Goldstone bosons in presence of charge density

Goldstone bosons in presence of charge density

Tomas Brauner*Department of Theoretical Physics, Nuclear Physics Institute ASCR, 25068 Rez, Czech Republic

(Received 15 January 2007; published 18 May 2007)

We investigate spontaneous symmetry breaking in Lorentz-noninvariant theories. Our general dis-cussion includes relativistic systems at finite density as well as intrinsically nonrelativistic systems. Themain result of the paper is a direct proof that nonzero density of a non-Abelian charge in the ground stateimplies the existence of a Goldstone boson with nonlinear (typically, quadratic) dispersion law. We showthat the Goldstone boson dispersion relation may in general be extracted from the current transitionamplitude and demonstrate on examples from recent literature, how the calculation of the dispersionrelation is utilized by this method. After then, we use the general results to analyze the nonrelativisticdegenerate Fermi gas of four fermion species. Because of its internal SU(4) symmetry, this systemprovides an analog to relativistic two-color quantum chromodynamics with two quark flavors. In the end,we extend our results to pseudo-Goldstone bosons of an explicitly broken symmetry.

DOI: 10.1103/PhysRevD.75.105014 PACS numbers: 11.30.Qc

I. INTRODUCTION

The basic result concerning the physics of spontaneousbreaking of global internal continuous symmetries inLorentz-noninvariant systems was achieved by Nielsenand Chadha 30 years ago [1]. They showed that, undercertain technical assumptions (the most important onebeing that of a sufficiently fast decrease of correlations,which ensures that no long-range forces are present), theenergy of the Goldstone boson (GB) of the spontaneouslybroken symmetry is proportional to some power of mo-mentum in the low-momentum limit, and classified theGBs as type-I and type-II according to whether this poweris odd or even, respectively. Their general counting rulestates that the number of type-I GBs plus twice the numberof type-II GBs is greater than or equal to the number ofbroken generators.

Two classic examples of systems where the type-II GBsoccur are the ferromagnet and the so-called A phase ofsuperfluid 3He. The issue of GBs in Lorentz-noninvarianttheories has regained considerable interest in the pastdecade due to the discovery of several other systems ex-hibiting the existence of type-II GBs. These fall into twowide classes: Relativistic systems at finite density andintrinsically nonrelativistic systems. The first class in-cludes various color-superconducting phases of densequark matter—the color-flavor-locked (CFL) phase witha kaon condensate [2,3] and the two-flavor color supercon-ductor [4]—as well as the Bose-Einstein condensation ofrelativistic Bose gases [5,6] and the relativistic nuclearferromagnet [7]. The second class covers the before men-tioned ferromagnet and superfluid helium, and the rapidlydeveloping field of the Bose-Einstein condensation of di-lute atomic gases with several internal degrees of freedom[8].

Besides the investigation of particular systems, a fewnew general results also appeared. Leutwyler [9] showed

within the low-energy effective field theory framework thatthe nonlinearity of the GB dispersion is tightly connectedto the fact that some of the conserved charges developnonzero density in the ground state. As a matter of fact,the nonzero charge density induces breaking of time-reversal invariance, which in turn leads to the appearanceof a term in the effective Lagrangian with a single timederivative. This term is then responsible for the modifica-tion of the GB dispersion. In addition to Leutwyler’sresults, Schaefer et al. [3] proved that nonzero ground-statedensity of a commutator of two broken charges is a neces-sary condition for an abnormal number of GBs.

In our previous work [10] we showed that, at least withina particular class of systems described by the relativisticlinear sigma model at finite density, and at tree level, theconverse to the theorem of Schaefer et al. is also true:Nonzero density of a commutator of two broken chargesimplies the existence of a single GB which has quadraticdispersion relation, i.e., is type-II. Moreover, there is a one-to-one correspondence between the pairs of broken gener-ators whose commutator has nonzero density, and the type-II GBs. In Ref. [11] we analyzed the particular model ofMiransky and Shovkovy [2] and Schaefer et al. [3] at theone-loop level and asserted that the one-loop correctionsdo not alter the qualitative conclusions achieved at the treelevel.

The purpose of this paper is to extend the previousresults in a model-independent way. We are going topresent a general argument, based on a Ward identity forthe broken symmetry, that nonzero ground-state density ofa non-Abelian charge leads to the existence of a GB withnonlinear (typically, quadratic) dispersion law. The plan ofthe paper is the following. In the next section we give adetailed derivation of the main result. In the subsequentsections we then apply it to several systems discussed inrecent literature, and show how it facilitates the calculationof the GB dispersion relations. Next we analyze in detailCooper pairing in a nonrelativistic gas of four degeneratefermion species. Finally, we show how the proposed*Electronic address: [email protected]

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http://dx.doi.org/10.1103/PhysRevD.75.105014

method may be generalized to include explicit symmetrybreaking.

II. GENERAL ARGUMENT

In Ref. [10] we provided a general argument that non-zero density of a non-Abelian charge leads to modified GBcounting. In particular, we invoked the standard proof ofthe Goldstone theorem [12] to show that

ifabch0jj0c�x�j0i � h0j�j

0a�x�; Qb�j0i

� 2i ImXn

h0jj0a�0�jnihnjj

0b�0�j0i; (1)

see Eq. (1) in Ref. [10] and the discussion below it. Herefabc is the set of structure constants of the symmetry groupand the index n runs over different zero-momentum statesin the spectrum. The vacuum expectation value h0jj0

cj0irepresents the order parameter for symmetry breaking,while both currents j0

a and j0b serve as interpolating fields

for the GBs. Qb is the charge operator associated with thecurrent j�b . Note that whereas in Lorentz-invariant systemsthere is a one-to-one correspondence between the GBs andthe broken currents, at nonzero charge density the situationchanges. As Eq. (1) clearly exhibits, a single GB nowcouples to the two broken currents whose commutatorhas nonzero vacuum expectation value. Before we expandthis simple observation into a more detailed argument, wediscuss what the involved current transition amplitudes tellus about the GB dispersion relation.

A. GB dispersion from transition amplitude

The matrix element which couples the GB state j��k�ito the broken current operator j�a �0�, may be generallyparametrized as

h0jj�a j��k�i � ik�Fa�jkj� � i��0Ga�jkj�: (2)

By the second term, Ga, we explicitly account for thepossibility of Lorentz violation. However, we still assumethat the rotational invariance remains intact. The currentconservation then implies the following equation for theGB dispersion (see also Ref. [13]), !�k�,

�!2 � k2�Fa �!Ga � 0: (3)

Let us now explicitly mention a few special cases, dis-tinguished by the limiting behavior of the functions Fa andGa at low momentum. First, when Ga � 0 for all values ofk, we recover the standard Lorentz-covariant parametriza-tion of the transition amplitude (2), and the standardLorentz-invariant dispersion relation, ! � jkj. Second,suppose that Ga=Fa is small in the limit jkj ! 0, of orderO�jkj�. Then the dispersion relation comes out linear, onlythe phase velocity is no more equal to one due to theLorentz-violating term Ga. Third and most importantlyfor the rest of the paper, when Ga has a finite limit,limjkj!0Ga � 0, the dispersion relation takes the low-

momentum form

!�k� � k2 FaGa

: (4)

Equation (4) implies that the GB dispersion relation isnonlinear. However, in order to determine the correspond-ing power of momentum and thus classify the GB accord-ing to the Nielsen-Chadha scheme, we would have tofurther know the limiting behavior of Fa.

