Counting Rule for Nambu-Goldstone Modes in Nonrelativistic Systems

Yoshimasa Hidaka

Mathematical Physics Laboratory, RIKEN Nishina Center, Saitama 351-0198, Japan(Received 16 March 2012; revised manuscript received 4 November 2012; published 27 February 2013)

The counting rule for Nambu-Goldstone modes is discussed using Mori’s projection operator method in

nonrelativistic systems at zero and finite temperatures. We show that the number of Nambu-Goldstone

modes is equal to the number of broken charges, Qa, minus half the rank of the expectation value

of [Qa, Qb].

DOI: 10.1103/PhysRevLett.110.091601 PACS numbers: 11.30.Qc

Introduction.—Symmetry and its spontaneous breakingare of basic importance for understanding the low-energyphysics in many-body systems. When a continuum sym-metry is spontaneously broken, there exist zero modescalled Nambu-Goldstone (NG) modes [1–3], e.g., pionsin hadron physics, spin waves in a ferromagnet, or phononsin a crystal. However, the number of NG modes associatedwith the spontaneous breaking that appear in a nonrelativ-istic system is an unsolved problem [4]. In the case ofrelativistic systems (more precisely, Lorentz invariant sys-tems), the number of NG modes coincides with the numberof broken symmetries, and their dispersion is linear, i.e.,! ¼ jkj, where ! and k are the energy and momentum ofthe NG modes. In contrast, in the case of nonrelativisticsystems, the number of NG modes is not necessarily equalto the number of broken symmetries, and the dispersionmay be nonlinear. Nielsen and Chadha (NC) [5] classifiedNG modes into two types: type-I and type-II NG modeswhose energies are odd and even powers of jkj, respec-tively. They showed that the number of type-I NG modesplus twice the number of type-II NG modes is greater thanor equal to the number of broken symmetries. In knownrealistic examples of type-II NG modes such as spin wavesand NG modes in the Kaon condensed phase [6,7], theinequality is saturated. The presence of an expectationvalue for charge densities implies a type-II dispersion forNG modes, as was clarified by Leutwyler in an effectivefield theory [8]. Schafer et al. pointed out that the expec-tation value of [Qa, Qb] rather than charge densities them-selves plays an important role, and if they all vanish,the number of NG modes is equal to the number ofbroken symmetries, where Qa are the broken charges [7].Recently, this theorem was generalized to the inequality,

NBS � NNG � 1

2rankh½Qa;Qb�i; (1)

by Watanabe and Brauner (WB) [9]. They conjectured thatthe inequality in Eq. (1) is saturated for general systems.These results were obtained at zero temperature.

In this Letter, we generalize the Nambu-Goldstone theo-rem in relativistic systems to that in nonrelativistic systems

at zero and finite temperatures, and we prove that theinequality in Eq. (1) is saturated. We give an alternativedefinition of type-I and type-II NG modes by using amatrix h½Qa;Qb�i, and show that the number of type-IING modes is equal to rank h½Qa;Qb�i=2, and the NCinequality is also saturated.For this purpose we apply Mori’s projection operator

method [10] to systems in which symmetry breakingoccurs spontaneously. In this approach, a Hamilton(Langevin) equation for NG modes is derived at zero(finite) temperature, in which the expectation value of thecommutation relations for NG fields and charge densitiesbecome the Poisson brackets for the fields; i.e., NGfields and charge densities are canonical variables. TheHamiltonian formalism for NG modes in nonrelativisticsystems at zero temperature was discussed in an analysis ofspecific models [11]. We develop the analysis in a modelindependent and nonperturbative way.In the following, we consider the case that the translation

symmetry is not spontaneously broken, and that the brokencharges do not explicitly depend on spacetime variables;i.e., the conserved charge densities are uniform [9],naðt; xÞ ¼ eiPxnað0; 0Þe�iPx, where P� are the spacetime-translation operators.Mori’s projection operator method.—Here, we briefly

review Mori’s projection operator method [10], which is apowerful tool for discussing the low-energy excitations,and essential for proving our theorem for nonrelativisticsystems. For readers who are unfamiliar with this method,see, e.g., Refs. [12–16] for more details of this method.Consider a set of operators fAnðt; xÞg corresponding to

the soft modes, chosen to be NG fields and broken chargedensities. First, we define the thermal average of an arbi-trary operator O as hOi � tr�eqO, where we choose

