nonrelativistic nambu-goldstone modes associated …michikaz/presentation/2015...2015/02/03 ·...
TRANSCRIPT
![Page 1: Nonrelativistic Nambu-Goldstone Modes Associated …michikaz/presentation/2015...2015/02/03 · Nambu-Goldstone modes and broken symmetries Watanabe-Brauner relation H. Watanabe and](https://reader033.vdocuments.mx/reader033/viewer/2022053020/5f2bcd15f8581741003dda39/html5/thumbnails/1.jpg)
Nonrelativistic Nambu-Goldstone Modes Nonrelativistic Nambu-Goldstone Modes Associated with Spontaneously BrokenAssociated with Spontaneously Broken
Space-Time and Internal SymmetrySpace-Time and Internal Symmetry
● Michikazu Kobayashi : Kyoto UniversityMichikazu Kobayashi : Kyoto University● Muneto Nitta : Keio UniversityMuneto Nitta : Keio University
Feb. 3, 2015, “Topological soliton”Feb. 3, 2015, “Topological soliton”
Nonrelativistic Nambu-Goldstone(NG) modeNonrelativistic Nambu-Goldstone(NG) mode
CPCP11 model model
NG modes associated with domain wallNG modes associated with domain wall
NG modes associated with baby skyrmion stringNG modes associated with baby skyrmion string
![Page 2: Nonrelativistic Nambu-Goldstone Modes Associated …michikaz/presentation/2015...2015/02/03 · Nambu-Goldstone modes and broken symmetries Watanabe-Brauner relation H. Watanabe and](https://reader033.vdocuments.mx/reader033/viewer/2022053020/5f2bcd15f8581741003dda39/html5/thumbnails/2.jpg)
Nonrelativistic Nambu-Goldstone Nonrelativistic Nambu-Goldstone modemode
![Page 3: Nonrelativistic Nambu-Goldstone Modes Associated …michikaz/presentation/2015...2015/02/03 · Nambu-Goldstone modes and broken symmetries Watanabe-Brauner relation H. Watanabe and](https://reader033.vdocuments.mx/reader033/viewer/2022053020/5f2bcd15f8581741003dda39/html5/thumbnails/3.jpg)
Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
Nambu-Goldstone modesNambu-Goldstone modesand broken symmetriesand broken symmetries
Watanabe-Brauner relationH. Watanabe and T. Brauner, Phys. Rev. D 84, 125013 (2011)H. Watanabe and H. Murayama, Phys. Rev. Lett. 108, 251602 (2012)
NBG: Number of broken symmetry generatorsNNG: Number of Nambu-Goldstone modesgi:Broken symmetry generators
NBG>NNG (Coupling of two NG modes to one NG mode) can occurs when two generators do not commute
![Page 4: Nonrelativistic Nambu-Goldstone Modes Associated …michikaz/presentation/2015...2015/02/03 · Nambu-Goldstone modes and broken symmetries Watanabe-Brauner relation H. Watanabe and](https://reader033.vdocuments.mx/reader033/viewer/2022053020/5f2bcd15f8581741003dda39/html5/thumbnails/4.jpg)
Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
Nambu-Goldstone modesNambu-Goldstone modesand broken symmetriesand broken symmetries
Nonlinear O(3) sigma model & Heisenberg Ferromagnet
![Page 5: Nonrelativistic Nambu-Goldstone Modes Associated …michikaz/presentation/2015...2015/02/03 · Nambu-Goldstone modes and broken symmetries Watanabe-Brauner relation H. Watanabe and](https://reader033.vdocuments.mx/reader033/viewer/2022053020/5f2bcd15f8581741003dda39/html5/thumbnails/5.jpg)
Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
Nambu-Goldstone modesNambu-Goldstone modesand broken symmetriesand broken symmetries
![Page 6: Nonrelativistic Nambu-Goldstone Modes Associated …michikaz/presentation/2015...2015/02/03 · Nambu-Goldstone modes and broken symmetries Watanabe-Brauner relation H. Watanabe and](https://reader033.vdocuments.mx/reader033/viewer/2022053020/5f2bcd15f8581741003dda39/html5/thumbnails/6.jpg)
CPCP11 model model
![Page 7: Nonrelativistic Nambu-Goldstone Modes Associated …michikaz/presentation/2015...2015/02/03 · Nambu-Goldstone modes and broken symmetries Watanabe-Brauner relation H. Watanabe and](https://reader033.vdocuments.mx/reader033/viewer/2022053020/5f2bcd15f8581741003dda39/html5/thumbnails/7.jpg)
Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
CPCP11 model model
U(1) gauge theory
![Page 8: Nonrelativistic Nambu-Goldstone Modes Associated …michikaz/presentation/2015...2015/02/03 · Nambu-Goldstone modes and broken symmetries Watanabe-Brauner relation H. Watanabe and](https://reader033.vdocuments.mx/reader033/viewer/2022053020/5f2bcd15f8581741003dda39/html5/thumbnails/8.jpg)
Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
CPCP11 model model
![Page 9: Nonrelativistic Nambu-Goldstone Modes Associated …michikaz/presentation/2015...2015/02/03 · Nambu-Goldstone modes and broken symmetries Watanabe-Brauner relation H. Watanabe and](https://reader033.vdocuments.mx/reader033/viewer/2022053020/5f2bcd15f8581741003dda39/html5/thumbnails/9.jpg)
Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
CPCP11 model model
![Page 10: Nonrelativistic Nambu-Goldstone Modes Associated …michikaz/presentation/2015...2015/02/03 · Nambu-Goldstone modes and broken symmetries Watanabe-Brauner relation H. Watanabe and](https://reader033.vdocuments.mx/reader033/viewer/2022053020/5f2bcd15f8581741003dda39/html5/thumbnails/10.jpg)
Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
CPCP11 model model
CP1 : Projection from south pole on S2 to complex plane
![Page 11: Nonrelativistic Nambu-Goldstone Modes Associated …michikaz/presentation/2015...2015/02/03 · Nambu-Goldstone modes and broken symmetries Watanabe-Brauner relation H. Watanabe and](https://reader033.vdocuments.mx/reader033/viewer/2022053020/5f2bcd15f8581741003dda39/html5/thumbnails/11.jpg)
Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
CPCP11 model model
ultrarelativistic limit : ¹! 0 massive O(3) sigma model
nonrelativistic limit : c! 1 Heisenberg ferromagnetwith one easy axis (Ising ferromagnet)
![Page 12: Nonrelativistic Nambu-Goldstone Modes Associated …michikaz/presentation/2015...2015/02/03 · Nambu-Goldstone modes and broken symmetries Watanabe-Brauner relation H. Watanabe and](https://reader033.vdocuments.mx/reader033/viewer/2022053020/5f2bcd15f8581741003dda39/html5/thumbnails/12.jpg)
Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
Internal symmetry of Internal symmetry of CPCP11 model model
![Page 13: Nonrelativistic Nambu-Goldstone Modes Associated …michikaz/presentation/2015...2015/02/03 · Nambu-Goldstone modes and broken symmetries Watanabe-Brauner relation H. Watanabe and](https://reader033.vdocuments.mx/reader033/viewer/2022053020/5f2bcd15f8581741003dda39/html5/thumbnails/13.jpg)
Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
Spacetime symmetry of Spacetime symmetry of CPCP11 model model
![Page 14: Nonrelativistic Nambu-Goldstone Modes Associated …michikaz/presentation/2015...2015/02/03 · Nambu-Goldstone modes and broken symmetries Watanabe-Brauner relation H. Watanabe and](https://reader033.vdocuments.mx/reader033/viewer/2022053020/5f2bcd15f8581741003dda39/html5/thumbnails/14.jpg)
Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
Spacetime symmetry of Spacetime symmetry of CPCP11 model model
![Page 15: Nonrelativistic Nambu-Goldstone Modes Associated …michikaz/presentation/2015...2015/02/03 · Nambu-Goldstone modes and broken symmetries Watanabe-Brauner relation H. Watanabe and](https://reader033.vdocuments.mx/reader033/viewer/2022053020/5f2bcd15f8581741003dda39/html5/thumbnails/15.jpg)
Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
Spacetime symmetry of Spacetime symmetry of CPCP11 model model
![Page 16: Nonrelativistic Nambu-Goldstone Modes Associated …michikaz/presentation/2015...2015/02/03 · Nambu-Goldstone modes and broken symmetries Watanabe-Brauner relation H. Watanabe and](https://reader033.vdocuments.mx/reader033/viewer/2022053020/5f2bcd15f8581741003dda39/html5/thumbnails/16.jpg)
Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
Ground-state symmetryGround-state symmetry
For internal symmetry
![Page 17: Nonrelativistic Nambu-Goldstone Modes Associated …michikaz/presentation/2015...2015/02/03 · Nambu-Goldstone modes and broken symmetries Watanabe-Brauner relation H. Watanabe and](https://reader033.vdocuments.mx/reader033/viewer/2022053020/5f2bcd15f8581741003dda39/html5/thumbnails/17.jpg)
Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
Ground-state symmetryGround-state symmetry
For spacetime symmetry
![Page 18: Nonrelativistic Nambu-Goldstone Modes Associated …michikaz/presentation/2015...2015/02/03 · Nambu-Goldstone modes and broken symmetries Watanabe-Brauner relation H. Watanabe and](https://reader033.vdocuments.mx/reader033/viewer/2022053020/5f2bcd15f8581741003dda39/html5/thumbnails/18.jpg)
Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
Nambu-Goldstone mode for Nambu-Goldstone mode for mm = 0 = 0
![Page 19: Nonrelativistic Nambu-Goldstone Modes Associated …michikaz/presentation/2015...2015/02/03 · Nambu-Goldstone modes and broken symmetries Watanabe-Brauner relation H. Watanabe and](https://reader033.vdocuments.mx/reader033/viewer/2022053020/5f2bcd15f8581741003dda39/html5/thumbnails/19.jpg)
Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
Nambu-Goldstone mode for Nambu-Goldstone mode for mm = 0 = 0
![Page 20: Nonrelativistic Nambu-Goldstone Modes Associated …michikaz/presentation/2015...2015/02/03 · Nambu-Goldstone modes and broken symmetries Watanabe-Brauner relation H. Watanabe and](https://reader033.vdocuments.mx/reader033/viewer/2022053020/5f2bcd15f8581741003dda39/html5/thumbnails/20.jpg)
Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
Nambu-Goldstone mode for Nambu-Goldstone mode for mm = 0 = 0
![Page 21: Nonrelativistic Nambu-Goldstone Modes Associated …michikaz/presentation/2015...2015/02/03 · Nambu-Goldstone modes and broken symmetries Watanabe-Brauner relation H. Watanabe and](https://reader033.vdocuments.mx/reader033/viewer/2022053020/5f2bcd15f8581741003dda39/html5/thumbnails/21.jpg)
NG modes associated withNG modes associated withdomain walldomain wall
![Page 22: Nonrelativistic Nambu-Goldstone Modes Associated …michikaz/presentation/2015...2015/02/03 · Nambu-Goldstone modes and broken symmetries Watanabe-Brauner relation H. Watanabe and](https://reader033.vdocuments.mx/reader033/viewer/2022053020/5f2bcd15f8581741003dda39/html5/thumbnails/22.jpg)
Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
Domain wall solutionDomain wall solution
![Page 23: Nonrelativistic Nambu-Goldstone Modes Associated …michikaz/presentation/2015...2015/02/03 · Nambu-Goldstone modes and broken symmetries Watanabe-Brauner relation H. Watanabe and](https://reader033.vdocuments.mx/reader033/viewer/2022053020/5f2bcd15f8581741003dda39/html5/thumbnails/23.jpg)
Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
Low energy dynamics ofLow energy dynamics ofdomain walldomain wall
Linear response theory
Bogoliubov-de Gennes equation
![Page 24: Nonrelativistic Nambu-Goldstone Modes Associated …michikaz/presentation/2015...2015/02/03 · Nambu-Goldstone modes and broken symmetries Watanabe-Brauner relation H. Watanabe and](https://reader033.vdocuments.mx/reader033/viewer/2022053020/5f2bcd15f8581741003dda39/html5/thumbnails/24.jpg)
Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
Low energy dynamics ofLow energy dynamics ofdomain walldomain wall
![Page 25: Nonrelativistic Nambu-Goldstone Modes Associated …michikaz/presentation/2015...2015/02/03 · Nambu-Goldstone modes and broken symmetries Watanabe-Brauner relation H. Watanabe and](https://reader033.vdocuments.mx/reader033/viewer/2022053020/5f2bcd15f8581741003dda39/html5/thumbnails/25.jpg)
Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
Nambu-Goldstone andNambu-Goldstone andgapful (Higgs) modesgapful (Higgs) modes
Coupling of translational and phase moduli
Inverse coupling of moduli to those of NG mode
![Page 26: Nonrelativistic Nambu-Goldstone Modes Associated …michikaz/presentation/2015...2015/02/03 · Nambu-Goldstone modes and broken symmetries Watanabe-Brauner relation H. Watanabe and](https://reader033.vdocuments.mx/reader033/viewer/2022053020/5f2bcd15f8581741003dda39/html5/thumbnails/26.jpg)
Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
Ultrarelativistic limit (Ultrarelativistic limit (¹¹ !! 0 0))
phase moduli translational moduli
![Page 27: Nonrelativistic Nambu-Goldstone Modes Associated …michikaz/presentation/2015...2015/02/03 · Nambu-Goldstone modes and broken symmetries Watanabe-Brauner relation H. Watanabe and](https://reader033.vdocuments.mx/reader033/viewer/2022053020/5f2bcd15f8581741003dda39/html5/thumbnails/27.jpg)
Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
Nonrelativistic limit (Nonrelativistic limit (cc !! 11))
