Nonrelativistic Nambu-Goldstone Modes Associated with Spontaneously BrokenSpace-Time and Internal Symmetries

Michikazu Kobayashi1 and Muneto Nitta21Department of Physics, Kyoto University, Oiwake-cho, Kitashirakawa, Sakyo-ku, Kyoto 606-8502, Japan

2Department of Physics and Research and Education Center for Natural Sciences, Keio University, Hiyoshi 4-1-1, Yokohama,Kanagawa 223-8521, Japan

(Received 19 March 2014; revised manuscript received 6 August 2014; published 18 September 2014)

We show that a momentum operator of a translational symmetry may not commute with an internalsymmetry operator in the presence of a topological soliton in nonrelativistic theories. As a strikingconsequence, there appears a coupled Nambu-Goldstone mode with a quadratic dispersion consisting oftranslational and internal zero modes in the vicinity of a domain wall in an O(3) σ model, a magneticdomain wall in ferromagnets with an easy axis.

DOI: 10.1103/PhysRevLett.113.120403 PACS numbers: 05.30.Jp, 03.75.Lm, 03.75.Mn, 11.27.+d

Introduction.—Symmetry is one of the most importantguiding principles in describing nature in quantum physics.In particular, relativistic quantum field theories rely onsymmetry principle and have been quite successful inunifying fundamental forces. Gauge symmetries of electro-magnetic, weak, and strong interactions can be unified toone group in grand unified theories. However, space-timesymmetry related to gravity could not be, because of theColeman-Mandula theorem [1] showing that internal sym-metry and space-time symmetry must be a direct product[2]. These are all based on relativistic field theories.In this Letter, we find a symmetry algebra including both

the space-time symmetry and an internal symmetry innonrelativistic field theory,

½P;Θ� ¼ W ≠ 0; ð1ÞwhereP is a translational operator,Θ is an internal symmetryoperator, and W is a “central” extension. We show that thecommutation relation, Eq. (1), is possible in the presence of atopological soliton in nonrelativistic theories. In a practicalmodel, the central charge W is a topological charge of adomain wall [3]. The central charge W vanishes in thecorresponding relativistic model. As a consequence of ournovel algebra, a Nambu-Goldstone (NG) mode for thetranslational symmetry (a ripple mode or ripplon if quan-tized) is coupled to that for the internal U(1) symmetry (amagnon) when the presence of a domain wall breaks boththe symmetries; They give rise to one NG mode with aquadratic dispersion relation, although two symmetry gen-erators are spontaneously broken. This is in contrast to thecorresponding relativistic model, in which these two modes,ripplon and magnon, appear independently with lineardispersion relations, which corresponds to the fact that Wvanishes. This phenomenon itself is already known as type-IIor type-B NG modes in nonrelativistic theories [4–8].However, previously known examples are either NG modesof both internal symmetries such as ferromagnets or those of

both space-time symmetry such as vortices and lumps(Skyrmions) [9,10], while our case mixes them togetherbecause of Eq. (1). We derive the dispersion relation in twoapproaches: the effective field theory for topological soli-tons and Bogoliubov theory. We also study a modelinterpolating between relativistic and nonrelativistic theoriesand find unexpectedly a coupled NG mode in an interpolat-ing region even though it has the Lorentz invariance.Models and a domain wall.—We start from the following

relativistic, and nonrelativistic CP1 Lagrangian densitiesLrel and Lnrel with an Ising-type potential,

Lrel ¼j _uj2 − j∇uj2 −m2juj2

ð1þ juj2Þ2 ;

Lnrel ¼iðu� _u − _u�uÞ2ð1þ juj2Þ −

j∇uj2 þm2juj2ð1þ juj2Þ2 ; ð2Þ

where u ∈ C is the complex projective coordinate definedas ϕT ¼ ð1; uÞT=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ juj2

pwith normalized two scalar

fields ϕ ¼ ðϕ1;ϕ2ÞT . Lrel and Lnrel are equivalent to O(3)nonlinear σ models,

Lrel ¼1

4fj _nj2 − j∇nj2 −m2ð1 − n23Þg;

Lnrel ¼_n1n2 − n1 _n22ð1þ n3Þ

−1

4fj∇nj2 þm2ð1 − n23Þg; ð3Þ

under the Hopf map for a three-vector of scalar fieldsn≡ ϕ†σϕ with the Pauli matrices σ. These models describeferromagnets with one easy axis.A Lagrangian density interpolating between Lrel and

