Nambu-Goldstone Bosons in Nonrelativistic Systems Haruki Watanabe - - PowerPoint PPT Presentation

Nambu-Goldstone Bosons in Nonrelativistic Systems

Haruki Watanabe MIT Pappalardo fellow

• Nov. 17, 2015 “Nambu and Science Frontier”@Osaka (room H701)

15:20-17:30: Session on topics from Nambu to various scales

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

\

This ability will be lost as she grows Either right- or left-handed → SSB Now she can use both hands equally well

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)

(θ, φ)

G = U(1) H = {e} G/H =U(1) = S1 G = SO(3) H = SO(2) G/H = S2

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)

(θ, φ)

G = U(1) H = {e} G/H =U(1) = S1 G = SO(3) H = SO(2) G/H = S2

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)

(θ, φ)

G = U(1) H = {e} G/H =U(1) = S1 G = SO(3) H = SO(2) G/H = S2

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)

(θ, φ)

G = U(1) H = {e} G/H =U(1) = S1 G = SO(3) H = SO(2) G/H = S2

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)

(θ, φ)

G = U(1) H = {e} G/H =U(1) = S1 G = SO(3) H = SO(2) G/H = S2

Absence of Lorentz invariance

Classic example: magnets

k ε(k)

k ε(k)

Ferromagnet Antiferromagnet SO(3) → SO(2) SO(3) → SO(2)

nBG nNG 2 2 2 1 dispersion k k2 G → H

• Antiferromagnet
• Ferromagnet

(θ, φ)

G = SO(3) H = SO(2) G/H = S2

nBG = 2

More recent examples in condensed matter physics

• Skyrmion crystal
• Y. Nii

π2(S2)= Z Zang, Mostovoy, Han, Nagaosa PRL (2011) Petrova, Tchernyshyov, PRB (2011)

Spinor BEC Skyrmion crystals U(1)×SO(3) → U(1)’ R3 → R1 (translation)

nBG nNG 3 2 2 1 dispersion

…

k and k2 k2 G → H

Ed Marti … D.M. Stamper-Kurn, PRL (2014)

• Spinor BEC

Higher energy examples

Miransky-Shavkovy PRL (2002) Schäfer et al. PLB (2001)

L = Dµψ†Dµψ − m2ψ†ψ − g 2(ψ†ψ)2

Dν = ∂ν + iµδν,0 ψ = (ψ1, ψ2)T

(μ: chemical potential)

U(2)→U(1) nBG = 3

hψi = v(0, 1)T

Higher energy examples

Miransky-Shavkovy PRL (2002) Schäfer et al. PLB (2001)

L = Dµψ†Dµψ − m2ψ†ψ − g 2(ψ†ψ)2

Dν = ∂ν + iµδν,0 ψ = (ψ1, ψ2)T

(μ: chemical potential)

U(2)→U(1) nBG = 3

hψi = v(0, 1)T

• nly two NGBs

→ →

Higher energy examples

Miransky-Shavkovy PRL (2002) Schäfer et al. PLB (2001)

L = Dµψ†Dµψ − m2ψ†ψ − g 2(ψ†ψ)2

Dν = ∂ν + iµδν,0 ψ = (ψ1, ψ2)T

(μ: chemical potential)

U(2)→U(1) nBG = 3

hψi = v(0, 1)T

• nly two NGBs

→ →

Higher energy examples

Miransky-Shavkovy PRL (2002) Schäfer et al. PLB (2001)

L = Dµψ†Dµψ − m2ψ†ψ − g 2(ψ†ψ)2

Dν = ∂ν + iµδν,0 ψ = (ψ1, ψ2)T

(μ: chemical potential)

U(2)→U(1) nBG = 3

hψi = v(0, 1)T

• nly two NGBs

→ →

Higher energy examples

Miransky-Shavkovy PRL (2002) Schäfer et al. PLB (2001)

L = Dµψ†Dµψ − m2ψ†ψ − g 2(ψ†ψ)2

Dν = ∂ν + iµδν,0 ψ = (ψ1, ψ2)T

(μ: chemical potential)