�eq as the canonical-ensemble-density operator, �eq �expð��HÞ=tr expð��HÞ, with the Hamiltonian H andthe inverse temperature � ¼ 1=T. One may consider thegrand canonical ensemble, by replacing H with H ��N,where � and N are the chemical potential and the con-served charge operator, respectively.Next, we introduce an inner product for arbitrary

operators O1 and O2:

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ðO1;O2Þ � 1

�

Z �

0d�he�HO1e

��HOy2 i; (2)

which satisfies positive definiteness, ðO1;O1Þ � 0, andconjugate symmetry, ðO1;O2Þ ¼ ðO2;O1Þ�. Using thisinner product, we define a metric:

gnmðx� yÞ � ðAnð0; xÞ; Amð0; yÞÞ: (3)

We also define gmlðy� zÞ as the inverse of gnmðx� yÞ, i.e.,Zd3ygnmðx� yÞgmlðy� zÞ ¼ �l

n�ð3Þðx� yÞ: (4)

Here, we usedEinstein notation: if an index appears twice ina single term, once as a superscript and once as subscript, asummation is assumed over all of its possible values. Anoperator with an upper index is defined as

Anðt; xÞ �Z

d3ygnmðx� yÞAmðt; yÞ: (5)

Since the metric is the two-point function of An, the inversemetric coincides with the second derivative of the effectiveaction, ��ðAnÞ, with respect to An:

gmlðx� yÞ ¼ �2��ðAnÞ�AlðyÞ�Ay

mðxÞ; (6)

where the effective action is given by the Legendre trans-formation of the generating functionalWðJnÞ:

� ðAnÞ ¼ WðJnÞ �Z

d3xJmðxÞ�WðJnÞ�JmðxÞ ; (7)

with

e��WðJnÞ ¼ tr exp

���H þ

Zd3xAnð0; xÞJnðxÞ

�: (8)

In order to decompose the degrees of freedom into theslow modes and others, we introduce the projection opera-tor P acting on a field B, given by

PBðt; xÞ �Z

d3yAnð0; yÞðBðt; xÞ; Anð0; yÞÞ: (9)

We also define Q � 1� P . These satisfy P 2 ¼ P ,Q2¼Q, and QP ¼ PQ ¼ 0.

Now, let us derive a generalized Langevin equation fromMori’s projection operator method by reconstructingthe Liouville equation, @0Anðt; xÞ ¼ iLAnðt; xÞ, whereLO � ½H;O�. The formal solution of the Liouville

equation is obtained as Anðt; xÞ ¼ eiLtAnð0; xÞ.First, we decompose @0e

iLt into

@0eiLt ¼ eiLtiL ¼ eiLtP iLþ eiLtQiL: (10)

Next, consider the Laplace transform of expðiLtÞ, whichcan be decomposed into

1

z� iL¼ 1

z�QiLþ 1

z� iLP iL

1

z�QiL: (11)

The inverse Laplace transform leads to

eiLt ¼ eQiLt þZ t

0dseiLðt�sÞPiLeQiLs: (12)

Substituting Eq. (12) into the second term in the right-handside of Eq. (10), we obtain the operator identity,

@0eiLt ¼ eiLtP iLþ

Z t

0dseiLðt�sÞP iLeQiLsQiL

þ eQiLtQiL: (13)

Multiplying Eq. (13) by Anð0; xÞ, we obtain the generalizedLangevin equation of motion [10],