Lnrel is given as the following form:

LG ¼ ϕ20

� j _uj2c2ð1þ juj2Þ2 −

j∇uj2ð1þ juj2Þ2

þ iMðu� _u − _u�uÞℏð1þ juj2Þ −

m2juj2ð1þ juj2Þ2

�; ð4Þ

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where ϕ20ð> 0Þ is a real positive (decay) constant having the

dimension of ½energy�=½length�. In the following, we omitϕ20 by measuring LG in the unit of ϕ2

0: LG → ϕ20LG. A

detailed derivation of the Lagrangian density LG is dis-cussed in the Supplemental Material [11]. Lrel is obtainedas the massless limit M → 0 of LG, while Lnrel is thenonrelativistic limit c → ∞ of LG [12].The actions S ¼ R

d4xLG are invariant under a globaldiscrete Z2 transformation: u↔1=u�, a global U(1) phaserotation: u → ueiα. S is also invariant under the Poincarétransformation as long as c is positive finite, or the Galileantransformation in the nonrelativistic limit c → ∞, as shownin the Supplemental Material [11]. There are two discretevacua juj ¼ 0 or juj → ∞, and m > 0 is defined as theenergy gap between them. For these vacua, the Z2

symmetry for the global discrete transformation is sponta-neously broken. In the framework of the nonlinear σmodels, Eq. (3), the vacua are expressed as n1 ¼ n2 ¼ 0,n3 ¼ �1, and the last m2ð1 − n23Þ terms are regarded as theIsing potential.Dynamics of u can be obtained by the Euler-Lagrange

equation for LG,

ð1þ juj2Þu − 2u� _u2

c2−2iMð1þ juj2Þ _u

ℏ¼ ð1þ juj2Þ∇2u − 2u�ð∇uÞ2 −m2ð1 − juj2Þu: ð5Þ

We next consider a static domain- or antidomain-wallsolution interpolating the two vacua. The flat and staticdomain-wall solution perpendicular to the z axis is [13] (seethe Supplemental Material [11])

u0 ¼ expfmðz − ZÞ þ iαg; ð6Þ

where α (0 ≤ α < 2π) and Z ∈ R are phase and transla-tional moduli of the domain wall. This is known as amagnetic domain wall in ferromagnets with an easy axis.In the presence of the domain wall, the H1 ≃ Uð1Þ ×R3

symmetry is further spontaneously broken, where U(1) isthe global symmetry for the internal phase rotation and R3

is the three-dimensional translational symmetry in a space.The remaining symmetry is H2 ≃ R2

xy, where R2xy indicates

the translation along the xy plane [14]. Breaking sym-metriesH1=H2 ≃ Uð1Þ ×Rz due to the domain wall are theinternal U(1) phase rotation and translation along the zdirection, and two moduli α and Z in Eq. (6) are regarded ascorresponding NG modes in the vicinity of the domainwall. The NG mode α is the phase mode known as amagnon localized in the domain wall. The other NG modefor Z is the translational surface mode of the domain wall,known as a ripple mode, or ripplon if quantized, incondensed matter physics. In the following, we show thatthe localized magnon and ripplon are coupled to each otherto become one “coupled ripplon” mode with fixeddispersion relation and amplitude.

Low-energy effective theory of a domain wall.—We nextconsider the NG modes excited along the domain wall byconstructing the effective theory on a domain wall by themoduli approximation [15]. Introducing r ¼ ðx; yÞ, and tdependences of two moduli α and Z as αðr; tÞ and Zðr; tÞ,we consider the ansatz u as

u ¼ exp½mfz − Zðr; tÞg þ iαðr; tÞ�: ð7Þ

Inserting Eq. (7) to Eq. (4), the effective Lagrangian LeffG

defined as LeffG ¼ limL→∞

RL−L dzLG becomes

LeffG ¼ m2ð _Z2=c2 − j∇rZj2Þ þ _α2=c2 − j∇rαj2

2m

þ 2MðZ − LÞ _αℏ

−mþ Oð∇3Þ; ð8Þ

up to the quadratic order in derivatives. Here,∇r ¼ ð∂x; ∂yÞis the derivative in the xy plane. The constant term m is thetension (the energy per unit area) of the static flat domainwall. The case in the massless limit M → 0 was alreadyobtained before [16].The low-energy dynamics of Z and α derived from the

