Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 054, 34 pages      arXiv:1703.04472      https://doi.org/10.3842/SIGMA.2017.054

Topological Phase Transition in a Molecular Hamiltonian with Symmetry and Pseudo-Symmetry, Studied through Quantum, Semi-Quantum and Classical Models

Guillaume Dhont a, Toshihiro Iwai b and Boris Zhilinskií a
a) Université du Littoral Côte d'Opale, Laboratoire de Physico-Chimie de l'Atmosphère, 189A Avenue Maurice Schumann, 59140 Dunkerque, France
b) Kyoto University, 606-8501 Kyoto, Japan

Received March 14, 2017, in final form July 04, 2017; Published online July 13, 2017

Abstract
The redistribution of energy levels between energy bands is studied for a family of simple effective Hamiltonians depending on one control parameter and possessing axial symmetry and energy-reflection symmetry. Further study is made on the topological phase transition in the corresponding semi-quantum and completely classical models, and finally the joint spectrum of the two commuting observables $(H=E,J_z)$ (also called the lattice of quantum states) is superposed on the image of the energy-momentum map for the classical model. Through these comparative analyses, mutual correspondence is demonstrated to exist among the redistribution of energy levels between energy bands for the quantum Hamiltonian, the modification of Chern numbers of eigenline bundles for the corresponding semi-quantum Hamiltonian, and the presence of Hamiltonian monodromy for the complete classical analog. In particular, as far as the band rearrangement is concerned, a fine agreement is found between the redistribution of the energy levels described in terms of joint spectrum of energy and momentum in the full quantum model and the evolution of singularities of the energy-momentum map of the complete classical model. The topological phase transition observed in the present semi-quantum and the complete classical models are analogous to topological phase transitions of matter.

Key words: energy bands; redistribution of energy levels; energy-reflection symmetry; Chern number; band inversion.

pdf (5013 kb)   tex (1508 kb)