One should keep in mind that so far, we have providedjust a parametrization of the GB dispersion relation interms of the current transition amplitude, which only as-sumed rotational invariance and current conservation. Inorder to decide which of the possibilities outlined above isactually realized, we need more information on the dy-namics of the system, i.e., on the functions Fa, Ga.

B. Transition amplitude from Ward identity

We now get back to Eq. (1). In order to turn it into aquantitative prediction of the GB dispersion relation, wereplace the commutator with the time ordering, that is,consider the correlator of two currents,h0jTfj�a �x�j�b�y�gj0i. Taking a divergence and using thecurrent conservation, we arrive at a simple Ward identity,

@x�h0jTfj�a �x�j�b�y�gj0i � �4�x� y�ifabch0jj

�c �x�j0i: (5)

Of course, with the assumed rotational invariance the right-hand side of Eq. (5) may only be nonzero for � � 0 so thatit exactly encompasses our order parameter—the chargedensity.

Throughout the paper we assume that the symmetry isnot anomalous so that, in particular, the naive Ward iden-tity (5) holds without any corrections due to, e.g.,Schwinger terms. This assumption is partially justified bythe models studied in the following sections, where ourresults prove equivalent to those obtained with differentmethods. In general, however, the validity of Eq. (5) shouldbe checked case by case.

Next we use the Kallen-Lehmann representation for thecurrent-current correlator in the momentum space, plug-ging in the parametrization (2). We find that the condition(3), as expected, ensures cancellation of the one-particlepoles in the correlator. Taking the limit jkj ! 0, it followsthat in order to get a finite nonzero limit as the right-handside of the Ward identity (5) requires, the functions Ga, Gbhave to have a nonzero limit as well. The residua of thepoles then yield the density rule GaG�b �G

�aGb �

ifabch0jj0cj0i, or,

ImGaG�b �

12fabch0jj

0cj0i; (6)

where the Gs now stand for the zero-momentum limits ofthe respective functions defined by Eq. (2). Equation (6)together with Eq. (4) constitute the main result of the paper.They say that nonzero density of a non-Abelian charge inthe ground state requires the existence of one GB with

TOMAS BRAUNER PHYSICAL REVIEW D 75, 105014 (2007)

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nonlinear dispersion relation. In general (in fact, in allcases the author is aware of), this dispersion is quadraticunless Fa vanishes at k � 0. We will speculate on thispossibility in the conclusions.

A few other comments are in order here. First, the factthat the function Ga has nonzero zero-momentum limitessentially means that the broken charge operator createsthe GB even at zero momentum with a finite probability.This is in contrast to the Lorentz-invariant case. Second, indense relativistic systems the function Ga must be propor-tional to the only Lorentz-violating parameter in the the-ory, the chemical potential. An example will be given in thenext section. Last, in view of Eqs. (4) and (6), it is temptingto conclude that the energy of the GB is inversely propor-tional to the charge density. This indeed happens in theferromagnet [9], but need not be true generally. Again inthe next section, we shall see that both Fa and Ga may beproportional to the charge density so that it drops in theratio.

III. LINEAR SIGMA MODEL

The first class of systems to which we shall now applyour general results will be that described by the relativisticlinear sigma model at finite density (or chemical potential).This was already studied in Ref. [10] where we found thatthe chemical potential induces mixing terms in theLagrangian and the excitation spectrum can be determinedonly upon an appropriate diagonalization of the matrixpropagator. In this section we shall see that this complica-tion may be circumvented by the use of Eq. (3), at least inthe case of type-II GBs. We start with a detailed analysis ofa simple example in order to illustrate the validity of theformulas (4) and (6).

A. Model with SU�2� U�1� symmetry

We recall the model that was used to describe kaoncondensation in the CFL phase of dense quark matter[2,3]. It is defined by the Lagrangian,

L � D��yD��M2�y�� y��2; (7)

where the scalar field� transforms as a complex doublet ofthe global SU(2) symmetry. In order to account for finitedensity, the chemical potential� associated with the globalU(1) invariance (particle number) is introduced via thecovariant derivative, D0� � �@0 � i��.

Once the value of the chemical potential exceeds themass parameter M, the scalar field condenses. The groundstate may be chosen so that only the lower component of�develops expectation value. At tree level, h�2i � v=

��2p

,

where v ��2 �M2�=�

p. This breaks the original

SU�2� U�1� symmetry to its U�1�Q subgroup, generatedby the matrix Q � 1

2 �1� �3�. The spectrum contains two(as opposed to three broken generators) GBs. The type-IIGB is annihilated by �1 and its exact (tree-level) disper-

sion relation is given by

! ��k2 ��2

q��: (8)

The type-I GB is annihilated by a (energy-dependent)linear combination of �2 and �y2 , and its dispersion rela-tion corresponds to the ‘‘�’’ case of

!2�k� � k2 � 3�2 �M2

��4�2k2 � �3�2 �M2�2

q:

(9)

After shifting the field �2 by its vacuum expectationvalue, the Noether currents j�a associated with the SU(2)symmetry of the model pick up terms linear in �. Fromthese, we may readily express the current transition ampli-tudes related to the type-II GB state jG�k�i as

h0jj�1 jG�k�i � �ih0jj�2 jG�k�i

�v��2p �k� � 2��0�h0j�1�0�jG�k�i: (10)

Note that this, according to Eq. (4), immediately leads tothe dispersion relation ! � k2=2�, which is indeed thelow-momentum limit of the exact result (8). Moreover,evaluating the amplitude h0j�1�0�jG�k�i explicitly gives

G1G�2 � �2i�2v2jh0j�1�0�jG�k�ij2 � �i�v2

at jkj � 0. This is in accord with the density rule (6) for theground-state isospin density is simply given by the con-stant term in j0

3, h0jj03j0i � ��v

2, and the relevant struc-ture constant of the SU(2) group in the basis of Paulimatrices is f123 � 2.

Analogously, the coupling of the type-I GB state j��k�ito the current j�3 is found to be

h0jj�3 j��k�i �v��2p ��k� � 2��0�h0j�y2 �0�j��k�i

� �k� � 2��0�h0j�2�0�j��k�i�:

The basic complication as opposed to the type-II GB is thatnow the current amplitude is not proportional to a singlescalar field amplitude as in Eq. (10), but contains twoentangled amplitudes. Therefore, these do not simplydrop in the ratio as before and have to be evaluated ex-plicitly. Using the expressions for the scalar field ampli-tudes derived in Ref. [11], the final result for the transitionamplitudes at jkj ! 0 becomes

h0jj03j��k�i � �

v��2p

��jkj

p ��2 �M2

3�2 �M2

��1=4

;

h0jj3j��k�i � �v��2p

k��jkj

p ��2 �M2

3�2 �M2

�1=4:

A simple ratio of the coefficients of the spatial and tempo-ral current transition amplitudes gives the dispersion rela-tion

GOLDSTONE BOSONS IN PRESENCE OF CHARGE DENSITY PHYSICAL REVIEW D 75, 105014 (2007)

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! �

��2 �M2

3�2 �M2

sjkj;

which is the correct low-momentum limit of the full dis-persion (9).

