A Novel (2+1)-Dimensional Model of Chiral Symmetry Breaking

A Novel (2+1)-Dimensional Model of Chiral Symmetry Breaking

Veselin G. Filev, a,b    Matthias Ihl, b    Dimitrios Zoakos, School of Theoretical Physics,
10 Burlington Road, Dublin 4, Ireland.Centro de Física do Porto e Departamento de Física e Astronomia,
Rua do Campo Alegre 687, 4169-007 Porto, Portugal.
Abstract

We propose a new model of flavour chiral symmetry breaking in a (2+1)-dimensional defect gauge theory of strongly coupled fermions by introducing probe -flavour branes on the conifold. After working out the flavour brane embeddings at zero temperature, we thoroughly investigate the spectra of small fluctuations on the world volume of the flavour branes (meson spectrum) and conclude that they are free of tachyons. Thus the proposed probe brane embedding is stable. Moreover, we introduce finite temperature and an external magnetic field and study the thermodynamics of the resulting configurations. Namely, we compute the free energies, entropies, heat capacities and magnetisations. The results are used to establish a detailed phase diagram of the model. We find that the effect of magnetic catalysis of chiral symmetry breaking is realised in our model and show that the meson-melting phase transition coincides with the chiral symmetry breaking phase transition. Furthermore, we show that the model is in a diamagnetic phase.

\preprint

DIAS-STP-13-11

1 Introduction

The idea of gauge/gravity correspondences is among the most impressive developments coming from string theory Maldacena:1997re . Since these dualities relate the strongly coupled regime of a gauge theory to the weakly coupled regime of a string theory, they evolved into powerful tools in the study of strongly interacting systems. Many of the holographic models that have been constructed over the years have common features with QCD at strong coupling, like a confinement/deconfinement phase transition and chiral symmetry breaking.
An important development in this line of research came from the Sakai-Sugimoto construction Sakai:2004cn ; Sakai:2005yt , which is realised through the addition of and -branes in a non-extremal -brane background Witten:1998zw . This model has very specific characteristics, both in the UV and in the IR. The separation between the branes in the UV gives rise to a flavour symmetry similar to the chiral symmetry of QCD, while the merging of the branes in the IR, spontaneously breaks chiral symmetry111The holographic realization of the chiral symmetry breaking first appeared in a different framework. When we embed only one flavour D7-brane in a confining geometry (like the Constable-Myers background) the axial can be broken spontaneously, and this is identified with a spontaneous chiral symmetry breaking Babington:2003vm ..
An alternative model to geometrically realise the chiral symmetry breaking was introduced by Kuperstein and Sonnenschein in Kuperstein:2008cq . This model is realised through the addition of and -branes on the conifold, namely the Klebanov-Witten background Klebanov:1998hh . The main advantage of the Kuperstein-Sonnenschein model compared to the Sakai-Sugimoto model is that the former is a genuine (3+1)-dimensional gauge theory, while the latter model is a (4+1)-dimensional gauge theory compactified on a circle and it turns out to be impossible to cleanly separate the mass scale of the glueballs from the mass scale of the KK modes.
In the present paper we propose a novel model of chiral symmetry breaking in a (2+1)-dimensional gauge theory of strongly coupled fermions, whose geometric realisation is inspired by the Sakai-Sugimoto and Kuperstein-Sonnenschein models. To implement this idea we introduce a pair of and probe branes into the Klebanov-Witten background. The dual gauge theory is a (2+1)-dimensional defect in the (3+1)-dimensional quiver gauge theory dual to the Klebanov-Witten model Klebanov:1998hh . The presence of the anti-brane completely breaks the supersymmetry of the background.
As was recently observed by Ben-Ami, Kuperstein and Sonnenschein Ben-Ami:2013lca , our construction is an example of a limited class of models that feature spontaneous conformal symmetry breaking. In addition, the proposed (2+1)-dimensional model has various potential condensed matter applications:
By turning on a non-trivial profile in the -direction, the model can be easily applied to holographic bilayers, following a recent paper by Evans and Kim Evans:2013jma . Moreover, by introducing a chemical potential, the model admits a holographic zero sound mode (for an overview of holographic zero sound, see e.g. Davison:2011ek ). At finite magnetic field, the model could serve as another top-down construction of type II Goldstone bosons (cf. Filev:2009xp ; for an example in a bottom-up approach, see Amado:2013xya ). Furthermore, as was discussed in the literature recently, one can also realize quantum Hall states Kristjansen:2012ny and quantum Hall ferromagnetism Kristjansen:2013hma .
An overview of the paper is as follows: In section 2 we analytically derive the probe and -brane embeddings. The brane wraps a maximal in the conifold and has a non-trivial profile along the direction of the fiber as a function of the holographic coordinate. The and -branes merge smoothly in the IR (see figures 1 and 2). This joint solution spontaneously breaks the chiral symmetry of the theory.
In section 3 we study the meson spectrum of the model, introducing Cartesian-like coordinates, in order to verify the stability of the brane profile under semiclassical fluctuations along the transverse directions and the gauge fields. The thorough analysis reveals a spectrum that is tachyon-free, with one massless vector and two massless scalar fields. The massless scalar fields are the Goldstone modes of the spontaneously broken conformal symmetry and chiral symmetry.
In section 4 we investigate the thermodynamics of the proposed model, after the addition of a finite temperature and an (external) magnetic field. As in the archetypal construction of Kuperstein-Sonnenschein Kuperstein:2008cq , the addition of any finite temperature immediately leads to chiral symmetry restoration. Turning on a magnetic field promotes the breaking of the global flavour symmetry, an effect known as magnetic catalysis of chiral symmetry breaking Gusynin:1994re 222For the holographic approach to magnetic catalysis, cf. Filev:2007gb . The competition between the dissociating effect of the temperature and the binding effect of the magnetic field results in an interesting non-trivial phase structure of first order phase transitions, presented in figure 7. The calculation of the free energy and the heat capacity for the different phases determines which of them are stable, unstable or metastable and in turn if chiral symmetry is spontaneously broken or restored. We also compute the entropy density and the magnetisation for the various phases. Across the phase transition, the entropy density features a finite jump corresponding to the released latent heat and we conclude that the chiral symmetry restoration phase is simultaneously a meson-melting phase transition. The theory has negative magnetisation suggesting a diamagnetic response which is stronger in the quark gluon plasma (melted mesons) phase. Thus it is also a conducting phase.

