Pierre Curie ( )

Transcript

1 前史 対称性と物理法則 結晶の分類 保存則 (conservation laws) Noether の定理 (1918) 選択則 (selection rules) Pierre Curie ( )

2 Wiedemann Effect (G. H. Wiedemann ) B, J t

3 P. Curie の理論 環境の対象性 =S env 現象の対象性 =S eff S env = S eff R z : B -> B J -> J t -> -t R x, R y : B -> -B J ->J t -> -t T: B -> -B J -> -J t -> -t

4 SSB: S L > S env = S eff 強磁性体 P. Weiss 1907 E. Ising 1925 W. Heisenberg 1928 Heisenberg 統一場理論 1959 H int = g J 5µ J 5 µ

5 SSBの一般的性質 Dynamical Degeneracy of the ground state symmetry Continuous symmetry continuous degeneracy Degrees of freedom N (thermodynamic limit) Superselection rule NG (Nambu-Goldstone) modes (restoration of lost symmetry) No. of NG modes = No. of broken symmetry operations (with exceptions) Spin wave 結晶の中の音波 (phonon)

6 Finite systems NG modes -> low lying states ( rotational band ) polyatomic molecules (Jahn-Teller 1937) strong coupling meson theory / Skyrme model (Wentzel 1940 / Skyrme 1961) BCS pairing in Nuclei

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8 BCS mechanism Fermion Cooper pairing -> mass gap formation original dynamics (high energy) scale SSB (low energy) scale NG (π) and Higgs (σ) modes Mass relations (quasi-supersymmetry) π: fermion: σ ~ 0 : 1 : 2 (weak coupling) (σ は一般一般に多数多数あってあって質量質量が異なる. sum rule: e.g. m(σ 1 2 ) + m(σ 2 ) 2 = 4m(f 2 ) Effective theories Ginzburg-Landau-Gell-Mann-Levy model Nonlinear σ model

9 Examples Superconductivity(SC) s wave d wave (high T c ) Superfluid 3 He p wave j = 0 (B phase), j = 2 (A phase) P-P and N-N pairing in nuclei s wave Interacting boson model (IBM) QCD-hadron chiral dynamics Weinberg-Salam electroweak theory Flavor dymanics (Higgs mechanism) Color SC

10 3 He (l =1, σ= 1) -> j = 0,1,2 B phase, j = 0 A phase j =2 l σ, l x σ, l i σ k IBM J = 3/2 shell (e.g. Pt), u(4) ~ O(6)xU(1) -> 6 bosons 3/2 x 3/2 φ,φ, j = 0, 5 -> a (φ φ) - v 2 ) 2 - b(φ m φ n - φ n φ m ) 2

11 Majorana mass term (ψ k ψ -k + ψ k ψ -k )

12 Superselection rule: Ψ=Π(α +β i i a i a -i ) 0 F >, α 2 +β 2 =1 (Ψ F,, Ψ) =Πα i 2 -> 0 >, -> massless (NG) excitations ω~1/λ

13 BCS Hamiltonian H int = -(G 2 /M²)(ψ ψ ψ ψ ) H int = -G(ψ ψ φ + ψ ψ φ) + M² φ φ Majorana mass (gap) term (ψ k ψ -k + ψ k ψ -k ) Gap equation 1 = (G 2 /M²)F(Λ,Δ), F BCS ~ CΔ 2 ln(λ/δ), Yukawa coupling 1/f² ~ F/ m 2 ~ C ln(λ/m) H = -G(ψ ψ φ + ψ ψ φ) + M² φ φ

