発表ファイル 数理物理・物性基礎論セミナー Watanabe 4 proof

Low-energy effective Lagrangians

(given) Hamiltonian, Lagrangian Þ (find) Ground state, symmetry breakig pattern.

(given) the symmetry breaking pattern Þ (find) the most general Lagrangian consistent with the symmetry.

NGBs as a local coordinate of the coset manifold G  H

- We want to discuss low-energy fluctuations inside the ground state manifold G_{H [}“flat directions” of the potential].

(Going outside of G_H Þ massive, amplitude

modes)

- The set of Goldstone fields Π^a [a = 1, 2, …, dimHGHL] is a local coordinate of GH.

- Π^a_{Hx, tL = C}^a (constant) corresponds to a different ground state. [condensation of Π^a shifts one ground state to another.]

- Inhomogeneous Π^a with long-wavelength modulation ® NGB.

The action of g = ã^{ä ΕiQi} on Π « transformation rule of Π under g Î G: Π^a® Π^a+ Εⁱhi^aHΠL.

This is a nonlinear realization of G. ¬ A hallmark of NGBs. hi= hi^a_{HΠL ¶}a, ¶a º ^¶

¶Π^a ¬ “vector” on the manifold G_H. Ahi^,hjE = fi j^khk^.

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H. Leutwyler, “Nonrelativistic effective Lagrangians” PRD 49, 3033

(1994)

Here we present the effective Lagrangian for global symmetries, not local symmetries. The original paper discussed the latter.

Ÿ Derivative expansion

We are interested in the low-energy, long-wavelength physics

® Expand the Lagrangian in the series of derivatives!

Ÿ Relativistic systems L_eff=¹

2^{ga bHΠL ¶ΜΠ}

a_¶Μ_Πb_{+ OI¶}4_M

- (internal) G invariance + (space-time) Poincare invariance

- Each Π^a corresponds to a mode Ω = c k. nNGB= dimHGHL is trivial. (An alternative “proof” of Nambu-Goldstone theorem)

Ÿ Nonrelativistic systems

Absence of Lorentz symmetry ® Looser constraint, more variety! key: Ω and k may scale differently.

We still demand space rotation for simplicity.

L_{eff= caHΠL Π} ^a₊ ¹

2^{ga bHΠL Π}

 a

Π ^b- ¹

2^{ga bHΠL ÑΠ}

a_{× Ñ Π}b_{+ OI¶}

t², ¶tѲ, Ñ⁴_M - (internal) G invariance + (space-time translation + space rotation).

- We have seen by examples that caHΠL Π ^a term may change the number and dispersion of NGBs.

Ÿ Symmetry requirement

What’s the condition for the invariance of L_eff under Π^a® Π^a+ Εⁱh_i^a_HΠL? For instance,

∆i_HcaΠ ^a_L

=_H∆ic_{aL Π} ^a+ c_b¶_t∆_iΠ^b

= ¶bcahi^bΠ ^a- ¶tcbhi^b+ ¶t_Ihi^bcb_M

=_H¶bca^{- ¶}acaL hi^bΠ

 a_{+ ¶} tIhi^bcbM º ¶t_Iei+ hi^bcb_M

Therefore, we need

H¶b^ca^{- ¶}a^cbL hi^{b= ¶}a^ei^.

In the same way, we can derive

- the differential equation for ei_HΠL

- the killing equation for g, g, but the calculation is tedious... Alternative way?

Ÿ Differential geometry HW&HM, PRX 2014

g_{HΠL º ga bHΠL d Π}^aÄ d Π^b, g_{ŽHΠL = ga bHΠL d Π}^aÄ d Π^b: G-invariant metrics on GH, i.e., Lhi^{g = L}hi^g

Ž= 0. c_{HΠL º c}aHΠL d Π^a: One form on G_H ® d c is a presymplectic two-form.

∆ ^L _{H L ¶}

d

L

e_L e

HΠL

M

e

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

L 

- ∆i is the Lie derivative L_hi with respect to the vector _hi= hi^a_{HΠL ¶}a. - Cartan's magic formula: L_hi= d Ι_hi+ Ι_hid (for arbitrary “forms”) - Interior product ΙX: W^r® W^r-1, ΙXΩ_HX1, …Xr-1_{L = ΩHX, X}1, …Xr-1_L.

