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”
Nonrelativistic Nambu-Goldstone Nonrelativistic Nambu-Goldstone mode mode
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
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
Nambu-Goldstone modes Nambu-Goldstone modes and broken symmetries and broken symmetries Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
CP 1 model CP 1 model
CP 1 model CP 1 model U (1) gauge theory Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
CP 1 model CP 1 model Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
CP 1 model CP 1 model Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
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
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
CP 1 Internal symmetry of CP model Internal symmetry of 1 model Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
CP 1 Spacetime symmetry of CP model Spacetime symmetry of 1 model Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
CP 1 Spacetime symmetry of CP model Spacetime symmetry of 1 model Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
CP 1 Spacetime symmetry of CP model Spacetime symmetry of 1 model Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
Ground-state symmetry Ground-state symmetry For internal symmetry Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
Ground-state symmetry Ground-state symmetry For spacetime symmetry Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
Nambu-Goldstone mode for m Nambu-Goldstone mode for m = 0 = 0 Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
Nambu-Goldstone mode for m Nambu-Goldstone mode for m = 0 = 0 Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
Nambu-Goldstone mode for m Nambu-Goldstone mode for m = 0 = 0 Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
NG modes associated with NG modes associated with domain wall domain wall
Domain wall solution Domain wall solution Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
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
Low energy dynamics of Low energy dynamics of domain wall domain wall Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
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
Ultrarelativistic limit ( ¹ ) Ultrarelativistic limit ( 0 ) ! 0 ¹ ! phase moduli translational moduli Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
Nonrelativistic limit ( c ) Nonrelativistic limit ( 1 ) ! c ! 1 Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
Commutation relation between Commutation relation between symmetric generators symmetric generators Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
NG modes associated with NG modes associated with baby skyrmion string baby skyrmion string
Baby skyrmion solution Baby skyrmion solution Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
Baby skyrmion solution Baby skyrmion solution Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
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
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
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
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
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
Ultrarelativistic limit Ultrarelativistic limit For l = 2 : bulk mode far from skyrmion Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
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
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
Commutation relation between Commutation relation between symmetric generators symmetric generators Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
Commutation relation between Commutation relation between symmetric generators symmetric generators Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
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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