin, 1996. |
[16] |
D. Fontanari and D. A. Sadovskií, Coherent states for the quantum complete rigid rotor, J. Geom. Phys., 129 (2018), 70–89.
doi: 10.1016/j.geomphys.2018.02.021. |
[17] |
J. N. Fuchs, F. Piéchon, M. O. Goerbig and G. Montambaux, Topological Berry phase and semiclassical quantization of cyclotron orbits for two dimensional electrons in coupled band models, Eur. Phys. J. B, 77 (2010), 351–362.
doi: 10.1140/epjb/e2010-10584-y. |
[18] |
F. D. M. Haldane, Model for a quantum Hall effect without Landau levels: Condensed-matter realization of the "parity anomaly", Phys. Rev. Lett., 61 (1988), 2015–2018.
doi: 10.1103/PhysRevLett.61.1029. |
[19] |
G. Herzberg and H. C. Longuet-Higgins, Intersection of potential energy surfaces in polyatomic molecules, Discuss. Faraday Soc., 35 (1963), 77–82.
doi: 10.1039/df9633500077. |
[20] |
T. Holstein and H. Primakoff, Field dependence of the intrinsic domain magnetization of a ferromagnet, Phys. Rev., 58 (1940), 1098–1113.
doi: 10.1103/PhysRev.58.1098. |
[21] |
T. Iwai and B. I. Zhilinskií, Energy bands: Chern numbers and symmetry, Ann. Phys. (NY), 326 (2011), 3013–3066.
doi: 10.1016/j.aop.2011.07.002. |
[22] |
T. Iwai and B. I. Zhilinskií, Qualitative feature of the rearrangement of molecular energy spectra from a wall-crossing perspective, Phys. Lett. A, 377 (2013), 2481–2486.
doi: 10.1016/j.physleta.2013.07.043. |
[23] |
T. Iwai and B. I. Zhilinskií, Local description of band rearrangements. Comparison of semi-quantum and full quantum approach, Acta Appl. Math., 137 (2015), 97–121.
doi: 10.1007/s10440-014-9992-y. |
[24] |
T. Iwai and B. I. Zhilinskií, Band rearrangement through the 2D-Dirac equation: Comparing the APS and the chiral bag boundary conditions, Indagationes Math., 27 (2016), 1081–1106.
doi: 10.1016/j.indag.2015.11.010. |
[25] |
T. Iwai and B. I. Zhilinskií, Chern number modification in crossing the boundary between different band structures: Three-band model with cubic symmetry, Rev. Math. Phys., 29 (2017), 1–91.
doi: 10.1142/S0129055X17500040. |
[26] |
M. Kohmoto, Topological invariant and the quantization of the Hall conductance, Ann. Phys., 160 (1985), 343–354.
doi: 10.1016/0003-4916(85)90148-4. |
[27] |
H. A. Kramers, Théorie générale de la rotation paramagnétique dans les cristaux, Proc. Konink. Akad. Wetensch., 33 (1930), 959-972. Google Scholar |
[28] |
J. S. W. Lamb, Reversing symmetries in dynamical systems, J. Phys. A, 25 (1992), 925–937.
doi: 10.1088/0305-4470/25/4/028. |
[29] |
J. S. Lamb and J. A. Roberts, Time-reversal symmetry in dynamical systems: A survey, Physica D, 112 (1998), 1–39.
doi: 10.1016/S0167-2789(97)00199-1. |
[30] |
Y. Lee Loh and M. Kim, Visualizing spin states using the spin coherent state representation, Am. J. Phys., 83 (2015), 30–35.
doi: 10.1119/1.4898595. |
[31] |
C. A. Mead, Molecular Kramers degeneracy and non-Abelian adiabatic phase factors, Phys. Rev. Lett., 59 (1987), 161–164.
doi: 10.1103/PhysRevLett.59.161. |
[32] |
C. A. Mead and D. G. Truhlar, On the determination of Born–Oppenheimer nuclear motion wave functions including complications due to conical intersections and identical nuclei, J. Chem. Phys., 70 (1979), 2284–2296.
doi: 10.1063/1.437734. |
[33] |
M. Moshinsky and A. Szczepaniak, The Dirac oscillator, J. Phys. A: Math. Gen., 22 (1989), L817–L819.
doi: 10.1088/0305-4470/22/17/002. |
[34] |
A. I. Neishtadt and A. A. Vasiliev, Destruction of adiabatic invariance at resonances in slow-fast Hamiltonian systems, Nucl. Instr. Meth. A, 561 (2006), 158–165.
