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

The power of symmetry

Yoshimasa Hidaka

RIKEN

What is symmetry

Figure

f(x) = 0

x = ±2

Function f(x) = x2 − 4

f(x) = f(−x)

Symmetry:

x y

• 3
• 2
• 1

1 2 3

• 4
• 2

2 4

Equilateral triangle

a

b

c

120 a b c 120 a b

c

Symmetry

Time translation Space translation Rotation U(1) symmetry

Conservation law

Energy Momentum Angular momentum Charge

∂tj0 + riji = 0

Noether's theorem

Classification

Atom periodic table Crystals

figure from Book by Chaikin and Lubensky

from wikipedia

Phase of matter

solid liquid gas

Pressure Temperature 100 1atm ( ℃ )

same symmetry different symmetry

Hydrodynamics

Mass conservation

Momentum conservation

∂tρ + r · (ρv) = 0

∂t(ρvi) = rip rj(ρvivj) + rjτji

wikipedia

Spontaneous symmetry breaking

Goldstone, Salam, Weinberg(’62)

Nambu-Goldstone theorem

Nambu(’60), Goldstone(61), Nambu, Jona-Lasinio(’61),

For Lorentz invariant systems,

# of broken symmetries # of Nambu-Goldstone modes

NBS NNG

=

ω = |k|

frequency wave number

Dispersion relation

Symmetry breaking in nature

chiral symm.

CC by-sa Aney

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

spacetime symm

longrange force

extended object

• pen system

Nonrelativistic

Nambu thoery

Nonrelativistic Systems

Puzzle

Ferromagnet

Anti-ferromagnet

Exception to NG theorem

SO(3) → SO(2)

Symmetry 
 breaking
 pattern

SO(3) → SO(2)

NBS

2

2

NNG

Dispersion
 relation

2

1

ω = c|k|

ω = c|k|2

Puzzle

Ferromagnet

Anti-ferromagnet

Exception to NG theorem

SO(3) → SO(2)

Symmetry 
 breaking
 pattern

SO(3) → SO(2)

NBS

2

2

NNG

Dispersion
 relation

2

1

ω = c|k|

ω = c|k|2

Nielsen - Chadha (’76), Schafer, Son, Stephanov, Toublan, and Verbaarschot (’01), Nambu (’04), Watanabe - Brauner (’11), ….

Generalization

Classification of NG modes was long standing problem…

Type-A Type-B

Watanabe, Murayama (’12), YH (’12)

Harmonic oscillation

Precession

• Ex. ) superfluid phonon
• Ex. ) magnon

Classification of NG modes

If the spin is not rotating, there are two independent oscillations. If the spin is rotating, one precession motion appears

{Lx, Ly} = Lz 6= 0

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 symmetry is two.

Intuitive example for type-B NG modes

Pendulum with a spinning top

NNG = NBS 1 2hi[Qa, Qb]i

NA = NBS rankh[iQa, Qb]i

NB = 1 2rankh[iQa, Qb]i

Type-A Type-B

Two types of excitations

Watanabe, Murayama (’12), YH (’12)

Harmonic oscillation

Precession

Type-A Type-B

gravity

ω ∼ √g

ω ∼ g

∼ √ k2

∼ k2

Watanabe, Murayama (’12), YH (’12)

ω = ak − ibk2

ω = a0k2 − ib0k4

At finite temperature

Hayata, YH (’14)

Two types of excitations

Harmonic oscillation

Precession

Open systems

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No energy-momentum conservation

Active “agents”(birds，fish) move using their internal energy, 
 and their energy and momentum diffuse by friction.

Example) Active matter

∂tv + (v · r)v = αv − βv2v − rP + DLr(r · v) + Dl(v · r)2v + f

∂ρ + r · (ρv) = 0

Ex) NG modes in active hydrodynamics

• J. Toner, and Y. Tu, PRE (1998)

noise

non-conserved part

ω = ck ω = iΓk2 gapless

Diffusive mode Propagating mode

O(3) → O(2) Steady solution: v = (v0 + δvx, δvy, δvz) Fluctuation: v2 = α/β ≡ v2 Symmetry breaking:

Conservation of birds:

Stochastic process

Symmetry of FP equation and its spontaneous breaking

NG modes Fokker-Planck (FP) equation

∂tP(t, v) = − ∂ ∂vi ⇣ Γij ∂ ∂vj − Fi ⌘ P(t, v)

Typical dispersion

• pen

Type-A Type-B closed

diffusive propagating propagating propagating

ω = ck2 − iγ0k4 ω = ck2 − iγk2

ω = ck − iγk2

ω = −iγk2

Minami, YH. (’15)

Generalization to extended objects

Generalized Global symmetries

Charged Object: Higher form symmetry Object: higher form

Line

1-form

Surface

2-form

p-dimensional object

p-form i.e., zero-form

Point

Gaiotto et al. (’15)

Conservation of 
 electric and flux

Qe = Z dS · E Qm = Z dS · B dQe dt = 0 dQm dt = 0

Gaiotto et al. (’15)

Photons as NG bosons

• cf. Ferrari, Picasso (’71), Hata (’82), Kugo, Terao, Uehara (’85)

Charged object

Electric and magnetic field LINES

These symmetry is spontaneously broken Photon = NG modes

Type-B Kelvon Type-B Ripplon-Magnon

Topological soliton

[Px, Py] ∝ N

1-form symmetry

y trans. x trans. z trans.

[Pz, Q] ∝ N

2-form symmetry

U(1)

Kobayashi, Nitta, 1403.4031 Kobayashi, Nitta, 1402.6826 c.f. Watanabe, Murayama 1401.8139

Nonrelativistic CP1 model

Summary

Type-A Type-B

Harmonic oscillation

Precession

• Higher form symmetry breaking
• Nonrelativistic systems

Topics in this talk

Symmetry is useful!

NG modes are classified as

• Open systems