![Page 28: Nonrelativistic Nambu-Goldstone Modes Associated …michikaz/presentation/2015...2015/02/03 · Nambu-Goldstone modes and broken symmetries Watanabe-Brauner relation H. Watanabe and](https://reader033.vdocuments.mx/reader033/viewer/2022053020/5f2bcd15f8581741003dda39/html5/thumbnails/28.jpg)
Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
Commutation relation between Commutation relation between symmetric generatorssymmetric generators
![Page 29: Nonrelativistic Nambu-Goldstone Modes Associated …michikaz/presentation/2015...2015/02/03 · Nambu-Goldstone modes and broken symmetries Watanabe-Brauner relation H. Watanabe and](https://reader033.vdocuments.mx/reader033/viewer/2022053020/5f2bcd15f8581741003dda39/html5/thumbnails/29.jpg)
NG modes associated withNG modes associated withbaby skyrmion stringbaby skyrmion string
![Page 30: Nonrelativistic Nambu-Goldstone Modes Associated …michikaz/presentation/2015...2015/02/03 · Nambu-Goldstone modes and broken symmetries Watanabe-Brauner relation H. Watanabe and](https://reader033.vdocuments.mx/reader033/viewer/2022053020/5f2bcd15f8581741003dda39/html5/thumbnails/30.jpg)
Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
Baby skyrmion solutionBaby skyrmion solution
![Page 31: Nonrelativistic Nambu-Goldstone Modes Associated …michikaz/presentation/2015...2015/02/03 · Nambu-Goldstone modes and broken symmetries Watanabe-Brauner relation H. Watanabe and](https://reader033.vdocuments.mx/reader033/viewer/2022053020/5f2bcd15f8581741003dda39/html5/thumbnails/31.jpg)
Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
Baby skyrmion solutionBaby skyrmion solution
![Page 32: Nonrelativistic Nambu-Goldstone Modes Associated …michikaz/presentation/2015...2015/02/03 · Nambu-Goldstone modes and broken symmetries Watanabe-Brauner relation H. Watanabe and](https://reader033.vdocuments.mx/reader033/viewer/2022053020/5f2bcd15f8581741003dda39/html5/thumbnails/32.jpg)
Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
Dilatation moduliDilatation moduli
stationary state
Dilatation symmetry is the symmetry of the Dilatation symmetry is the symmetry of the stationary state of the dynamical equationstationary state of the dynamical equation
![Page 33: Nonrelativistic Nambu-Goldstone Modes Associated …michikaz/presentation/2015...2015/02/03 · Nambu-Goldstone modes and broken symmetries Watanabe-Brauner relation H. Watanabe and](https://reader033.vdocuments.mx/reader033/viewer/2022053020/5f2bcd15f8581741003dda39/html5/thumbnails/33.jpg)
Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
Low energy dynamics ofLow energy dynamics ofdomain walldomain wall
Linear response theory
Bogoliubov-de Gennes equation
![Page 34: Nonrelativistic Nambu-Goldstone Modes Associated …michikaz/presentation/2015...2015/02/03 · Nambu-Goldstone modes and broken symmetries Watanabe-Brauner relation H. Watanabe and](https://reader033.vdocuments.mx/reader033/viewer/2022053020/5f2bcd15f8581741003dda39/html5/thumbnails/34.jpg)
Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
Low energy dynamics ofLow energy dynamics ofdomain walldomain wall
For arbitrary l
![Page 35: Nonrelativistic Nambu-Goldstone Modes Associated …michikaz/presentation/2015...2015/02/03 · Nambu-Goldstone modes and broken symmetries Watanabe-Brauner relation H. Watanabe and](https://reader033.vdocuments.mx/reader033/viewer/2022053020/5f2bcd15f8581741003dda39/html5/thumbnails/35.jpg)
Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
Ultrarelativistic limitUltrarelativistic limit
For l = 0
Translational mode along x direction
Translational mode along y direction
Two NG modesTwo NG modes
![Page 36: Nonrelativistic Nambu-Goldstone Modes Associated …michikaz/presentation/2015...2015/02/03 · Nambu-Goldstone modes and broken symmetries Watanabe-Brauner relation H. Watanabe and](https://reader033.vdocuments.mx/reader033/viewer/2022053020/5f2bcd15f8581741003dda39/html5/thumbnails/36.jpg)
Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
Ultrarelativistic limitUltrarelativistic limit
For l = 1
Phase mode : NG mode Dilatation mode : quasi NG mode
One NG andOne NG andone QNG modesone QNG modes