In this language,

∆ic = L_hic =_{Hd Ιhi}+ Ι_hid_{L c = d HΙhi}c_{L + Ιhi}d c º d_HΙhic + ei_L.

Thus

Ι_hid c = d ei. ... (1)

Very simple! Let’s keep going...

The function eiHΠL introduced in Eq. (1) should satisfy

d_ILhiej_{M = LhiId e}j_{M = LhiIΙhj}d c_{M = f}i j^k_HΙhkd c_{L + ΙhjHLhi}d c_{L = f}i j^k_{Hd e}k_{L = dIf}i j^kek_M.

In the derivation, we used L_h

i^{d c =}Hd Ιhi^{+ Ι}hi^dL d c = d HΙhi^{d c}L + Ιhi^d

2_{c = d}2_e

i^{+ Ι}_hi^{d2c = 0.}

Hence, ei should obey L_h

i^{ej= Ι}hi^{d ej= fi j}

k_e

k+ zi j. ... (2) Combining Eqs. (1) and (2), we find

Ι_hiΙ_hjd c = Ι_hid e_j= f_{i j}^ke_k+ z_{i j}. ... (3)

Ÿ Noether current The Noether theorem:

ji⁰_{HxL =}

¶L

¶Π ^a∆iΠ

a_{- X}i_{= e}

i_{HΠL - g}a bHΠL Π ^bhiaHΠL, where ∆_iL = ¶_tIei+ h_i^bcbM = ¶tXⁱ.

Ρi jº -limV®¥ä¹

V ^{X0¤AQi, Qj}E  0\ = -X0¤Aä Qi, jj⁰HxLE  0\ = X0¤ ∆ijj⁰HxL  0\

= L_hiej_HΠL Π=0= fi j^kek_{H0L + z}i j,

... (4)

where we neglected quantum correlations, i.e., _XΠHxLⁿ\ = 0. (Will verify later; XΠHxL\ = 0 follows by definition.) - Eqs. (3) and (4) tell us that the c term is indeed related to e_{iHΠL and Ρa b}= f_{a b}ⁱe_{iH0L + za b}.

- In the original paper by Leutwlyer, zi j was not allowed because it was a local (gauge) theory. Here, we demand only the global invariance.

HW and Hitoshi Murayama, PRL (2012)

Ÿ Solution of Eqs. (3) and (4) around the origin Eq. (3) at Π = 0 and Eq. (4):

hi^a_{H0L h}j^b_{H0L H¶}bca- ¶acb_{L H0L = f}i j^kek_{H0L + z}i j= Ρi j,

which fixes the antisymmetric combination Ab a= ¶bca_{H0L - ¶}acbH0L. What’s the role of A? ca_{HΠL = c}a_{H0L + ¶}bca_{H0L Π}^b+ O_IΠ²_M

= caH0L +¹₂@H¶bca+ ¶acbL H0L + H¶bca- ¶acbL H0LD Π^{b+ O}IΠ²M º caH0L + ¹₂HSb a+ Ab aL Π^{b+ O}IΠ²M

L œ caHΠL Π ^{a= 1}₂^Ab aΠ^bΠ ^a+ ¶tBcaH0L Π^{a+ 1}₂^Sb a Π^bΠ^a

2 ^{F + OIΠ} 3_M

® A is the combination of our interest!

H0L ^dL

d

¹ 0

a

HŽ L

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I M

® A is the combination of our interest!

If we define Mi^{a= h}i^aH0L, Eq. (3) for broken components Hi = c, j = dL reads Mc^aMd^bAa b= -Ρc d ® Aa b= -Ρc dM^-1a

cM^-1b d. (We’ll explain why det M ¹ 0 below.)

