Nambu-Goldstone bosons with fractional-power dispersion relations

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  • Nambu-Goldstone bosons with fractional-power dispersion relations

    Haruki Watanabe1,* and Hitoshi Murayama1,2,3,1Department of Physics, University of California, Berkeley, California 94720, USA

    2Theoretical Physics Group, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA3Kavli Institute for the Physics andMathematics of the Universe (WPI), Todai Institutes for Advanced Study,

    University of Tokyo, Kashiwa 277-8583, Japan(Received 13 March 2014; published 1 May 2014)

    We pin down the origin of a peculiar dispersion relation of domain wall fluctuation, the so-called ripplon,in a superfluid-superfluid interface. A ripplon has a dispersion relation k3=2 due to the nonlocality ofthe effective Lagrangian, which is mediated by gapless superfluid phonons in the bulk. We point out theanalogy to the longitudinal phonon in the two-dimensional Wigner crystal.

    DOI: 10.1103/PhysRevD.89.101701 PACS numbers: 11.30.Qc, 14.80.Va


    The dispersion relation of Nambu-Goldstone bosons(NGBs) determines the low-energy property of systemswith spontaneous symmetry breaking. It is directly con-nected to the temperature dependence of thermodynamicquantities, such as heat capacitance. The softness of NGBsalso sets the severeness of infrared divergence, from whichone can determine the stability of the symmetry-breakingground state [1].The dispersion relation of NGBs is usually an integer

    power in the long-wavelength limit; i.e., jkijni withni Z for ~k kixi (i 1;; d; no sum over i), with anappropriate choice of the axes. For example, phonons in an

    ordinary crystal have a linear dispersion relation of j~kj(ni 1), and magnons in a Heisenberg ferromagnet have aquadratic dispersion relation, j~kj2 (ni 2). In a helicalmagnet with the spiral vector along the z axis, thedispersion relation of helimagnons takes an anisotropic

    form, ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2z Ck2x k2y2

    q[2]. Namely, it is linear in

    the z direction (nz 1) and quadratic in x, y direc-tions (nx ny 2).We can actually prove the integer power by assuming a

    local effective Lagrangian. In Fourier space, the qua-dratic effective Lagrangian can be expressed as

    1=2a~k;G1ab ~k;b~k;, where as are Nambu-Goldstone fields (see Ref. [1] for more details). For the

    wave vector ~k kixi, one finds

    G1ab gab2 iab giabk2i g0iabk4i ; (1)

    where is an antisymmetric matrix and g, gi, gi0 aresymmetric matrices. For example, when 0, all NGBshave the usual linear dispersion relation (type A). When is

    nonzero and full rank, we may neglect the gab term, and allNGBs have a quadratic dispersion relation (type B). Whensome components of giab somehow vanish, these NGBsmay have softer dispersion relations, but their power is stillan integer [3]. Crucially, Eq. (1) does not have a termCiabki; as such a term would cause an instability toward a

    ground state with a nonzero ~k. Even in this case, we can re-

    expand G1ab from the momentum minimum ~k0 with noresulting linear term in ~k ~k0.Nevertheless, it is known that the fluctuation of a certain

    domain wall has a dispersion relation with a fractionalpower, k3=2 [4]. Although there are several analyticaland numerical studies that support this weird dispersionrelation [59], the physically intuitive picture behind itremains unclear in the existing literature. In this paper, wepin down its origin as the breakdown of the locality of theeffective Lagrangian, resulting from integrating out gaplessmodes in the bulk. In general, when the microscopicLagrangian of a system is local, its effective Lagrangianobtained by integrating out only higher-energy modes isstill local. However, when the system contains an inter-action of two subsystems SA;B of different dimensionality,it may be useful to integrate out SBregardless of its gapto get an effective theory of SA alone. A nonlocal effectiveLagrangian in real space is equivalent to a nonanalyticdependence on momentum in Fourier space, which maylead to a dispersion relation with a fractional power, as weshall see below.


    We consider a domain wall in a two-component BoseEinstein condensate. When the intercomponent repulsionis much stronger than the intracomponent one, a sponta-neous phase separation occurs and a domain wall isformed between the components. Since the domain wallspontaneously breaks the translational symmetry in thedirection perpendicular to the plane, there should be agapless Nambu-Goldstone mode corresponding to the


    PHYSICAL REVIEW D 89, 101701(R) (2014)

    1550-7998=2014=89(10)=101701(6) 101701-1 2014 American Physical Society


  • fluctuation (ripple) of the interface. Such a mode isreferred to as a ripplon. Here we rederive its dispersionrelation by deriving an effective Lagrangian written solelyin terms of the displacement field u.