References

  1. Abrikosov A.A., Beneslavskii S.D., Possible existence of substances intermediate between metals and dielectrics, JETP 32 (1971), 699-708.
  2. Altland A., Zirnbauer M.R., Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures, Phys. Rev. B 55 (1997), 1142-1161.
  3. Arnold V.I., Remarks on eigenvalues and eigenvectors of Hermitian matrices, Berry phase, adiabatic connections and quantum Hall effect, Selecta Math. (N.S.) 1 (1995), 1-19.
  4. Arnold V.I., Symplectization, complexification and mathematical trinities, in The Arnoldfest (Toronto, ON, 1997), Fields Inst. Commun., Vol. 24, Amer. Math. Soc., Providence, RI, 1999, 23-37.
  5. Arnold V.I., Polymathematics: is mathematics a single science or a set of arts?, in Mathematics: Frontiers and Perspectives, Amer. Math. Soc., Providence, RI, 2000, 403-416.
  6. Atiyah M.F., Patodi V.K., Singer I.M., Spectral asymmetry and Riemannian geometry. III, Math. Proc. Cambridge Philos. Soc. 79 (1976), 71-99.
  7. Avron J.E., Sadun L., Segert J., Simon B., Chern numbers, quaternions, and Berry's phases in Fermi systems, Comm. Math. Phys. 124 (1989), 595-627.
  8. Avron J.E., Seiler R., Simon B., Homotopy and quantization in condensed matter physics, Phys. Rev. Lett. 51 (1983), 51-53.
  9. Bernevig B.A., Topological insulators and topological superconductors, Princeton University Press, Princeton, NJ, 2013.
  10. Child M.S., Quantum states in a champagne bottle, J. Phys. A: Math. Gen. 31 (1998), 657-670.
  11. Cooper F., Khare A., Sukhatme U., Supersymmetry and quantum mechanics, Phys. Rep. 251 (1995), 267-385, hep-th/9405029.
  12. Correa F., Dunne G.V., Plyushchay M.S., The Bogoliubov-de Gennes system, the AKNS hierarchy, and nonlinear quantum mechanical supersymmetry, Ann. Physics 324 (2009), 2522-2547, arXiv:0904.2768.
  13. Cushman R.H., Bates L.M., Global aspects of classical integrable systems, Birkhäuser Verlag, Basel, 1997.
  14. Cushman R., Zhilinskií B., Monodromy of a two degrees of freedom Liouville integrable system with many focus-focus singular points, J. Phys. A: Math. Gen. 35 (2002), L415-L419.
  15. De Nittis G., Gomi K., Classification of ''quaternionic" Bloch-bundles: topological quantum systems of type  AII, Comm. Math. Phys. 339 (2015), 1-55, arXiv:1404.5804.
  16. Duistermaat J.J., On global action-angle coordinates, Comm. Pure Appl. Math. 33 (1980), 687-706.
  17. Duistermaat J.J., Heckman G.J., On the variation in the cohomology of the symplectic form of the reduced phase space, Invent. Math. 69 (1982), 259-268.
  18. Dunne G.V., Shifman M., Duality and self-duality (energy reflection symmetry) of quasi-exactly solvable periodic potentials, Ann. Physics 299 (2002), 143-173, hep-th/0204224.
  19. Dyson F.J., The threefold way. Algebraic structure of symmetry groups and ensembles in quantum mechanics, J. Math. Phys. 3 (1962), 1199-1215.
  20. Efstathiou K., Metamorphoses of Hamiltonian systems with symmetries, Lecture Notes in Math., Vol. 1864, Springer-Verlag, Berlin, 2005.
  21. Efstathiou K., Sadovskií D.A., Normalization and global analysis of perturbations of the hydrogen atom, Rev. Modern Phys. 82 (2010), 2099-2154.
  22. Egea J., Ferrer S., van der Meer J.C., Bifurcations of the Hamiltonian fourfold $1:1$ resonance with toroidal symmetry, J. Nonlinear Sci. 21 (2011), 835-874.
  23. Faure F., Zhilinskií B., Topological Chern indices in molecular spectra, Phys. Rev. Lett. 85 (2000), 960-963, quant-ph/9912091.
  24. Gat O., Robbins J.M., Topology of time-reversal invariant energy bands with adiabatic structure, arXiv:1511.08994.
  25. Grinevich P.G., Mironov A.E., Novikov S.P., On the non-relativistic two-dimensional purely magnetic supersymmetric Pauli operator, Russian Math. Surveys 70 (2015), 299-329, arXiv:1101.5678.
