Nambu-Goldstone Bosons in Nonrelativistic Systems

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Nov. 17, 2015 Nambu and Science Frontier @Osaka

1 Nov. 17, 2015 Nambu and Science (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

2 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

3 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

4 Spontaneous Symmetry Breaking \ Now she can use both hands equally well

$Spontaneous Symmetry Breaking \ Now$

5 Spontaneous Symmetry Breaking \ Now she can use both hands equally well

$Spontaneous Symmetry Breaking \ Now$

6 Spontaneous Symmetry Breaking \ Now she can use both hands equally well

7 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

8 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 (nbg) = the number of flat directions the number of Nambu-Goldstone Bosons (nng) G = U(1) H = {e} G/H =U(1) = S 1 (, ) G = SO(3) H = SO(2) G/H = S 2

9 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 (nbg) = the number of flat directions the number of Nambu-Goldstone Bosons (nng) G = U(1) H = {e} G/H =U(1) = S 1 (, ) G = SO(3) H = SO(2) G/H = S 2

10 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 (nbg) = the number of flat directions the number of Nambu-Goldstone Bosons (nng) G = U(1) H = {e} G/H =U(1) = S 1 (, ) G = SO(3) H = SO(2) G/H = S 2

11 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 (nbg) = the number of flat directions the number of Nambu-Goldstone Bosons (nng) G = U(1) H = {e} G/H =U(1) = S 1 (, ) G = SO(3) H = SO(2) G/H = S 2

12 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) Absence of Lorentz invariance = the number of broken symmetry generators (nbg) = the number of flat directions the number of Nambu-Goldstone Bosons (nng) G = U(1) H = {e} G/H =U(1) = S 1 (, ) G = SO(3) H = SO(2) G/H = S 2

13 Classic example: magnets (, ) G = SO(3) H = SO(2) G/H = S 2 nbg = 2 Antiferromagnet Ferromagnet ε(k) ε(k) k k G H nbg nng dispersion Antiferromagnet SO(3) SO(2) 2 2 k Ferromagnet SO(3) SO(2) 2 1 k 2

14 More recent examples in condensed matter physics Spinor BEC Skyrmion crystal π2(s 2 )= Z Y. Nii Ed Marti D.M. Stamper-Kurn, PRL (2014) Zang, Mostovoy, Han, Nagaosa PRL (2011) Petrova, Tchernyshyov, PRB (2011) G H nbg nng dispersion Spinor BEC U(1) SO(3) U(1) 3 2 k and k 2 Skyrmion crystals R 3 R 1 (translation) 2 1 k 2

15 Miransky-Shavkovy PRL (2002) Schäfer et al. PLB (2001) Higher energy examples L = D µ D µ m 2 g 2 ( ) 2 =( 1, 2 ) T D + iµ,0 (μ: chemical potential) h i = v(0, 1) T U(2) U(1) nbg = 3

16 Miransky-Shavkovy PRL (2002) Schäfer et al. PLB (2001) Higher energy examples L = D µ D µ m 2 g 2 ( ) 2 =( 1, 2 ) T D + iµ,0 (μ: chemical potential) h i = v(0, 1) T U(2) U(1) nbg = 3 only two NGBs

17 Miransky-Shavkovy PRL (2002) Schäfer et al. PLB (2001) Higher energy examples L = D µ D µ m 2 g 2 ( ) 2 =( 1, 2 ) T D + iµ,0 (μ: chemical potential) h i = v(0, 1) T U(2) U(1) nbg = 3 only two NGBs

18 Miransky-Shavkovy PRL (2002) Schäfer et al. PLB (2001) Higher energy examples L = D µ D µ m 2 g 2 ( ) 2 =( 1, 2 ) T D + iµ,0 (μ: chemical potential) h i = v(0, 1) T U(2) U(1) nbg = 3 only two NGBs

19 Miransky-Shavkovy PRL (2002) Schäfer et al. PLB (2001) Higher energy examples 1, T ) 2 D + iµ m 2 g ( 2 )2,0 (μ: chemical potential) h i = v(0, 1)T U(2) U(1) nbg = 3 only two NGBs =( D µ L = Dµ

20 Questions

21 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?

22 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) h[q a,q b ]i6=0 Their zero modes are canonical conjugate (not independent) But how do we prove this?

23 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) h[q a,q b ]i6=0 Their zero modes are canonical conjugate (not independent) But how do we prove this? We clarified all of these points using low-energy effective Lagrangian.

