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Number of Nambu-Goldstone bosons and its relation to charge densities

Haruki Watanabe1,2,* and Tomas Brauner3,4,

1Department of Physics, University of Tokyo, Hongo, Tokyo 113-0033, Japan2Department of Physics, University of California, Berkeley, California 94720, USA

3Faculty of Physics, University of Bielefeld, 33615 Bielefeld, Germany4Department of Theoretical Physics, Nuclear Physics Institute ASCR, 25068 Rez, Czech Republic

(Received 30 September 2011; published 9 December 2011)

The low-energy physics of systems with spontaneous symmetry breaking is governed by the associated

Nambu-Goldstone (NG) bosons. While NG bosons in Lorentz-invariant systems are well understood, the

precise characterization of their number and dispersion relations in a general quantum many-body system

is still an open problem. An inequality relating the number of NG bosons and their dispersion relations to

the number of broken symmetry generators was found by Nielsen and Chadha. In this paper, we give a

presumably first example of a system in which the Nielsen-Chadha inequality is actually not saturated. We

suggest that the number of NG bosons is exactly equal to the number of broken generators minus the

number of pairs of broken generators whose commutator has a nonzero vacuum expectation value. This

naturally leads us to a proposal for a different classification of NG bosons.

DOI: 10.1103/PhysRevD.84.125013 PACS numbers: 11.30.Qc, 14.80.Va

I. INTRODUCTION AND SUMMARY

Spontaneous symmetry breaking (SSB) is a ubiquitousphenomenon in nature. In quantum many-body theory, itssignificance is further boosted by the fact that it provides uswith a rare example of a general exact result: the Goldstonetheorem [1,2]. The low-energy physics of systems exhib-iting SSB is governed by the associated soft excitations, theNambu-Goldstone (NG) bosons (see Ref. [3] for a com-prehensive early review). While NG bosons in Lorentz-invariant systems are well understood, the precise charac-terization of their number and dispersion relations in ageneral quantum many-body system is still an openproblem.The systematic study of NG bosons in Lorentz-

noninvariant systems was initiated by Nielsen andChadha in their seminal work [4]. They showed that undercertain technical assumptions, including rotational invari-ance and the requirement of absence of long-range inter-actions, the energy of the NG boson is proportional to aninteger power of momentum in the long-wavelength limit,"k / jkjn. The NG boson is then classified as type I if n isodd, and as type II if n is even. Nielsen and Chadha provedthat the number of type I NG bosons plus twice the numberof type II NG bosons is greater than or equal to the numberof broken symmetry generators.In Lorentz-invariant systems, the Nielsen-Chadha (NC)

inequality is trivial since it is well-known that the numberof NG bosons, all being naturally type I, equals the numberof broken generators. In Lorentz-noninvariant systems, theNC inequality allows the number of NG bosons to besmaller than the number of broken generators provided

some of them have nonlinear dispersion relation. A renownexample is the ferromagnet where two generators brokenby the spontaneous magnetization give rise to only one NGbosonthe magnonhaving a quadratic dispersion rela-tion at low momentum.It is perhaps worth emphasizing that the NC counting

rule is formulated as an inequality and that it does notconstrain in any way the power of momentum appearing inthe dispersion relation. In fact, a quick sociological surveyreveals that this generality of the Nielsen-Chadha theoremis not very well appreciated in publications citing theoriginal work [4]. The reason for this probably is that inthe concrete systems studied in literature, type I and type IING bosons always have linear and quadratic dispersionrelations, respectively, and the NC inequality is alwayssaturated. In Appendix A of this paper, we point out arather trivial class of theories where NG bosons withenergy proportional to an in principle arbitrarily highpower of momentum can appear, and the NC inequalityis not necessarily saturated [5]. This ultimately leads us tothe proposal for a different classification of NG bosons,and the evidence that this provides an exact equality fortheir count.Our work is based on the dependence of the number of

NG bosons and their dispersion relations on expectationvalues of conserved charges, which was first investigated atthe general level by Leutwyler [7] and Schafer et al. [8](see also Ref. [9] for a recent review). Leutwyler [7] usedeffective field theory to argue that, in the absence ofquantum anomalies, nonzero density of a non-Abelianconserved charge leads to the appearance of NG bosonswith a quadratic dispersion relation. This connection wasfurther elaborated in Ref. [10] where en exact equality(saturating the NC inequality) was established for the classof linear sigma models with chemical potential at tree