2 General setup

In this article we are investigating the addition of a flavour sector to the Klebanov-Witten background Klebanov:1998hh that geometrically realises chiral symmetry breaking in the holographic dual of a dimensional gauge theory of strongly coupled fermions.

The Klebanov-Witten background is the near horizon limit of the geometry generated by coincident -branes at the tip of a conical singularity. The resulting geometry is an AdS supergravity background with a metric given by333For a discussion of the dual field theory, see Klebanov:1998hh ; Kuperstein:2008cq ; for other aspects of these models, see also Bayona:2010bg ; Ihl:2010zg . :

 ds2 =r2L2(−dt2+dx21+dx22+dx23) +L2r2⎡⎣dr2+r26(2∑i=1dθ2i+sin2θidϕ2i)+r29(dψ+2∑i=1cosθidϕi)2⎤⎦, (1)

where .
The introduction of flavour probe -brane pairs adds (2+1)-dimensional fundamental degrees of freedom to the quiver diagram of the theory. To stay in the probe approximation we consider . This corresponds on the field theory side to the quenched approximation, when fundamental loops are suppressed.
Our ansatz for the profile of the D5–branes is inspired by the classification of the supersymmetric embeddings of D5–branes in the Klebanov-Witten background performed in Arean:2004mm . A supersymmetric probe D5–brane necessarily forms a (2+1)-dimensional defect in the worldvolume of the D3–branes. It also extends along the holographic coordinate and wraps a maximal in the part of the geometry, which is orthogonal to the fiber (parametrised by in equation (2)) and has projections on both ’s (parametrised by in equation (2)). The kappa-symmetry requires that either of the following conditions are satisfied Arean:2004mm :

 θ2 =θ1,ϕ2=2π−ϕ1andx3=const. (2) θ2 =π−θ1,ϕ2=ϕ1andx3=const. (3)

Alternatively, one can define (cf. (2)),

 θ±=θ1±θ22andϕ±=ϕ1±ϕ22 (4)

and fix or . Fixing, without loss of generality, and the metric reads

 ds2=r2L2(−dt2+dx21+dx22)+L2r2(dr2+19dψ2)+13dΩ22, (5)

Note that the presence of -branes will break supersymmetry completely in our setup. Nevertheless the “straight” embeddings satisfying (2) continue to extremise the DBI action of the probe branes. The two pairs of “straight” probe -branes meet at the origin of the AdS. This configuration is analogous to the V-shaped embeddings of Kuperstein:2008cq , thus corresponding to a phase in which the chiral symmetry of the theory is preserved.
However, there is also the possibility of a joined (U-shaped in the terminology of Kuperstein:2008cq ) solution which breaks the chiral symmetry of the theory down to the diagonal . In general the profile of these U-shaped embeddings would describe a two-surface in the part of the geometry, which changes at different holographic slices (as a function of ). It turns out that the intuitive configuration in which the probe brane still wraps the same maximal as the straight embeddings, but has the position in the direction of the fiber running with the holographic coordinate, namely ), is a solution to the general equations of motion. This is why we consider the following ansatz for the U-shaped embeddings:

 x0x1x2x3rθ−ϕ+θ+ϕ−ψD3××××⋅⋅⋅⋅⋅⋅D5/¯¯¯¯¯¯¯D5×××⋅×××⋅⋅ψ(r)