24 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) (, )

25 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) SO(3,1): L = 1 2 g ab( )@ µ µ b (, )

26 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) SO(3,1): SO(3): L = 1 2 g ab( )@ µ µ b (, ) L = c a ( ) a + 1 2ḡab( ) a b 1 2 g ab( )r a r b

27 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) linearize SO(3,1): L = 1 2 g ab( )@ µ µ b (, ) L = c a ( ) a + 2ḡab( ) 1 a b 1 SO(3): 2 g ab( )r a r b 1 2 ab a b + O( 3 ) ρ: real and skew matrix

28 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) linearize SO(3,1): L = 1 2 g ab( )@ µ µ b (, ) L = c a ( ) a + 2ḡab( ) 1 a b 1 SO(3): 2 g ab( )r a r b 1 2 ab a b + O( 3 ) ρ: real and skew matrix i ab = h[q a,j 0 b (~x, t)]i = lim!1 1 h[q a,q b ]i Ω: volume of the system

29 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) linearize SO(3,1): L = 1 2 g ab( )@ µ µ b L = c a ( ) a + 2ḡab( ) 1 a b 1 SO(3): p 2 g ab( )r a r b b = 1 2 ab a 1 2 ab a b + O( 3 ) ρ: real and skew matrix (, ) i ab = h[q a,j 0 b (~x, t)]i = lim!1 1 h[q a,q b ]i Ω: volume of the system

linearize SO(3,1): L = 1 2 g ab( )@ µ a @ µ b L = c a ( ) a + 2ḡab( ) 1 a b 1 SO(3): p 2 g ab( )r a r b b = @L @ b = 1 2 ab a 1 2 ab a b + O( 3 ) ρ: real and

30 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) linearize SO(3,1): L = 1 2 g ab( )@ µ µ b L = c a ( ) a + 2ḡab( ) 1 a b 1 SO(3): p 2 g ab( )r a r b b = 1 2 ab a 1 2 ab a b + O( 3 ) ρ: real and skew matrix (, ) i ab = h[q a,j 0 b (~x, t)]i = lim!1 1 h[q a,q b ]i number Ω: volume of the system dispersion type-b (paired) NGBs type-a (unpaired) NGBs (total) NGBs nb = (1/2) rank[ρ] na = dim(g/h) - rank[ρ] na + nb = dim(g/h) (1/2)rank[ρ]

31 Dispersion relation Linearized Lagrangian L = 1 2 ab a b + 1 2ḡab(0) a b 1 2 g ab(0)r a r b + ω ω 2 k 2 c.f. Nielsen-Chadha counting rule na+2nb dim(g/h), Nucl. Phys. B(1976) number dispersion type-b (paired) NGBs type-a (unpaired) NGBs (total) NGBs nb = (1/2) rank[ρ] na = dim(g/h) - rank[ρ] na + nb = dim(g/h) (1/2)rank[ρ]

32 Dispersion relation Linearized Lagrangian L = 1 2 ab a b + 1 2ḡab(0) a b 1 2 g ab(0)r a r b + ω ω 2 k 2 c.f. Nielsen-Chadha counting rule na+2nb dim(g/h), Nucl. Phys. B(1976) number dispersion type-b (paired) NGBs type-a (unpaired) NGBs (total) NGBs nb = (1/2) rank[ρ] na = dim(g/h) - rank[ρ] na + nb = dim(g/h) (1/2)rank[ρ] k 2

33 Dispersion relation Linearized Lagrangian L = 1 2 ab a b + 1 2ḡab(0) a b 1 2 g ab(0)r a r b + ω ω 2 k 2 c.f. Nielsen-Chadha counting rule na+2nb dim(g/h), Nucl. Phys. B(1976) number dispersion type-b (paired) NGBs type-a (unpaired) NGBs (total) NGBs nb = (1/2) rank[ρ] na = dim(g/h) - rank[ρ] na + nb = dim(g/h) (1/2)rank[ρ] k 2 k

34 Examples i ab = h[q a,j 0 b (~x, t)]i = lim!1 1 h[q a,q b ]i Ω: volume of the system <[Sx,Sy]> = i<sz> 0 (ferro) <[Sx,Sy]> = i<sz> = 0 (antiferro) Similar for Spinor BEC and Kaon condensation <[Px,Py]> = 0 (ordinary crystals) <[Px,Py]> = i W[n] (skyrmion crystals) number Y. Nii HW, Hitoshi Murayama, PRL (2014) dispersion type-b (paired) NGBs type-a (unpaired) NGBs (total) NGBs nb = (1/2) rank[ρ] na = dim(g/h) - rank[ρ] na + nb = dim(g/h) (1/2)rank[ρ] k 2 k