  26. Grinevich P.G., Volovik G.E., Topology of gap nodes in superfluid $^3\mathrm{He}$: $\pi_4$ homotopy group for $^3\mathrm{He}$-B disclination, J. Low Temp. Phys. 72 (1988), 371-380.
  27. Grondin L., Sadovskií D.A., Zhilinskií B.I., Monodromy as topological obstruction to global action-angle variables in systems with coupled angular momenta and rearrangement of bands in quantum spectra, Phys. Rev. A 65 (2001), 012105, 15 pages.
  28. Guillemin V., Moment maps and combinatorial invariants of Hamiltonian $T^n$-spaces, Progress in Mathematics, Vol. 122, Birkhäuser Boston, Inc., Boston, MA, 1994.
  29. Haldane F.D.M., Model for a quantum Hall effect without Landau levels: condensed-matter realization of the ''parity anomaly'', Phys. Rev. Lett. 61 (1988), 2015-2018.
  30. Hansen M.S., Faure F., Zhilinskií B.I., Fractional monodromy in systems with coupled angular momenta, J. Phys. A: Math. Theor. 40 (2007), 13075-13089, quant-ph/0702227.
  31. Hasan M.Z., Kane C.L., Colloquium: Topological insulators, Rev. Modern Phys. 82 (2010), 3045-3067, arXiv:1002.3895.
  32. Herman M., Campargue A., El Idrissi M.I., Vander Auwera J., Vibrational spectroscopic database on acetylene, $\tilde{\rm X}^1\Sigma_g^+$ ($\rm {}^{12}C_2H_2$, $\rm {}^{12}C_2D_2$, and $\rm {}^{13}C_2H_2$), J. Phys. Chem. Ref. Data 32 (2003), 921-1361.
  33. Herring C., Accidental degeneracy in the energy bands of crystals, Phys. Rev. 52 (1937), 365-373.
  34. Herzberg G., Longuet-Higgins H.C., Intersection of potential energy surfaces in polyatomic molecules, Discuss. Faraday Soc. 35 (1963), 77-82.
  35. Hořava P., Stability of Fermi surfaces and $K$ theory, Phys. Rev. Lett. 95 (2005), 016405, 4 pages, hep-th/0503006.
  36. Iwai T., Zhilinskií B., Energy bands: Chern numbers and symmetry, Ann. Physics 326 (2011), 3013-3066.
  37. Iwai T., Zhilinskií B., Qualitative features of the rearrangement of molecular energy spectra from a ''wall-crossing'' perspective, Phys. Lett. A 377 (2013), 2481-2486, arXiv:1307.7277.
  38. Iwai T., Zhilinskií B., Topological phase transitions in the vibration-rotation dynamics of an isolated molecule, Theor. Chem. Acc. 133 (2014), 1501, 13 pages.
  39. Iwai T., Zhilinskií B., Chern number modification in crossing the boundary between different band structures: three-band models with cubic symmetry, Rev. Math. Phys. 29 (2017), 1750004, 91 pages.
  40. Kitaev A., Periodic table for topological insulators and superconductors, AIP Conf. Proc. 1134 (2009), 22-30, arXiv:0901.2686.
  41. Lifshitz I.M., Anomalies of electron characteristics of a metal in the high pressure, JETP 11 (1960), 1130-1135.
  42. Matveev V.S., Integrable Hamiltonian systems with two degrees of freedom. Topological structure of saturated neighborhoods of points of focus-focus and saddle-saddle types, Sb. Math. 187 (1996), 495-524.
  43. Michel L., Zhilinskií B.I., Symmetry, invariants, topology. Basic tools, Phys. Rep. 341 (2001), 11-84.
  44. Nakahara M., Geometry, topology and physics, Graduate Student Series in Physics, Adam Hilger, Ltd., Bristol, 1990.
  45. Nekhoroshev N.N., Sadovskií D.A., Zhilinskií B.I., Fractional Hamiltonian monodromy, Ann. Henri Poincaré 7 (2006), 1099-1211.
  46. Nielsen H.B., Ninomiya M., Absence of neutrinos on a lattice. I. Proof by homotopy theory, Nuclear Phys. B 185 (1981), 20-40.
  47. Nielsen H.B., Ninomiya M., Absence of neutrinos on a lattice. II. Intuitive topological proof, Nuclear Phys. B 193 (1981), 173-194.
  48. Nielsen H.B., Ninomiya M., The Adler-Bell-Jackiw anomaly and Weyl fermions in a crystal, Phys. Lett. B 130 (1983), 389-396.
  49. Novikov S.P., Magnetic Bloch functions and vector bundles. Typical dispersion laws and their quantum numbers, Soviet Math. Dokl. 23 (1981), 298-303.