    A. Model

    Let us take a domain wall at z 0 in the equilibrium anddescribe its fluctuation by a displacement field u u~x; twith ~x x; yT . A superfluid 1 (2) fills the spacez > u (z < u). Each superfluid is described by the

    standard Lagrangian PaXa X2a

    2ga, with Xa a _a

    2ma1 ~a2 za2 for a 1, 2. (Here and here-after, there is no sum over a and 1.) Thus thephenomenological Lagrangian for this system should begiven by [6,9]

    Leff Z



    dzP2X2 S; (2)

    where S ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 ~u2

    qis the area element and is the

    tension of the domain wall. Here we take the space to beinfinitely extended. a 0 and u 0 characterize theground state without a superflow. We consider smallfluctuations above it at zero temperature; hence, there isno normal component of the fluid.The variation of the Lagrangian (2) with respect to u

    gives Laplaces law, P1 P2jzu ~ S1 ~u,which relates the pressure difference across the surfaceto the surface tension. The pressure balance at the groundstate requires 21=2g1 22=2g2.Our goal is to derive an effective field theory of the

    surface fluctuation in terms of the displacement fieldu~x; t. Our strategy is simply to integrate out bulk degreesof freedom 1;2.

    B. Equation of motion of superfluid phonons

    Let us first clarify the equation of motion for 1;2 fieldsthat describe Bogoliubov phonons in superfluids. Thevariation of the Lagrangian (2) with respect to a requiresextra attention to the u dependence of the integrationdomain. For example, when we integrate by parts the timederivative in a combination

    R dt

    Ru dzfz; ttz; t,t may act on u~x; t, as well as on fz; t in the integrand.

    Therefore, the variation of the action with respect to a is

    Seff 1aZ

    dtd2xjza ujaajzuZ

    dtd2xdzjaa: (3)

    The first line gives the boundary condition jza ujaat z u. Here, ja Pa=a is the conserved

    U(1) current. Using its linearized form, the boundarycondition is

    _u zama


    : (4)

    The physical meaning of this condition is clear: the domainwall and the superfluids have the same z component of thevelocity at the boundary. This boundary condition isnecessary for the conservation of U(1) charges,

    Q1 Z



    udzj01; Q2





    The second line of Eq. (3) gives the equation ofcontinuity ja 0. It can be linearized as

    a v2a2 2za 0; v2a ama : (6)

    We solve this equation in the form ~x; z; t a~k;ei~k~x1aazit, with a

    ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 =va2

    qand k j~kj, assuming the fluctuation is localized onthe domain wall. Combined with the condition in Eq. (4),we find an expression of in terms of u:

    a~k; 1a maia u~k;: (7)

    C. The effective domain wall Lagrangian

    To get the effective Lagrangian in terms of the displace-ment field u only, we substitute the solution of in Eq. (7)back into the Lagrangian. This is equivalent to integrating

    out a at the tree level. The bulk term _2a ~2 z2=2ga does not contribute since this vanishes thanksto the equation of motion. The crucial contribution comesfrom


    udzn1 _1


    dzn2 _2 (8)

    in the Lagrangian (2), where na a=ga is the superfluid(number) density. After integrating by parts, we obtain

    n11 n22jz0 _u (9)at the linearized level. This combination is quite intriguingsince it makes _u and n11 n22jz0 canonicallyconjugate to each other (at least when we neglecthigher-order terms _2a). Such a conjugate relation usuallyleads to noncommuting symmetry algebra, although all




  • symmetries under consideration [U1a and translations]are naively Abelian. We will further discuss this point later.Putting the pieces together, we find

    Leff 1


    m1n12ffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 2v2


    q m2n22ffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 2v2


    q k2!u (10)

    in Fourier space [10]. In the long-wavelength limit, theLagrangian correctly describes the known dispersion rela-tion 2 k3=m1n1 m2n2. The 2=v2a term in thedenominator can be consistently neglected for a dispersionrelation kn (n > 1) in the long-wavelength limit.The effective Lagrangian in real space reads

    Leff 1


    Zd2xd2y _u~x; t

    m1n1 m2n22j~x ~yj _u~y; t



    2 ~u~x; t2 (11)

    to leading order in derivatives. One can observe aCoulomb-type long-range coupling between velocities ofthe domain wall. Hence, the effective Lagrangian is non-local, invalidating our proof of the integer power for alocal effective Lagrangian. It may be an interesting futurework to examine the Coulomb-type long-range couplingfrom the dual picture of superfluids. This peculiar form ofthe time-derivative term may also have other interestingphysical consequences, e.g., response to an external force.

