Nambu Goldstone theorem in nonrelativistic systems QCD in a - - PowerPoint PPT Presentation

Yoshimasa Hidaka

(RIKEN)

Nambu Goldstone theorem in nonrelativistic systems QCD in a magnetic field

1

Keyword: Symmetry breaking

Explicit Quantum Spontaneous

2

Keyword: Symmetry breaking

Explicit Quantum Spontaneous

Magnetic field

Nambu-Goldstone theorem

Chiral anomaly

2

Nambu Goldstone theorem in nonrelativistic systems

Yoshimasa Hidaka

(RIKEN)

3

Zero modes in nature

4

Zero modes in nature

Light (Photon)

4

Zero modes in nature

Light (Photon)

4

Zero modes in nature

Light (Photon)

Crystal Vibrations (Phonon)

4

Zero modes in nature

Light (Photon)

Edge modes in topological insulator

Topological Insulator

Electron(down spin) Electron(up spin)

Crystal Vibrations (Phonon)

4

Zero modes in nature

Light (Photon)

Edge modes in topological insulator

Topological Insulator

Electron(down spin) Electron(up spin)

Crystal Vibrations (Phonon)

Gauge symmetry

Spontaneous symmetry breaking of translation

Topological

4

Spontaneous Symmetry breaking

Unbroken Broken V (φ) V (φ) φ φ

5

Nambu-Goldstone bosons

Examples in hadron physics Pions and Kaons

NG mode in Kaon condensed color flavor locking phase

Chiral symmetry breaking

Schafer, Son, Stephanov, Toublan, and Verbaarschot (’01) Miransky, Shovkovy (’02)

E = p k2 + m2

E = ak2

Type-I NG modes

Type-II NG modes

6

Nambu-Goldstone bosons

Phonon in crystal (Galilean, rotational, translational symmetries) Phonon in superfluid (U(1) symmetry) Acoustic phonon? (Galilean symmetry) Magnon (Rotational symmetry)

Other Examples

7

NBS

NNG

: the number of broken symmetries : the number of NG modes

Dispersion relation

Nambu-Goldstone theorem

in Lorentz invariant systems

NNG = NBS

Nambu(’60), Goldstone(61), Nambu Jona-Lasinio(’61), Goldstone, Salam, Weinberg(’62).

Ek = |k|

8

Nielsen and Chadha (’76)

Ntype-I + 2Ntype-II ≥ NBS

Generalization

9

Nielsen and Chadha (’76)

Ntype-I + 2Ntype-II ≥ NBS

Schafer, Son, Stephanov, Toublan, and Verbaarschot

NNG = NBS

(’01)

h[Qa, Qb]i = 0

Generalization

9

Nielsen and Chadha (’76)

Ntype-I + 2Ntype-II ≥ NBS

Schafer, Son, Stephanov, Toublan, and Verbaarschot

NNG = NBS

(’01)

h[Qa, Qb]i = 0

Watanabe and Brauner (’11)

NBS NNG 1 2rank[Qa, Qb]

Generalization

9

Example of type-II

NBS Ntype-I Ntype-II

1 2rankh[Qa, Qb]iNBS − NNG Magnon in Ferromagnet

O(3)→O(2)

NG mode in Kaon condensed color flavor locked phase

SU(2)xSU(1)Y→SU(2)em

Kelvon in Vortex of superfluid

Translation Px, Py breaking

2 1 1 2 3 1 1 1 3 2 1 1 2

NBS Ntype-I Ntyep-II

1 2rankh[Qa, Qb]i Ntype-I + 2Ntype-II

Ntype-I + 2Ntype-II = NBS NBS NNG = 1

2rankh[Qa, Qb]i

Known examples satisfy the equalities

10

The recent results

Ntype-I + 2Ntype-II = NBS

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

Ntype-II = 1 2rankh[Qa, Qb]i

Watanabe, Murayama (’12), YH (’12)

11

zero mode

Spontaneous symmetry breaking (SSB)

NG fields Conserved charges

a = 1, · · · , NBS h[φi, Qa]i = trρ [φi, Qa] trρ 6= 0 ρ = |ΩihΩ| ρ = exp(−β(H − µN))