Euler-Lagrange equation reads

mZc2

¼ 2M _α

ℏþm∇2

rZ;α

mc2¼ −

2M _Zℏ

þ∇2rα

m: ð9Þ

In the massless limit M → 0, the dynamics of Z and α areindependent of each other, giving linear dispersions:

ω ¼ �cjkj; ð10Þ

with the frequencies ω both for Z and α, and the wavenumber k ¼ ðkx; kyÞ. Waves for Z and α independentlypropagate as a ripplon and a localized magnon in thevicinity of the domain wall.As long as M ≠ 0, the dynamics of Z and α couple to

each other. There are four typical solutions of Eq (9),

Z�1 ¼ A�

1

msinðk · r∓ω1tþ δ�1 Þ;

α�1 ¼ �A�1 cosðk · r∓ω1tþ δ�1 Þ; ð11aÞ

Z�2 ¼ A�

2

msinðk · r� ω2tþ δ�2 Þ;

α�2 ¼ �A�2 cosðk · r� ω2tþ δ�2 Þ; ð11bÞ

where A�1;2 ∈ R and δ�1;2 ∈ R are arbitrary constants. Waves

of Z and α couple to each other and propagate as a coupledripplon with dispersions

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ω1¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2c4þℏ2c2k2

p−Mc2

ℏ¼ℏk2

2MþOðk4Þ;

ω2¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM2c4þℏ2c2k2

pþMc2

ℏ¼2Mc2

ℏþℏk2

2MþOðk4Þ: ð12Þ

For the solutions of (Z�1 , α�1 ), the coupled ripplons

propagate in the direction parallel (for þ sign) andantiparallel (for − sign) to k with a gapless quadraticdispersion ω1 showing type-II NG modes [17]. Left andright panels of Fig. 1 show the schematic pictures ofcoupled ripplons for (Zþ

1 , αþ1 ) and (Z

−1 , α

−1 ), respectively. In

contrast to a quantized vortex in superfluids in which aKelvin wave is a combination of two translational modes inreal space, the coupled ripplon is a combination of thetranslational mode in real space and the phase mode of theinternal degree of freedom. For the solutions of (Z�

2 , α�2 ),

on the other hand, the coupled ripplons propagate in theopposite directions to (Z�

1 , α�1 ), respectively, with a gapped

dispersion ω2, and do not behave as NG modes. In thenonrelativistic limit c → ∞, the gap in ω2 diverges and thesolutions (Z�

2 , α�2 ) disappear, and only (Z�

1 , α�1 ) for NG

modes remain as solutions. In the massless limit M → 0,the gap in ω2 disappears and two dispersions ω1;2 becomethe same linear one: ω ¼ cjkj.Linear response theory.—Dynamics of a ripplon can also

be analyzed by the linear response theory. We consider theansatz as the static domain-wall solution and its fluctuation:u ¼ u0 þ δu ¼ u0 þ aþeiðk·r−ωtÞ þ a�−e−iðk·r−ωtÞ. Insertingthis ansatz into the dynamical equation (5), we can obtainthe Bogoliubov–de Gennes equation,

�ω2

c2�2Mω

ℏ

�a�

¼�ðk2−∂2

zÞþ4me2mz∂z−m2ð3e2mz−1Þ

1þe2mz

�a�þOða2�Þ;

ð13Þ

up to the linear order of a�. The normalizable solution isa� ∝ emz and corresponding ω takes the value ω1;2 shownin Eq. (12). In the massless limit M → 0, ω1 ¼ ω2 ¼ cjkjgives the general solution δu ¼ emzfgþ1 eiðk·r−cjkjtÞ þg−1 e

−iðk·r−cjkjtÞ þ gþ2 eiðk·rþcjkjtÞ þ g−2 e

−iðk·rþcjkjtÞg with arbi-trary constants g�1;2 ∈ C. The localized magnon is obtainedby taking gþ1 ¼ −g−1 ¼ g0eiδ and g�2 ¼ 0 (parallel directionto k), or g�1 ¼ 0 and gþ2 ¼ −g−2 ¼ g0eiδ (antiparalleldirection to k) with g0, δ ∈ R. The ripplon is obtainedby taking g1 ¼ g�2 ¼ g0eiδ and g3 ¼ g4 ¼ 0 (paralleldirection to k), or g1 ¼ g2 ¼ 0 and g3 ¼ g�4 ¼ g0eiδ

(antiparallel direction to k). For M ≠ 0 case, the solutionis δu1 ¼ emzfgþ1 eiðk·r−ω1tÞ þ g−1 e

−iðk·rþω1tÞg and δu2 ¼emzfgþ2 eiðk·rþω2tÞ þ g−2 e

−iðk·r−ω2tÞg with arbitrary constantsg�1;2 ∈ C. gþ1;2 ¼ iAþ

1;2eiδþ

1;2 and g−1;2 ¼ 0 (gþ1;2 ¼ 0 andg−1;2 ¼ −iA1;2e

iδ−1;2) correspond to the coupled ripplon

solution (Zþ1;2, α

þ1;2) [(Z

−1;2, α

−1;2)] in Eq. (11).