    D. Discussions

    The nonlocality of the effective Lagrangian clearlyoriginates from integrating out gapless bulk modes.When the bulk mode is gapless, a low-energy fluctuationof the domain wall excites bulk modes and, in turn, the bulkoscillation affects a different part of the domain wall, asillustrated in Fig. 1. This is the physical picture behindthe nonlocal term mediated by gapless bulk modes. Thenthe Fourier component of the nonlocal termRddxddya~x; tfab~x ~yb~x; t has a singularity at

    ~k 0. For the existence of k1 knfab~kj~k0, we needj R ddxfab~xx1 xnj < ; thus, fab~x should behave asxr (r > n d) as x .If we explicitly break the U(1) symmetry to open the

    bulk gap by adding M2a2a=2ga to the Lagrangian (2), thefirst term of Eq. (10) is replaced by

    m1n12ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM21 k2 2v2


    q m2n22ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM22 k2 2v2


    q : (12)

    Then the k dependence in the denominator becomessubleading, and the usual linear dispersion relation

    k is recovered. In real space, the induced term islocal _u2~x; t to leading order.This understanding helps us to generalize our analysis.

    For example, the domain wall of the Z2 symmetry-brokenphase of the real scalar 4 theory should have the ordinarylinear dispersion relation since the bulk is gapped. Indeed,such domain wall fluctuation can be well described by the(local) Nambu-Goto action [11].Even when the bulk is gapped, if there are additional

    gapless degrees of freedom on the domain wall, type-BNambu-Goldstone bosons [12] are possible, as recentlydiscussed in Ref. [13]. However, they cannot have afractional dispersion relation unless one introduces a non-local term by hand.Another essential ingredient in the above derivation is a

    finite particle density. That is, the same phenomenoncannot be realized by relativistic superfluids at the zerochemical potential. This can be easily seen by settingna 0 (or equivalently a 0) in Eqs. (9) and (10). In theabsence of the 2=k term in Eq. (10), we add the leadingorder term O2 to the Lagrangian and again find a lineardispersion relation. This basically means that, when theLagrangian does not have the na _a term, the domain wallfluctuation u and the bulk phase fluctuation a arecompletely decoupled to quadratic order in the fluctuations.As pointed out before by many authors [4,8,9,14], the

    fluctuation of a fluid surface with the dispersion relation k3=2 is known to occur even in classical hydrody-namics [15]. In this context, the mode is called a capillarywave. In the classical fluid mechanics, the velocity field ofan irrotational flow can be written as ~v ~, where is called the velocity potential. Further assuming theincompressibility and neglecting the dissipation and the

    Bogoliubov phonons in the bulk

    fluctuation of the domain wall



    photons in the bulk

    lattice vibration0

    FIG. 1. (Top panel) Fluctuation of the domain wall (ripple)excites Bogoliubov phonons in the bulk, and the bulk excitationin turn affects the domain wall fluctuation, inducing an effectivenonlocal interaction. (Bottom panel) The system of latticevibration (phonons) of a Wigner crystal and gauge photons inthe bulk is perfectly analogous to the ripple example.




  • gravitational potential, the pressure of a classical fluid canbe expressed as [15]

    P p0 _ 2 ~2 ; (13)

    where is the mass density of the fluid. This should becompared to the pressure of the superfluid,

    P 2

    2g n _ n

    2m ~2 n

    2mv2_2 : (14)

    We notice the formal correspondence m. Therefore,the above derivation goes without changes for classicalfluids, except that (i) we replace P in Eq. (2) with thepressure [Eq. (13)] for a classical fluid and (ii) we take theinfinite speed of sound limit v in, e.g., Eq. (10) to beconsistent with the assumption of incompressibility. Forexample, it is known that the potential on the surface jz0and the position of the surface u are canonically conjugateto each other [16]. This fact can be understood by Eq. (9)and the correspondence =m.Finally, let us comment on the effect of the gravitational

    potential. Naively, it explicitly breaks the translationalsymmetry in the z direction; hence, the NGB associatedwith the translational symmetry should open a gap. Indeed,the effective Lagrangian may obtain a mass termM2u2=2 with M2m1n1m2n2g>0 [17]. However,it turns out that the domain wall fluctuation remainsmassless, thanks to the interplay with bulk gapless modes.Let us first discuss the incompressible limit va .

