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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

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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 University

Michikazu Kobayashi : Kyoto University

  • Muneto Nitta : Keio University

Muneto Nitta : Keio University

  • Feb. 3, 2015, “Topological soliton”
  • Feb. 3, 2015, “Topological soliton”

Nonrelativistic Nambu-Goldstone(NG) mode Nonrelativistic Nambu-Goldstone(NG) mode CP CP1

1 model

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

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Nonrelativistic Nambu-Goldstone Nonrelativistic Nambu-Goldstone mode mode

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Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

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)

NBG： Number of broken symmetry generators NNG： Number of Nambu-Goldstone modes gi：Broken symmetry generators

NBG>NNG (Coupling of two NG modes to one NG mode) can occurs when two generators do not commute

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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

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Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

Nambu-Goldstone modes Nambu-Goldstone modes and broken symmetries and broken symmetries

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CP CP1

1 model

model

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Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

CP CP1

1 model

model

U(1) gauge theory

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Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

CP CP1

1 model

model

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Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

CP CP1

1 model

model

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Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

CP CP1

1 model

model

CP1 : Projection from south pole on S2 to complex plane

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Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

CP CP1

1 model

model

ultrarelativistic limit : ¹! 0 massive O(3) sigma model nonrelativistic limit : c! 1 Heisenberg ferromagnet with one easy axis (Ising ferromagnet)

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Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

Internal symmetry of Internal symmetry of CP CP1

1 model

model

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Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

Spacetime symmetry of Spacetime symmetry of CP CP1

1 model

model

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Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

Spacetime symmetry of Spacetime symmetry of CP CP1

1 model

model

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Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

Spacetime symmetry of Spacetime symmetry of CP CP1

1 model

model

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Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

Ground-state symmetry Ground-state symmetry

For internal symmetry

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Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

Ground-state symmetry Ground-state symmetry

For spacetime symmetry

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Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

Nambu-Goldstone mode for Nambu-Goldstone mode for m m = 0 = 0

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Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

Nambu-Goldstone mode for Nambu-Goldstone mode for m m = 0 = 0

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Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

Nambu-Goldstone mode for Nambu-Goldstone mode for m m = 0 = 0

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NG modes associated with NG modes associated with domain wall domain wall

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Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

Domain wall solution Domain wall solution

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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

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Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

Low energy dynamics of Low energy dynamics of domain wall domain wall

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Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

Nambu-Goldstone and Nambu-Goldstone and gapful (Higgs) modes gapful (Higgs) modes

Coupling of translational and phase moduli Inverse coupling of moduli to those of NG mode

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Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

Ultrarelativistic limit ( Ultrarelativistic limit (¹ ¹ ! ! 0 0) )

phase moduli translational moduli

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Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

Nonrelativistic limit ( Nonrelativistic limit (c c ! ! 1 1) )

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Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

Commutation relation between Commutation relation between symmetric generators symmetric generators

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NG modes associated with NG modes associated with baby skyrmion string baby skyrmion string

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Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

Baby skyrmion solution Baby skyrmion solution

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Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

Baby skyrmion solution Baby skyrmion solution

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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

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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

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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

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Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

Ultrarelativistic limit Ultrarelativistic limit

For l = 0

Translational mode along x direction Translational mode along y direction

Two NG modes Two NG modes

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Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

Ultrarelativistic limit Ultrarelativistic limit

For l = 1

Phase mode : NG mode Dilatation mode : quasi NG mode

One NG and One NG and

  • ne QNG modes
  • ne QNG modes
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Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

Ultrarelativistic limit Ultrarelativistic limit

For l = 2 : bulk mode far from skyrmion

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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

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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

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Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

Commutation relation between Commutation relation between symmetric generators symmetric generators

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Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

Commutation relation between Commutation relation between symmetric generators symmetric generators

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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

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Nonrelativistic NG Modes Associated with Spontaneously Broken Space-Time and Internal Symmetries

Summary Summary

  • 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 dispersion couple to one NG with quadratic dispersion

  • For domain wall case in massive

For domain wall case in massive CP CP1

1 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 mode NG mode

  • For skyrmion case in massless

For skyrmion case in massless CP CP1

1 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 mode mode) and one NG-QNG mode

  • Any couplings of NG modes satisfy Watanabe-

Any couplings of NG modes satisfy Watanabe- Brauner relation Brauner relation