doi: 10.1007/978-1-4020-6964-2_3. |
[35] |
A. I. Neishtadt, Averaging method and adiabatic invariants, in Hamiltonian Dynamical Systems and Applications (ed. W. Craig), NATO science for peace and security series B: Physics and Biophysics, Springer Science, Netherlands, 2008, 53–66.
doi: 10.1007/978-1-4020-6964-2_3. |
[36] |
V. B. Pavlov-Verevkin, D. A. Sadovskií and B. I. Zhilinskií, On the dynamical meaning of the diabolic points, Europhysics Letters, 6 (1988), 573–8.
doi: 10.1209/0295-5075/6/7/001. |
[37] |
D. A. Sadovskií and B. I. Zhilinskií, Monodromy, diabolic points, and angular momentum coupling, Phys. Lett. A, 256 (1999), 235–44.
doi: 10.1016/S0375-9601(99)00229-7. |
[38] |
B. Simon, Holonomy, the quantum adiabatic theorem, and Berry's phase, Phys. Rev. Lett., 51 (1983), 2167–2170.
doi: 10.1103/PhysRevLett.51.2167. |
[39] |
D. J. Thouless, Topological Quantum Numbers in Nonrelativistic Physics, World Scientific, Singapore, 1998.
doi: 10.1142/3318. |
[40] |
D. J. Thouless, M. Kohmoto, M. P. Nightingale and M. den Nijs, Quantized Hall conductance in a two-dimensional periodic potential, Phys. Rev. Lett., 49 (1982), 405–408.
doi: 10.1103/PhysRevLett.49.405. |
[41] |
J. von Neumann and E. P. Wigner, Über merkwürdige diskrete Eigenwerte. Über das Verhalten von Eigenwerten bei adiabatischen Prozessen, Physicalische Z., 30 (1929), 467-470. Google Scholar |
[42] |
E. P. Wigner,
Über die Operation der Zeitumkehr in der Quantenmechanik, Nachr. Akad. Ges. Wiss. Göttingen, 31 (1932), 546-559.
doi: 10.1007/978-3-662-02781-3_15. |
[43] |
F. Wilczek and A. Shapere, Geometric Phases in Physics, Advanced Series in Mathematical Physics, 5, World Scientific, 1989.
doi: 10.1142/0613. |
[44] |
R. N. Zare, Angular Momentum: Understanding Spatial Aspects in Chemistry and Physics, Wiley, New York, 1988.
doi: 10.1063/1.2811251. |
[45] |
W.-M. Zhang, D. H. Feng and R. Gilmore, Coherent states: Theory and some applications, Rev. Mod. Phys., 62 (1990), 867–927.
doi: 10.1103/RevModPhys.62.867. |
show all references
References:
[1] |
V. I. Arnold, Remarks on eigenvalues and eigenvectors of Hermitian matrices, Berry phase, adiabatic connections and quantum Hall effect, Select. Math., 1 (1995), 1–19.
doi: 10.1007/BF01614072. |
[2] |
M. F. Atiyah, V. K. Patodi and I. M. Singer, Spectral asymmetry and Riemannian geometry. I, Math. Proc. Cambridge Phil. Soc., 77 (1975), 49–69.
doi: 10.1017/S0305004100049410. |
[3] |
M. F. Atiyah, V. K. Patodi and I. M. Singer, Spectral asymmetry and Riemannian geometry. Ⅱ, Math. Proc. Cambridge Phil. Soc., 78 (1975), 405–432.
doi: 10.1017/S0305004100051872. |
[4] |
M. F. Atiyah, V. K. Patodi and I. M. Singer, Spectral asymmetry and Riemannian geometry. Ⅲ, Math. Proc. Cambridge Phil. Soc., 79 (1976), 71–99.
doi: 10.1017/S0305004100052105. |
[5] |
J. E. Avron, L. Sadun, J. Segert and B. Simon, Topological invariants in Fermi systems with time-reversal invariance, Phys. Rev. Lett., 61 (1988), 1329–1332.
doi: 10.1103/PhysRevLett.61.1329. |
[6] |
J. E. Avron, L. Sadun, J. Segert and B. Simon, Chern numbers, quaternions, and Berry's phases in Fermi systems, Commun. Math. Phys., 124 (1989), 595–627.