Commutation relation.—We obtain gapless and local-ized magnon and ripplon with linear dispersions, Eq. (10),only in the massless limit M → 0 and the coupled ripplonwith quadratic dispersion, Eq. (11a), with M ≠ 0 in thevicinity of the domain wall. These modes are type-I (forM → 0) and type-II (for M > 0) NG modes as a conse-quence of the spontaneous breaking of the U(1) symmetryfor the phase rotation and the translational symmetry:H1 ≃Uð1Þ ×R3 → H2 ≃R2 under the appearance of thedomain wall. In both cases, our dispersions saturate theequality of the Nielsen-Chadha inequality [4]:NI þ 2NII ≥ NBG, where NI, NII, and NBG are the totalnumbers of the type-I NG modes, the type-II NG modes,and broken generators (BG) which correspond to sponta-neously broken symmetries. Recently, it has been shown inRefs. [7,8] that for internal symmetry the equality of theNielsen-Chadha inequality is saturated as the Watanabe-Brauner’s relation [6],

NBG − NNG ¼ 1

2rankρ;

ρi;j ¼ limV→∞

1

V

Zd3xð−i½Ωi;Ωj�Þ

����u¼u0

; ð14Þ

where NNG ¼ NI þ NII is the number of NG modes, V isthe volume of the system, Ωi is the Noether’s charge or agenerator of broken symmetries, and [·; ·] is a commutatoror the Poisson bracket in classical level. According to thisrelation, NBG ≠ NNG takes place when commutators ofbroken generators are nonvanishing. This relation has been

FIG. 1. Schematic pictures of coupled ripplons and theirpropagating directions for solutions (Zþ

1 , αþ1 ) (left) and (Z−1 ,

α−1 ) (right). The middle shaded area shows the region of thedomain wall juj ≈ 1 (nz ≈ 0) and its tone shows the phase α of u[direction of (nx, ny)]. The þ and − signs show the areas forα > 0 and α < 0, respectively. The vertical axis is the z axis andthe horizontal axis shows the direction of the wave vector k in thexy plane. For left (right) figures, the coupled ripplon propagatesin the right (left) direction. See the Supplemental Material foranimations of their dynamics [11].

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proven for internal symmetries such as the Heisenbergferromagnet and has been confirmed for space-time sym-metries such as a quantized vortex in superfluid and a two-dimensional Skyrmion in ferromagnets [10]. In our case, onthe other hand, broken generators consist of one internalsymmetry and one spatial symmetry which intuitivelycommute because underlying symmetries are the directproduct and are independent of each other, i.e.,H1=H2 ≃ Uð1Þ ×R. To check whether the relation,Eq. (14), also holds in our case or not, we directly calculatethe commutation relation between symmetry generators ofthe internal U(1) phase rotation and the translation.Defining the momenta v conjugate to u as

v ¼ ∂LG

∂ _u ¼ _u�

c2ð1þ juj2Þ2 þiMu�

ℏð1þ juj2Þ ; ð15Þ

the Noether’s charges for the phase rotation and thetranslation along z axis are obtained as

Θ ¼Z

dzJ0α; J0α ¼ iuv; ð16Þ

P ¼Z

dzJ0Z; J0Z ¼ ð∂zuÞv; ð17Þ

respectively. The commutator between P and Θ can becalculated from ½uðz1Þ; vðz2Þ� ¼ iδðz1 − z2Þ, to yield

½P;Θ� ¼Z

dz1

Zdz2½J0Zðz1Þ; J0αðz2Þ�

¼ iZ

dz1

Zdz2½∂z1uðz1Þvðz1Þ; uðz2Þvðz2Þ�

¼ iZ

dz1

Zdz2f∂z1uðz1Þ½vðz1Þ; uðz2Þ�vðz2Þ þ uðz2Þ∂z1 ½uðz1Þ; vðz2Þ�vðz1Þg

¼Z

dz1

Zdz2f∂z1uðz1Þvðz2Þ þ uðz2Þ∂z1vðz1Þgδðz1 − z2Þ

¼Z

dz∂z½uðzÞvðzÞ�: ð18Þ

For the static domain-wall solution, the first term inEq. (15) does not contribute to the commutator because of_u ¼ 0. Consequently, the commutator becomes

