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

Nambu Goldstone theorem in nonrelativistic systems QCD in a magnetic field Yoshimasa Hidaka (RIKEN) 1

Keyword: Symmetry breaking Spontaneous Explicit Quantum 2

Keyword: Symmetry breaking Spontaneous Nambu-Goldstone theorem Explicit Magnetic field Quantum 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) Crystal Vibrations (Phonon) Edge modes in topological insulator Topological Insulator Electron(up spin) Electron(down spin) 4

Zero modes in nature Light (Photon) Gauge symmetry Crystal Vibrations (Phonon) Spontaneous symmetry breaking of translation Edge modes in topological insulator Topological Insulator Electron(up spin) Electron(down spin) Topological 4

Spontaneous Symmetry breaking V ( φ ) V ( φ ) φ φ Unbroken Broken 5

Nambu-Goldstone bosons Examples in hadron physics Pions and Kaons Chiral symmetry breaking Type-I NG modes p k 2 + m 2 E = NG mode in Kaon condensed color flavor locking phase Schafer, Son, Stephanov, Toublan, and Verbaarschot (’01) Miransky, Shovkovy (’02) E = a k 2 Type-II NG modes 6

Nambu-Goldstone bosons Other Examples Phonon in crystal ( Galilean, rotational, translational symmetries ) Phonon in superfluid (U(1) symmetry) Acoustic phonon? (Galilean symmetry) Magnon (Rotational symmetry) 7

Nambu-Goldstone theorem in Lorentz invariant systems Nambu(’60), Goldstone(61), Nambu Jona-Lasinio(’61), Goldstone, Salam, Weinberg(’62). N NG = N BS N BS : the number of broken symmetries N NG : the number of NG modes Dispersion relation E k = | k | 8

Generalization Nielsen and Chadha (’76) N type-I + 2 N type-II ≥ N BS 9

Generalization Nielsen and Chadha (’76) N type-I + 2 N type-II ≥ N BS Schafer, Son, Stephanov, Toublan, and Verbaarschot (’01) N NG = N BS h [ Q a , Q b ] i = 0 9

Generalization Nielsen and Chadha (’76) N type-I + 2 N type-II ≥ N BS Schafer, Son, Stephanov, Toublan, and Verbaarschot (’01) N NG = N BS h [ Q a , Q b ] i = 0 Watanabe and Brauner (’11) N BS � N NG � 1 2rank � [ Q a , Q b ] � 9

Example of type-II 1 1 2rank h [ Q a , Q b ] i N BS − N NG N BS N type-I N type-II N BS N type-I N tyep-II 2rank h [ Q a , Q b ] i N type-I + 2 N type-II Magnon 2 0 1 1 2 in Ferromagnet O(3) → O(2) NG mode in Kaon condensed color 3 1 1 1 3 flavor locked phase SU(2)xSU(1) Y → SU(2) em Kelvon in Vortex of 2 0 1 1 2 superfluid Translation P x , P y breaking Known examples satisfy the equalities N type-I + 2 N type-II = N BS N BS � N NG = 1 2rank h [ Q a , Q b ] i 10

The recent results Watanabe, Murayama (’12), YH (’12) N BS � N NG = 1 2rank h [ Q a , Q b ] i N type-I + 2 N type-II = N BS N type-II = 1 2rank h [ Q a , Q b ] i 11

Spontaneous symmetry breaking (SSB) Γ [ φ ] h [ φ i , Q a ] i = tr ρ [ φ i , Q a ] 6 = 0 tr ρ NG fields Conserved charges a = 1 , · · · , N BS ρ = | Ω ih Ω | Vacuum: φ Matter: ρ = exp( − β ( H − µN )) zero mode 12

Suppose the classical action is invariant under The effective action satisfies � i → � i + ✏ a [ Q a , � i ] Γ [ φ ] d d x δΓ [ φ ] Z δφ i ( x ) h [ Q a , φ i ( x )] i = 0 13

Suppose the classical action is invariant under The effective action satisfies � i → � i + ✏ a [ Q a , � i ] Γ [ φ ] d d x δΓ [ φ ] Z δφ i ( x ) h [ Q a , φ i ( x )] i = 0 δ 2 Γ [ φ ] Z d d x δφ j ( y ) δφ i ( x ) h [ Q a , φ i ( x )] i = 0 13

Suppose the classical action is invariant under The effective action satisfies � i → � i + ✏ a [ Q a , � i ] Γ [ φ ] d d x δΓ [ φ ] Z δφ i ( x ) h [ Q a , φ i ( x )] i = 0 δ 2 Γ [ φ ] Z d d x δφ j ( y ) δφ i ( x ) h [ Q a , φ i ( x )] i = 0 δ 2 Γ [ φ ] Inverse of propagators D − 1 ji ( y, x ) = φ j ( y ) δφ i ( x ) have zero eigenvalues. 13

Suppose the classical action is invariant under The effective action satisfies � i → � i + ✏ a [ Q a , � i ] Γ [ φ ] d d x δΓ [ φ ] Z δφ i ( x ) h [ Q a , φ i ( x )] i = 0 δ 2 Γ [ φ ] Z d d x δφ j ( y ) δφ i ( x ) h [ Q a , φ i ( x )] i = 0 δ 2 Γ [ φ ] Inverse of propagators D − 1 ji ( y, x ) = φ j ( y ) δφ i ( x ) have zero eigenvalues. The number coincides with the number of independent eigenvectors h [ Q a , φ i ( x )] i 13

For the Lorentz invariant system Goldstone, Salam, Weinberg (’62) h [ Q a , φ i ( x )] i ⌘ M ( a ) independent of x i ji ( p 2 = 0) M ( a ) D − 1 = 0 i N NG = the number of independent eigenvectors = N BS 14