Vacuum: Matter:

Γ[φ]

φ

12

Suppose the classical action is invariant under

Z ddx δΓ[φ] δφi(x)h[Qa, φi(x)]i = 0

The effective action satisfies

i → i + ✏a[Qa, i] Γ[φ]

13

Suppose the classical action is invariant under

Z ddx δΓ[φ] δφi(x)h[Qa, φi(x)]i = 0

The effective action satisfies

i → i + ✏a[Qa, i] Z ddx δ2Γ[φ] δφj(y)δφi(x)h[Qa, φi(x)]i = 0 Γ[φ]

13

Suppose the classical action is invariant under

Z ddx δΓ[φ] δφi(x)h[Qa, φi(x)]i = 0

The effective action satisfies

i → i + ✏a[Qa, i] D−1

ji (y, x) =

δ2Γ[φ] φj(y)δφi(x)

Inverse of propagators have zero eigenvalues.

Z ddx δ2Γ[φ] δφj(y)δφi(x)h[Qa, φi(x)]i = 0 Γ[φ]

13

Suppose the classical action is invariant under

Z ddx δΓ[φ] δφi(x)h[Qa, φi(x)]i = 0

The effective action satisfies

i → i + ✏a[Qa, i] D−1

ji (y, x) =

δ2Γ[φ] φj(y)δφi(x)

Inverse of propagators have zero eigenvalues. The number coincides with the number of independent eigenvectorsh[Qa, φi(x)]i

Z ddx δ2Γ[φ] δφj(y)δφi(x)h[Qa, φi(x)]i = 0 Γ[φ]

13

For the Lorentz invariant system

h[Qa, φi(x)]i ⌘ M (a)

i

D−1

ji (p2 = 0)M (a) i

= 0

NNG = the number of independent eigenvectors

independent of x

Goldstone, Salam, Weinberg (’62)

= NBS

14

h[Qa, φi(x)]i is not always the eigenvector. h[Qa, φi(x)]i should be eigenvector of unbroken translation

In general, NBS ≠ the number of eigenvectors

Low, and Manohar (’02)

15

h[Qa, φi(x)]i is not always the eigenvector. h[Qa, φi(x)]i should be eigenvector of unbroken translation

Example: Domain wall Broken symmetry: Translation (Px) Rotation (Ly,Lz)

h[Px, φ]i = ∂xhφi 6= 0

h[Lz, φ]i = y∂xhφi 6= 0

h[Ly, φ]i = z∂xhφi 6= 0

One NG mode exists associated with Px.

h[Ly,z, φ]i are not eigenvectors of Py, Pz

In general, NBS ≠ the number of eigenvectors

Low, and Manohar (’02)

15

The number of independent eigenvectors is not always equal to the number of NG modes

Example: Ferromagnet

O(3)→ O(2)

Spin rotation

Broken generator: Sx, Sy Two eigenvector:

One spin wave appears.

✏ijzhszi

16

Intuitive example for type-II 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 symmetry is two.

17

Pendulum has two oscillation motions if the top is not spinning.

Intuitive example for type-II NG modes

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If the top is spinning,

the only one rotation motion (Precession) exists.

In this case,

{Lx, Ly}P = Lz 6= 0

Intuitive example for type-II NG modes

19

NG theorem in

Hamiltonian formalism

20

A simple Hamiltonian system

{x, p}P = 1

H = a(k) 2 p2 + b(k) 2 x2

21

A simple Hamiltonian system

{x, p}P = 1

H = a(k) 2 p2 + b(k) 2 x2

∂tx = {x, H}P = a(k)p ∂tp = {p, H}P = −b(k)x

21

A simple Hamiltonian system

{x, p}P = 1

H = a(k) 2 p2 + b(k) 2 x2

∂tx = {x, H}P = a(k)p ∂tp = {p, H}P = −b(k)x

∂2

t x + a(k)b(k)x = 0

21

∂2

t x + a(k)b(k)x = 0

22

a(k) = a0 + a2k2

b(k) = b0 + b2k2

a(k)b(k) = a0b0 + (a0b2 + a2b0)k2 + a2b2k4

∂2

t x + a(k)b(k)x = 0

22

a(k) = a0 + a2k2

b(k) = b0 + b2k2

a(k)b(k) = a0b0 + (a0b2 + a2b0)k2 + a2b2k4

∂2

t x + a(k)b(k)x = 0

a0 b0 p a0b0 ∼ |k| ∼ k2

Energy

gapped

nonzero nonzero

gapless (type-I)

zero nonzero

gapless (type-II)

zero zero

• cf. Nambu (’04)