Therefore, ca_{HΠL Π}

 a

=¹

2 ^{Ab aΠ} b_Π ^a_{+ ¶}

t_{@…D + OIΠ}³_M

= ¹

2 ^{Ρc dIΠ} b_M-1

b

dM IΠ ^aM-1acM + ¶^t@…D + OIΠ³M

= ¹

2 ^{Ρc dΠ}

Žd_Ў ^c

+ ¶_{t@…D + OIΠ}³_M, where Π^Ža= Π^bM^-1b

a. The field is now normalized as

∆bΠ^Ža= ∆bΠ^cM^-1c

a= hb^c_{HΠL M}^-1c

a=_@Mb^c+ O_{HΠLD M}^-1c

a= ∆b^a+ O_{HЎ L}.

Ÿ Number and dispersion of NGBs

To discuss “modes”, we focus on the quadratic part of L_eff. L_{free part=}¹

2 ^{Ρa bΠ} b_Π a

+¹

2 ^{ga bH0L Π}

 a

Π ^b-¹

2^{ga bH0L ÑΠ}

a_{× Ñ Π}b_{+ OIΠ}3_{, ¶}

t², ¶tѲ, Ñ⁴_M

Ρ =

0 -Λ₁ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ

Λ₁ 0 ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ

ƒ ƒ 0 -Λ2 ƒ ƒ ƒ ƒ ƒ ƒ

ƒ ƒ Λ2 0 ƒ ƒ ƒ ƒ ƒ ƒ

ƒ ƒ ƒ ƒ _{¸ ƒ} ƒ ƒ ƒ ƒ

ƒ ƒ ƒ ƒ ƒ 0 -Λm ƒ ƒ ƒ

ƒ ƒ ƒ ƒ ƒ Λm 0 ƒ ƒ ƒ

ƒ ƒ ƒ ƒ ƒ ƒ ƒ 0 ƒ ƒ

ƒ ƒ ƒ ƒ ƒ ƒ ƒ _{ƒ ¸ ƒ}

ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ 0

We classify NGBs into two classes: type A and type B.

- Type B NGBs are those produced by canonically-conjugate pairs: _IΠ¹, Π²_{M, IΠ}³, Π⁴_{M, …, IΠ}^{2 m-1}, Π^{2 m}_M. - The others are type A; i.e., each of Π^{2 m+1}, Π^{2 m+2}, …, Π^dimHGML produce their own NGB.

It follows by definition that

nB= number of pairs = m = ¹

2^{rank Ρ}

n_A= dimHGHL - 2 m = dimHGHL - rank Ρ Therefore,

nNGB^{= n}A^{+ n}B^{= dim}HGHL -¹₂^{rank Ρ.}

Ÿ Dispersion NGBs Type-B NGBs

L_eff=¹

2 ^{Ρa bΠ} b_Π a₊1

2^{ga bH0L Π}

 a_Π b_-1

2^{ga bH0L ÑΠ}

a_{× Ñ Π}b_+OIΠ3_{, ¶}

t², ¶t^Ñ2, Ñ⁴_M Balancing OHΩL term and OIk²M terms ® Ω µ k^2.

Type-A NGBs L_eff=¹

2 ^{Ρa bΠ} b_Π ^a₊¹

2^{ga bH0L Π}

 a

Π ^b-¹

2^{ga bH0L ÑΠ}

a_{× Ñ Π}b_+OIΠ3_{, ¶}

t^{2, ¶}t^{Ñ2, Ñ4}M Balancing O_IΩ²M term and OIk²M terms ® Ω µ k.

Equality version of Nielsen-Chadha n_A+ 2 n_B= dim_HGHL.

n _{L ¹ 0}

+ 1 M

H L

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nA+ 2 nB= dim_HGHL.

nA= nI and nB= n_{I I} provided that det gH0L ¹ 0. Even if we fine tune this parameter, this term will be automatically generated by changing scales (integrating out higher energy modes) unless it is prohibited by symmetry.

e.g., vortex lattice in a rotating superfluid in 2 + 1 D.

U(1) and (magnetic) 2D translations are spontaneously broken. L_eff=¹

2^{g Θ}

 2

-¹

2^{g 'IÑ} 2_ΘM²_.

HÑΘL² term is prohibited by the magnetic translation Θ ® Θ + W z^`_{´ x + …}

HW and Hitoshi Murayama, PRX 2014

- The analytic solution of c_HΠL - Scaling of interacitons, fluctuations

Ÿ Peparation 1: useful parametrization of GH U_{HΠL º ã}^{ä ΠaTa}.

