Unified Description of Nambu-Goldstone Bosons without Lorentz Invariance

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  • Unified Description of Nambu-Goldstone Bosons without Lorentz Invariance

    Haruki Watanabe1,2,* and Hitoshi Murayama1,3,4,

    1Department of Physics, University of California, Berkeley, California 94720, USA2Department of Physics, University of Tokyo, Hongo, Tokyo 113-0033, Japan

    3Theoretical Physics Group, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA4Kavli Institute for the Physics and Mathematics of the Universe (WPI), Todai Institutes for Advanced Study,

    University of Tokyo, Kashiwa 277-8583, Japan(Received 3 March 2012; published 21 June 2012)

    Using the effective Lagrangian approach, we clarify general issues about Nambu-Goldstone bosons

    without Lorentz invariance. We show how to count their number and study their dispersion relations. Their

    number is less than the number of broken generators when some of them form canonically conjugate pairs.

    The pairing occurs when the generators have a nonzero expectation value of their commutator. For non-

    semi-simple algebras, central extensions are possible. The underlying geometry of the coset space in

    general is partially symplectic.

    DOI: 10.1103/PhysRevLett.108.251602 PACS numbers: 11.30.Qc, 14.80.Va

    Introduction.Spontaneous symmetry breaking (SSB)is ubiquitous in nature. The examples include magnets,superfluids, phonons, Bose-Einstein condensates (BECs),neutron stars, and cosmological phase transitions. Whencontinuous and global symmetries are spontaneously bro-ken, the Nambu-Goldstone theorem [13] ensures the ex-istence of gapless excitation modes, i.e., Nambu-Goldstonebosons (NGBs). Since the long-distance behavior of sys-tems with SSB is dominated by NGBs, it is clearly impor-tant to have general theorems on their number of degrees offreedom and dispersion relations.

    In Lorentz-invariant systems, the number of NGBs nNGBis always equal to the number of broken generators nBG.All of them have the identical linear dispersion ! cjkj.However, once we discard the Lorentz invariance, thesituation varies from one system to another.

    Until recently, systematic studies on NGBs withoutLorentz invariance have been limited. (See Ref. [4] for arecent review.) Nielsen and Chadha [5] classified NGBsinto two types: type-I (II) NGBs have dispersion relationsproportional to odd (even) powers of their momenta in thelong-wavelength limit. They proved nI 2nII nBG,where nI (nII) is the number of type-I (II) NGBs. Schaferet al. [6] showed that nNGB is exactly equal to nBG ifh0jQi;Qjj0i vanishes for all pairs of the symmetry gen-eratorsQi. A similar observation is given in Ref. [7]. Giventhese results, Brauner and one of us (H.W.) [8] conjectured

    nBG nNGB 12 rank; (1)

    ij lim!1


    h0jQi;Qjj0i; (2)

    where is the spatial volume of the system.In this Letter, we clarify these long-standing questions

    about the NGBs in Lorentz-noninvariant systems by proving

    the conjecture and showing the equality in the Nielsen-Chadha theorem with an improved definition using effectiveLagrangiansLeff . We also clarify how the central extensionof the Lie algebra makes a contribution to [9].Coset space.When a symmetry group G is sponta-

    neously broken into its subgroup H, the set of groundstates forms the coset space G=H where two elementsof G are identified if g1 g2h for 9h 2 H. Every pointon this space is equivalent under the action of G, andwe pick one as the origin. The unbroken group H leavesthe origin fixed, while the broken symmetries move theorigin to any other point. The infinitesimal action of Gis given in terms of vector fields hi hi a@a i 1; . . . ; dimG on G=H, where @a @@a with the localcoordinate system fag (a 1; . . . ; nBG dimGdimH) around the origin. The infinitesimal transforma-tions hi satisfy the Lie algebra hi;hj fkijhk. Wecan always pick the coordinate system such that astransform linearly under H, namely, that hi bRpTiab@a, where RpTi is a representation of H[10]. On the other hand, the broken generators arerealized nonlinearly, hb hba@a with hba0 Xba.Since broken generators form a basis of the tangentspace at the origin, the matrix X must be full-rankand hence invertible.The long-distance excitations are described by the

    NGB fields ax that map the space-time into G=H.We now write down its Leff in a systematic expansionin powers of derivatives, because higher derivative termsare less important at long distances.Effective Lagrangians without Lorentz invariance.We

    discuss theLeff for the NGB degrees of freedom followingRefs. [11,12]. Under global symmetry G, the NGBs trans-form as a ihia where i are infinitesimal parame-ters. However, we do not make i local (gauge) unlike inthese papers because it puts unnecessary restrictions on

    PRL 108, 251602 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending22 JUNE 2012

    0031-9007=12=108(25)=251602(5) 251602-1 2012 American Physical Society

  • possible types of symmetries and their realizations, as wewill see below.