B. General case

We checked the formulas (3) and (6) on the simpleexample of the linear sigma model with an SU�2� U�1�symmetry. We saw that their application is particularlysimple for the type-II GBs, which carry unbroken charge.On the other hand, the application to type-I GBs is ob-scured by the mixing of scalar field transition amplitudes.

With this experience we are ready to investigate thegeneral linear sigma model with arbitrary symmetry. InRef. [10], we dealt with the Lagrangian

L � D��yD�� V��;

where V�� is the most general static potential containingterms up to fourth order in � that is invariant under theprescribed symmetry. The covariant derivative, D�� @� � iA��, includes the chemical potential(s) assignedto one or more mutually commuting generators of thesymmetry group.

Once the chemical potential is large enough, the scalarfield develops nonzero expectation value, �0, and must bereparametrized as

��x� � ei��x��0 �H�x��: (11)

The field ��x� is a linear combination of the brokengenerators and includes the Goldstone degrees of freedom.The ‘‘radial’’ degrees of freedom are described by H�x�.

The Noether current associated with a particular con-served charge Ta reads

j�a � �2 Im�yTaD��

� �2 Im�yTa@�� 2�yTaA

��:

Now we just have to insert the parametrization (11) andretain the terms linear in the field �,

j�a;�;lin � ��y0 fTa; @

��g�0 � 2i�y0A��Ta;��0:

According to Eq. (4), this is all we need to calculate thedispersions of the type-II GBs present in the system. Thecommutator and anticommutator will give factors that arepurely geometrical, i.e., determined by the structure of thesymmetry group and the symmetry-breaking pattern.Without having to evaluate them explicitly, we simplyobserve that the constant (Ga) term is proportional to thechemical potential. This means that the dispersion relationalways comes out as ! � ck2=�, where c is some, yetunknown �-independent constant.

This constant can be determined in a different manner.Recall that (at least at tree level), once there is a (at least

discrete) symmetry that prevents the potential V�� frompicking a cubic term, the phase transitions between thenormal and the Bose-Einstein condensed phases as well asthose between two ordered phases are of second order. Thismeans that the dispersion relations of the excitations mustbe continuous across the phase transitions. Hence, due totheir particularly simple form, they can be matched to thedispersion in the normal phase. In the normal phase, aparticle carrying charge Q of the symmetry equippedwith the chemical potential �, has dispersion ! ��

k2 �M2p

��Q, M being its gap at � � 0. At M ��Q the particle becomes a GB of the spontaneously bro-ken symmetry and its low-momentum dispersion relationreads

! �k2

2�Q: (12)

Because of the argument sketched above, this dispersionrelation then persists to the broken-symmetry phase forarbitrary values of the chemical potential.

IV. NAMBU-JONA-LASINIO MODEL

Now we turn our attention to models of the Nambu-Jona-Lasinio (NJL) type, i.e., purely fermionic modelswith local, nonderivative four-fermion interaction. There,symmetry is broken dynamically by a vacuum expectationvalue of a composite operator. At the same time, the GB ofthe spontaneously broken symmetry is a bound state of theelementary fermions, and it manifests itself as a pole in thecorrelation functions of the appropriate compositeoperators.

The standard way to calculate the dispersion relation ofthe GB is as follows (see e.g. Ref. [4]). One first performsthe Hubbard-Stratonovich transformation, including thecorresponding pairing channels. The purely scalar actionis next expanded to second order in the auxiliary scalarfield so as to get the GB propagator and find its pole or,equivalently, the zero of its inverse. However, the inversepropagator contains a constant term which prevents usfrom identifying the gapless pole directly. This term isremoved with the help of the gap equation. Only afterthen may the inverse propagator be expanded to secondpower in momentum in order to establish the dispersionrelation.

The approach proposed in this paper simplifies the cal-culation of the GB dispersion relation in three aspects.First, there is no need to use the gap equation in order toestablish the existence of a gapless excitation. Indeed,since the only assumption, besides rotational invariance,is the current conservation, the broken symmetry is alreadybuilt into the formalism. Second, the loop integrals are onlyexpanded to first order in the momentum, again since onepower of momentum is already brought about by thecurrent conservation. This saves us one order in the


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Taylor expansion and thus reduces the analytic formulasconsiderably. (Of course, the final result must be the samewhatever method we use. We just choose the less elaborateone.) Third, at least when calculating the dispersion rela-tion of a type-II GB, we may take the zeroth component ofthe external momentum equal to zero from the very begin-ning. This means that, at finite temperature, we get a directexpression of the GB dispersion in terms of thermal loops,without having to analytically continue the result back toMinkowski spacetime afterwards.

A. Dispersion relation from fermion loops

In the models of the NJL type, the Noether currents arefermion bilinears. Their couplings to the Goldstone statesare therefore given by the one-loop diagrams, symboli-cally,

0| j µ |π (k) =

The solid circles denote the full fermion propagators,which are in turn obtained by solving the gap equation,while the empty circle stands for the yet unknown vertexcoupling the two fermions to the GB state. In case of atype-II GB the dispersion relation is then, according toEqs. (2) and (4), given by the ratio,

ω (k) =

k · ja

j 0a

, (13)

where the symbols standing in front of the loops denote theoperators inserted in place of the crosses.

Before we explain in detail how the Goldstone-fermionvertex may be extracted from another Ward identity, let usnote that the explicit evaluation of this vertex may in factoften be circumvented. This will be demonstrated later onin explicit examples. In general, the auxiliary scalar fieldsintroduced by the Hubbard-Stratonovich transformationmix so that one has to look for poles of a matrix propagator.However, it seems to be a generic feature of type-II GBsthat they carry charge of some unbroken symmetry. As aresult, they may couple to just one of the auxiliary scalarfields and no mixing appears. (For instance, recall themodel of Sec. III A where the type-II GB couples just to�1, while the type-I GB couples to both �2 and �y2 .) Inother words, we may devise an interpolating field yX , Xbeing some matrix in the flavor space, and calculate insteadof the transition amplitude h0jj�j��k�i the correlationfunction of j� and yX . The point is that, by theKallen-Lehmann representation, this correlation functionis proportional to the desired transition amplitude, and that

the GB propagator as well as the unknown amplitudeh��k�j yX j0i drop out in the ratio (13).

In practice this means that, instead of the unknownGoldstone-fermion vertex in the loops in Eq. (13), wemay simply insert the operator yX . In particular, inthe nonrelativistic NJL model, the current j�a associatedto the charge Ta reads

j0a � yTa ; ja �

1

2im� yTar �r yTa �;

where m is the common mass of the degenerate fermionflavors. Inserting these operators into the fermion loops inEq. (13), and making the Taylor expansion of the spatialloop to the first order in the external momentum, we arriveat the result

!�k� �k2

2m

�1�

4

3

R d4q�2��4

Tr�q2 @S@q2 �q�XS�q�Ta�R d4q

�2��4 Tr�S�q�XS�q�Ta�

�; (14)

where the trace is performed over the flavor space and Sdenotes the full fermion propagator. At finite temperature1=� the integral over q0 in Eq. (14) becomes a sum overfermion Matsubara frequencies, �n � �2n� 1��=�. TheGB thus naturally remains gapless up to a possible thermalsymmetry restoring phase transition, but otherwise its dis-persion can depend on temperature.