U(2)→U(1) nBG = 3

hψi = v(0, 1)T

• nly two NGBs

→ →

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 canonical conjugate (not independent) But how do we prove this?

h[Qa, Qb]i 6= 0

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? We clarified all of these points using low-energy effective Lagrangian. Partial result in Y. Nambu, J. Stat. Phys. (2004) → Their zero modes are canonical conjugate (not independent) But how do we prove this?

h[Qa, Qb]i 6= 0

Effective Lagrangian

• Non-Linear sigma model with the target space G/H
• Capture low-energy, long-wavelength physics (derivative expansion)

θ (θ, φ)

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 (θ, φ)

L = 1 2gab(π)∂µπa∂µπb

SO(3,1):

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 (θ, φ)

L = 1 2gab(π)∂µπa∂µπb

SO(3,1):

L = ca(π) ˙ πa + 1 2 ¯ gab(π) ˙ πa ˙ πb 1 2gab(π)rπa · rπb

SO(3):

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 (θ, φ)

L = 1 2gab(π)∂µπa∂µπb

SO(3,1):

L = ca(π) ˙ πa + 1 2 ¯ gab(π) ˙ πa ˙ πb 1 2gab(π)rπa · rπb

SO(3): = −1 2ρabπa ˙ πb + O(π3)

ρ: real and skew matrix

linearize

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 (θ, φ)

L = 1 2gab(π)∂µπa∂µπb

SO(3,1):

L = ca(π) ˙ πa + 1 2 ¯ gab(π) ˙ πa ˙ πb 1 2gab(π)rπa · rπb

SO(3): = −1 2ρabπa ˙ πb + O(π3)

ρ: real and skew matrix

i⇢ab = h[Qa, j0

b (~

x, t)]i = lim

Ω→∞

1 Ωh[Qa, Qb]i

Ω: volume of the system

linearize

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 (θ, φ)

L = 1 2gab(π)∂µπa∂µπb

SO(3,1):

L = ca(π) ˙ πa + 1 2 ¯ gab(π) ˙ πa ˙ πb 1 2gab(π)rπa · rπb

SO(3): = −1 2ρabπa ˙ πb + O(π3)

ρ: real and skew matrix

pb = ∂L ∂ ˙ πb = −1 2ρabπa i⇢ab = h[Qa, j0

b (~

x, t)]i = lim

Ω→∞

1 Ωh[Qa, Qb]i

Ω: volume of the system

linearize

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 (θ, φ)

L = 1 2gab(π)∂µπa∂µπb

SO(3,1):

L = ca(π) ˙ πa + 1 2 ¯ gab(π) ˙ πa ˙ πb 1 2gab(π)rπa · rπb

SO(3): = −1 2ρabπa ˙ πb + O(π3)

ρ: real and skew matrix

pb = ∂L ∂ ˙ πb = −1 2ρabπa

type-A (unpaired) NGBs type-B (paired) NGBs

dispersion number

nA = dim(G/H) - rank[ρ] nB = (1/2) rank[ρ] nA + nB = dim(G/H) – (1/2)rank[ρ] (total) NGBs

i⇢ab = h[Qa, j0

b (~

x, t)]i = lim

Ω→∞

1 Ωh[Qa, Qb]i

Ω: volume of the system

linearize

HW, Tomas Brauner PRD (2014) HW, Hitoshi Murayama PRL (2012), PRX (2014) c.f. Hidaka PRL (2013)

Dispersion relation

L = 1 2ρabπa ˙ πb + 1 2 ¯ gab(0) ˙ πa ˙ πb 1 2gab(0)rπa · rπb + · · ·

ω k2 ω2

type-A (unpaired) NGBs type-B (paired) NGBs

dispersion number

nA = dim(G/H) - rank[ρ] nB = (1/2) rank[ρ] nA + nB = dim(G/H) – (1/2)rank[ρ] (total) NGBs

Linearized Lagrangian

c.f. Nielsen-Chadha counting rule nA+2nB ≥ dim(G/H), Nucl. Phys. B(1976)