![Page 37: Nonrelativistic Nambu-Goldstone Modes Associated …michikaz/presentation/2015...2015/02/03 · Nambu-Goldstone modes and broken symmetries Watanabe-Brauner relation H. Watanabe and](https://reader033.vdocuments.mx/reader033/viewer/2022053020/5f2bcd15f8581741003dda39/html5/thumbnails/37.jpg)
Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
Ultrarelativistic limitUltrarelativistic limit
For l = 2 : bulk mode far from skyrmion
![Page 38: Nonrelativistic Nambu-Goldstone Modes Associated …michikaz/presentation/2015...2015/02/03 · Nambu-Goldstone modes and broken symmetries Watanabe-Brauner relation H. Watanabe and](https://reader033.vdocuments.mx/reader033/viewer/2022053020/5f2bcd15f8581741003dda39/html5/thumbnails/38.jpg)
Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
Relativistic modeRelativistic mode
For l = 0
Coupling of two Coupling of two translational translational (NG) modes(NG) modes
![Page 39: Nonrelativistic Nambu-Goldstone Modes Associated …michikaz/presentation/2015...2015/02/03 · Nambu-Goldstone modes and broken symmetries Watanabe-Brauner relation H. Watanabe and](https://reader033.vdocuments.mx/reader033/viewer/2022053020/5f2bcd15f8581741003dda39/html5/thumbnails/39.jpg)
Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
Relativistic modeRelativistic mode
For l = 1
Coupling of Coupling of phase (NG) and phase (NG) and dilatation (QNG) dilatation (QNG) modesmodes
![Page 40: Nonrelativistic Nambu-Goldstone Modes Associated …michikaz/presentation/2015...2015/02/03 · Nambu-Goldstone modes and broken symmetries Watanabe-Brauner relation H. Watanabe and](https://reader033.vdocuments.mx/reader033/viewer/2022053020/5f2bcd15f8581741003dda39/html5/thumbnails/40.jpg)
Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
Commutation relation between Commutation relation between symmetric generatorssymmetric generators
![Page 41: Nonrelativistic Nambu-Goldstone Modes Associated …michikaz/presentation/2015...2015/02/03 · Nambu-Goldstone modes and broken symmetries Watanabe-Brauner relation H. Watanabe and](https://reader033.vdocuments.mx/reader033/viewer/2022053020/5f2bcd15f8581741003dda39/html5/thumbnails/41.jpg)
Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
Commutation relation between Commutation relation between symmetric generatorssymmetric generators
![Page 42: Nonrelativistic Nambu-Goldstone Modes Associated …michikaz/presentation/2015...2015/02/03 · Nambu-Goldstone modes and broken symmetries Watanabe-Brauner relation H. Watanabe and](https://reader033.vdocuments.mx/reader033/viewer/2022053020/5f2bcd15f8581741003dda39/html5/thumbnails/42.jpg)
Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
Commutation relation between Commutation relation between symmetric generatorssymmetric generators
All other commutation relations vanish
![Page 43: Nonrelativistic Nambu-Goldstone Modes Associated …michikaz/presentation/2015...2015/02/03 · Nambu-Goldstone modes and broken symmetries Watanabe-Brauner relation H. Watanabe and](https://reader033.vdocuments.mx/reader033/viewer/2022053020/5f2bcd15f8581741003dda39/html5/thumbnails/43.jpg)
Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries
SummarySummary
● For systems with spontaneously broken Lorentz For systems with spontaneously broken Lorentz symmetry (or without Lorentz symmetry), two NG may symmetry (or without Lorentz symmetry), two NG may couple to one NG with quadratic dispersioncouple to one NG with quadratic dispersion
● For domain wall case in massive For domain wall case in massive CPCP11 model, model,
translational NG mode (spacetime symmetry) and translational NG mode (spacetime symmetry) and phase NG mode (internal symmetry) couple to one phase NG mode (internal symmetry) couple to one NG modeNG mode
● For skyrmion case in massless For skyrmion case in massless CPCP11 model, two model, two
translational NG modes, and phase NG mode and translational NG modes, and phase NG mode and dilatation QNG mode couple to one NG mode(kelvin dilatation QNG mode couple to one NG mode(kelvin mode) and one NG-QNG modemode) and one NG-QNG mode
● Any couplings of NG modes satisfy Watanabe-Any couplings of NG modes satisfy Watanabe-Brauner relationBrauner relation