B. GB-fermion coupling from Ward identity

As already stressed in the discussion below Eq. (13), theformula (14) has a limited use. It applies only to the casesin which the GB couples to just one of the auxiliary scalarfields introduced by the Hubbard-Stratonovich transforma-tion (no mixing). Unless this happens, the (unknown)couplings of the GB to the fermion bilinear operators donot drop in the ratio (13).

We also noted that this no-mixing property is most likelyto be generic for type-II GBs, which carry charge of someunbroken symmetry. Nevertheless, in any case, the GB-fermion coupling can be evaluated directly by using yetanother Ward identity [14]. Let us introduce the connectedvertex function, G�

a �x; y; z� � h0jTfj�a �x� �y� y�z�gj0i.

Taking the divergence and stripping off the full fermionpropagators corresponding to the external legs, we find theWard identity for the proper vertex,

k��a �p� k; p� � TaS�1�p� k� � S�1�p�Ta: (15)

This vertex exhibits a pole due to the propagation of theintermediate GB state. Near the GB mass shell, it may beapproximated by a sum of the bare vertex and the polecontribution [15]. Writing the inverse nonrelativistic fer-mion propagator as S�1�!;p� � !� p2

2m�� !;p�,where � is the self-energy, the bare part of the propagatorexactly cancels against the bare vertex in the Ward identity(15). The remaining term expresses the residuum at the GBpole in the vertex function—which is proportional to the


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desired GB-fermion coupling—in terms of the fermionself-energy, so that

=1N

[Σ(p)Ta − TaΣ( p + k)] . (16)

The normalization constant N is proportional to the cou-pling of the GB to the broken current itself, but this is notimportant since this factor only appears in the ratio (13).The GB-fermion coupling is apparently determined by thesymmetry-breaking part of the fermion self-energy. Oncethis is calculated, e.g., in the mean-field approximation, itmay be plugged into the loops in Eq. (13) to determine theGB dispersion.

In conclusion of our general discussion let us remarkthat all the results presented so far in Sec. IV extendstraightforwardly to cases including Cooper pairing offermions. Once the GB is annihilated by a difermionoperator instead of a fermion-antifermion one, we justhave to flip the direction of some of the arrows in thefermion loops. The S�q� in Eq. (14) then denotes the matrixpropagator in both the flavor and the Nambu ( ; y) space.

In addition, Eqs. (15) and (16) modify accordingly: Thematrix Ta must be replaced with the block matrix in theNambu space,

Ta 00 �TTa

� �:

In the end, the GB-fermion coupling vertex comes out as

= −1N

Σψψ (p)T Ta + TaΣψψ (p + k) , (17)

where � denotes the anomalous (pairing) part of thefermion self-energy.

C. Three-flavor degenerate Fermi gas

The nonrelativistic Fermi gas consisting of three degen-erate fermion species has been investigated theoretically ingreat detail [8,16,17]. The reason is that such a system hasalready been successfully realized in experiments usingatoms of 6Li and 40K. At the same time it provides aninteresting analogy with quantum chromodynamics(QCD). A recent theoretical analysis suggests that undercertain conditions, bound clusters of three atoms mayform, being singlets with respect to the global SU(3)symmetry, very much like baryons in QCD [18]. Such athree-flavor atomic Fermi gas thus creates a natural play-ground for testing properties of quark matter.

In the following, we are going to illustrate the utility ofour general formulas and rederive the results of Ref. [8]concerning the type-II GBs in the three-flavor system. Tothat end, recall that with a proper choice of the interaction,the global SU(3) symmetry is broken by a difermioncondensate, h i ji. Since this is antisymmetric under the

exchange of i, j, it transforms in the �3 representation ofSU(3). This leads to the symmetry-breaking patternSU�3� U�1� ! SU�2� U�1�.

Once we choose the condensate to point in the thirddirection, the generator �8 acquires nonzero vacuum ex-pectation value. Based on the general discussion in Sec. II,we anticipate that the four broken generators, �4, �5, �6,�7, couple to a doublet of type-II GBs with quadraticdispersion relations. Because of the unbroken symmetry,these two GBs have the same dispersion. For the explicitcalculation, we choose the GB in the sector ��4; �5�.

To determine its dispersion relation, we need to evaluatethe loop in Fig. 1 at nonzero external momentum k. [Inorder to make contact with literature [8], we do not performthe Taylor expansion in the external momentum as inEq. (14).] Note that since the type-II GB couples to twobroken currents, j�4 and j�5 , we are free to choose any ofthem to calculate its dispersion. Here, we take up j�4 . Thetwo propagators that appear in the loop are given by thefinite-temperature expressions (see Ref. [8]) G33�i�n;k� �1=�i�n � k�, F 12�i�n;k� � �i�=��i�n�2 � E2

k�, where� is the gap parameter, k � k2=2m�� is the barefermion energy measured from the Fermi surface, and

Ek ��2

k � �2q

. The interpolating field for the GB ischosen as T�7 . This may be understood as follows.By our choice, the ground-state condensate carries the third(anti)color of the representation �3. The current j�4 whosetransition amplitude we calculate, then excites a difermionstate with the first (anti)color. The GB thus carries thequantum numbers of a pair of fermions with the secondand the third color, respectively. The same conclusionfollows immediately from Eq. (17): The fermion self-energy is proportional to �2 and f�2; �4g � ��7.

The point of using Eq. (13) to calculate the GB disper-sion relation is that all common algebraic factors includingthe trace over the flavor space cancel in the ratio. Afterstripping off such trivial factors, we are thus left with theloop integral

FIG. 1. Fermion loop for the calculation of the type-II GBdispersion in the three-flavor Fermi gas. The numbers adjacent tothe fermion propagators denote the respective flavors. The upperline represents the normal propagator of the (ungapped) thirdfermion flavor, while the lower line represents the pairing of thefirst two flavors.


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J�k� �1

�

Xn

Z d3q�2��3

�1

�k�2q2m

�1

�i�n�2 � E2q�k

1

i�n � q;

(18)

where the two options in the curly brackets stand for theinsertion of the temporal and spatial components of thecurrent, respectively. Note that the Matsubara frequencyentering the loop graphs has been set to zero because, asalready remarked above, in case of a type-II GB thetransition amplitude of j0 is nonzero even at vanishingenergy.

After performing the Matsubara sum, Eq. (18) acquiresthe form

J�k� �Z d3q�2��3

� 1�k�2q

2m

�1

2Eq�k

�1� f�Eq�k� � f�q�

Eq�k � q

�f�Eq�k� � f�q�

Eq�k � q

�; (19)

where f�x� � 1=�e�x � 1� is the Fermi-Dirac distribution.This result is completely equivalent to Eq. (A9) in Ref. [8],though not identical, of course, for we calculate a differentloop graph here. The GB dispersion may be directly in-ferred from here by means of Eq. (13). We omit the explicitexpression since it is rather cumbersome and not particu-larly revealing.