Dispersion relation

L = 1 2ρabπa ˙ πb + 1 2 ¯ gab(0) ˙ πa ˙ πb 1 2gab(0)rπa · rπb + · · ·

ω k2 ω2

type-A (unpaired) NGBs type-B (paired) NGBs

dispersion number

nA = dim(G/H) - rank[ρ] nB = (1/2) rank[ρ] nA + nB = dim(G/H) – (1/2)rank[ρ] (total) NGBs k2

Linearized Lagrangian

c.f. Nielsen-Chadha counting rule nA+2nB ≥ dim(G/H), Nucl. Phys. B(1976)

Dispersion relation

L = 1 2ρabπa ˙ πb + 1 2 ¯ gab(0) ˙ πa ˙ πb 1 2gab(0)rπa · rπb + · · ·

ω k2 ω2

type-A (unpaired) NGBs type-B (paired) NGBs

dispersion number

nA = dim(G/H) - rank[ρ] nB = (1/2) rank[ρ] nA + nB = dim(G/H) – (1/2)rank[ρ] (total) NGBs k2 k

Linearized Lagrangian

c.f. Nielsen-Chadha counting rule nA+2nB ≥ dim(G/H), Nucl. Phys. B(1976)

Examples

• <[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)

i⇢ab = h[Qa, j0

b (~

x, t)]i = lim

Ω→∞

1 Ωh[Qa, Qb]i

Ω: volume of the system

• Y. Nii

HW, Hitoshi Murayama, PRL (2014)

type-A (unpaired) NGBs type-B (paired) NGBs

dispersion number

nA = dim(G/H) - rank[ρ] nB = (1/2) rank[ρ] nA + nB = dim(G/H) – (1/2)rank[ρ] (total) NGBs k2 k

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)
• …

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)
• …

NGB Matter field

• NGBs of internal symmetry

→ well-defined irrespective of any details of the system

v⃗

k′,⃗ k

⃗ k′, ω′ ⃗ k, ω ⃗ q, ν

lim

~ k0→~ k

v~

k0,~ k= 0

e.g. scattering of electrons off of NGBs

Decouple at low-E limit!

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)
• …

NGB Matter field

• NGBs of internal symmetry

→ well-defined irrespective of any details of the system

v⃗

k′,⃗ k

⃗ k′, ω′ ⃗ k, ω ⃗ q, ν

lim

~ k0→~ k

v~

k0,~ k= 0

e.g. scattering of electrons off of NGBs

Decouple at low-E limit!

Oganesyan-Kivelson-Fradkin, PRB (2001)

General condition for anomalous coupling: [Qa, P] ≠ 0

Nematic Fermi fluid

HW, Ashvin Vishwanath, PNAS (2014)

• 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

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

Question

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

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?

• A partial answer for systems with finite density of particles
• Not necessarily continuous symmetries

c.f. Absence of Quantum Time crystals: HW & Masaki Oshikawa, PRL (2015) finite density QCD (Fukushima / Masuda)

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)

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)

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)

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)

Oshikawa-Hasting theorem

(Lieb-Shultz-Mattis in higher dimension)

Filling ν: # of particles per unit cell Assume

• U(1) charge conservation
• Lattice translation ei Pa

}

Oshikawa-Hasting theorem

(Lieb-Shultz-Mattis in higher dimension)

If ν is not an integer, the system cannot realize a gapped unique ground state Filling ν: # of particles per unit cell Assume

• U(1) charge conservation
• Lattice translation ei Pa

}

Oshikawa-Hasting theorem

(Lieb-Shultz-Mattis in higher dimension)

If ν is not an integer, the system cannot realize a gapped unique ground state Filling ν: # of particles per unit cell Assume

• U(1) charge conservation
• Lattice translation ei Pa

}

Δ E Gapped: degeneracies due to

• Spontaneous symmetry breaking
• (Topological order (exotic))

frustrated magnet

Oshikawa-Hasting theorem

(Lieb-Shultz-Mattis in higher dimension)

If ν is not an integer, the system cannot realize a gapped unique ground state E Gapless:

• Metal (Fermi surface)
• Goldstone modes
• Gauge bosons
• (Fractional excitation (exotic))