@tAnðt; xÞ ¼Z

d3yi�nmðx� yÞAmðt; yÞ

�Z 1

0dsZ

d3yKnmðt� s; x� yÞAmðs; yÞ

þ Rnðt; xÞ: (14)

Here i�nmðx� yÞ is the streaming term, Km

n ðt� s; x� yÞis the memory function, and Rnðt; xÞ is the noise termdefined as

i�nmðx� yÞ � ðiLAnð0; xÞ; Amð0; yÞÞ

¼ � i

�h½Anð0; xÞ; Amyð0; yÞ�i; (15)

Knmðt�s;x�yÞ���ðt�sÞðiLRnðt;xÞ;Amðs;yÞÞ; (16)

Rnðt; xÞ � eitQLQiLAnð0; xÞ; (17)

where �ðtÞ is the step function. Note that this equation is anoperator identity, and thus Eq. (14) is equivalent to theLiouville equation. The noise term is orthogonal toAmð0; xÞ, i.e., ðAmð0; xÞ; Rnðt; yÞÞ ¼ 0, so that it drops outwhen the two-point function (Anðt; xÞ, Amð0; yÞ) is consid-ered. Therefore, the dispersion relation for An can bedetermined by the linear equation with i�n

mðx� yÞ andKn

mðt; x� yÞ. In momentum space, we can write theequation as

@tAðt;kÞ ¼ i�ðkÞAðt;kÞ�ZdsKðt� s;kÞAðs;kÞþRðt;kÞ;

(18)

in matrix notation. This is the basic equation used in ouranalysis.Nambu-Goldstone theorem.—Let us start with the

Nambu-Goldstone theorem for the effective potential [3].Consider a set of conserved charges,

Qa ¼Z

d3xnaðt; xÞ; (19)

which commutes with the Hamiltonian: ½Qa;H� ¼ 0.When the symmetry is spontaneously broken, there existsa matrix,

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�ih½Qa;iðt; xÞ�i � ½Mn�ai; (20)

such thatMn is regular, where i and a run from one to the

number of broken symmetries NBS. We assume that iðxÞare not conserved charges. Without loss of generality, i

can be chosen to be a real field.Here, let us consider the effective potential defined as

V ði; NaÞ � �ðAnÞ=V by choosing An ¼ ði; NaÞ thatcontain NG fields and broken charge densities, where Vdenotes the space volume. We parametrize ¼ ði; ’jÞandN ¼ ðna; n0bÞ, wherei (’j) and na (n

0b) are (non-) NG

fields and (un-) broken charge densities.The effective potential is invariant under the infinitesi-

mal symmetry transformations:

�i ¼ �i�ah½Qa;i�i � �a½MN�ai; (21)

�Nb ¼ �i�ah½Qa;Nb�i � �a½MNN�ab; (22)

that is,

½MN�ai �V�i

þ ½MNN�ac �V�Nc

¼ 0: (23)

This is an identity of the effective potential. If i and Na

belong to linear representations of a Lie group, theybecome ½MN�ai ¼ �i½Ta�i, and ½MNN�ac ¼ fab

cNc,

where Ta and fabc are the generator and the structure

constant, respectively. Differentiating Eq. (23) with respectto i or Na, we obtain

�½MN�ai�j

�V�i

þ ½MN�ai �2V�i�j

þ ½MNN�ac �2V�Nc�j

¼ 0; (24)

�½MNN�ac�Nb

�V�Nc

þ ½MN�ai �2V�i�Nb

þ ½MNN�ac �2V�Nc�Nb

¼ 0: (25)

At the stationary point, the first terms in Eqs. (24) and (25)drop, and only terms with NG fields and broken chargessurvive. Equations (24) and (25) become

MnV þMnnV n ¼ 0;