35 NGB for space-time symmetries? Active ongoing researches T. Hayata, Y. Hidaka, Broken spacetime symmetries and elastic variables, PLB (2014) Y. Hidaka, T. Noumi and G. Shiu, Effective field theory for spacetime symmetry breaking, PRD (2015)

36 NGB for space-time symmetries? Active ongoing researches T. Hayata, Y. Hidaka, Broken spacetime symmetries and elastic variables, PLB (2014) Y. Hidaka, T. Noumi and G. Shiu, Effective field theory for spacetime symmetry breaking, PRD (2015) NGBs of internal symmetry well-defined irrespective of any details of the system NGB Matter field Decouple at low-e limit! k, ω k, ω v k, k q, ν lim ~ k0! ~ k v ~k0, ~ k =0 e.g. scattering of electrons off of NGBs

NGB for space-time symmetries? Active ongoing researches T. Hayata, Y. Hidaka, Broken spacetime symmetries and elastic variables, PLB (2014) Y. Hidaka, T. Noumi and G.

37 NGB for space-time symmetries? Active ongoing researches T. Hayata, Y. Hidaka, Broken spacetime symmetries and elastic variables, PLB (2014) Y. Hidaka, T. Noumi and G. Shiu, Effective field theory for spacetime symmetry breaking, PRD (2015) NGBs of internal symmetry well-defined irrespective of any details of the system NGB Matter field Decouple at low-e limit! k, ω k, ω v k, k q, ν lim ~ k0! ~ k v ~k0, ~ k =0 e.g. scattering of electrons off of NGBs This nice property is lost if symmetries are space-time Goldstone modes: overdamped. does not propagate as particles electrons: lose properties of Fermi liquid Non-Fermi liquid Nematic Fermi fluid Oganesyan-Kivelson-Fradkin, PRB (2001) General condition for anomalous coupling: [Qa, P] 0 HW, Ashvin Vishwanath, PNAS (2014)

38 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

39 Question Nambu-Goldstone theorem: As a result of spontaneous breaking of continuous symmetries, massless particles appear

c.f. Absence of Quantum Time crystals: HW & Masaki Oshikawa, PRL (2015) A partial answer for systems

40 Question Nambu-Goldstone theorem: As a result of spontaneous breaking of continuous symmetries, massless particles appear But in what condition can we expect symmetries to be spontaneously broken? c.f. Absence of Quantum Time crystals: HW & Masaki Oshikawa, PRL (2015) A partial answer for systems with finite density of particles Not necessarily continuous symmetries finite density QCD (Fukushima / Masuda)

41 Systems with finite particle density Bosons at finite density <ψ (x) ψ(x)> in free space U(1) symmetry breaking <ψ(x)> 0 Superfluid, BEC Electrons at Half filling (i.e. particles per site) charge density wave (on lattice)

42 Systems with finite particle density Bosons at finite density <ψ (x) ψ(x)> in free space U(1) symmetry breaking <ψ(x)> 0 Superfluid, BEC Electrons at Half filling (i.e. particles per site) charge density wave (on lattice)

43 Systems with finite particle density Bosons at finite density <ψ (x) ψ(x)> in free space U(1) symmetry breaking <ψ(x)> 0 Superfluid, BEC Electrons at Half filling (i.e. particles per site) charge density wave (on lattice)

44 Systems with finite particle density Bosons at finite density <ψ (x) ψ(x)> in free space U(1) symmetry breaking <ψ(x)> 0 Superfluid, BEC Electrons at Half filling (i.e. particles per site) charge density wave (on lattice)

45 Oshikawa-Hasting theorem (Lieb-Shultz-Mattis in higher dimension) Assume U(1) charge conservation Lattice translation e i Pa } Filling ν: # of particles per unit cell

46 Oshikawa-Hasting theorem (Lieb-Shultz-Mattis in higher dimension) Assume U(1) charge conservation Lattice translation e i Pa } Filling ν: # of particles per unit cell If ν is not an integer, the system cannot realize a gapped unique ground state

47 Oshikawa-Hasting theorem (Lieb-Shultz-Mattis in higher dimension) Assume U(1) charge conservation Lattice translation e i Pa } Filling ν: # of particles per unit cell If ν is not an integer, the system cannot realize a gapped unique ground state Gapped: degeneracies due to Spontaneous symmetry breaking (Topological order (exotic)) frustrated magnet E Δ

48 Oshikawa-Hasting theorem (Lieb-Shultz-Mattis in higher dimension) Assume U(1) charge conservation Lattice translation e i Pa } Filling ν: # of particles per unit cell If ν is not an integer, the system cannot realize a gapped unique ground state Gapped: degeneracies due to Spontaneous symmetry breaking (Topological order (exotic)) frustrated magnet Gapless: Metal (Fermi surface) Goldstone modes Gauge bosons (Fractional excitation (exotic)) E E Δ