doi: 10.1007/BF01218452. |
[7] |
M. Baake, A brief guide to reversing and extended symmetries of dynamical systems, in Ergodic Theory and Dynamical Systems in Their Interactions with Arithmetics and Combinatorics (eds. S. Ferenczi, J. Kulaga-Przymus and M. Lemanczyk), Lect. Notes Math., 2213, Springer-Verlag, Berlin, Heidelberg, 2018, 35–40. |
[8] |
N. Berglund and B. Gentz, Deterministic slow-fast systems, in Noise-induced Phenomena in Slow-fast Dynamical Systems, Probability and its Applications, Springer Verlag, London, UK, 2006, 17–49. |
[9] |
B. A. Bernevig and T. L. Hughes, Topological Insulators and Topological Superconductors, Princeton Univ. Press, Princeton, NJ, 2013.
doi: 10.1515/9781400846733.![]() ![]() |
[10] |
M. V. Berry, Quantal phase factors accompanying adiabatic changes, Proc. Royal Soc. Lond. A, 392 (1984), 45–57.
doi: 10.1098/rspa.1984.0023. |
[11] |
F. Faure and B. I. Zhilinskií, Topological Chern indices in molecular spectra, Phys. Rev. Lett., 85 (2000), 960–963.
doi: 10.1103/PhysRevLett.85.960. |
[12] |
F. Faure and B. I. Zhilinskií, Topological properties of the Born–Oppenheimer approximation and implications for the exact spectrum, Lett. Math. Phys., 55 (2001), 239–247.
doi: 10.1023/A:1010912815438. |
[13] |
F. Faure and B. I. Zhilinskií, Qualitative features of intra-molecular dynamics. What can be learned from symmetry and topology, Acta Appl. Math., 70 (2002), 265–282.
doi: 10.1023/A:1013986518747. |
[14] |
F. Faure and B. I. Zhilinskií, Topologically coupled energy bands in molecules, Phys. Lett. A, 302 (2002), 985–988.
doi: 10.1016/S0375-9601(02)01171-4. |
[15] |
B. Fedosov, Deformation Quantization and Index Theory, Academie Verlag, Berlin, 1996. |
[16] |
D. Fontanari and D. A. Sadovskií, Coherent states for the quantum complete rigid rotor, J. Geom. Phys., 129 (2018), 70–89.
doi: 10.1016/j.geomphys.2018.02.021. |
[17] |
J. N. Fuchs, F. Piéchon, M. O. Goerbig and G. Montambaux, Topological Berry phase and semiclassical quantization of cyclotron orbits for two dimensional electrons in coupled band models, Eur. Phys. J. B, 77 (2010), 351–362.
doi: 10.1140/epjb/e2010-10584-y. |
[18] |
F. D. M. Haldane, Model for a quantum Hall effect without Landau levels: Condensed-matter realization of the "parity anomaly", Phys. Rev. Lett., 61 (1988), 2015–2018.
doi: 10.1103/PhysRevLett.61.1029. |
[19] |
G. Herzberg and H. C. Longuet-Higgins, Intersection of potential energy surfaces in polyatomic molecules, Discuss. Faraday Soc., 35 (1963), 77–82.
doi: 10.1039/df9633500077. |
[20] |
T. Holstein and H. Primakoff, Field dependence of the intrinsic domain magnetization of a ferromagnet, Phys. Rev., 58 (1940), 1098–1113.
doi: 10.1103/PhysRev.58.1098. |
[21] |
T. Iwai and B. I. Zhilinskií, Energy bands: Chern numbers and symmetry, Ann. Phys. (NY), 326 (2011), 3013–3066.
doi: 10.1016/j.aop.2011.07.002. |
[22] |
T. Iwai and B. I. Zhilinskií, Qualitative feature of the rearrangement of molecular energy spectra from a wall-crossing perspective, Phys. Lett. A, 377 (2013), 2481–2486.
doi: 10.1016/j.physleta.2013.07.043. |
[23] |
T. Iwai and B. I. Zhilinskií, Local description of band rearrangements. Comparison of semi-quantum and full quantum approach, Acta Appl. Math., 137 (2015), 97–121.
doi: 10.1007/s10440-014-9992-y. |
[24] |
T. Iwai and B. I. Zhilinskií, Band rearrangement through the 2D-Dirac equation: Comparing the APS and the chiral bag boundary conditions, Indagationes Math., 27 (2016), 1081–1106.
doi: 10.1016/j.indag.2015.11.010. |
[25] |
T. Iwai and B. I. Zhilinskií, Chern number modification in crossing the boundary between different band structures: Three-band model with cubic symmetry, Rev. Math. Phys., 29 (2017), 1–91.