V. FOUR-FLAVOR DEGENERATE FERMI GAS

In this section we are going to discuss the nonrelativisticdegenerate Fermi gas with four fermion flavors which, toour best knowledge, has not been analyzed in literature sofar. Besides being the next-to-simplest nontrivial case of anonrelativistic fermionic system exhibiting type-II GBs, italso provides an interesting analogy with the two-colorQCD with two quark flavors [19–21].

We start by assuming the existence of four degeneratefermion species with a common chemical potential �,which interact by a contact four-fermion interaction invari-ant under the global SU�4� U�1� symmetry. For the timebeing, all we need to know is that this interaction desta-bilizes the Fermi sea by the formation of Cooper pairs. Thedetailed form of the interaction will be specified later.

A. Symmetry-breaking patterns

First we analyze the symmetry properties of the orderedground state. Analogously to the three-flavor case, theoriginal SU�4� U�1� symmetry is spontaneously brokenby a difermion condensate h i ji, which transforms as anantisymmetric rank-2 tensor of SU(4).

The analysis of the symmetry-breaking patterns is muchsimplified by noting the Lie algebra isomorphism SU�4� ’SO�6�. The six-dimensional antisymmetric tensor repre-sentation of SU(4) is equivalent to the vector representationof SO(6). This correspondence is worked out in detail in

the appendix. The order parameter is thus represented by acomplex 6-vector, �, or equivalently, by two real 6-vectors, �R � Re� and �I � Im�. The situation is prettymuch similar to that in spin-one color superconductorswhere the order parameter is a complex 3-vector ofSO(3) [22–24].

There are two qualitatively different possibilities, dis-tinguished by the relative orientation of�R and�I. First, if�R and �I are parallel, there is obviously an unbrokenSO(5) rotational invariance. The symmetry-breaking pat-tern is SO�6� U�1� ! SO�5�, or SU�4� U�1� ! Sp�4�,with six broken generators, transforming as a 5-plet and asinglet of the unbroken SO(5), respectively. Note that inthis case, the U(1) symmetry of the Lagrangian may beused to even eliminate entirely the imaginary part of�, i.e.,to set �I � 0. Let us remark that it is this phase of thenonrelativistic four-flavor Fermi gas that is analogous tothe two-flavor two-color QCD. There, the reality of � isenforced by C and P conservation. Also, there is no globalU(1) symmetry in two-color QCD as the correspondingphase transformations are the axial ones, being broken bythe axial anomaly [20,21].

The more interesting, but also more complicated possi-bility is that with nonzero �R and �I pointing in differentdirections. Then, the SO�6� U�1� symmetry may be ex-ploited to cast the order parameter in the form � ��;�� i�; 0; 0; 0; 0�T , with real , �, �, and both and� nonzero. In general the SO�6� U�1� symmetry is nowbroken down to SO(4), thus breaking 10 generators, form-ing two vectors of the unbroken SO(4) as well as twosinglets. However, when � � 0 and jj � j�j, there is anadditional unbroken U(1) symmetry generated by a com-bination of a rotation in the ��R;�I� plane and the originalU(1) phase transformation. There is thus one less brokengenerator and one less GB.

B. Fermion propagator and excitation dispersions

According to our general discussion above, we need toknow the charge densities in the ground state in order todetermine the spectrum of GBs of the broken symmetry.The charge densities, h yTa i, are obtained from the traceof the fermion propagator with the charge matrix Ta. Atthis place, we have to specify the Lagrangian of the system.

Anticipating the Cooper pairing of the fermions, wechoose the interaction Lagrangian in a form suitable for amean-field calculation,

L � y�i@t �

r2

2m��

� �G

X6

i�1

j T�i j2; (20)

where is the four-component fermion field and �i are aset of six basic antisymmetric 4 4 matrices, transformingas a vector of SO(6), see Eqs. (A1) and (A2). Thus, theLagrangian (20) is manifestly SO�6� ’ SU�4� invariant.

Upon introduction of an auxiliary scalar field �i by theHubbard-Stratonovich transformation, the Lagrangian be-


105014-7

comes

L � y�i@t �

r2

2m��

� �

1

4G�yi �i �

1

2 T�i �

yi

� H:c: (21)

The vacuum expectation value of � provides the orderparameter for symmetry breaking. In the mean-field ap-proximation, � is simply replaced with its vacuum expec-tation value and the semibosonized Lagrangian (21) thenyields the inverse fermion propagator in the Nambu space,

G�1 ��G�0 �

�1 ��

�� G�0 ��1

� �;

where G0 are the bare propagators and �� i�yi ,

�� y. Inverting this block 2 2 matrix, we obtain[24]

G �G� ��

�� G�

� �;

where

G � ��G0 ��1 ��G�0 ��1;

� � �G�0 �G:(22)

At finite temperature, the bare thermal propagators aresimply G�0 �i�n;k� � 1=�i�n � k� and G�0 �i�n;k� �1=�i�n � k�. As explained in the previous section, wecan exploit the SO(6) symmetry of the Lagrangian (20)so that only two components of the 6-vector � are actuallynonzero. Let us denote them generally as a, b (a � b), sothat �� a�ya ��b�yb . In the previous section wesimply took a � 1 and b � 2, but when later calculatingthe charge densities, it will be more convenient to set a � 1and b � 6.

According to Eq. (22), the full mean-field fermionpropagator G� reads

G��i�n;k� � ��i�n � k��2n �

2k ��1: (23)

Using the properties of the �i matrices we have

�� j�aj2 � j�bj

2 � 2i Im��a��b��a�yb

� j�aj2 � j�bj

2 � 2 Im��a��b��P

��ab � P

��ab �:

The projectors P��ab are defined in Eq. (A5). With projec-tors in the denominator, the matrix in the expression (23)for G� can easily be inverted and yields the final result

G��i�n;k� � ��i�n � k��2n �

2k � j�aj

2 � j�bj2 � 2 Im��a�

�b��P

��ab � P

��ab �

��2n �

2k � j�aj

2 � j�bj2�2 � 4�Im��a�

�b��

2

� ��i�n � k�

�P��ab

�2n � 2

k � j�aj2 � j�bj

2 � 2 Im��a��b��

P��ab

�2n � 2

k � j�aj2 � j�bj

2 � 2 Im��a��b�

�:

(24)

From Eq. (24) we can see that there are two doublets offermionic excitations over the ordered ground state, withgenerally nonzero gaps. Their dispersion relations are mostconveniently written in the form

!��k� ��2

k � j�a � i�bj2

q: (25)

Two special cases deserve particular attention. First, in theSO(5) symmetric phase �b is set equal to zero so that allfour excitations are degenerate, forming a fundamentalquadruplet of SO�5� ’ Sp�4�. Second, in the SO�4� U�1� symmetric phase, j�aj � j�bj with their complexphases differing by �=2. Equation (25) then implies thattwo of the fermions are actually gapless. Altogether wehave two pairs of degenerate fermions, which transform asdoublets under the respective SU(2) factors of SO�4� ’SU�2� SU�2�.