    Adding the mass term to the effective Lagrangian (10), onefinds [5,18]

    ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim1n1 m2n2gk k3

    m1n1 m2n2

    s: (15)

    The dispersion is proportional to k1=2 in the long-wave-length limit. This mode is called the gravity wave of a fluidsurface and exists in the incompressible classical fluid aswell [15]. For a finite va, one can no longer neglect 2=v2asin the denominator of (10) for a small k limit [9]. Thedispersion in Eq. (15) is valid only for the wave vector thatis sufficiently bigger than


    m1n1 m2n2gm1n1 m2n2v2a

    ; (16)

    but smaller than mina2=a, with a 2maa1 beingthe coherence length.

    E. Commutation relations

    Now let us come back to the consequence of Eq. (9).This type of quadratic term linear in the time derivative

    usually implies that conserved charges associated withthese fields do not commute [1,12].Indeed, from Noethers theorem, the z component of the

    momentum operator Pz is found to be

    Pz Z







    d2xn11 n22jz0: (17)

    The formula for U(1) charges Qa in Eq. (5) can also belinearized as

    Q1 Z

    d2xn1u; Q2 Z

    d2xn2u: (18)

    The canonical commutation relation mentioned belowEq. (9) then suggests that the algebra of symmetry gen-erators is centrally extended due to the presence of thedomain wall:

    Pz;Q1 iAn1; pz;Q2 iAn2; (19)where A R d2x is the total area of the domain wall thatdiverges in the thermodynamic limit. This result is counter-intuitive since Qa and Pz naively commute. Also, theJacobi identity among Jx, Py, Qa (~J is the angularmomentum) appears to prohibit such central extensions.However, in the presence of the domain wall, Jx is nolonger well defined, as it changes the field configuration atthe spatial boundary and we do not have to impose theJacobi identity (as discussed in Ref. [19]).The noncommuting algebra (central extensions) as a

    consequence of topological solitons or domain walls arediscussed in Refs. [13,19]. Here we confirmed that thesuperfluid-superfluid interface of phase-separated two-component BoseEinstein condensates is an example ofsuch an extension.Note that the counting rule of NGBs discussed in

    Refs. [1,12,20,21] can be violated when the effectiveLagrangian fails to be local because the proof assumes alocal effective Lagrangian.


    The longitudinal phonon in a two-dimensional crystalsheet of electrons (Wigner crystal) embedded in three-dimensional space is completely analogous to the ripplonexample. It has the dispersion relation


    p, due to the

    long-range Coulomb interaction [22]. Here we revisit thiswell-known example to clarify the similarity.The long-range Coulomb interaction is mediated by

    photons propagating in the three-dimensional space. Toexplicitly deal with this physics, let us take the free photonLagrangian plus the interaction to charges in the system:




  • Z



    16FF Aje jp


    where F A A andje n0ez1 ~u; _ux; _uy; 0; (21)

    jp n0ez1; 0; 0; 0 (22)

    are the electron current including the effect of the latticedeformation and the background positive charge thatneutralizes the total electric charge density at the equilib-rium. Here we consider only the in-plane fluc-tuation ~u ux; uy.In the Lorentz gauge, the equation of motion of the

    gauge field can be solved as

    A~k; kz; 4

    k2 k2z 2c2je jp~k;: (23)

    Hence integrating out photons induces a term,





    4kikj ij 2c2k2 k2z 2c2





    2kikj ij 2c2ffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 2c2

    q uj: (24)After taking the limit c , this term reduces to theCoulomb interaction between excess charges n0e ~u:


    Zd2xd2y ~~x ~u~x; t n



    j~x ~yj ~~y ~u~y; t (25)

    in real space, which is again nonlocal.To discuss the dispersion relation of the longitudinal

    mode uL~k ~k ~u=k, we introduce the kinetic termRd2xmn0=2 _~u2~x; t to the Lagrangian (20). Then, we

    find the well-known dispersion relation 2 2n0k=m,as k in the denominator of Eq. (24) reduces the one powerof k in the numerator. This derivation again clarifies theimportance of the gapless nature of the surrounding modes.When the difference of the dimensionality between the bulkand the crystal subspace is m, integrating out photonsproduces an interaction km2 among the longitudinalcomponent kuL, ending up with a k

    m=2 dispersion [23].

    Our analysis above corresponds to the case m 1. Thethree-dimensional Wigner crystal, whose longitudinal pho-non is gapped, corresponds to the case m 0. Specifically,m has to be an odd integer to get a fractional power fromthis mechanism.