In general, N BS ≠ the number of eigenvectors Low, and Manohar (’02) h [ Q a , φ i ( x )] i should be eigenvector of unbroken translation h [ Q a , φ i ( x )] i is not always the eigenvector. 15

In general, N BS ≠ the number of eigenvectors Low, and Manohar (’02) h [ Q a , φ i ( x )] i should be eigenvector of unbroken translation h [ Q a , φ i ( x )] i is not always the eigenvector. Example: Domain wall Broken symmetry: Translation ( P x ) h [ P x , φ ] i = ∂ x h φ i 6 = 0 h [ L z , φ ] i = � y ∂ x h φ i 6 = 0 Rotation ( L y , L z ) h [ L y , φ ] i = z ∂ x h φ i 6 = 0 h [ L y,z , φ ] i are not eigenvectors of P y , P z One NG mode exists associated with P x . 15

The number of independent eigenvectors is not always equal to the number of NG modes Example: Ferromagnet Spin rotation O(3) → O(2) Broken generator: S x , S y Two eigenvector: ✏ ijz h s z i One spin wave appears. 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

Intuitive example Pendulum has two oscillation motions if the top is not spinning. for type-II NG modes 18

Intuitive example If the top is spinning, the only one rotation motion (Precession) exists. In this case, for type-II NG modes { L x , L y } P = L z 6 = 0 19

NG theorem in Hamiltonian formalism 20

A simple Hamiltonian system { x, p } P = 1 H = a ( k ) 2 p 2 + b ( k ) 2 x 2 21

A simple Hamiltonian system { x, p } P = 1 H = a ( k ) 2 p 2 + b ( k ) 2 x 2 ∂ t x = { x, H } P = a ( k ) p ∂ t p = { p, H } P = − b ( k ) x 21

A simple Hamiltonian system { x, p } P = 1 H = a ( k ) 2 p 2 + b ( k ) 2 x 2 ∂ t x = { x, H } P = a ( k ) p ∂ t p = { 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

∂ 2 t x + a ( k ) b ( k ) x = 0 a ( k ) = a 0 + a 2 k 2 b ( k ) = b 0 + b 2 k 2 a ( k ) b ( k ) = a 0 b 0 + ( a 0 b 2 + a 2 b 0 ) k 2 + a 2 b 2 k 4 22

Energy gapped zero zero gapless (type-II) nonzero zero gapless (type-I) nonzero nonzero ∂ 2 t x + a ( k ) b ( k ) x = 0 a ( k ) = a 0 + a 2 k 2 b ( k ) = b 0 + b 2 k 2 a ( k ) b ( k ) = a 0 b 0 + ( a 0 b 2 + a 2 b 0 ) k 2 + a 2 b 2 k 4 b 0 a 0 p a 0 b 0 ∼ | k | ∼ k 2 cf. Nambu (’04) 22

{ x i , p j } P = δ ij ( i, j = 1 , · · · , N BS ) H = 1 2 p t H pp p + 1 2 x t H xx x 23

{ x i , p j } P = δ ij ( i, j = 1 , · · · , N BS ) H = 1 2 p t H pp p + 1 2 x t H xx x m 2 = H pp H xx Mass matrix: N massive = rank( m 2 ) The number of NG modes: N NG = N BS − N massive 23

✓ x ◆ ✓ x ◆ ∂ t = M H p p ✓ 0 ✓ H xx ◆ ◆ 0 1 where M = H = H pp 0 − 1 0 24

✓ x ◆ ✓ x ◆ ∂ t = M H p p ✓ 0 ✓ H xx ◆ ◆ 0 1 where H = M = H pp 0 − 1 0 half of the number of nonzero N massive = eigenvalues of M H The number of NG modes N NG = N BS − N massive 24

✓ x ◆ ✓ x ◆ ∂ t = M H p p ✓ M xx ✓ ◆ H xx H xp ◆ M xp where H = M = ( H xp ) t H pp − M t M pp xp half of the number of nonzero N massive = eigenvalues of M H The number of NG modes N NG = N BS − N massive 25

What are canonical variables? What is the Poisson bracket? What is the Hamiltonian? 26

Projection operator method Mori (’65) Operator set { A n } A m (0) A n (0) 27

Projection operator method Mori (’65) Operator set { A n } A m (0) QA n ( t ) A n ( t ) P A n ( t ) A n (0) 27

Projection operator method Mori (’65) ∂ t A n ( t, k ) = i [ H, A n ( t, k )] 28

Projection operator method Mori (’65) ∂ t A n ( t, k ) = i [ H, A n ( t, k )] Generalized Langevin equation ∂ t A n ( t, k ) = M nm ( k ) Γ ml ( k ) A m ( t, k ) streaming � ∞ dsK nm ( t − s, k ) Γ ml ( k ) A l ( s, k ) − 0 dissipation + R n ( t, k ) noise 28

Expectation value: Inner product: Projection operator method Mori (’65) h O i ⌘ tr e − β H O tr e − β H Z β ( O 1 , O 2 ) ⌘ 1 d τ h e τ H O 1 e − τ H O † 2 i β 0 g nm ( x − y ) ≡ ( A n (0 , x ) , A m (0 , y )) Z d 3 yg nm ( x − y ) g ml ( y − z ) = δ l n δ (3) ( x − y ) . δ 2 βΓ ( A n ) g ml ( x − y ) = ≡ Γ ml ( x − y ) , δ A l ( y ) δ A † m ( x ) � d 3 yg nm ( x − y ) A m ( t, y ) A n ( t, x ) ≡ 29