C. Charge densities and GBs

The ground-state densities of the conserved charges areobtained from the propagator G� as

h yTa i � �1

�

Xn

Z d3k�2��3

Tr�TTaG��i�n;k��:

For the sake of this section, it is convenient to fix thedirection of the order parameter � in the SO(6) space.As already remarked above, we choose a � 1 and b � 6.With the explicit form of the �i matrices given in Eq. (A2),we find

i�1�y6 � P��16 � P��16 � �

�3 00 �3

� �:

Since i�1�y6 � M16 is one of the SO(6) generators Mij

defined in Eq. (A6), the orthogonality relation (A8) impliesthat it can also be the only one which acquires nonzerodensity in the ground state,


105014-8

hM16i � 8 Im��1��6�

1

�

Xn

Z d3k�2��3

�i�n � k�

��2n � 2

k � j�1j2 � j�6j

2�2 � 4�Im��1��6��2

� �Z d3k�2��3

k

�1� 2f�!��k��

!��k�

�1� 2f�!��k��

!��k�

�;

after the Matsubara summation.The spectrum of GBs may be understood as follows.

First, in the SO(5) symmetric phase,�6 � 0 and there is nocharge density in the ground state. According to our gen-eral results, there are then six type-I GBs—five corre-sponding to the coset SO�6�=SO�5�, and one to thebroken U(1).

From now on, we shall analyze the SO(4) symmetricphase of the model (20), i.e., assume that both �1 and �6

are nonzero and have different complex phases. The con-densate �1 breaks all SO(6) generators of the form M1i,where i � 2; . . . ; 6. On the other hand, the condensate �6

breaks all SO(6) generators of the form Mj6, where j �1; . . . ; 5. This is altogether nine spontaneously broken gen-erators of SO(6). One of them, M16, is a singlet of theunbroken SO(4). The remaining eight ones group into twoSO(4) vectors, M1i and Mi6, with i � 2; . . . ; 5. Note that,by Eq. (A7), �M1i;Mi6� � 2iM16 (no summation over i).Since M16 has nonzero density in the ground state, the twobroken generators M1i andMi6 couple to a type-II GB witha quadratic dispersion relation. These eight broken gener-ators are thus associated to an SO(4) vector of type-II GBs.

Finally, for �6 � i�1 there is a residual unbrokenU(1) symmetry generated by 1M16. Note that the com-binations M1i iMi6 are charged under this unbrokensymmetry, carrying opposite charges. It may also be illu-minating to observe that, for instance, when �6 � �i�1,the matrix �� has the form

�� 2�1

0 0 1 00 0 0 0�1 0 0 00 0 0 0

0BBB@

1CCCA:

In other words, only the first and third fermion flavorsparticipate in the pairing. The second and fourth do not,and represent the two gapless excitations predicted byEq. (25).

Let us now calculate the dispersion relation of the type-II GBs in this SO�4� U�1� symmetric phase. (As will beshown in the next section, the relation �6 � i�1 is oneof the two possible solutions of the mean-field gap equa-tion, the other one corresponding to the SO(5) symmetricphase.) As already noted above, there are four type-II GBsthat couple to the broken generators M1i and Mi6. Since

these form a multiplet of the unbroken SO(4), they have allthe same dispersions. We choose i � 3 for which thebroken generators have a particularly simple form,

M13 � ��1 00 �1

� �; M36 �

�2 00 �2

� �:

Together with M16 these generate an SU(2) subalgebra ofSU(4).

The generators M13 and M36 must both couple to thesought type-II GB. When acting on the ground state, theyboth excite the third component of �, so the interpolatingfield for the GB will be T�3 . In order to determine theGB dispersion, we evaluate its coupling to the brokencurrent of M13, which is given by the loop graph inFig. 2, analogous to that for the three-flavor case inFig. 1. After performing the Matsubara sum, and throwingaway trivial numerical factors, we arrive at the loop inte-gral, analogous to Eq. (19),

J�k� �Z d3q�2��3

f1

�k�2q2mg

q

2!��q�k!��q

�1� f�!��q�k� � f�!��q �

!��q�k �!��q

�f�!��q�k� � f�!

��q �

!��q�k �!��q

�;

where the upper and lower signs were obtained with �6 �i�1, respectively. Ultimately, both yield the same dis-persion, which is found by the Taylor expansion to firstorder in momentum and plugging the result into Eq. (4).

D. Gap equation

The actual value of the order parameter � is determinedby the gap equation. To that end, we need to know theanomalous part of the matrix propagator which followsfrom Eq. (22),

��i�n;k� � �1

i�n � k��a�ya ��b�yb �G

��i�n;k�:

FIG. 2. Fermion loop for the calculation of the type-II GBdispersion in the four-flavor Fermi gas described by theLagrangian (20).


105014-9

In the mean-field approximation, we require the self-consistency at the one-loop level, which leads to the gapequation

�� 2G1

�

Xn

Z d3k�2��3

�yi Tr��i��i�n;k��:

After working out the algebraic trace and the Matsubarasum, this set of gap equations for �a and �b may berewritten in a particularly symmetric form,

�a � i�b � 4G��a � i�b�I��;

�a � i�b � 4G��a � i�b�I��;

(26)

where

I�� Z d3k�2��3

1� 2f�!��k��

!��k�:

The gap equations (26) apparently admit two qualita-tively different solutions. The first one corresponds toIm��a�

�b� � 0, i.e., the SO(5) symmetric phase. The gap

equation then simply reads

1 � 4GI��: (27)

Either sign can be used in the integrals since, in this case,the !�� dispersions are degenerate.

The second solution has Im��a��b� � 0. Equation (26)then implies that�a � i�b. For the two possibilities, thegap equations are

1 � 4GI�� for �a � �i�b;

1 � 4GI�� for �a � �i�b:(28)

Note that this solution of the gap equations (26) pre-cisely corresponds to the SO�4� U�1� phase discussed inSec. VA. In either case, �a � i�b, only the two gappedfermion flavors contribute to the gap equation (28).

It is interesting to observe that the mean-field analysisdoes not support the most general form of the order pa-rameter �. Rather, either the real and imaginary parts of �are parallel, breaking the SO�6� U�1� symmetry toSO(5), or they are perpendicular with the same magnitude,breaking the symmetry to SO�4� U�1�.

Of course, which of these two possibilities is the actualground state may depend on the magnitude of the couplingconstant and the chemical potential, and can only be de-cided by solving the gap equations (27) and (28) andcomparing their free energies. This task is, however, be-yond the scope of the present paper where the model (20)only serves as an illustration of the general features ofsymmetry breaking in the presence of charge density.

VI. PSEUDO-GOLDSTONE BOSONS

So far we have been dealing exclusively with exactsymmetries for which, once spontaneously broken, theGoldstone theorem ensures the existence of exactly gapless

excitations. Real world is, however, more complicated.Almost as a rule, global internal symmetries are usuallyonly approximate, leading to approximately conservedcurrents and, when spontaneously broken, to the existenceof excitations with a small gap, so called pseudo-Goldstonebosons (pGBs). (In the following, we shall keep using theterm Goldstone boson unless the existence of a nonzerogap is important for our discussion).

