Nov. 17, 2015 “Nambu and Science Frontier”@Osaka (room H701) 15:20-17:30: Session on topics from Nambu to various scales Nambu-Goldstone Bosons in Nonrelativistic Systems Haruki Watanabe MIT Pappalardo fellow
Plan 1. Nambu-Goldstone modes in nonrelativistic systems (20 mins) 2. Extension of Oshikawa-Hastings-Lieb-Schultz- Mattis theorem (10 mins) ✔ Spontaneous symmetry breaking ✖ Nambu-Goldstone modes
Plan 1. Nambu-Goldstone modes in nonrelativistic systems (20 mins) 2. Extension of Oshikawa-Hastings-Lieb-Schultz- Mattis theorem (10 mins) ✔ Spontaneous symmetry breaking ✖ Nambu-Goldstone modes
Spontaneous Symmetry Breaking \ Now she can use both hands equally well
Spontaneous Symmetry Breaking \ Now she can use both hands equally well
Spontaneous Symmetry Breaking \ Now she can use both hands equally well
Spontaneous Symmetry Breaking \ Now she can use both hands equally well This ability will be lost as she grows Either right- or left-handed → SSB
Spontaneous breaking of continuous symmetry → ‘Flat’ directions • Lie group G: symmetry of Lagrangian (‘laws of physics’) • Lie group H: symmetry of ground state (state realized in Nature) • Coset space G/H: the manifold of degenerate ground states • dim(G/H) = dim(G) – dim(H) = the number of broken symmetry generators (n BG ) = the number of flat directions ≠ the number of Nambu-Goldstone Bosons (n NG ) G = U(1) G = SO(3) H = {e} H = SO(2) G/H =U(1) = S 1 G/H = S 2 ( θ , φ )
Spontaneous breaking of continuous symmetry → ‘Flat’ directions • Lie group G: symmetry of Lagrangian (‘laws of physics’) • Lie group H: symmetry of ground state (state realized in Nature) • Coset space G/H: the manifold of degenerate ground states • dim(G/H) = dim(G) – dim(H) = the number of broken symmetry generators (n BG ) = the number of flat directions ≠ the number of Nambu-Goldstone Bosons (n NG ) G = U(1) G = SO(3) H = {e} H = SO(2) G/H =U(1) = S 1 G/H = S 2 ( θ , φ )
Spontaneous breaking of continuous symmetry → ‘Flat’ directions • Lie group G: symmetry of Lagrangian (‘laws of physics’) • Lie group H: symmetry of ground state (state realized in Nature) • Coset space G/H: the manifold of degenerate ground states • dim(G/H) = dim(G) – dim(H) = the number of broken symmetry generators (n BG ) = the number of flat directions ≠ the number of Nambu-Goldstone Bosons (n NG ) G = U(1) G = SO(3) H = {e} H = SO(2) G/H =U(1) = S 1 G/H = S 2 ( θ , φ )
Spontaneous breaking of continuous symmetry → ‘Flat’ directions • Lie group G: symmetry of Lagrangian (‘laws of physics’) • Lie group H: symmetry of ground state (state realized in Nature) • Coset space G/H: the manifold of degenerate ground states • dim(G/H) = dim(G) – dim(H) = the number of broken symmetry generators (n BG ) = the number of flat directions ≠ the number of Nambu-Goldstone Bosons (n NG ) G = U(1) G = SO(3) H = {e} H = SO(2) G/H =U(1) = S 1 G/H = S 2 ( θ , φ )