Filling ν: # of particles per unit cell Assume

• U(1) charge conservation
• Lattice translation ei Pa

}

Δ E Gapped: degeneracies due to

• Spontaneous symmetry breaking
• (Topological order (exotic))

frustrated magnet

Oshikawa-Hasting theorem

(Lieb-Shultz-Mattis in higher dimension)

If ν is not an integer, the system cannot realize a gapped unique ground state E Gapless:

• Metal (Fermi surface)
• Goldstone modes
• Gauge bosons
• (Fractional excitation (exotic))

Filling ν: # of particles per unit cell Assume

• U(1) charge conservation
• Lattice translation ei Pa

}

Δ E Gapped: degeneracies due to

• Spontaneous symmetry breaking
• (Topological order (exotic))

frustrated magnet

Flux threading argument

Oshikawa PRL (2000)

|ψθ = 0〉 Hθ = 0 y x

Ny Nx

Flux threading argument

Oshikawa PRL (2000)

|ψθ = 0〉 Hθ = 0 y x

Ny Nx

ーAdiabatic flux threading→

∂x → ∂x − θ/Nx

θ: fictitious magnetic flux

Flux threading argument

Oshikawa PRL (2000)

|ψθ = 0〉 Hθ = 2π |ψθ = 2π〉 Hθ = 0 y x

Ny Nx

ーAdiabatic flux threading→

∂x → ∂x − θ/Nx

θ: fictitious magnetic flux

Flux threading argument

Oshikawa PRL (2000)

|ψθ = 0〉 Hθ = 2π |ψθ = 2π〉 Hθ = 0 y x

Ny Nx 2π flux is physically equivalent with 0

ーAdiabatic flux threading→

∂x → ∂x − θ/Nx

θ: fictitious magnetic flux

Flux threading argument

Oshikawa PRL (2000)

|ψθ = 0〉 Hθ = 2π |ψθ = 2π〉 Hθ = 0 y x

Ny Nx

←Gauge transform backー

U Hθ = 2π U-1 =Hθ = 2π

2π flux is physically equivalent with 0

ーAdiabatic flux threading→

∂x → ∂x − θ/Nx

θ: fictitious magnetic flux

Flux threading argument

Oshikawa PRL (2000)

|ψθ = 0〉 Hθ = 2π |ψθ = 2π〉 U|ψθ = 2π〉 Hθ = 0 y x

Ny Nx

←Gauge transform backー

U Hθ = 2π U-1 =Hθ = 2π

2π flux is physically equivalent with 0

ーAdiabatic flux threading→

∂x → ∂x − θ/Nx

θ: fictitious magnetic flux

Flux threading argument

|ψθ = 0〉and U|ψθ = 2π〉

• 1. Both are ground states of Hθ = 0
• 2. Different eigenvalues of translation ei Pa unless ei 2πν Ny = 1

U = e2πi ∫dxdy x n(x,y)/ Nx eiPa U e-iPa = e2πi ∫dxdy (x+1) n(x,y) / Nx = U e2πi ν Ny ⇨ Degeneracy unless ν is an integer!

Oshikawa PRL (2000)

|ψθ = 0〉 Hθ = 2π |ψθ = 2π〉 U|ψθ = 2π〉 Hθ = 0 y x

Ny Nx

←Gauge transform backー

U Hθ = 2π U-1 =Hθ = 2π

2π flux is physically equivalent with 0

ーAdiabatic flux threading→

∂x → ∂x − θ/Nx

θ: fictitious magnetic flux

Our work: Extension of Oshikawa-Hastings theorem

Further assume some symmetries e.g. Time reversal symmetries (Non-Symmorphic) space groups If ν is not an integer, the system cannot realize a gapped unique ground state

HW, HC Po, A. Vishwanath, M. Zalatel, PNAS (2015)

Our work: Extension of Oshikawa-Hastings theorem

Further assume some symmetries e.g. Time reversal symmetries (Non-Symmorphic) space groups If ν is not an integer, the system cannot realize a gapped unique ground state

even integer, 4×integer, … depending on additional symmetries

HW, HC Po, A. Vishwanath, M. Zalatel, PNAS (2015)

Summary

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)

Nambu-Goldstone theorem:

• Non-perturbative general arguments based on symmetry
• Applicable to any quantum many-body interacting systems

Cool to find more.