Let us just note that, in fact, the only generic type of anexact global symmetry is the particle number conservation.This is, however, only Abelian and hence does not lead tothe nontrivial phenomena studied in this paper. In addition,gauge symmetries are always exact. Here, once the gaugeis appropriately fixed and the remaining global symmetryspontaneously broken, there are no physical GB states asthey are ‘‘eaten’’ by the gauge bosons, making them mas-sive by the Higgs mechanism. It turns out that the numberof massive gauge bosons is always equal to the number ofbroken generators even if the number of would-be GBs issmaller [25].

A. Counting of pseudo-Goldstone bosons

It should be stressed that for approximate symmetries,there is no rigorous theorem analogous to the Goldstone’s,which would guarantee the existence of ‘‘approximatelygapless’’ excitations. On a heuristic level, however, theconclusions of the theorems of Goldstone and of Nielsenand Chadha may be carried over to this less perfect, yetmore common and physical situation. Before a more de-tailed discussion of approximate symmetries, let us illus-trate this on a simple example.

Both Goldstone and Nielsen-Chadha theorems makepredictions about the low-momentum limit of the GBdispersion relation without actually specifying what thelow momentum means. What is the type and number ofpGBs associated with a spontaneously broken approximatesymmetry, therefore depends on the scale at which thesymmetry is broken explicitly. Consider the simple modelstudied in Sec. III A. The type-II GB has a massive partnerwhich is annihilated by �y1 , i.e., its antiparticle, and theirexact tree-level dispersions may be jointly expressed as

! ��k2 ��2

q�;

see Eq. (8). At low momentum, we clearly see a gaplessmode with a quadratic dispersion and a mode with the gap2�. On the other hand, at high momentum both modeshave almost linear dispersion.

So, once the SU�2� U�1� symmetry of the model (7) isexplicitly broken, the resulting pGB spectrum depends onwhether the scale of the explicit breaking is smaller orlarger than the chemical potential �. For weak breaking,we still see an approximately gapless mode with a qua-dratic dispersion and a mode with a gap approximately 2�.On the other hand, for strong explicit breaking, larger than


105014-10

�, we can no longer resolve the quadratic onset of thedispersion, and instead find two pGBs with linear disper-sion relations. Note that in both cases the Nielsen-Chadhacounting rule is satisfied even for pGBs.

B. Ward identity and dispersion relations

Let us now consider a symmetry which is explicitlybroken by adding a small term to the Lagrangian of thetheory. The associated current then becomes only approxi-mately conserved and we find @�j

�a � �aL, where �aL is

the variation of the Lagrangian under the symmetry trans-formation. The basic assumption of an ‘‘approximate’’spontaneously broken symmetry is that there is still amode j��k�iwhich couples to the current j�a . The parame-trization (2), which is based just on rotational invariance,thus still holds. What does change is, of course, Eq. (3)which now becomes

�!2 � k2�Fa �!Ga � h0j�aLj��k�i: (29)

Up to this change, we proceed along the same line ofargument as in Sec. II. The Ward identity (5) acquires anew term that accounts for the explicit symmetry breaking,

@x�h0jTfj�a �x�j�b�y�gj0i � h0jTf�aL�x�j�b�y�gj0i

� �4�x� y�ifabch0jj�c �x�j0i:

Since both the broken currents and �aL couple to the GBstate, both correlators can be expressed via the Kallen-Lehmann representation. We find that cancellation of theone-particle poles exactly requires the energy and momen-tum to fulfill the dispersion relation (29). In the limit jkj !0, the Ward identity leads to the modified density rule,

Im �GaG�b �!�0�GaF�b� �12fabch0jj

0cj0i;

cf. Eq. (6). The explicit breaking of the symmetry entershere in terms of the (nonzero) energy gap, !�0�.

Equation (29) may be used to give a precise meaning tothe otherwise vague term ‘‘approximately gapless mode,’’as well as to distinguish type-I and type-II GBs in the caseof an explicitly broken symmetry. Let us thus, again,assume that Ga�jkj� has a nonzero limit as jkj ! 0, anddenote h0j�aLj��k�i � Ha�jkj�, for sake of brevity.

According to Eq. (29), the energy gap!�0� of the pGB isgiven by the solution of the equation

!2�0�Fa �!�0�Ga � Ha;

which has exactly one positive solution provided all coef-ficients, Fa, Ga, Ha are positive,

!�0� �1

2Fa��Ga �

��G2a � 4FaHa

q�:

The low-momentum behavior of the GB dispersion isapparently determined by the relative value of G2

a andFaHa. If the explicit symmetry breaking is weak enough,i.e., if FaHa � G2

a, the term quadratic in ! may be ne-

glected in comparison with that linear in !, and the low-momentum dispersion relation of the pGB reads

!�k� �Ha

Ga� k2 Fa

Ga; (30)

as opposed to Eq. (4). We thus find a type-II pGB.If, on the other hand, FaHa G2

a, the !2 term domi-nates over the ! term in Eq. (29), and the resulting low-momentum dispersion relation,

!�k� �

��Ha

Fa� k2

s; (31)

is one of a type-I pGB with nonzero mass squared Ha=Fa.

C. Nambu-Jona-Lasinio model

The formulas (29)–(31) are absolutely general but, un-fortunately, not of much use unless we are somehow able tocalculate the new amplitude, h0j�aLj��k�i. Nevertheless,there is a wide class of systems where this can be deter-mined in the same way as the current amplitudes Fa andGa: They are the models of the NJL type where thesymmetry-breaking operator is bilinear in the fermionfields.

First of all, note that this is all we need, for instance, toprovide a realistic description of dense quark matter sinceboth quark masses and the various chemical potentials canbe taken into account by adding appropriate bilinear op-erators into the Lagrangian.

The point is that in such a case, the dispersion relation ofthe GB (30) [or (31) as well] is given by the ratio offermion loops, just like in Eq. (13), which now becomes

ω (k) =

∆a + k · ja

j 0a

.

All that was said in Sec. IVA then holds without changealso in the presence of explicit symmetry breaking.

VII. CONCLUSIONS

In this paper, we analyzed the general problem of spon-taneous symmetry breaking in the presence of nonzerocharge density. Based on a Ward identity for the brokensymmetry, we showed in Sec. II that nonzero density of acommutator of two broken currents implies the existenceof a GB with nonlinear dispersion relation. Under fairlygeneral circumstances, this dispersion is quadratic. Theonly assumption is the existence of a nonzero low-momentum limit of the transition amplitude of the spatialcurrent, Fa.

It is interesting to compare this result to that ofLeutwyler [9]. In his approach, nonzero density of a non-


105014-11

Abelian charge leads to the presence of a term with a singletime derivative in the effective Lagrangian. Together withthe standard two-derivative kinetic term, this results in aquadratic dispersion of the GB. In fact, however,Leutwyler’s kinetic term is proportional to the same con-stant Fa that we use here just to parametrize the currenttransition amplitude. Thus, regarding the possibility of theexistence of GBs with energy proportional to some higherpower of momentum, our results are equivalent.Nevertheless, the comparison with Leutwyler’s approachgives us new insight into this open question: In order tohave a GB with the power of momentum higher than two inthe dispersion relation, the standard kinetic term in thelow-energy effective Lagrangian would have to vanish. Itis hard to imagine how such a theory would be defined evenperturbatively, yet such a possibility is not theoreticallyruled out.