−i½P;Θ� ¼ Mℏ

Zdz∂z

� juj21þ juj2

�¼ M

ℏ

� juj21þ juj2

z¼þ∞

z¼−∞

¼ 2Mℏ

�1

2½1 − nz�z¼þ∞

z¼−∞

�≡ 2MW

ℏ: ð19Þ

W is precisely the topological charge of the domain walland is proportional to the tension of the domain wall[13,16] (see the Supplemental Material [11]). Evaluatingthis in the domain-wall background u ¼ u0, we findW ¼ 1. As a result, two generators P and Θ do notcommute as long as M ≠ 0, giving NBG − NNG ¼ 1 andone type-II NG mode, or commute in the massless limitM → 0 giving two type-I NG modes, which is consistentwith our result.Conclusion.—In conclusion, we have considered NG

modes excited on one flat domain wall in the CP1 modelswith the Ising potential. NG modes in the relativistic modelare the localized magnon for the U(1) phase rotation and thetranslational ripplon which are independent of each otherand have linear dispersions. In the nonrelativistic limit, onthe other hand, there is one coupled ripplon with a quadraticdispersion as the combination of the localized magnon and

the ripplon. We also find the coupled localized magnon andthe ripplon in the interpolating model connecting therelativistic and nonrelativistic theories even though it hasthe Lorentz invariance. The numbers of NG modes saturatethe equality of the Nielsen-Chadha inequality, and alsosatisfy the Watanabe-Brauner’s relation in which thecommutator between two generators of the internal phasemode and spatial translational mode gives the topologicaldomain-wall charge.Quantum effects on localized type-II NG modes remain

as an important problem, which was studied for a vortexwith non-Abelian localized modes [18].The term juj2=ð1þ juj2Þ ¼ ð1=2Þð1 − nzÞ in Eq. (19) is

known as the momentum map in symplectic geometry andthe D-term in supersymmetric gauge theory. Therefore, ourmodel can be extended to the CPn model, Grassmann σmodel [19,20], σ models on more general Kähler targetmanifolds, and non-Abelian gauge theories. A domain wallin two-component Bose-Einstein condensates has a differ-ent structure of NG modes [21], although there are alsotranslational and internal U(1) zero modes [22]. This maybe because the U(1) zero mode is non-normalizable in theircase. If one couples a gauge field, their model reduces toours in the strong gauge coupling limit (see theSupplemental Material [11]), with the internal U(1) modebecoming normalizable.

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We thank H. Takeuchi for useful discussions and H.Watanabe for explaining their results. We also thank theanonymous referees for the helpful suggestions and com-ments. This work is supported in part by Grant-in-Aid forScientific Research [Grants No. 26870295 (M. K.) andNo. 25400268 (M. N.]) and the work of M. N. is alsosupported in part by the “Topological QuantumPhenomena” Grant-in-Aid for Scientific Research onInnovative Areas (No. 25103720) from the Ministry ofEducation, Culture, Sports, Science, and Technology(MEXT) of Japan.

[1] S. R. Coleman and J. Mandula, Phys. Rev. 159, 1251 (1967).[2] If we relax one of the assumptions that the Coleman-

Mandula theorem is based on, supersymmetry (which is afermionic symmetry) can combine space-time symmetryand internal symmetry, as shown by R. Haag, J. T.Łopuszański, and M. Zohnius in Nucl. Phys. B88, 257(1975). Namely, the supersymmetry algebra fQI

α;QJ_βg ∼

δIJPμσμ

α_βand fQI

α;QJβg ¼ ϵαβCIJ contains the translational

operator P and the indices I, J for an internal symmetry,where CIJ is a central extension.