MnVn þMnnV nn ¼ 0;(26)

where we used matrix notation, and V �� ��2V =ð����Þ with �;� ¼ ði; naÞ. Therefore, one finds

that ððbÞi ; nðbÞa Þ ¼ ð½Mn�ib; ½Mnn�abÞ are eigenvectors of

V �� with the vanishing eigenvalue. The number of eigen-vectors is equal to the number of broken symmetries. InLorentz invariant systems, Mnn ¼ 0 because non-Lorentzscalar operators cannot condense. In this case, the inverseof the propagator at k� ¼ 0,V, vanishes from Eq. (26),

which implies the propagator of have poles at k2 ¼ 0.The number of NG modes coincides with the number ofeigenvectors, i.e., the number of broken symmetries, andtheir dispersion is ! ¼ jkj. This is the NG theorem inLorentz invariant systems. In the case of nonrelativisticsystems, however, the number of NG modes may differfrom the number of broken symmetries.To derive the counting rule for NG modes in nonrela-

tivistic systems, we choose the operators in Eq. (14) as

AnðxÞ ¼ ð ~iðxÞ; ~naðxÞÞ with ~iðxÞ � iðxÞ � hiðxÞi and~naðxÞ � naðxÞ � hnaðxÞi. We assume that no other zeromodes that couple to NG modes exist in the system. Ifthis is not the case, we must add a field coupled to the zeromode to An as a slow variable and solve the coupledequations.Now, let us consider the Laplace transformation of the

memory function, Kðz;kÞ, which gives the dissipation ofNG modes. We are interested in the behavior of the mem-ory function at low energy and low momentum, so wetake z ! 0 and k ! 0. Since L~naðz;kÞ ¼ �k � jaðz;kÞ,where jaðz;kÞ is the current operator, Knnðz; kÞ � k2,Knðz; kÞ � k, they vanish at k ¼ 0. Only Kðz; kÞ cansurvive at k ¼ 0. Kðz; 0Þ can be expanded as

�Kðz; 0Þ ¼ �M þ L þ z�Z þOðz2Þ: (27)

The first and second terms �M and L, are called the

Onsager coefficients. Both are real, and �MT ¼ ��M

and LT ¼ LT

are satisfied. �M gives a correction of

M. L, which represents dissipation of the field is

related to a spectral function at the zero frequency, and itvanishes at zero temperature because there is no spectrumat the zero frequency and the zero momentum. SinceMðkÞ and Kðz; kÞ appear as a combinationMðkÞ �Kðz; kÞ in the Langevin equation [see Eq. (18)], �M

can be renormalized to M by the shift M ! M ��M and �K!�K��M. The third term �Z

denotes the correction of the wave function. In the follow-ing we drop �Z, which will be justified later in the

analysis of zero modes. First let us consider the case atzero temperature dropping the memory function. Then, theequation of motion becomes

@0~ðt;kÞ~nðt; kÞ

!¼ i�

ðkÞ i�nðkÞ

i�n ðkÞ i�n

nðkÞ

0@

1A ~ðt;kÞ

~nðt;kÞ

!; (28)

with i���ðkÞ ¼ M� ðkÞ� �ðkÞ, where M��ðkÞ and

���ðkÞ is defined as the Fourier transform of�ih½�ð0; xÞ; �ð0; 0Þ�i and �2�=ð��ðxÞ��ð0ÞÞ, whichcoincides with M�� and V �� at k ¼ 0, respectively.

Equation (28) has a form of a Hamilton equation.M��ðkÞ can be identified as the Poisson bracket for �

and �. We assume detM��ðkÞ � 0, which ensures ~ and

~n are independent canonical variables. ���ðkÞ correspondsto the Hessian of the Hamiltonian, whose eigenvalues are

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all positive at k � 0, and NBS of them becomes zero atk ¼ 0, i.e., rankðV ��Þ ¼ NBS. The eigenmodes areobtained by detði!þ i�Þ ¼ 0. ! have always realEn pairs (En � 0, n ¼ 1; . . . ; NBS) since this is theHamiltonian system with nonnegative Hessian.