In Sec. III we demonstrated the general results of Sec. IIon the linear sigma model, which we studied in detail inour previous work [10,11]. We were thus able to prove on avery general ground, that the tree-level dispersion of type-II GBs in such a class of models is of the form (12).

While in the framework of the linear sigma model wemerely verified results that could be obtained in a differentmanner, in the models of the NJL type our general resultsare practically useful. In Sec. IV we suggested to calculatethe GB dispersion relation by a direct evaluation of itscoupling to the broken current. This procedure consider-ably reduces both the length and the complexity of theanalytic computation as compared with the conventionalapproach using the Hubbard-Stratonovich transformationand the consecutive diagonalization of the quadratic part ofthe action.

In Sec. V we analyzed a particular example of a systemexhibiting type-II GBs: The nonrelativistic gas of fourfermion flavors with an SU�4� U�1� global symmetry.We showed that the mean-field gap equation admits twoqualitatively different solutions, i.e., two different orderedphases. In the first one, all fermion flavors pair in a sym-metric way, leaving unbroken an SO(5) subgroup of theoriginal symmetry. The six broken generators give rise tosix type-I GBs.

In the other phase, only two of the four fermion flavorsparticipate in the pairing. The other two flavors remaingapless. In this case, the unbroken symmetry is SO�4� U�1� and one of the broken charges—corresponding to thedifference of the number of the paired and unpaired fermi-ons—acquires nonzero density. Consequently, eight of thenine broken generators give rise to four type-II GBs with aquadratic dispersion. Which of these two phases is theactual ground state of the system can only be decidedupon the solution of the gap equation.

Finally, in Sec. VI we extended the results achieved inthe preceding parts of the paper to spontaneously brokenapproximate symmetries. We discovered that once thenotions of the type and low-momentum dispersion relation

of the GB are defined and treated carefully, the generalresults remain valid. In particular, in the NJL model thisallows their application to the realistic description of densequark matter.

ACKNOWLEDGMENTS

The author is indebted to M. Buballa and J. Hosek forfruitful discussions and critical reading of the manuscript.He is also obliged to J. Noronha, D. Rischke, and A.Smetana for discussions. Furthermore, he gratefully ac-knowledges the hospitality offered him by ECT� Trentoand by the Frankfurt Institute for Advanced Studies, whereparts of the work were performed. The present work was inpart supported by the Institutional Research PlanNo. AV0Z10480505 and by the GA CR Grant No. 202/06/0734.

APPENDIX A: ALGEBRA OF ANTISYMMETRIC4 4 MATRICES

In Ref. [19] we showed that a general unitary antisym-metric 4 4 matrix � with unit determinant may be cast inthe form � � ni�i, i � 1; . . . ; 6, where �i is a set of sixbasis matrices, satisfying the constraint

�yi �j � �yj �i � 2�ij1; (A1)

and ni is an either real or pure imaginary unit vector. Bythis construction we provided an explicit mapping betweenthe cosets SU�4�=Sp�4� and SO�6�=SO�5�. We also found aparticular explicit solution to Eq. (A1),

�1 �0 �11 0

� �; �2 �

�2 00 �2

� �;

�3 �0 i�1

�i�1 0

� �; �4 �

i�2 00 �i�2

� �;

�5 �0 i�2

i�2 0

� �; �6 �

0 i�3

�i�3 0

� �:

(A2)

By a simple complex conjugation, using the antisymme-try of the �is, Eq. (A1) translates to

�i�yj � �j�

yi � 2�ij1: (A3)

In the context of the present paper, ni is no longer a unitvector, but rather an arbitrary complex six-dimensionalvector. We start by showing that its real and imaginaryparts realize independent real representations of the groupSU�4� ’ SO�6�. This acts on � as �! U�UT . For aninfinitesimal transformation, U � 1� iX with a traceless,Hermitian X. Using the orthogonality relation

Tr �yi �j � 4�ij; (A4)

we derive the change of the coordinate ni induced by thetransformation U,

�ni �inj4

Tr��yi �X�j � �jXT��

inj2

Tr�X�j�yi �:


105014-12

(We transposed the second term and used the antisymmetryof �i.) Now for traceless X, the matrix iTr�X�j�

yi � is,

with a little help from Eq. (A3), readily seen to be real. Asan immediate consequence, the real part of ni transformsinto the real part and analogously for the imaginary part, aswas to be proven.

In fact, just the six matrices �i generate the wholealgebra of SU(4). To see this, note that i�i�

yj , i � j, are

15 independent matrices with the following properties:(1) They are Hermitian; (2) Their trace is zero; (3) Theirsquare is equal to unit matrix.

Proof.—Hermiticity follows from �i�i�yj �y �

�i�j�yi � i�i�

yj , by Eq. (A3). The same identity to-

gether with the unitarity of �i implies �i�i�yj �

2 �

��i�yj �i�

yj � �i�

yj �j�

yi � 1. The tracelessness is an

immediate consequence of Eq. (A4).The fact that the matrices i�i�

yj square to one tells us

that their eigenvalues are 1 and, moreover, both with thetwofold degeneracy, in order to ensure zero trace. We maythen write i�i�

yj � P��ij � P

��ij , where the projectors P��ij

are conveniently expressed as

P��ij � ��i � i�j��i � i�j�y;

P��ij � ��i � i�j��i � i�j�y:

(A5)

The identity (A3) implies that the product i�i�yj is, for

i � j, antisymmetric under the exchange of i and j. Tomake this antisymmetry explicit, let us define a new set of

matrices,

Mij �i2��i�

yj ��j�

yi � � i��i�

yj � �ij1�: (A6)

By means of Eqs. (A1) and (A3), these may be shown tosatisfy the commutation relation

�Mij;Mkl� � 2i��ilMjk � �jkMil � �ikMjl � �jlMik�:

(A7)

The matrices Mij thus provide an explicit mapping be-tween the Lie algebras of SU(4) and SO(6).

Finally, Eqs. (A1), (A3), and (A4) imply the identity,analogous to that for the Dirac � matrices,

Tr ��i�yj �k�

yl � � 4��ij�kl � �ik�jl � �il�jk�:

As a consequence we have the orthogonality relation forthe SO(6) generators Mij,

Tr �MijMkl� � 4��ik�jl � �il�jk�: (A8)

Note also that none of the identities proven in thisappendix relies on the explicit representation of the matri-ces �i, Eq. (A2). All the results thus hold independently ofthe particular realization of �i. In a sense, the constructionworked out here is similar to that of the Clifford algebra ofthe spinor representations of the orthogonal groups.However, in case of SO�2n�, this is realized by matricesof rank 2n [26], while here we need just 4 4 matrices forthe group SO(6).

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Goldstone bosons in presence of charge density