14 Self-energy δm = mcg 2 ln(λ/m) If δm = m m = mcg 2 ln(λ/m) either m = 0, or 1 = Cg 2 ln(λ/m) δm = = mcg 2 ln(λ/m)

15 Effective σ model L= L fv + φ φ + f (φ ψ k ψ -k + φ ψ k ψ -k ) - f 2 (φ φ v 2 ) 2, fv = Δ

16 Effective GL-GL (σ) model = φ φ φ = f 4 φ 4 φ φ

17 Name v =<σ>(ev) Δ(ev) f supercondictor ~10³ 10 ³ ev ~ 0.1 ³He ~ ~ 0.05 QCD-chiral dynamics 100Mev 1Gev 10 nuclear pairing ~ 100Mev 1Mev ~ 10 ² standard model 250Gev 174Gev(t) 0.68

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19 Color superconductivity in high density quark matter B. Barrois (1977); S. Frautchi (1978) D. Bailin and A. Love(1981) M. Alford, K. Rajagopal, and F. Wilczek (1998) E_F = μ Majorana mass term qq (vector and scalar) 0 quark mass term qq Chiral SSB -> NG bosons = mesons Effect of (current) quark masses ->μ -> 2nd SSB (meson condensation) -> new NG bosons

20 Nielsen anomaly: No. of zero modes (NG bosons) N_z N = No. of generators in the coset G/H, ω = ck γ N = Σ i γ i H. B. Nielsen and S. Chadha, Nuc. Phys. B105: 445 (1976) SSB by a chemical potential H = H 0 μn, i Ψ Ψ/ t = (H 0 μn)ψ L: / t -> / t iμ V. A. Miransky and I. A. Shovkovy, hep-ph/ ; Phys. Rev. Lett. 88: (2002)

21 [Q i, Q j ] = iq 0 Q 0 = C [Z i, Z j ] = ic A U(2) model : Φ = (Φ (1), Φ (2) ), (K meson or Higgs ) L = ( ₀ + iμ)φ ( ₀ - iμ)φ - Φ Φ - m²φ Φ -λ(φ Φ)², H = Π Π + Φ Φ + m²φ Φ - μq₀ +λ(φ Φ)², Π = ₀ + iμ)φ, Π = ( ₀ - iμ)φ, Q₀ = - i(π Φ - Φ Π ) Φ(¹) = (x + iy)/ 2, Π(¹) = (p x + ip y )/ 2, Φ(²) = (u + iw)/ 2, Π(²) = (p u + ip w )/ 2.

22 H = 1/2[(p x +μy)² + (p y -μx)² + (p u +μw)² + (p w -μu)²] + (m²-μ²)(x² + y² + u² + w²)] +λ(x² + y² + u² + w²)²/4, p x +μy = 0, p y -μx = 0, p u +μw = 0, p w -μu = 0, m²x-μp y = m²y+μp x = m²u μp u = m²w+μp_u = -λ(x² + y² + u² + w²) -λr² or λr² =μ²- m² (>0), x -> v +x, p y -> μv + p y, H = (1/2)[(p x +μy)² + (p y -μx)²+ (p u + μw)² + (p w -μu)² + k²(x² + y² + u² + w²)](μ²- m²)x² -μ⁴/(4λ) + nonlinear terms = (p x ² + p y ² + p u ² + p w ²)/2 + (μ² + k²)(x² + y² + u² + w²)/2 + (μ²- m²)x² - p y - yp x + up w - wp u ) - μ⁴/(4λ) +

23 Q i Q i = Q i + Z i, Z₀ = Z₃ = v(p y +μx)+μv², Z₁ = v(p w +μu), Z₂ = v(p u -μw), [Z₁, Z₂] = 2iv²μ

24 The dispersion law Assumed: H_{col}l = A(p,q) + k²b(p,q), [A,B] 0 (unless A=0). (The kinetic energy causes excitations.) Then there are three possibilities: 1) A depends on both p and q: A = A(p,q), 2) A = A(p) (or equivalently A = A(q)), 3) A = 0. Case 1) At k = 0, H takes the form of a harmonic oscillator, so the frequency ω will go as const.+ O(k²) for small k. This is the massive (Higgs) mode. Case 2) A(p) (or A(q)) and B(p,q) together essentially form a harmonic oscillator, so ω k. This is the regular case of the NG mode with phononlike dispersion law. In the above example it holds for the charges Z₀ = Z₃ coming from the Φ(¹) component.