At k ¼ 0, i�nð0Þ and i�n

nð0Þ vanish from Eq. (26),which is equivalent to the conservation law dQa=dt ¼ 0.We also obtain

i�nð0Þ ¼ F�1; i�

ð0Þ ¼ �F�1G; (29)

where F�1 � ½Mn �MðMnÞ�1Mnn�V nn can be

identified as the inverse decay constant (matrix) for theNG modes up to a renormalization factor, and G ¼MnnMn

�1. F�1 is a regular matrix, which can be con-

firmed as follows: Consider V 0 ¼ M��V � M �. Since

M�� is a regular matrix, rankðV 0Þ ¼ rankðV ��Þ ¼ NBSis satisfied. V 0 is explicitly obtained as

V 0 ¼ F�1ð�GM þMnÞ 0

0 0

!: (30)

Then, rankðV 0Þ ¼ rankðF�1ð�GM þMnÞÞ ¼ NBS;

therefore, rankðF�1Þ ¼ NBS. The rank is equal to thematrix size, so that F�1 is a regular matrix.

The equations of motion become

@0 ~ðt; 0Þ ¼ �F�1G ~ðt; 0Þ þ F�1~nðt; 0Þ; (31)

@0~nðt; 0Þ ¼ 0: (32)

Then, the eigenvalue equation is detði!þ i�Þ ¼ði!ÞNBS detði!� F�1GÞ ¼ 0. The number ofmassive modes coincides with rankðF�1GÞ=2 ¼rankðF�1MnnM

�1nÞ=2 ¼ rankðMnnÞ=2. Therefore, we

arrive at

NBS � NNG ¼ 1

2rankðMnnÞ: (33)

Equation (33) is nothing but the WB conjecture.At a finite temperature, the Onsager coefficient, L,

contributes to the equation of motion. This effect can beincluded by replacingM withM � L. In this case,

the above counting rule does not change as long asdetM�� � 0.

Next, let us classify the NG modes. DifferentiatingEq. (31) by @0, we obtain

@20~ðt; 0Þ ¼ �F�1G@0 ~ðt; 0Þ: (34)

We define the NG fields satisfying G ~I ¼ 0 as type-I NG

fields, and others that are linearly independent of ~I as

type-II NG fields ~II. Obviously, the number of type-I NGfields is equal to NBS � rankðMnnÞ, which coincide withthe number of type-I NG bosons, Ntype-I, because

@20~I ¼ 0. On the other hand, the number of type-II NG

modes is equal to rankðMnnÞ=2 � Ntype-II. Since NNG ¼Ntype-I þ Ntype-II, we find

NBS ¼ Ntype-I þ 2Ntype-II; (35)

from Eq. (33). This is the saturation of the NC inequality[5]. Equations (33) and (35) are the main results in thisLetter. Here, let us discuss whether the correctionof the wave function �Z in Eq. (27) is relevant to

the counting rule of the zero modes. �Z changes @20 to

ð1� �ZGTV nnGÞ@20. For the type-I NG modes, this

correction does not contribute to the counting rule because

G ~I ¼ 0. For the type-II NG modes, the second derivativeis higher order; therefore, �Z does not change the

counting rule.Explicit breaking and mass formulas.—Here we discuss

the masses of the NG modes when an explicit breakingterm is added. Suppose the symmetry is explicitly brokenby an interaction term �V ¼ ih

i. In this case, thecontribution from the explicit breaking term is obtainedfrom Eq. (24):

½i�n ð0Þ�ai ¼ ��½Mn�aj

�i

hj; (36)

where we used �V =�i ¼ hi. Therefore, the equation ofmotion becomes

@20~ðt; 0Þ ¼ �F�1G@0 ~ðt; 0Þ þ F�1i�

n ð0Þ ~ðt; 0Þ: (37)

