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Nonrelativistic Nambu-Goldstone Modes Nonrelativistic Nambu-Goldstone Modes Associated with Spontaneously Broken Associated with Spontaneously Broken Space-Time and Internal Symmetry Space-Time and Internal Symmetry Michikazu Kobayashi : Kyoto

  1. Nonrelativistic Nambu-Goldstone Modes Nonrelativistic Nambu-Goldstone Modes Associated with Spontaneously Broken Associated with Spontaneously Broken Space-Time and Internal Symmetry Space-Time and Internal Symmetry Michikazu Kobayashi : Kyoto University Michikazu Kobayashi : Kyoto University ● Muneto Nitta : Keio University Muneto Nitta : Keio University ● Nonrelativistic Nambu-Goldstone(NG) mode Nonrelativistic Nambu-Goldstone(NG) mode CP 1 CP model 1 model NG modes associated with domain wall NG modes associated with domain wall NG modes associated with baby skyrmion string NG modes associated with baby skyrmion string Feb. 3, 2015, “Topological soliton” Feb. 3, 2015, “Topological soliton”

  2. Nonrelativistic Nambu-Goldstone Nonrelativistic Nambu-Goldstone mode mode

  3. Nambu-Goldstone modes Nambu-Goldstone modes and broken symmetries and broken symmetries Watanabe-Brauner relation H. Watanabe and T. Brauner, Phys. Rev. D 84 , 125013 (2011) H. Watanabe and H. Murayama, Phys. Rev. Lett. 108 , 251602 (2012) N BG ： Number of broken symmetry generators N NG ： Number of Nambu-Goldstone modes g i ：Broken symmetry generators N BG >N NG ( Coupling of two NG modes to one NG mode) can occurs when two generators do not commute Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

  4. Nambu-Goldstone modes Nambu-Goldstone modes and broken symmetries and broken symmetries Nonlinear O (3) sigma model & Heisenberg Ferromagnet Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

  5. Nambu-Goldstone modes Nambu-Goldstone modes and broken symmetries and broken symmetries Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

  6. CP 1 model CP 1 model

  7. CP 1 model CP 1 model U (1) gauge theory Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

  8. CP 1 model CP 1 model Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

  9. CP 1 model CP 1 model Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

  10. CP 1 model CP 1 model CP 1 : Projection from south pole on S 2 to complex plane Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

  11. CP 1 model CP 1 model ultrarelativistic limit : ¹ ! 0 massive O (3) sigma model nonrelativistic limit : c ! 1 Heisenberg ferromagnet with one easy axis (Ising ferromagnet) Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

  12. CP 1 Internal symmetry of CP model Internal symmetry of 1 model Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

  13. CP 1 Spacetime symmetry of CP model Spacetime symmetry of 1 model Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

  14. CP 1 Spacetime symmetry of CP model Spacetime symmetry of 1 model Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

  15. CP 1 Spacetime symmetry of CP model Spacetime symmetry of 1 model Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

  16. Ground-state symmetry Ground-state symmetry For internal symmetry Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

  17. Ground-state symmetry Ground-state symmetry For spacetime symmetry Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

  18. Nambu-Goldstone mode for m Nambu-Goldstone mode for m = 0 = 0 Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

  19. Nambu-Goldstone mode for m Nambu-Goldstone mode for m = 0 = 0 Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

  20. Nambu-Goldstone mode for m Nambu-Goldstone mode for m = 0 = 0 Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

  21. NG modes associated with NG modes associated with domain wall domain wall

  22. Domain wall solution Domain wall solution Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

  23. Low energy dynamics of Low energy dynamics of domain wall domain wall Linear response theory Bogoliubov-de Gennes equation Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

  24. Low energy dynamics of Low energy dynamics of domain wall domain wall Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

  25. Nambu-Goldstone and Nambu-Goldstone and gapful (Higgs) modes gapful (Higgs) modes Inverse coupling of moduli Coupling of translational to those of NG mode and phase moduli Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

  26. Ultrarelativistic limit ( ¹ ) Ultrarelativistic limit ( 0 ) ! 0 ¹ ! phase moduli translational moduli Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

  27. Nonrelativistic limit ( c ) Nonrelativistic limit ( 1 ) ! c ! 1 Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

  28. Commutation relation between Commutation relation between symmetric generators symmetric generators Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

  29. NG modes associated with NG modes associated with baby skyrmion string baby skyrmion string

  30. Baby skyrmion solution Baby skyrmion solution Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

  31. Baby skyrmion solution Baby skyrmion solution Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

  32. Dilatation moduli Dilatation moduli stationary state Dilatation symmetry is the symmetry of the Dilatation symmetry is the symmetry of the stationary state of the dynamical equation stationary state of the dynamical equation Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

  33. Low energy dynamics of Low energy dynamics of domain wall domain wall Linear response theory Bogoliubov-de Gennes equation Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

  34. Low energy dynamics of Low energy dynamics of domain wall domain wall For arbitrary l Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

  35. Ultrarelativistic limit Ultrarelativistic limit For l = 0 Two NG modes Two NG modes Translational mode Translational mode along x direction along y direction Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

  36. Ultrarelativistic limit Ultrarelativistic limit For l = 1 One NG and One NG and one QNG modes one QNG modes Phase mode : NG mode Dilatation mode : quasi NG mode Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

  37. Ultrarelativistic limit Ultrarelativistic limit For l = 2 : bulk mode far from skyrmion Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

  38. Relativistic mode Relativistic mode For l = 0 Coupling of two Coupling of two translational translational (NG) modes (NG) modes Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

  39. Relativistic mode Relativistic mode For l = 1 Coupling of Coupling of phase (NG) and phase (NG) and dilatation (QNG) dilatation (QNG) modes modes Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

  40. Commutation relation between Commutation relation between symmetric generators symmetric generators Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

  41. Commutation relation between Commutation relation between symmetric generators symmetric generators Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

  42. Commutation relation between Commutation relation between symmetric generators symmetric generators All other commutation relations vanish Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

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