    Note that long-range interactions between spatial deriv-atives of fields [e.g., Eq. (25)] make dispersion relationsharder, while those among time derivatives [e.g., Eq. (10)]have the opposite effect.


    In this paper, we clarified the origin of the fractional-power dispersion relation of ripplons in a superfluid-superfluid interface. Gapless modes in the bulk induce anonlocal interaction, which takes a nonanalytic form inFourier space, leading to a fractional-power dispersionrelation of the domain wall fluctuation.When there are gapless modes localized on the domain

    wall other than the ripple mode of the domain wall or thein-plane phonon modes, bulk gapless excitation can, inprinciple, induce a nonlocal coupling for them as well.Softer modes in the bulk tend to have a more drastic

    effect on modes localized on the domain wall. It might bean interesting future work to look for a realistic example ofan interface of two bulk systems which support NGBs witha softer dispersion relation.The low-energy physics of a system can be qualitatively

    modified only by coupling it to other gapless degrees offreedom. When Goldstone modes associated with space-time symmetries are coupled to a Fermi surface, forexample, they get overdamped and can no longer be agood particlelike excitation [24]. This is again a result ofthe interaction with a continuous spectrum of electron-holepair excitations.


    We thank Tom Brauner for stimulating discussion andfor critical reading of the draft, and Yoshimasa Hidaka,Hiromitsu Takeuchi, Muneto Nitta, and MichikazuKobayashi for a useful discussion of the ripplon. H.W.appreciates the financial support of the Honjo InternationalScholarship Foundation. The work of H.M. was supportedby the U.S. DOE under Contract No. DE-AC03-76SF00098, by the NSF under Grants No. PHY-1002399 and No. PHY-1316783, by JSPS Grant No. (C)23540289, and by WPI, MEXT, Japan.

    Note added in proof.After the competition of our work,we have been informed of several related studies incompletely different contexts. Reference [25] discusseda fractional-power spectrum in the entanglementHamiltonian, which is a result of a nonlocal interactiongenerated by tracing out gapless modes. Reference [26]obtained a NGB with a fractional-power dispersion relationin the context of holography.




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    051701 (2011).[3] We may add a term C0iki, but the conclusion does not

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    University Press, New York, 2003).[6] H. Takeuchi, N. Suzuki, K. Kasamatsu, H. Saito, and

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    M. Marklund, Phys. Rev. A 82, 043608 (2010).[8] D.Kobyakov,V.Bychkov, E.Lundh,A.Bezett,V.Akkerman,

    and M. Marklund, Phys. Rev. A 83, 043623 (2011).[9] H. Takeuchi and K. Kasamatsu, Phys. Rev. A 88, 043612

    (2013).[10] Reference Volovik presented a very similar expression,

    which neglects the 2=v2a term in the denominator. Thisneglected term becomes important when we take gravityinto account.

    [11] M. Shifman, Advanced Topics in Quantum Field Theory:A Lecture Course (Cambridge University Press, Cambridge,England, 2012).

    [12] H. Watanabe and H. Murayama, Phys. Rev. Lett. 108,251602 (2012).

    [13] M. Kobayashi and M. Nitta, arXiv:1402.6826.[14] T. Hayata and Y. Hidaka, arXiv:1312.0008.[15] L. Landau and E. Lifshitz, Fluid Mechanics (Pergamon

    Press, Oxford, 1987), 2nd ed.[16] V. E. Zakharov, J. Appl. Mech. Tech. Phys. 9, 190 (1968).[17] Gravity also creates a density gradient. However, its effect

    on the mass term is higher order in g and we ignore itduring our discussions.

    [18] C. Pethick and H. Smith, Bose-Einstein Condensation inDilute Gases (Cambridge University Press, Cambridge,England, 2008), 2nd ed.

    [19] H. Watanabe and H. Murayama, arXiv:1401.8139.[20] H. Watanabe and T. Brauner, Phys. Rev. D 84, 125013

    (2011).[21] Y. Hidaka, Phys. Rev. Lett. 110, 091601 (2013).[22] G. Giuliani and G. Vignale,Quantum Theory of the Electron

    Liquid (Cambridge University Press, Cambridge, England,2005).

    [23] This evaluation is based on scaling and might misslogarithmic corrections.

    [24] H. Watanabe and A. Vishwanath, arXiv:1404.3728.[25] R. Lundgren, Y. Fuji, S. Furukawa, and M. Oshikawa, Phys.

    Rev. B 88, 245137 (2013).[26] M. Edalati and J. F. Pedraza, Phys. Rev. D 88, 086004






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