    It is well known that a symmetry transformation canchange the Lagrangian density by a total derivative. Theexamples include space-time translations, supersymmetry,and gauge symmetry in the ChernSimons theory [13]. Weallow for this possibility in the Leff of the NGB fields. Weassume spatial translational invariance and rotational in-variance at sufficiently long distances in the continuumlimit, while we can still discuss their SSB.

    If Lorentz invariant, the Leff is highly constrained,

    L eff 12 gab@a@b O@4: (3)

    The invariance of the Lagrangian underG requires that gabis a G-invariant metric on G=H, namely @cgabhi

    c gac@bhi

    c gcb@ahic 0. When the coordinates a arereducible under H, the metric g is a direct sum of irreduc-ible components gab PpF2ppab where pab vanishes out-side the irreducible representation p with arbitraryconstants Fp for each of them.

    On the other hand, once we drop Lorentz invariance, thegeneral Leff has substantially more freedom,

    Leff ca _a 12 gab _a _b 1


    O@3t ; @t@2r ; @4r; (4)where gab is also G invariant. Here and hereafter, r 1; . . . ; d refers to spatial directions.

    Note that the spatial isotropy does not allow terms withfirst derivatives in space in the Leff . Therefore, the spatialderivatives always start with at least the second powerO@2r. (Actually, it is not critical for us whether there areterms of O@2r; it may as well start at O@4r withoutaffecting our results, as we will see below.)

    The Lagrangian density changes by a total derivativeunder the infinitesimal transformation a ihia iff

    @bca @acbhib @aei: (5)The functions ei introduced in this way are actuallyrelated to the charge densities of the system. By payingattention to the variation of the Lagrangian by the surfaceterm

    Leff i@tcahia ei; (6)we can derive the Noether current for the global symmetryj0i ei gabhia _b. Since the ground state is time inde-pendent _b 0,

    ei0 h0jj0i xj0i: (7)It must vanish in the Lorentz-invariant case, which can beunderstood as the special situation where ca and ei vanish,and gab c2 gab.

    Before presenting the proof, we explain the advantage innot gauging the symmetry. A tedious calculation verifies@bhia@aej fkijek 0, with a general solution,

    hia@aej fkijek cij: (8)

    Therefore, eis transform as the adjoint representationunder G, up to possible integration constants cij cji.These constants play important roles as seen below.In the presence of such constants, the global symmetry

    cannot be gauged [11]. This is reminiscent of the WessZumino term that also changes by a surface term under aglobal symmetry and produces an anomaly upon gauging[14,15]. It is known that the constants can be chosen tovanish with suitable definitions of ei for semisimple Liealgebras, while a nontrivial second cohomology of the Liealgebra presents an obstruction [16].Proof of the conjecture.The basic point to show is that

    when ij 0, the NGB fields for the generators i and j are

    canonically conjugate to each other.From Eq. (7) and the assumed translational symmetry,

    the formula for in Eq. (2) is reduced to

    ij ih0jQi; j0j j0i hia@aejj0: (9)

    Obviously, this must vanish for unbroken generators bydefinition. Combining this with Eq. (5), we have


    b@bca @acbj0 ij: (10)

    We now solve this differential equation around the origin.The Taylor expansion of ca can be written as ca ca0 Sab Aabb O2, where Sab and Aab standfor the symmetric and antisymmetric parts of the derivative@bcaj0. Obviously ca0 and Sab lead to only totalderivative terms in the Leff and thus will be dropped later:

    ca _a Aab _ab @tca0a 12 Sab


    O3: (11)

    The equation for the antisymmetric part 2XcaXd

    bAab cd has a unique solution which gives

    ca _a 12ab _~a ~b O ~3; (12)

    where ~a bX1ba. Since the matrix is realand antisymmetric, we can always transform it into thefollowing form by a suitable orthogonal transformation~Qi OijQj:

    PRL 108, 251602 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending22 JUNE 2012


  • M1

    . ..



    . ..








    Here, 0 for 1; . . . ; m 12 rank, while the re-maining elements identically vanish.

    The most important step in the proof is to write down theexplicit expression of the Leff in Eq. (4),

    ca _a Xm1


    2 ~2 _~21 _~2 ~21; (14)

    which is in the familiar form of the Lagrangian on thephase space L pi _qi H [17]. Namely, ~21 and ~2are canonically conjugate variables, and they togetherrepresent one degree of freedom rather than two degreesof freedom. Hereafter we call the first set of ~a s (a 1; . . . ; 2m) type B, and the rest type A. Hence, nA 2nB nBG with nA nBG 2m and nB m. Thus we provedthe conjecture Eq. (1).

    The definition of a degree of freedom here is the con-ventional one in physics; i.e., one needs to specify both theinstantaneous value and its time derivative for each degreeof freedom as initial conditions. This definition does notdepend on the terms with spatial derivatives in theLagrangian.