Starting from a groud-state expectation value of the order parameter _XΦ\,

one can explore all possible ground-state expectation values by changing Π: XΦ\ HΠL = UHΠL XΦ\. Here, Ti is a matrix representation of g. We require the following normalization and the orthogonality:

T_i=Ρ: unbroken Ti=a: broken tr_ATΡ, TΣ_{E = Λ ∆}Α Β, tr_@Ta, TbD = Λ ∆a b, tr_ATΡ, Tb_{E = 0.}

- The unbroken generators are closed: _ATΡ, TΣ_{E = ä f}Ρ Σ^ΛTΛ

- Furthermore, in this choice of basis, _@TΣ, TaD is broken (i.e., @T^{Σ, Ta}D = ä f^{Σ abTb)} because

tr_A@TΣ, Ta_{D, T}Ρ_E

= tr_AHTΣTa-TaTΣ_{L T}Ρ- TΡ_HTΣTa-TaTΣ_LE

= tr_AT_ΣT_aT_Ρ-T_aT_ΣT_Ρ-T_ΡT_ΣT_a+T_ΡT_aT_ΣE

= tr_AT_ΡT_ΣT_a-T_ΣT_ΡT_a-T_aT_ΡT_Σ+T_aT_ΣT_ΡE

= tr_AITΡTΣ^-TΣTΡM^Ta^{- T}aI^TΡTΣ^-TΣTΡME

= tr_AATΡ, TΣE, TaE

= fΡ Σ^Λtr_@TΛTa_D

= 0.

- Thus broken generators form a (reducible) representation of H.

h Tah^-1= Da^b_{HhL T}b=

D^r=1_HhL ƒ ƒ ƒ

ƒ D^r=2_{HhL ƒ} ƒ

ƒ ƒ _¸ ƒ

ƒ ƒ ƒ D^r=R_{HhL a b} Tb.

- Transformation of Π’s under g Î G is fixed by g UHΠL = UHΠ'L hHΠ, gL, ^hHΠ, gL Î H. (Recall that hHΠ, gL XΦ\ = XΦ\.)

NGBs transform linearly under unbroken symmetry:

H L ^a

H L

HΠ ΕL ^{+ Ε}

¹ 0

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X \

NGBs transform linearly under unbroken symmetry: h U_{HΠL = h ã}^{ä ΠaTa}h^-1h = ã^{ä Πah Tah-1}h = ã^{ä ΠaDHhLabTb}h

® Π^a' = Π^bD_HhLb^a.

NGBs transform nonlinearly under broken symmetry: g U_{HΠL = g}^{i ΕaTa}ã^{ä ΠaTa}= ã^{äHΠa+Εa+OHΠ ΕLL Ta}ã^{i OHΕL TΑ}

® Π^a' = Π^a+Ε^a+ O_{HΠ ΕL.} [Recall Θ ® Θ + Ε under U(1)] This gives at least one example of parametrizations that satisfy det M ¹ 0.

Ÿ Peparation 2: Maurer-Cartan one-form

ΩHΠL º -ä U¾ HΠL d UHΠL ¬ building blocks of the effective Lagrangian. In component,

Ω_{HΠL = Ω}ⁱ_{HΠL Ti}= Ω_aⁱ_{HΠL d Π}^aT_i. Ω =_{A-ä ã}^{-ä s P}d ã^{ä s P}_Es=0^s=1, P = Π^aTa

= -ä_Ùs=0^s=1d s ^d

d s ^ã

-ä s P_{d ã}ä s P

= -ä_Ùs=0^s=1d s_9ã^{-ä s P}_{H-ä PL d ã}^{ä s P}+ ã^{-ä s P}d_{Iä P ã}^{ä s P}_M=

= _Ùs=0^s=1d s ã^{-ä s P}d P ã^{ä s P}

= _Ún=0^{¥ H-äLn}

n! JÙ0

1d s sⁿN @P, @P, …, @P, d P DDDHnL

= _Ún=0^{¥ H-äL}_Hn+1L!ⁿ @P, @P, …, @P, d P DDDHnL

=_9∆aⁱ+¹

2^Π b_f

b aⁱ+ O_IΠ²_{M= d Π}^aTi

Transformation rule of MC form:

ΩHΠL ® ΩHΠ'L = -ä @h UHΠL¾ g¾D d@g UHΠL h¾D

= -ä h UHΠL¾ Hg¾ d gL UHΠL h¾ - ä h @UHΠL¾ d UHΠLD h¾ - ä h d h¾

= h ΩHΠL h¾ - ä h d h¾ If we further define

- Ω¦= Ω^aTa: perpendicular to H (broken) - Ω_þ= Ω^ΑTΑ: parallel to H (unbroken) then,

Ω¦_{HΠ'L = h Ω}¦_{HΠL h¾},

Ω_{þHΠ'L = h Ωþ}HΠL h¾ - ä h d h¾^.

For example, we see

tr_@Ω¦_HΠLÄΩ¦HΠLD ® tr@h Ω^¦HΠL h¾Äh Ω^¦HΠL h¾D = tr@Ω^¦HΠLÄ Ω^¦HΠLD. are invariant under g. When Ta is reducible under H,

h Tah^-1= Da^b_{HhL T}b=

D^r=1_HhL ƒ ƒ ƒ

ƒ D^r=2_{HhL ƒ} ƒ

ƒ ƒ _¸ ƒ

ƒ ƒ ƒ D^r=R_HhL a b

Tb.

Then we have as many parameters as the number of blocks R: e.g. g =_Úr=1^Rf_r²tr_@Ω¦^HrL_HΠLÄΩ¦^HrL_HΠLD.

This is the lowest order effective Larangian in the field-theory textbook. (We need at least two Ω’s to contract Lorentz indices Μ and Ν.)

Ÿ Are there any other invariants?

We can guess based on the transformation rule: Ω¦_{HΠ'L = h Ω}¦_{HΠL h¾,}

h¾ HΠLD

H

Χ H2L

H 8e<

6 4_proof.nb

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Ω¦_{HΠ'L = h Ω}¦_{HΠL h¾,}

Ω_{þHΠ'L = h Ωþ}HΠL h¾ - ä h d h¾.

- tr_@Ω¦HΠLD is invariant, but usually vanishes (if semisimple ® traceless) [might be nonzero for the U(1) factor of a non- semisimple group, but then becomes a total derivative...]

- If H is abelian,

Ω_{þHΠL ® h Ωþ}HΠL h¾ - ä h d h¾ = ΩþHΠL + d Χ invariant up to a total derivative!

eg. ferromagnet [H = SO_H2L]

- More generally, generators that commute with H should work... e.g. SOH3L ® 8e< (I don’t know the model that realizes this pattern)

Ÿ Analitic solutions in terms of the Maurer-Cartan one-form Our answer for the case zi j= 0 (no central extentions)

c_{HΠL = -Ω}ⁱ_{HΠL e}i_{H0L + d Χ,}

Parameters eiH0L should satisfy

fΡ i^je_{jH0L = 0} ¬ invariance of e_jH0L under H.

This is needed because fΡ i^jejH0L = -ä YAQ^{Ρ, ji}H0LE] ¹ 0 would imply Q^Ρ is broken.

e.g. ferromagnet L œ ^S

a^dHcosΘ - 1L Φ  ⁼ _a^Sd

n^yn ^x-n^xn ^y

1+n^{z = -Xsz\ Ω} z_HΠL

The number of parameters ejH0L:

NC= the number of Cartan generators of G that commute with H. Therefore, rank Ρ cannot explore all possible values in the range 0 £ rank Ρ £¹

2^dimHGHL.

Ÿ Presymplectic two-form d Ω = -ä U¾_{HΠL d UHΠL}

= -äH-U¾ d UHΠL U¾Lïd UHΠL

= -äH-äL U¾ d UHΠLïH-äL U¾ d UHΠL

= -ä Ω_ïΩ

= ¹

2 ^{fj k}

i_Ωj_ïΩk_T i

Therefore, d c = -¹

Λ^{d Tr}@ΩHΠL eH0LD, eH0L º eⁱH0L Tⁱ

= -¹

Λ^Tr@d ΩHΠL eH0LD

= -¹

Λ^TrA 1 2 ^{fj k}

i_Ωj_ïΩk_T ie_H0LE

= -¹

2 ^{fj k} i_e

i_{H0L Ω}^j_ïΩ^k

= -¹

2 ^{Ρi jΩ} i_ïΩj

= -¹

2 ^{Ρa bΩ} a_ïΩb

= -¹

2 ^{Ρa bd Π}

a_{ïd Π}b_{+ OIΠ}3_M

Partially symplectic two-form. Reminiscent of the symplectic two-form W =_Úad q^a_{ïd p}b that specifies

@q^{a, p}bD = ä ∆^ab.