Spontaneous breaking of continuous symmetry → ‘Flat’ directions • Lie group G: symmetry of Lagrangian (‘laws of physics’) • Lie group H: symmetry of ground state (state realized in Nature) • Coset space G/H: the manifold of degenerate ground states • dim(G/H) = dim(G) – dim(H) = the number of broken symmetry generators (n BG ) = the number of flat directions Absence of Lorentz invariance ≠ the number of Nambu-Goldstone Bosons (n NG ) G = U(1) G = SO(3) H = {e} H = SO(2) G/H =U(1) = S 1 G/H = S 2 ( θ , φ )
Classic example: magnets G = SO(3) H = SO(2) G/H = S 2 ( θ , φ ) n BG = 2 Antiferromagnet Ferromagnet • • ε ( k ) ε ( k ) k k G → H n BG n NG dispersion SO(3) → SO(2) k 2 2 Antiferromagnet SO(3) → SO(2) 2 k 2 Ferromagnet 1
More recent examples in condensed matter physics Skyrmion crystal • Spinor BEC • π 2 (S 2 )= Z Y. Nii Zang, Mostovoy, Han, Nagaosa PRL (2011) Ed Marti … D.M. Stamper-Kurn, PRL (2014) Petrova, Tchernyshyov, PRB (2011) G → H n BG n NG dispersion U(1) × SO(3) → U(1)’ k and k 2 3 2 Spinor BEC R 3 → R 1 (translation) k 2 2 1 Skyrmion crystals …
Miransky-Shavkovy PRL (2002) Schäfer et al. PLB (2001) Higher energy examples L = D µ ψ † D µ ψ − m 2 ψ † ψ − g 2( ψ † ψ ) 2 h ψ i = v (0 , 1) T U(2) → U(1) ψ = ( ψ 1 , ψ 2 ) T n BG = 3 D ν = ∂ ν + iµ δ ν , 0 ( μ : chemical potential)
Miransky-Shavkovy PRL (2002) Schäfer et al. PLB (2001) Higher energy examples L = D µ ψ † D µ ψ − m 2 ψ † ψ − g 2( ψ † ψ ) 2 h ψ i = v (0 , 1) T U(2) → U(1) ψ = ( ψ 1 , ψ 2 ) T → only two NGBs n BG = 3 → D ν = ∂ ν + iµ δ ν , 0 ( μ : chemical potential)
Miransky-Shavkovy PRL (2002) Schäfer et al. PLB (2001) Higher energy examples L = D µ ψ † D µ ψ − m 2 ψ † ψ − g 2( ψ † ψ ) 2 h ψ i = v (0 , 1) T U(2) → U(1) ψ = ( ψ 1 , ψ 2 ) T → only two NGBs n BG = 3 → D ν = ∂ ν + iµ δ ν , 0 ( μ : chemical potential)
Miransky-Shavkovy PRL (2002) Schäfer et al. PLB (2001) Higher energy examples L = D µ ψ † D µ ψ − m 2 ψ † ψ − g 2( ψ † ψ ) 2 h ψ i = v (0 , 1) T U(2) → U(1) ψ = ( ψ 1 , ψ 2 ) T → only two NGBs n BG = 3 → D ν = ∂ ν + iµ δ ν , 0 ( μ : chemical potential)
Miransky-Shavkovy PRL (2002) Schäfer et al. PLB (2001) Higher energy examples L = D µ ψ † D µ ψ − m 2 ψ † ψ − g 2( ψ † ψ ) 2 h ψ i = v (0 , 1) T U(2) → U(1) ψ = ( ψ 1 , ψ 2 ) T → only two NGBs n BG = 3 → D ν = ∂ ν + iµ δ ν , 0 ( μ : chemical potential)
Questions
Questions • In general, how many NGBs appear as a result of G → H? • How many linear and quadratic modes? • What is the necessary input to predict the number and dispersion?
Questions • In general, how many NGBs appear as a result of G → H? • How many linear and quadratic modes? • What is the necessary input to predict the number and dispersion? Partial result in Y. Nambu, J. Stat. Phys. (2004) → Their zero modes are h [ Q a , Q b ] i 6 = 0 canonical conjugate (not independent) But how do we prove this?
Questions • In general, how many NGBs appear as a result of G → H? • How many linear and quadratic modes? • What is the necessary input to predict the number and dispersion? Partial result in Y. Nambu, J. Stat. Phys. (2004) → Their zero modes are h [ Q a , Q b ] i 6 = 0 canonical conjugate (not independent) But how do we prove this? We clarified all of these points using low-energy effective Lagrangian.
