The power of symmetry Yoshimasa Hidaka RIKEN What is symmetry a - PowerPoint PPT Presentation

The power of symmetry Yoshimasa Hidaka RIKEN

What is symmetry a a c 120 � 120 � Figure b b c b a c Equilateral triangle Function f ( x ) = x 2 − 4 y 4 Symmetry: f ( x ) = f ( − x ) 2 - 3 - 2 - 1 1 2 3 x f ( x ) = 0 - 2 x = ± 2 - 4

Noether's theorem Symmetry Conservation law ∂ t j 0 + r i j i = 0 Time translation Energy Space translation Momentum Angular momentum Rotation U(1) symmetry Charge

figure from Book by Chaikin and Lubensky from wikipedia Classification Crystals Atom periodic table

Phase of matter Pressure di ff erent symmetry same symmetry liquid 1atm solid gas Temperature 0 100 ( ℃ )

Hydrodynamics Mass conservation ∂ t ρ + r · ( ρ v ) = 0 Momentum conservation ∂ t ( ρ v i ) = �r i p � r j ( ρ v i v j ) + r j τ ji wikipedia

Spontaneous symmetry breaking For Lorentz invariant systems, # of broken symmetries # of Nambu-Goldstone modes frequency wave number Dispersion relation Nambu-Goldstone theorem Nambu(’60), Goldstone(61), Nambu, Jona-Lasinio(’61), Goldstone, Salam, Weinberg(’62) N BS N NG = ω = | k |

chiral symm. spin symm. U(1) symm. translation symm. U(1) (1form) symm rotation symm. pion superfluid phonon magnon photon surface wave Nambu-Goldstone modes diffusive mode Symmetry breaking in nature CC by-sa Aney

longrange force Nonrelativistic Nambu thoery spacetime symm open system extended object

Nonrelativistic Systems

Puzzle Dispersion 
 Symmetry 
 breaking 
 pattern relation Exception to NG theorem Ferromagnet Anti-ferromagnet SO (3) → SO (2) SO (3) → SO (2) N BS 2 2 N NG 2 1 ω = c | k | 2 ω = c | k |

Puzzle Dispersion 
 Symmetry 
 breaking 
 pattern relation Exception to NG theorem Ferromagnet Anti-ferromagnet SO (3) → SO (2) SO (3) → SO (2) N BS 2 2 N NG 2 1 ω = c | k | 2 ω = c | k |

Generalization Nielsen - Chadha (’76), Schafer, Son, Stephanov, Toublan, and Verbaarschot (’01), Nambu (’04), Watanabe - Brauner (’11), …. Classification of NG modes was long standing problem…

Classification of NG modes Watanabe, Murayama (’12), YH (’12) Type-B Type-A Harmonic oscillation Precession Ex. ) superfluid phonon Ex. ) magnon

Intuitive example for type-B NG modes Pendulum with a spinning top Rotation symmetry is explicitly 
 broken by a weak gravity Rotation along with z axis is unbroken. Rotation along with x or y is broken. The number of broken If the spin is not rotating, there are If the spin is rotating, one symmetry is two. two independent oscillations. precession motion appears { L x , L y } = L z 6 = 0

Two types of excitations Watanabe, Murayama (’12), YH (’12) Type-A Type-B Harmonic oscillation Precession N B = 1 N A = N BS � rank h [ iQ a , Q b ] i 2rank h [ iQ a , Q b ] i N NG = N BS � 1 2 h i [ Q a , Q b ] i

gravity Two types of excitations Watanabe, Murayama (’12), YH (’12) Type-A Type-B Harmonic oscillation Precession √ ω ∼ √ g ∼ k 2 k 2 ω ∼ g ∼ At finite temperature ω = ak − ibk 2 ω = a 0 k 2 − ib 0 k 4 Hayata, YH (’14)

CC BY-SA 2.0 Open systems Example) Active matter Active “agents”(birds ， fish) move using their internal energy, 
 and their energy and momentum di ff use by friction. No energy-momentum conservation

Ex) NG modes in active hydrodynamics J. Toner, and Y. Tu, PRE (1998) Conservation of birds: ∂ρ + r · ( ρ v ) = 0 ∂ t v + ( v · r ) v = α v − β v 2 v − r P + D L r ( r · v ) + D l ( v · r ) 2 v + f noise non-conserved part Steady solution: v 2 = α / β ≡ v 2 0 Symmetry breaking: O (3) → O (2) Fluctuation: v = ( v 0 + δ v x , δ v y , δ v z ) ω = ck ω = i Γ k 2 gapless Propagating mode Diffusive mode

Stochastic process Fokker-Planck (FP) equation ∂ t P ( t, v ) = − ∂ ∂ ⇣ ⌘ P ( t, v ) Γ ij − F i ∂ v i ∂ v j Symmetry of FP equation and its spontaneous breaking NG modes

Typical dispersion Minami, YH. (’15) open closed Type-A diffusive propagating ω = − i γ k 2 ω = ck − i γ k 2 propagating propagating Type-B ω = ck 2 − i γ k 2 ω = ck 2 − i γ 0 k 4

Generalization to extended objects

Generalized Global symmetries Gaiotto et al. (’15) Charged Object: Point i.e., zero-form Higher form symmetry Object: higher form Line Surface p-dimensional object 2-form 1-form p-form

cf. Ferrari, Picasso (’71), Hata (’82), Kugo, Terao, Uehara (’85) Photons as NG bosons Gaiotto et al. (’15) Charged object Conservation of 
 Electric and magnetic field LINES electric and flux dQ e Z Q e = d S · E = 0 dt dQ m Z Q m = d S · B = 0 dt These symmetry is spontaneously broken Photon = NG modes

Topological soliton Kobayashi, Nitta, 1403.4031 Kobayashi, Nitta, 1402.6826 c.f. Watanabe, Murayama 1401.8139 Nonrelativistic CP1 model Type-B Kelvon Type-B Ripplon-Magnon [ P z , Q ] ∝ N [ P x , P y ] ∝ N 2-form symmetry U(1) z trans. 1-form symmetry x trans. y trans.

Summary Symmetry is useful! NG modes are classified as Type-A Type-B Precession Harmonic oscillation Topics in this talk ● Nonrelativistic systems ● Open systems ● Higher form symmetry breaking