22

{xi, pj}P = δij

(i, j = 1, · · · , NBS)

H = 1 2ptHppp + 1 2xtHxxx

23

{xi, pj}P = δij

(i, j = 1, · · · , NBS)

H = 1 2ptHppp + 1 2xtHxxx

Mass matrix:

NNG = NBS − Nmassive

The number of NG modes:

m2 = HppHxx

Nmassive = rank(m2)

23

∂t ✓x p ◆ = MH ✓x p ◆

where M = ✓ 0 1 −1 ◆

H = ✓Hxx Hpp ◆

24

∂t ✓x p ◆ = MH ✓x p ◆

where

Nmassive =

MH

half of the number of nonzero eigenvalues of

NNG = NBS − Nmassive

The number of NG modes M = ✓ 0 1 −1 ◆

H = ✓Hxx Hpp ◆

24

∂t ✓x p ◆ = MH ✓x p ◆

where

Nmassive =

MH

half of the number of nonzero eigenvalues of

NNG = NBS − Nmassive

The number of NG modes

M = ✓ Mxx Mxp −M t

xp

Mpp ◆

H = ✓ Hxx Hxp (Hxp)t Hpp ◆

25

What are canonical variables? What is the Poisson bracket? What is the Hamiltonian?

26

Projection operator method

Operator set {An}

Am(0) An(0)

Mori (’65)

27

Projection operator method

Operator set {An}

Am(0) An(0) An(t) QAn(t) PAn(t)

Mori (’65)

27

Projection operator method

Mori (’65)

∂tAn(t, k) = i[H, An(t, k)]

28

Projection operator method

Mori (’65)

+Rn(t, k)

streaming dissipation noise

Generalized Langevin equation

∂tAn(t, k) = Mnm(k)Γ ml(k)Am(t, k)

− ∞ dsKnm(t − s, k)Γ ml(k)Al(s, k)

∂tAn(t, k) = i[H, An(t, k)]

28

Projection operator method

Mori (’65)

(O1, O2) ⌘ 1 β Z β dτheτHO1e−τHO†

2i

hOi ⌘ tre−βHO tre−βH

gnm(x − y) ≡ (An(0, x), Am(0, y))

Z d3ygnm(x − y)gml(y − z) = δl

nδ(3)(x − y).

Expectation value: Inner product:

An(t, x) ≡

• d3ygnm(x − y)Am(t, y)

gml(x − y) = δ2βΓ(An) δAl(y)δA†

m(x)

≡ Γ ml(x − y),

29

Projection operator method

Mori (’65)

Projection operator

Rn(t, x) ≡ eitQLQiLAn(0, x) Kn

m(t, x − y) ≡ θ(t)(Rn(t, x), Rm(0, x)))

Memory function Noise

with Q = 1 − P

PB(t, x) ≡ Z d3yAn(0, y)(B(t, x), An(0, y))

30

∂ ∂teiLt = eiLtiL = eiLtPiL + eiLtQiL

1.

31

∂ ∂teiLt = eiLtiL = eiLtPiL + eiLtQiL

1.

1 z − iL = 1 z − iL(z − QiL) 1 z − QiL = 1 z − iL(z − iL + PiL) 1 z − QiL = 1 z − QiL + 1 z − iLPiL 1 z − QiL

2.

31

∂ ∂teiLt = eiLtiL = eiLtPiL + eiLtQiL

1.

1 z − iL = 1 z − iL(z − QiL) 1 z − QiL = 1 z − iL(z − iL + PiL) 1 z − QiL = 1 z − QiL + 1 z − iLPiL 1 z − QiL

2.

eiLt = eQiLt + Z t dseiL(t−s)PiLeQiLs

31

∂ ∂teiLt = eiLtiL = eiLtPiL + eiLtQiL

1.