HW, Tomas Brauner PRD (2014) HW, Hitoshi Murayama PRL (2012), PRX (2014) c.f. Hidaka PRL (2013) Effective Lagrangian Non-Linear sigma model with the target space G/H - Capture low-energy, long-wavelength physics (derivative expansion) - ( θ , φ ) θ
HW, Tomas Brauner PRD (2014) HW, Hitoshi Murayama PRL (2012), PRX (2014) c.f. Hidaka PRL (2013) Effective Lagrangian Non-Linear sigma model with the target space G/H - Capture low-energy, long-wavelength physics (derivative expansion) - L = 1 2 g ab ( π ) ∂ µ π a ∂ µ π b SO(3,1): ( θ , φ ) θ
HW, Tomas Brauner PRD (2014) HW, Hitoshi Murayama PRL (2012), PRX (2014) c.f. Hidaka PRL (2013) Effective Lagrangian Non-Linear sigma model with the target space G/H - Capture low-energy, long-wavelength physics (derivative expansion) - L = 1 2 g ab ( π ) ∂ µ π a ∂ µ π b SO(3,1): ( θ , φ ) θ π a + 1 π b � 1 π a ˙ 2 g ab ( π ) r π a · r π b SO(3): L = c a ( π ) ˙ 2 ¯ g ab ( π ) ˙
HW, Tomas Brauner PRD (2014) HW, Hitoshi Murayama PRL (2012), PRX (2014) c.f. Hidaka PRL (2013) Effective Lagrangian Non-Linear sigma model with the target space G/H - Capture low-energy, long-wavelength physics (derivative expansion) - L = 1 2 g ab ( π ) ∂ µ π a ∂ µ π b SO(3,1): ( θ , φ ) θ π a + 1 π b � 1 π a ˙ 2 g ab ( π ) r π a · r π b SO(3): L = c a ( π ) ˙ 2 ¯ g ab ( π ) ˙ = − 1 2 ρ ab π a ˙ π b + O ( π 3 ) linearize ρ : real and skew matrix
HW, Tomas Brauner PRD (2014) HW, Hitoshi Murayama PRL (2012), PRX (2014) c.f. Hidaka PRL (2013) Effective Lagrangian Non-Linear sigma model with the target space G/H - Capture low-energy, long-wavelength physics (derivative expansion) - L = 1 2 g ab ( π ) ∂ µ π a ∂ µ π b SO(3,1): ( θ , φ ) θ π a + 1 π b � 1 π a ˙ 2 g ab ( π ) r π a · r π b SO(3): L = c a ( π ) ˙ 2 ¯ g ab ( π ) ˙ = − 1 2 ρ ab π a ˙ π b + O ( π 3 ) linearize ρ : real and skew matrix 1 i ⇢ ab = h [ Q a , j 0 b ( ~ x, t )] i = lim Ω h [ Q a , Q b ] i Ω : volume of the system Ω →∞
HW, Tomas Brauner PRD (2014) HW, Hitoshi Murayama PRL (2012), PRX (2014) c.f. Hidaka PRL (2013) Effective Lagrangian Non-Linear sigma model with the target space G/H - Capture low-energy, long-wavelength physics (derivative expansion) - L = 1 2 g ab ( π ) ∂ µ π a ∂ µ π b SO(3,1): ( θ , φ ) θ π a + 1 p b = ∂ L π b � 1 π b = − 1 π a ˙ 2 g ab ( π ) r π a · r π b 2 ρ ab π a SO(3): L = c a ( π ) ˙ 2 ¯ g ab ( π ) ˙ ∂ ˙ = − 1 2 ρ ab π a ˙ π b + O ( π 3 ) linearize ρ : real and skew matrix 1 i ⇢ ab = h [ Q a , j 0 b ( ~ x, t )] i = lim Ω h [ Q a , Q b ] i Ω : volume of the system Ω →∞
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