  50. Pavlov-Verevkin V.B., Sadovskií D., Zhilinskií B.I., On the dynamical meaning of the diabolic points, Europhys. Lett. 6 (1988), 573-578.
  51. Pérez-Bernal F., Iachello F., Algebraic approach to two-dimensional systems: shape phase transitions, monodromy, and thermodynamic quantities, Phys. Rev. A 77 (2008), 032115, 21 pages.
  52. Ryu S., Schnyder A.P., Furusaki A., Ludwig A.W.W., Topological insulators and superconductors: tenfold way and dimensional hierarchy, New J. Phys. 12 (2010), 065010, 60 pages, arXiv:0912.2157.
  53. Sadovskií D.A., Zhilinskií B.I., Monodromy, diabolic points, and angular momentum coupling, Phys. Lett. A 256 (1999), 235-244.
  54. Schnyder A.P., Ryu S., Furusaki A., Ludwig A.W.W., Classification of topological insulators and superconductors in three spatial dimensions, Phys. Rev. B 78 (2008), 195125, 22 pages, arXiv:0803.2786.
  55. Schwinger J., On angular momentum, in Quantum Theory of Angular Momentum, Editors L.C. Biedenharn, H. van Dam, Academic Press, New York, 1965, 229-279.
  56. Shapere A., Wilczek F. (Editors), Geometric phases in physics, Advanced Series in Mathematical Physics, Vol. 5, World Scientific Publishing Co., Inc., Teaneck, NJ, 1989.
  57. Simon B., Holonomy, the quantum adiabatic theorem, and Berry's phase, Phys. Rev. Lett. 51 (1983), 2167-2170.
  58. Stránský P., Macek M., Cejnar P., Excited-state quantum phase transitions in systems with two degrees of freedom: level density, level dynamics, thermal properties, Ann. Physics 345 (2014), 73-97.
  59. Thouless D., Kohmoto M., Nightingale M.P., den Nijs M., Quantized Hall conductance in a two-dimensional periodic potential, Phys. Rev. Lett. 49 (1982), 405-408.
  60. Tyng V., Kellman M.E., Critical points bifurcation analysis of high-$l$ bending dynamics in acetylene, J. Chem. Phys. 131 (2009), 244111, 11 pages.
  61. Volovik G.E., The universe in a helium droplet, International Series of Monographs on Physics, Vol. 117, The Clarendon Press, Oxford University Press, New York, 2003.
  62. von Neumann J., Wigner E.P., Über das Verhalten von Eigenwerten bei adiabatischen Prozessen, Phys. Z. 30 (1929), 467-470.
  63. Vũ Ngoc S., Moment polytopes for symplectic manifolds with monodromy, Adv. Math. 208 (2007), 909-934, math.SG/0504165.
  64. Wigner E.P., Group theory and its application to the quantum mechanics of atomic spectra, Pure and Applied Physics, Vol. 5, Academic Press, New York - London, 1959.
  65. Winkler R., Zülicke U., Time reversal of pseudo-spin $1/2$ degrees of freedom, Phys. Lett. A 374 (2010), 4003-4006, arXiv:0909.2169.
  66. Winnewisser B.P., Winnewisser M., Medvedev I.R., Behnke M., De Lucia F.C., Ross S.C., Koput J., Experimental confirmation of quantum monodromy: the millimeter wave spectrum of cyanogen isothiocyanate NCNCS, Phys. Rev. Lett. 95 (2005), 243002, 4 pages.
  67. Yarkony D.R., Diabolical conical intersections, Rev. Modern Phys. 68 (1996), 985-1013.
  68. Zhang C.L., Xu S.Y., Belopolski I., Yuan Z., Lin Z., Tong B., Bian G., Alidoust N., Lee C.C., Huang S.M., Chang T.R., Chang G., Hsu C.H., Jeng H.T., Neupane M., Sanchez D.S., Zheng H., Wang J., Lin H., Zhang C., Lu H.Z., Shen S.Q., Neupert T., Hasan M.Z., Jia S., Signatures of the Adler-Bell-Jackiw chiral anomaly in a Weyl fermion semimetal, Nat. Commun. 7 (2016), 10735, 9 pages, arXiv:1601.04208.
  69. Zhilinskií B., Symmetry, invariants, and topology in molecular models, Phys. Rep. 341 (2001), 85-171.
  70. Zhilinskií B., Hamiltonian monodromy as lattice defect, in Topology in Condensed Matter, Springer Series in Solid-State Sciences, Vol. 150, Springer, Berlin - Heidelberg, 2006, 165-186, quant-ph/0303181.

Previous article  Next article   Contents of Volume 13 (2017)