[3] This is similar to supersymmetry algebra in the presence ofBogomol’nyi-Prasad-Sommerfield (BPS) kinks, as shownin E. Witten and D. I. Olive, Phys. Lett. 78B, 97 (1978).More precisely, W can be a tensorial charge rather than acentral charge for extended objects as supersymmetryalgebra with BPS extended topological solitons andp-branes in supergravity and string theory; See G. R. Dvaliand M. A. Shifman, Nucl. Phys. B504, 127 (1997); Phys.Lett. B 396, 64 (1997); , 407, 452(E) (1997); A. Gorsky andM. A. Shifman, Phys. Rev. D 61, 085001 (2000), for BPSextended topological solitons; J. A. de Azcarraga, J. P.Gauntlett, J. M. Izquierdo, and P. K. Townsend, Phys.Rev. Lett. 63, 2443 (1989); E. R. C. Abraham and P. K.Townsend, Nucl. Phys. B351, 313 (1991); P. K. Townsend,in The World in Eleven Dimensions: Supergravity, Super-membranes, and M-Theory, Series in High Energy Physics,Cosmology, and Gravitation, edited by M. J. Duff (CRCPress, Boca Raton, FL, 1999), pp. 375–389, for p-branes.

[4] H. B. Nielsen and S. Chadha, Nucl. Phys. B105, 445 (1976).[5] Y. Nambu, J. Stat. Phys. 115, 7 (2004).[6] H.Watanabe and T. Brauner, Phys. Rev. D 84, 125013 (2011).[7] H. Watanabe and H. Murayama, Phys. Rev. Lett. 108,

251602 (2012).[8] Y. Hidaka, Phys. Rev. Lett. 110, 091601 (2013).[9] M. Kobayashi and M. Nitta, Prog. Theor. Exp. Phys.

021B01 (2014).

[10] H. Watanabe and H. Murayama, Phys. Rev. Lett. 112,191804 (2014).

[11] See the Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevLett.113.120403 for math-ematical details and animations of coupled ripplons.

[12] The massless limitM → 0 is often called the ultrarelativisticlimit and Lrel may have to be called an ultrarelativisticmodel. In Refs. [4–10], however, similar ultrarelati-vistic models with only a second time derivative term arereferred as relativistic models. We therefore refer to Lrel as arelativistic model too.

[13] E. R. C. Abraham and P. K. Townsend, Phys. Lett. B 291, 85(1992); 295, 225 (1992).

[14] In the existence of the domain wall, the three-dimensionalSO(3) rotational symmetry in space is also broken to thetwo-dimensional SO(2) rotational symmetry along the zaxis. The breaking rotational symmetry along the direc-tion perpendicular to the z axis corresponding toSOð3Þ=SOð2Þ≃ S2, however, does not contribute as addi-tional Nambu-Goldstone modes, because the rotationalmodes can be constructed from the local translational mode,as discussed in E. A. Ivanov and V. I. Ogievetskii, Theor.Math. Phys. 25, 1050 (1975); I. Low and A. V. Manohar,Phys. Rev. Lett. 88, 101602 (2002).

[15] N. S. Manton, Phys. Lett. 110B, 54 (1982); M. Eto,Y. Isozumi, M. Nitta, K. Ohashi, and N. Sakai, Phys.Rev. D 73, 125008 (2006).

[16] M. Arai, M. Naganuma, M. Nitta, and N. Sakai, Nucl.Phys. B652, 35 (2003); M. Nitta, Phys. Rev. D 86, 125004(2012).

[17] Here, we note that the term of type I or II is defined inRef. [4], that the exponent of the dispersion is odd or even,whereas that of type A or B is defined in Ref. [7] from thecommutation relations of the corresponding broken gener-ators. In our case, type I (II) and type A (B) coincide, and weuse the terminology type I or II.

[18] M. Nitta, S. Uchino, and W. Vinci, arXiv:1311.5408[J. High Energy Phys. (to be published)].

[19] Y. Isozumi, M. Nitta, K. Ohashi, and N. Sakai, Phys. Rev.Lett. 93, 161601 (2004); Phys. Rev. D 70, 125014 (2004);71, 065018 (2005); M. Eto, Y. Isozumi, M. Nitta, K. Ohashi,K. Ohta, and N. Sakai, Phys. Rev. D 71, 125006 (2005).

[20] M. Eto, Y. Isozumi, M. Nitta, K. Ohashi, and N. Sakai,J. Phys. A 39, R315 (2006).

[21] H. Takeuchi and K. Kasamatsu, Phys. Rev. A 88, 043612(2013).

[22] The dispersion relation is ω ∼ k3=2 for large k while it isquadratic for small momenta with a finite system size. SeeRef. [21] and H. Watanabe and H. Murayama, Phys. Rev. D89, 101701 (2014); D. A. Takahashi and M. Nitta,arXiv:1404.7696.

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