doi: 10.1142/S0129055X17500040. |
[26] |
M. Kohmoto, Topological invariant and the quantization of the Hall conductance, Ann. Phys., 160 (1985), 343–354.
doi: 10.1016/0003-4916(85)90148-4. |
[27] |
H. A. Kramers, Théorie générale de la rotation paramagnétique dans les cristaux, Proc. Konink. Akad. Wetensch., 33 (1930), 959-972. Google Scholar |
[28] |
J. S. W. Lamb, Reversing symmetries in dynamical systems, J. Phys. A, 25 (1992), 925–937.
doi: 10.1088/0305-4470/25/4/028. |
[29] |
J. S. Lamb and J. A. Roberts, Time-reversal symmetry in dynamical systems: A survey, Physica D, 112 (1998), 1–39.
doi: 10.1016/S0167-2789(97)00199-1. |
[30] |
Y. Lee Loh and M. Kim, Visualizing spin states using the spin coherent state representation, Am. J. Phys., 83 (2015), 30–35.
doi: 10.1119/1.4898595. |
[31] |
C. A. Mead, Molecular Kramers degeneracy and non-Abelian adiabatic phase factors, Phys. Rev. Lett., 59 (1987), 161–164.
doi: 10.1103/PhysRevLett.59.161. |
[32] |
C. A. Mead and D. G. Truhlar, On the determination of Born–Oppenheimer nuclear motion wave functions including complications due to conical intersections and identical nuclei, J. Chem. Phys., 70 (1979), 2284–2296.
doi: 10.1063/1.437734. |
[33] |
M. Moshinsky and A. Szczepaniak, The Dirac oscillator, J. Phys. A: Math. Gen., 22 (1989), L817–L819.
doi: 10.1088/0305-4470/22/17/002. |
[34] |
A. I. Neishtadt and A. A. Vasiliev, Destruction of adiabatic invariance at resonances in slow-fast Hamiltonian systems, Nucl. Instr. Meth. A, 561 (2006), 158–165.
doi: 10.1007/978-1-4020-6964-2_3. |
[35] |
A. I. Neishtadt, Averaging method and adiabatic invariants, in Hamiltonian Dynamical Systems and Applications (ed. W. Craig), NATO science for peace and security series B: Physics and Biophysics, Springer Science, Netherlands, 2008, 53–66.
doi: 10.1007/978-1-4020-6964-2_3. |
[36] |
V. B. Pavlov-Verevkin, D. A. Sadovskií and B. I. Zhilinskií, On the dynamical meaning of the diabolic points, Europhysics Letters, 6 (1988), 573–8.
doi: 10.1209/0295-5075/6/7/001. |
[37] |
D. A. Sadovskií and B. I. Zhilinskií, Monodromy, diabolic points, and angular momentum coupling, Phys. Lett. A, 256 (1999), 235–44.
doi: 10.1016/S0375-9601(99)00229-7. |
[38] |
B. Simon, Holonomy, the quantum adiabatic theorem, and Berry's phase, Phys. Rev. Lett., 51 (1983), 2167–2170.
doi: 10.1103/PhysRevLett.51.2167. |
[39] |
D. J. Thouless, Topological Quantum Numbers in Nonrelativistic Physics, World Scientific, Singapore, 1998.
doi: 10.1142/3318. |
[40] |
D. J. Thouless, M. Kohmoto, M. P. Nightingale and M. den Nijs, Quantized Hall conductance in a two-dimensional periodic potential, Phys. Rev. Lett., 49 (1982), 405–408.
doi: 10.1103/PhysRevLett.49.405. |
[41] |
J. von Neumann and E. P. Wigner, Über merkwürdige diskrete Eigenwerte. Über das Verhalten von Eigenwerten bei adiabatischen Prozessen, Physicalische Z., 30 (1929), 467-470. Google Scholar |
[42] |
E. P. Wigner,
Über die Operation der Zeitumkehr in der Quantenmechanik, Nachr. Akad. Ges. Wiss. Göttingen, 31 (1932), 546-559.
doi: 10.1007/978-3-662-02781-3_15. |
[43] |
F. Wilczek and A. Shapere, Geometric Phases in Physics, Advanced Series in Mathematical Physics, 5, World Scientific, 1989.
doi: 10.1142/0613. |
[44] |
R. N. Zare, Angular Momentum: Understanding Spatial Aspects in Chemistry and Physics, Wiley, New York, 1988.
doi: 10.1063/1.2811251. |
[45] |
W.-M. Zhang, D. H. Feng and R. Gilmore, Coherent states: Theory and some applications, Rev. Mod. Phys., 62 (1990), 867–927.
doi: 10.1103/RevModPhys.62.867. |












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