For general cases, it is not easy to derive the dispersionrelation from Eq. (37), so here we consider a specialcase: the symmetry breaking term does not mix thetype-I and the type-II fields, so we can decompose theexplicit breaking term to F�1i�n

ð0Þ¼ ½F�1i�n�type-I

½F�1i�n�type-II. Then, the equation of motion for ~I reads

@20~Iðt; 0Þ ¼ ½F�1i�n

�type-I ~Iðt; 0Þ; (38)

and thus, the squared mass matrix for type-I NG modesbecomes m2

type-I ¼ �½F�1i�n�type-I ¼ OðhÞ, which cor-

responds to a generalized Gell-Mann-Oakes-Renner rela-tion [17];m2

� ¼ �mqh �c c i��1I , where h �c c i, ��1

I5 , andmq

correspond to Mn, V nn, and i�n, respectively.

On the other hand, the equation of motion for ~II is

F�1G@0 ~IIðt; 0Þ ¼ i½F�1i�n�type-II ~IIðt; 0Þ; (39)

where we dropped the @20~IIðt; 0Þ term because it is higher

order in h. Since the @0 � h in Eq. (39), the squared massbecomes m2

type-II ¼ Oðh2Þ.The NG modes of type I and type II have a different type

of mass formula: the squared mass is proportional to the hand h2 for type-I and type-II NG modes, respectively. Theorder counting of the mass matrices for type-I and type-IING modes does not change even if the mixing term

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between their modes exists in F�1i�n [18]. For a typical

case at finite momentum, the order of the momentumdependence corresponds to h� k2, then the dispersionof type-I NG modes is !� jkj, and that of type-II NGmodes is !� k2. In this case, the classification of type Iand type II in this Letter is the same as that by Nielsen-Chadha.

Summary and outlook.—We have derived the countingrule for the NG modes, Eqs. (33) and (35), in nonrelativ-istic systems at zero and finite temperatures. Our method ismodel independent; the details of the theory are reflectedby the expectation value of commutation relations for theNG fields and the broken charges, as well as the secondderivative of the effective action. To derive the countingrule, we employed the following assumptions: (1) Thebroken charges are uniform; (2) Translation symmetry isnot broken; (3) detMn � 0 and detM�� � 0 exist. The

assumption (1) implies that the local operator does notexplicitly depend on space-time variables. The assumption(2) is needed because NG modes must be eigenmodes ofmomentum. More generally, the unbroken translation sym-

metry may be discrete. The assumption (3) ensures that ~and ~n are independent canonical variables. In this Letterwe assumed that deth½Qa;i�i � 0 and i are not con-served charges. If a set of conserved charges Q0

a exist suchthat h½Q0

a;i�i¼h½Q0a;Qb�i¼0 for all i and Qb (�Q0

a)but h½Q0

a;Q0b�i � 0,Q0

a are also type-II NG fields [19]. This

can be derived using the same reasoning used in this work.The number of these modes is equal to rank½Q0

a;Q0b�=2.

Therefore, our results in Eqs. (33) and (35) remainunchanged.

Our method can also apply to the systems where space-time symmetry is spontaneously broken; this is beyond thescope of this work.

We thank H. Abuki, T. Brauner, Y. Hama, K. Hashimoto,T. Higaki, T. Kimura, M. Murata, and K. Yazaki for useful

discussions. This work was supported by JSPS KAKENHIGrant No. 23340067.Note added.—After finishing this work we became

aware of similar work by Watanabe and Murayama [20].Although their analysis is different from ours, the obtainedresults are consistent with our work.

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965 (1962).[4] T. Brauner, Symmetry 2, 609 (2010).[5] H. B.Nielsen and S. Chadha, Nucl. Phys.B105, 445 (1976).[6] V. A. Miransky and I. A. Shovkovy, Phys. Rev. Lett. 88,

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field is a conserved charge, and there is no NG field that isnot a conserved charge.

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

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