49 Oshikawa-Hasting theorem (Lieb-Shultz-Mattis in higher dimension) Assume U(1) charge conservation Lattice translation e i Pa } Filling ν: # of particles per unit cell If ν is not an integer, the system cannot realize a gapped unique ground state Gapped: degeneracies due to Spontaneous symmetry breaking (Topological order (exotic)) frustrated magnet Gapless: Metal (Fermi surface) Goldstone modes Gauge bosons (Fractional excitation (exotic)) E E Δ

50 Flux threading argument Oshikawa PRL (2000) Hθ = 0 ψθ = 0 y Ny x Nx

51 Flux threading argument Oshikawa PRL (2000) Hθ = 0 y Ny θ: fictitious magnetic flux x Nx ψθ = 0 x x θ/nx Adiabatic flux threading

52 Flux threading argument Oshikawa PRL (2000) Hθ = 0 y Ny θ: fictitious magnetic flux x Nx Hθ = 2π ψθ = 0 x x θ/nx Adiabatic flux threading ψθ = 2π

53 Flux threading argument Oshikawa PRL (2000) Hθ = 0 y Ny θ: fictitious magnetic flux x Nx Hθ = 2π ψθ = 0 x x θ/nx Adiabatic flux threading ψθ = 2π 2π flux is physically equivalent with 0

54 Flux threading argument Oshikawa PRL (2000) Hθ = 0 y Ny θ: fictitious magnetic flux x Nx Hθ = 2π ψθ = 0 x x θ/nx Adiabatic flux threading Gauge transform back U Hθ = 2π U -1 =Hθ = 2π ψθ = 2π 2π flux is physically equivalent with 0

55 Flux threading argument Oshikawa PRL (2000) Hθ = 0 y Ny θ: fictitious magnetic flux x Nx Hθ = 2π ψθ = 0 U ψθ = 2π x x θ/nx Adiabatic flux threading Gauge transform back U Hθ = 2π U -1 =Hθ = 2π ψθ = 2π 2π flux is physically equivalent with 0

56 Flux threading argument Oshikawa PRL (2000) Hθ = 0 y Ny θ: fictitious magnetic flux x Nx Hθ = 2π ψθ = 0 U ψθ = 2π ψθ = 0 and U ψθ = 2π x x θ/nx Adiabatic flux threading Gauge transform back U Hθ = 2π U -1 =Hθ = 2π ψθ = 2π 2π flux is physically equivalent with 0 1. Both are ground states of Hθ = 0 2. Different eigenvalues of translation e i Pa unless e i 2πν N y = 1 U = e 2πi dxdy x n(x,y)/ N x e ipa U e -ipa = e 2πi dxdy (x+1) n(x,y) / N x = U e 2πi ν N y Degeneracy unless ν is an integer!

57 Our work: Extension of Oshikawa-Hastings theorem Further assume some symmetries e.g. Time reversal symmetries (Non-Symmorphic) space groups HW, HC Po, A. Vishwanath, M. Zalatel, PNAS (2015) If ν is not an integer, the system cannot realize a gapped unique ground state

58 Our work: Extension of Oshikawa-Hastings theorem Further assume some symmetries e.g. Time reversal symmetries (Non-Symmorphic) space groups HW, HC Po, A. Vishwanath, M. Zalatel, PNAS (2015) even integer, 4 integer, depending on additional symmetries If ν is not an integer, the system cannot realize a gapped unique ground state

59 Summary Nambu-Goldstone theorem: - Non-perturbative general arguments based on symmetry - Applicable to any quantum many-body interacting systems Cool to find more. References [1] Nambu-Goldstone bosons: HW & Hitoshi Murayama, PRL (2012), PRX (2014), [2] Non-Fermi liquid in systems with broken space-time symmetry: HW & Ashvin Vishwanath, PNAS (2014) [3] Absence of Quantum Time crystals: HW & Masaki Oshikawa, PRL (2015) [4] Extension of Oshikawa-Hastings theorem: HW, H.C. Po, A. Vishwanath, M. Zaletel, PNAS (2015)

Formula for the Number of Nambu-Goldstone Modes

Formula for the Number of Nambu-Goldstone Modes Haruki Watanabe Department of Applied Physics, University of Tokyo, Tokyo 113-8656, Japan; email: haruki.watanabe@ap.t.u-tokyo.ac.jp arxiv:1904.00569v1 [cond-mat.other]