    Now we are in the position to prove that the equality issatisfied in the Nielsen-Chadha theorem if the term withtwo spatial derivatives exists with a nondegenerate metricgab. Then Eq. (4) implies that the type A NGB fields havelinear dispersion relations ! / k, while the type B NGBfields have quadratic dispersions ! / k2. In this case, ourtype A (B) coincides with their type I (II), respectively, andthe Nielsen-Chadha inequality is saturated.

    On the other hand, if we allow the second-order termO@2r to vanish accidentally but the fourth-order termO@4r to exist, the unpaired (type A) NGBs happen tohave a quadratic dispersion (!2 / k4, and hence type II)yet count as independent degrees of freedom each [8].Therefore, the Nielsen-Chadha theorem is still an inequal-ity in general. In contrast, our distinction between type Aand type B NGBs is clearly determined by the first twotime derivatives, and defines the number of degrees offreedom unambiguously. Therefore, the classification be-tween odd and even powers in the dispersion relation is notan essential one, and our theorem is stronger than that byNielsen and Chadha.

    Note that the Lagrangian formalism is mandatory in ourdiscussion, because the presence of the first-order deriva-

    tives in time essentially affects the definition of the canoni-cal momentum, while a Hamiltonian is written with a fixeddefinition of the canonical momentum.Examples.The simplest and most famous example of a

    type B NGB is the Heisenberg ferromagnet H JPhi;jisi sj with J > 0 on a d-dimensional square lattice(d > 1). In this case, the original symmetry group O3 isspontaneously broken down into the subgroup O2. Thecoset space is O3=O2 ffi S2. We assume that the groundstate has all spins lined up along the positive z directionwithout a lack of generality. Even though there are twobroken generators, there is only one NGB with the qua-dratic dispersion relation ! / k2.The coset space can be parametrized as nx; ny; nz

    1; 2; ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 12 22p . The O3 transformationhi

    a iajnj (i, j x, y, z; a 1, 2) is realized linearlyfor the unbroken generator hz

    a abb, while non-linearly for broken ones Xa

    b ab. One can show that theLeff consistent with the O3 symmetry up to O@2 is

    L eff mny _nx nx _ny

    1 nz 1

    2F2 _n2 1

    2F2@rn@rn: (15)

    Comparing to Eqs. (4) and (5), we can read off ca and ei asc1 mny1nz , c2

    mnx1nz and ei mni. Hence m hj0zi

    represents the magnetization of the ground state. It is clearthat there is only one type B NGB because 1 and 2 arecanonically conjugate to each other, with a quadratic dis-persion ! / k2.However, for an antiferromagnet, J < 0, the overall

    magnetization cancels between sublattices, and thereforeei 0, which in turn requires ca 0. As a consequence,the lowest order term in the time derivative expansion hastwo powers, and we find that both 1 and 2 representindependent type A NGBs with linear dispersions ! / jkj.The generalization to the ferrimagnetic case isstraightforward.Another example is the spontaneously broken transla-

    tional invariance that leads to acoustic phonons in anisotropic medium [18]. The displacement vector ux rep-resents the NGBs under the spatial translation u ! u ,hence G R3 and H 0. Then with O3 symmetry ofspatial rotations, the most general form of the continuumLeff is

    L eff 12 _u2 c


    2r u2 c


    2r u2: (16)

    We recover the usual result of one longitudinal and twotransverse phonons with linear dispersions ! ck and! ctk, respectively (type A). When the O3 symmetryis reduced to SO2 Z2 for rotation in the xy plane andthe reflection z ! z, there are considerably more termsone can write down. Using the notation c ux iuy, @ 12 @x i@y, and c and @ for their complex conjugates, themost general Leff is

    PRL 108, 251602 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending22 JUNE 2012


  • Leff icxy2

    c _c 12_u2z _c _c F20 @c @ c

    12F21@zuz2 @uz; @zc

    F22 F23F23 F


    @uz@z c

    12F25@c 2 c:c:: (17)

    With cxy 0, we find there is one type A NGB with a

    linear dispersion, and one type B NGB with a quadratic

    dispersion. The first termicxy2

    c _c 12 cxyuy _ux ux _uyimplies

    xy ih0jPx; j0yj0i cxy 0: (18)Namely, this Lie algebra is a central extension of theAbelian algebra of the translation generators, i.e.,Pi; Pj cij. As pointed out in Ref. [19], when themedium is electrically charged, an external magnetic fieldalong the z axis precisely leads to this behavior with cxy 2!c (the cyclotron frequency), because the gauge-invarianttranslations in a magnetic field are generated by Pi i@@i ec Ai, which satisfy h0jPx; Pyj0i i @ec BzN withthe number of particles N. This would not be possible withthe gauged Leff in Ref. [11] that does not allow for thecentral extension.

    As a more nontrivial example, let us consider a spinorBEC with F 1. The symmetry group is G SO3 U1, where SO3 rotates three components of F 1states, while U1 symmetry gives the number conserva-tion. The Lagrangian is written using a three-componentcompl...


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