25 Case 3) This applies to charges Z₁ and Z₂ coming from Φ(²). If present, they would appear as conjugates and massive like the Case 1, which is a contradiction for zero modes. Hence A = 0, B = B(p,q), and ω k² trivially.

26 Examples: Ferromagnetism and antiferromagnetism S i = σ i /2, i = 1..3, O i = even σ i /2 - odd σ i /2, O₃ = C, S₃ =0. [S i,s j ] = [O i,o j ] = iɛ ijk S k, [S i,o j ] = iɛ ijk O k ω S k 2, ω O k

27 Breaking of Lorentz symmetry: D. Collady and V. A. Kostelecky (1998) (γ p m + gvγ₅γ z )ψ=0 The dispersion law (p+²-m²)(p ²-m²)= 4g²m²v², p ± z = p z ± gv, ω² = p²+m²+g²v²±2gv(p z ²+m²) 1/2 = p x ²+p y ²+((m²+k z ²) 1/2 ±gv)²

28 Dispersion law for a quasiparticle y x 2 5 4

29 Collective modes: broken gauge field L col = L μν V μ V ν, L μν = Ag μν k²+bk μ k ν +Cg μν k₃ + Dg 3μ g k²+eg 3ν 3μg 3ν k₃² + F(g 3μ k₃k ν +g 3ν k₃k μ ) + igɛ 3μνλ - M²g μν -ΔM²g 3μ g 3ν Induced Chern-Simons term: Gɛ 3μνλ μνλ k λ μνλ k λ S.M.Carrol, G.B.Fried, and R.Jackiw (90) Induced Schwinger term: [j₀(0),j_{i}(x)] = Cε 30ik k δ³(x), C gvλ

30 H = ap⁴ - bp² + V(x) dx/dt = 4ap³ - 2bp dp/dt = - dv/dx

31 f = ap⁴ - bp², a, b> p

32 f = ap⁴ - bp² p

33 Plot of x(t), H = ap⁴ - bp² + gx, p

34 The Unruh problem (G. W. Unruh 1976, S. Hawking 1974) Ω = ωv = ω(1 + gz/c²), or H =H(1+μz) This form of the Hamiltonian is for an observer moving with uniform acceleration μ in the z direction. The term μzh may be regarded as a "chemical potential" for Lorentz boost iµ(z t + t z ), t being set to 0. In quantum field theory it leads to the wave equation in the Unruh coordinates, leading to a thermal medium of T = 2πgµ. The present anomay would also apply to a coaccelerating medium and observer.

35 Unruh problem t=ξsinh(aη),x=ξcosh(aη)=ξ (1+(t/ξ)²) dt/dη=aξcosh(aη)=ax,dx/dη=aξsinh(aη)=at ds²=-dξ²+a²ξ²dη² exp( iet+ikx+iωη)dtdη = exp[ imx₀sinh(η±θ)+iωη]dη = exp[ imx₀sinhη+iωη]dη = exp[ imx₀sinhη+iωη]dη =2exp(±ωπ/2)K_{±iω}(mx₀) = real, K_{ν}=K_{-ν}

36 (γ p m + gv μ γ μ γ₅)ψ = 0 I. If V is spacelike: V μ =(0,0,0,v), ω²`= p x ² + p y ² + ((p z ²+m²) 1/2 ±gv)² II. If V is timelike: V μ = (v,0,0,0), ω² = m² + ( p ±gv)² III. If V is lightlike: V μ = (v,0,0,v): ω² = [±gv + (m² + p x ² + p y ² + (p z + gv)²) ½ ]²

37 Unruh coordinates x =ξsinhη, t = ξ sinhη