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Ÿ ^Fibration

See

- T. Brauner & S. Moroz, “Topological interactions of Nambu-Goldstone bosons in quantum many-body systems” PRD (2014)

- J.O. Andersen, T. Brauner, C. P. Hofmann, A. Vuorinen, “Effective Lagrangians for quantum many-body systems” JHEP (2014)

for more details on the (i) topological terms and (ii) higher order (derivative) terms in the effective Lagrangian.

Scaling of the interaction

Why is it OK to neglect interactions among NGBs?

Ÿ Scaling of type-A NGBs

S =_{Ù d}^dx d t_A¹₂g_{a bH0L Π} ^aΠ ^b- ¹

2^{ga bH0L ÑΠ}

a_{× Ñ Π}b_{E + …}

To keep the free part of the action invariant, we have to scale field as Π 'HΑ x, Α tL = Α^{1-d2 Π}Hx, tL.

Then, the most relevant interactions like _{Ù d}^dx d t Ñ²Π³, _{Ù d}^dx d t ¶t²Π³ scale as Α^-

d-1 2 _.

- When d > 1, the interaction becomes smaller and smaller as Α increases.

= 1

= 0 > 0

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- When d > 1, the interaction becomes smaller and smaller as Α increases.

- d = 1 is the strongly interacting case ® symmetry might be restored / gap might open. (1+1D Heisenberg AF)

- consistent with Coleman’s theorem in 1+1 D at T = 0, (Hohenberg-)Mermin-Wagner theorem for T > 0 in 2 spatial dimensions.

Ÿ Scaling of type-B NGBs S =_{Ù d}^dx d t_A¹

2 ^{Ρa bΠ} b_Π a

- ¹

2^{ga bH0L ÑΠ}

a_{× Ñ Π}b_{E + …}

To keep the free part of the action invariant, we have to scale field as Π '_{IΑ x, Α}²t_{M = Α}^-

d 2_{ΠHx, tL.}

Then, the most relevant interactions like _{Ù d}^dx d t Ñ²Π³, _{Ù d}^dx d t ¶tΠ³ scale as Α^-

d 2_.

- When d > 0, the interaction becomes smaller and smaller as Α increases. - Consistent with the fact that ferromagnet exists even in 1+1 D.

What if type-A,B are mixed? The velocity of type-A NGBs diverges??

Other related topics

Ÿ ⁿNGB for space-time symmetries

Crystals break spatial rotation. Why no rotational modes?

BEC breaks not only U(1), but also Galilean boosts. Why the latter does not produce independent NGBs? References

- Low&Manohar, “Spontaneously Broken Spacetime Symmetries and Goldstone’s Theorem” PRL (2012) - HW&HM, “Redundancies in Nambu-Goldstone Bosons” PRL (2013)

- Hayata&Hidaka, “Broken spacetime symmetries and elastic variables” PLB (2014) - …

Ÿ Higgs mechanism in nonrelativisic systems

When the symmetry is local, NGBs are “eaten” by gauge fields, which - becomes gapped.

- acquires the longitudinal dof.

How is this famous story modified in nonrelativistic systems in general? References

- V. Gusynin, V. Miransky, and I. Shovkovy, Phys. Lett. B 581, 82 (2004) / Mod. Phys. Lett. A 19, 1341 (2004). - Y. Hama, T. Hatsuda, and S. Uchino, Phys. Rev. D 83, 125009 (2011).

- S. Gongyo and S. Karasawa, Phys. Rev. D 90, 085014 (2014).

- HW&HM, “Spontaneously broken non-Abelian gauge symmetries in nonrelativistic systems” PRD(R) (2014)

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