*hwatanabe@berkeley.edutbrauner@physik.uni-bielefeld.de

PHYSICAL REVIEW D 84, 125013 (2011)

1550-7998=2011=84(12)=125013(9) 125013-1 2011 American Physical Society

level. Every pair of broken generators whose commutatorhas a nonzero expectation value was found to give rise toone type II NG boson with a quadratic dispersion relation.On the other hand, Schafer et al. [8] related charge den-sities to the number of NG bosons by showing that thisnumber equals the number of broken generators providedthe expectation values of commutators of all pairs ofbroken generators vanish.Our present proposal can in a sense be understood as a

synthesis of these previous works. We argue that given thethree characteristics of SSB, that is, the number of NGbosons, their dispersion relations, and the expectation val-ues of charge densities, an exact equality can be obtainedby focusing on the relation between the number of NGbosons and the charge densities. Thus, our work general-izes the theorem of Schafer et al. [8]. This is in contrast toNielsen and Chadha [4] as well as Leutwyler [7] whoemphasized the role of the NG boson dispersion relations.In order that the generality of our proposal is not ob-

scured by the technical details, we formulate it here. LetQa be the set of (spontaneously broken) conserved chargesof the theory. The number of NG bosons nNG is related tothe number of broken symmetry generators nBS by theequality

nBS nNG 12 rank; (1)where the matrix of commutators is defined by iab lim!1 1 h0jQa;Qbj0i and is the space volume. Notethat rank is always even since is real and antisymmet-ric. Unfortunately, we were not able to prove Eq. (1) in thefull generality so it remains a conjecture at the moment.However, we can prove a one-sided inequality and provideevidence that the opposite inequality holds as well. Weshould also emphasize that our argument applies to con-tinuum field theories as well as to models defined on a(space) lattice.The paper is organized as follows. In Sec. II we make

some general remarks on SSB, basically setting up thestage for the discussion of our main result. We also in-troduce a class of uniform symmetries to which thestandard derivation of the Goldstone theorem applies. InSec. III we give a partial proof of the conjecture (1) andprovide evidence for the missing part of the proof. Finally,in Sec. IV we conclude and afford some speculations.Some technical details as well as specific examples thatdo not pertain to the general argument are relegated to theappendixes.

II. GENERAL REMARKS ON SSB AND THEGOLDSTONE THEOREM

In this section we will briefly review the notion of SSBand the Goldstone theorem. Although we will not repeat indetail its proof, we would like to make a number of com-ments explaining under what conditions and technicalassumptions this proof applies. This may at times look

like purely academic pedantry, yet the example of a sharpinequality in the NC counting rule presented inAppendix A shows that one should be prepared for theunexpected.Let us first, following Nielsen and Chadha [4], define

what we mean by the number of broken generators nBS. Wedemand that there is a set of conserved chargesQa and a setof (quasi)local fieldsax, or, on a space lattice, operatorsat; xi, where a 1; . . . ; nBS, such that the matrix

Mab h0jQa;b0j0i (2)is nonsingular (i.e., has a nonzero determinant). Moreover,we assume that the conserved charges can be expressed asa spatial integral or sum of local charge densities,

Qa Zddxj0at; x; or Qa

Xxi2lattice

at; xi: (3)

Strictly speaking, the broken charge operators are onlywell defined in a finite volume. Nevertheless, in the formalderivations they only appear in commutators which have awell-defined infinite-volume limit [3].

A. Uniform symmetries

The proof of the Goldstone theorem essentially proceedsby finding the spectral representation of the commutator(2). This heavily relies on the integral representation (3)and the following translational property of the chargedensities,

j0ax eiPxj0a0eiPx; orat; xi eiHtTyxia0Txi eiHt;

(4)

where P is the four-momentum operator, H theHamiltonian, and Txi is the operator of a finite (lattice)

translation by xi. Even though this property is usuallytaken for granted, it only holds provided that the chargedensity operator, as constructed in terms of the local op-erators of the theory, does not depend explicitly on thecoordinate. We will call symmetries whose charge den-sities satisfy this condition uniform.As an example, consider the simplest field theory exhib-

iting SSB and a NG boson: the free massless relativisticscalar field theory, defined by the Lagrangian

L 12@@: (5)The action of this theory is invariant under the set ofglobal transformations x ! x x with x a bx, where