1 z − iL = 1 z − iL(z − QiL) 1 z − QiL = 1 z − iL(z − iL + PiL) 1 z − QiL = 1 z − QiL + 1 z − iLPiL 1 z − QiL

2.

eiLt = eQiLt + Z t dseiL(t−s)PiLeQiLs

∂ ∂teiLt = eiLtPiL + Z t dseiL(t−s)PiLeQiLsQiL + eQiLtQiL.

• 1. and 2.

31

∂ ∂teiLt = eiLtiL = eiLtPiL + eiLtQiL

1.

1 z − iL = 1 z − iL(z − QiL) 1 z − QiL = 1 z − iL(z − iL + PiL) 1 z − QiL = 1 z − QiL + 1 z − iLPiL 1 z − QiL

2.

eiLt = eQiLt + Z t dseiL(t−s)PiLeQiLs

∂ ∂teiLt = eiLtPiL + Z t dseiL(t−s)PiLeQiLsQiL + eQiLtQiL.

• 1. and 2.

An(0) Multiplying Generalized Langevin Eq.

31

{An, Am}P

ih[An, Am]i H(An)

Γ(An)

Correspondence

Classical Hamiltonian formalism Projection

• perator

method

32

Nambu-Goldstone Theorem

33

Spontaneous symmetry breaking

det Mnφ = 0

: broken charges : would-be NG fields

symmetry breaking We choose

i[Qa, φi(t, x)] [Mnφ]ai,

Qa =

• d3xna(t, x)

φi(t, x)

Am = {δna(t, x), δφi(t, x)}

34

∂0 φ n

• =

Mφφ Mφn Mnφ Mnn Γ φφ Γ φn Γ nφ Γ nn φ n

• At k = 0

35

∂0 φ n

• =

Mφφ Mφn Mnφ Mnn Γ φφ Γ φn Γ nφ Γ nn φ n

• At k = 0

35

∂0 φ n

• =

Mφφ Mφn Mnφ Mnn Γ φφ Γ φn Γ nφ Γ nn φ n

• At k = 0

=

• MφnΓ nφ

MφnΓ nn MnφΓ φφ + MnnΓ nφ MnφΓ φn + MnnΓ nn φ n

• 35

∂0 φ n

• =

Mφφ Mφn Mnφ Mnn Γ φφ Γ φn Γ nφ Γ nn φ n

• At k = 0

=

• MφnΓ nφ

MφnΓ nn MnφΓ φφ + MnnΓ nφ MnφΓ φn + MnnΓ nn φ n

• F −1 = MφnΓ nn

G = MnnM −1

φn

and where

=

• −F −1G

F −1 φ n

• 35

∂0 φ n

• =

Mφφ Mφn Mnφ Mnn Γ φφ Γ φn Γ nφ Γ nn φ n

• At k = 0

=

• MφnΓ nφ

MφnΓ nn MnφΓ φφ + MnnΓ nφ MnφΓ φn + MnnΓ nn φ n

• F −1 = MφnΓ nn

G = MnnM −1

φn

and where

=

• −F −1G

F −1 φ n

• Nmassive = 1

2rank(F −1G) = 1 2rank(Mnn)

35

∂0 φ n

• =

Mφφ Mφn Mnφ Mnn Γ φφ Γ φn Γ nφ Γ nn φ n

• At k = 0

=

• MφnΓ nφ

MφnΓ nn MnφΓ φφ + MnnΓ nφ MnφΓ φn + MnnΓ nn φ n

• F −1 = MφnΓ nn

G = MnnM −1

φn

and where

=

• −F −1G

F −1 φ n

• Nmassive = 1

2rank(F −1G) = 1 2rank(Mnn)

Watanabe Brauner’s conjecture

NBS NNG = Nmassive = 1 2rank([Qa, Qb])

35

∂0 φ n

• =

−F −1G F −1 φ n

• 36

∂2

0φ = F −1G∂0φ ∂0 φ n

• =

−F −1G F −1 φ n

• 36

∂2

0φ = F −1G∂0φ

F −1GφI = 0

Ntype-I = NBS − rank(Mnn)

∂2

0φI = 0

Type-I NG modes:

∂0 φ n

• =

−F −1G F −1 φ n

• 36

∂2

0φ = F −1G∂0φ

F −1GφI = 0

Ntype-I = NBS − rank(Mnn)

∂2

0φI = 0

Type-I NG modes:

∂0 φ n

• =

−F −1G F −1 φ n

• thers

Ntype-II = 1 2rank(Mnn)

Type-II NG modes:

G∂0φII = 0

36

∂2

0φ = F −1G∂0φ

F −1GφI = 0

Ntype-I = NBS − rank(Mnn)

∂2

0φI = 0

Type-I NG modes:

Saturation of Nielsen and Chadha inequality

Ntype-I + 2Ntype-II = NBS

∂0 φ n

• =

−F −1G F −1 φ n

• thers

Ntype-II = 1 2rank(Mnn)

Type-II NG modes:

G∂0φII = 0

36

Mass relation

Explicit breaking term VI = φihi

∂0φ = F −1Gφ + F −1n, ∂0na = δ[Mnφ]aj δφi hj,

37

Mass relation

Explicit breaking term VI = φihi

∂0φ = F −1Gφ + F −1n, ∂0na = δ[Mnφ]aj δφi hj,

Type-I sector

m2

type-I = F −1iΩn φ = O(h)

Type-II sector

m2

type-II = O(h2)

37

{An, Am}P = Mnm

Finite Temperature

38

{An, Am}P = Mnm

Finite Temperature

Onsager coefficient

Lnm(k) = Z ∞ dt Z β dτ Z d3xe−ix·khRn(t iτ, x)Rm(0, 0)i

Rn(t, x) ≡ eitQLQiLAn(0, x)

{An, Am}T ≡ Mnm − Lnm

LAn ≡ [H, An]

38

G = MnnM −1

φn

and where

if

Finite Temperature

= ✓ − ˜ F −1G ˜ F −1 ◆ ✓ φ n ◆

At k = 0

˜ F −1 = (Mφn − Lφφ(Mnφ)−1Mnn)Γ nn

NBS NNG = Nmassive = 1 2rank([Qa, Qb])

39

Effective Lagrangian approach

Watanabe, Murayama (’12) Leutwyler(’94)

Writing down all possible terms

L = 1 2ρabπa ˙ πb + ¯ gab 2 ˙ πa ˙ πb − gab 2 ∂iπa∂iπb

+ higher oders

40

Effective Lagrangian approach

Watanabe, Murayama (’12) Leutwyler(’94)

Writing down all possible terms

For nonrelativstic case, One time derivative term may appear

L = 1 2ρabπa ˙ πb + ¯ gab 2 ˙ πa ˙ πb − gab 2 ∂iπa∂iπb

+ higher oders

40

Effective Lagrangian approach

Watanabe, Murayama (’12) Leutwyler(’94)

Writing down all possible terms

For nonrelativstic case, One time derivative term may appear

L = 1 2ρabπa ˙ πb + ¯ gab 2 ˙ πa ˙ πb − gab 2 ∂iπa∂iπb

+ higher oders

Lagrangian is invariant under symmetry transformation up to surface term

Watanabe, Murayama (’12)

ρab / ih[Qa, j0

b (x)]i

40

Analogy between type-II NG mode and classical dynamics with Berry phase

{xi, pj}P = δij

{pi, pj}P = 0

{xi, xj}P = ✏ijkΩk

δna ↔ xi

δφi ↔ pi

S = Z dt( ˙ x · p − ˙ p · A(p) − H(x, p))

Berry curvature

Berry phase

41

Summary

Ntype-I + 2Ntype-II = NBS

NBS NNG = 1 2rankh[Qa, Qb]i Ntype-II = 1 2rankh[Qa, Qb]i

42

Examples

QCD at finite T

L = ¯ ψ(iγµ(∂µ − igAµ) − mq)ψ − 1 2trFµνF µν

43

Examples

SU(2)L × SU(2)R → SU(2)V

QCD at finite T

L = ¯ ψ(iγµ(∂µ − igAµ) − mq)ψ − 1 2trFµνF µν

43

Examples

SU(2)L × SU(2)R → SU(2)V

QCD at finite T

Broken charge densities : n5a ≡ − ¯

ψγ0γ5 τa 2 ψ

Order parameter : NG fields :

h ¯ ψψi

πa ⌘ ¯ ψiγ5τaψ h ¯ ψψi

L = ¯ ψ(iγµ(∂µ − igAµ) − mq)ψ − 1 2trFµνF µν

43

(∂2

t v2 πr2 Γr2∂t + m2 π)πa = 0

QCD at finite T

44

(∂2

t v2 πr2 Γr2∂t + m2 π)πa = 0

v2

π = f 2

χ5

• cf. screening mass: m2 = mqh ¯

ψψi f 2

Γ = D5 + Dπ

Son and Stephanov (’01)

Pion mass: Velocity: Damping:

QCD at finite T

44

(∂2

t v2 πr2 Γr2∂t + m2 π)πa = 0

1 f 2 δab k2 + m2 = Z β dτ Z d3xe−ik·xhπa(iτ, x)πb(0, 0)i

v2

π = f 2

χ5

• cf. screening mass: m2 = mqh ¯

ψψi f 2

Γ = D5 + Dπ

Son and Stephanov (’01)

Pion mass: Velocity: Damping:

QCD at finite T

44

(∂2

t v2 πr2 Γr2∂t + m2 π)πa = 0

1 f 2 δab k2 + m2 = Z β dτ Z d3xe−ik·xhπa(iτ, x)πb(0, 0)i

D5 = 1 9χ5 X

i,a

Z ∞ dt Z β dτ Z d3xhji

5a(t iτ, x)ji 5a(0, 0)i

Dπ = f 2 3 X

a

Z ∞ dt Z β dτ Z d3xhRπa(t iτ, x)Rπa(0, 0)i

v2

π = f 2

χ5

• cf. screening mass: m2 = mqh ¯

ψψi f 2

Γ = D5 + Dπ

Son and Stephanov (’01)

Pion mass: Velocity: Damping:

QCD at finite T

44

Example

U(2) model at finite density

L = |(∂0 − iµ)φ|2 + |∂iφ|2 + m2|φ|2 − λ|φ|4 − hϕ0

nµ = i(∂0τµφ†)φ − iφ†(τµ∂0φ) τµ = (1, σi) Fields: Charge densities

φ = 1 √ 2 ϕ0 + iϕ3 ϕ2 + iϕ1

• 45

Example

U(2) model at finite density

U(2) → U(1)

NG fields: ϕ1, ϕ2, ϕ3

n0

3 = 1

2(n0 + n3) n1, n2,

Order parameters:

hϕ0i = ν

hn0

3i = µν2

Symmetry breaking:

L = |(∂0 − iµ)φ|2 + |∂iφ|2 + m2|φ|2 − λ|φ|4 − hϕ0

nµ = i(∂0τµφ†)φ − iφ†(τµ∂0φ) τµ = (1, σi) Fields: Charge densities

φ = 1 √ 2 ϕ0 + iϕ3 ϕ2 + iϕ1

• 45

ih[ϕa(x), nb(y)]i = νδabδ(x y) ih[n1(x), n2(y)]i = µν2δ(x y)

At tree level: commutation relation

+1 2 X

a

h (∂iϕa)2 + m2

hϕ2 a

i

m2

h = h

ν

Heff = 1 2ν2 h (n1 − 2µνϕ2)2 + (n2 + 2µνϕ1)2 + µ2 + m2 3µ2 + m2 n2

3

i

46

ih[ϕa(x), nb(y)]i = νδabδ(x y) ih[n1(x), n2(y)]i = µν2δ(x y)

At tree level: commutation relation

+1 2 X

a

h (∂iϕa)2 + m2

hϕ2 a

i

m2

h = h

ν

Type-I: Type-II:

Ek = ✓ µ2 + m2 3µ2 + m2 (k2 + m2

h)

◆1/2

Ek = k2 + m2

h

2µ

Heff = 1 2ν2 h (n1 − 2µνϕ2)2 + (n2 + 2µνϕ1)2 + µ2 + m2 3µ2 + m2 n2

3

i

46