The -branes will follow a non-trivial trajectory in the -subspace, as determined by minimizing the DBI world volume action of the -branes. Using (5) we arrive at the following one-dimensional Lagrangian,

 S=−τ5∫dξ6√detP[g]=−2N∫drr2√1+r29(∂ψ∂r)2, (6)

where and the factor of two reflects that it describes a configuration. This leads straightforwardly to the equation of motion

 r49ψ′√1+r29ψ′2=c0, (7)

in which the constant can be determined from the physical requirement that :

 c0=r303. (8)

The solution to the equation of motion is given by

 ψ(0)±(r)=±arctan⎛⎝√(rr0)6−1⎞⎠. (9)

A qualitative visualisation of all the possible types of embeddings is presented in figure 2.

The asymptotic expansion of is given by . In the (2+1)-dimensional defect field theory the non-trivial profile of corresponds to the insertion of a dimension three operator with expectation value proportional to . This condensate breaks the chiral symmetry of the theory spontaneously, thus it can be used as an order parameter of the chiral phase transition. Note that when we introduce more scales to the problem (such as temperature and external magnetic field) the asymptotic expansion of will be , where and will vary with the extra scale of the theory. Furthermore and will be thermodynamically conjugated and we will use them to characterise the different phases of the theory.

Before we continue with the addition of temperature and external magnetic field we have to verify that the U-shaped embedding in figure 1 is stable under semiclassical fluctuations or equivalently we have to explore the meson spectrum of the theory and verify that it is tachyon free.

3 Meson Spectrum

In this section we study the meson spectrum of our model. To this end will study the quadratic fluctuations of the D5-brane along the transverse directions and the gauge fields.

Following the approach of Kuperstein:2008cq we perform a change of coordinates in the plane, convenient for the parametrization of the U-shaped embedding:

 y=r3cosψ ,   z=r3sinψ . (10)

In these coordinates the relevant part of the metric (2) transforms to:

 L2r2[dr2+r29dψ2]=19L2z2+y2(dz2+dy2) . (11)

Remarkably in the coordinates the two branches of the U-shaped embedding described by (9) are covered by for . Let us choose a classical embedding corresponding to and . We are now ready to fluctuate our probe brane. We select the following ansatz for the scalars:

 y=r30+(2πα′)δy(t,z,θ+,ϕ−),θ−=(2πα′)δθm(t,z,θ+,ϕ−), ϕ+=π+(2πα′)δϕp(t,z,θ+,ϕ−),x3=(2πα′)δx3(t,z,θ+,ϕ−). (12)

In addition we turn on the gauge field of the D5–brane , which enters in the DBI action through the term and thus contributes to the quadratic order of the expansion. We introduce the symmetric matrix in the following way:

 ||E0ab||−1=S, (13)

while the non-zero elements are

 Stt=G−100,S11=S22=G−111,Szz=G−1zz, S++=G−1θ+θ+,S−−=G−1ϕ−ϕ−, (14)

with

 −G00=G11=(r60+z2)1/3L2,Gzz=L29(r60+z2), Gθ+θ+=L23,Gϕ−ϕ−=L23sin2θ+. (15)

The non-cross terms in the quadratic expansion of the action are

 −L(2)δθmδθm=12√−E0g(0)θ−θ−Sab∂aδθm∂bδθm+f(z)δθ2m, −L(2)δyδy=12√−E0g(0)yySab∂aδy∂bδy, (16) −L(2)δϕpδϕp=12√−E0g(0)ϕ+ϕ+[1−g(0)2zϕ+g(0)ϕ+ϕ+Szz]Sab∂aδϕp∂bδϕp, −L(2)δx3δx3=12√−E0g(0)33Sab∂aδx3∂bδx3,−L(2)δFδF=14√−E0SmpSnqFpqFmn,

with

 f(r)≡14√−E0S−−[g(0)′′ϕ−ϕ−−2Szz(g(0)′zϕ−)2]. (17)

while the cross terms are

 −L(2)δϕpδy=√−E0g(0)ϕ+ySab∂aδϕp∂bδy+√−E0∂y(g(0)zϕ+Szz)y=r30δy∂zδϕ+, −L(2)δθmδϕp=√−E0S−−[g(0)′ϕ+ϕ−−g(0)Vol(R2,1)zϕ+g(0)′zϕ−Szz]δθm∂ϕ−δϕp, (18) −L(2)δθmδy=√−E0S−−g(0)′yϕ−δθm∂ϕ−δy ,

where are the components of the ten dimensional metric as functions of and .

3.1 Spectrum of δx3

Looking at (3) it is clear that the scalar modes decouple from the rest, and it is possible to solve them separately. Applying the usual ansatz

 δx3=eiMth3(z)Θ(θ+)Φ(ϕ−), (19)

separating variables and defining and , we have

 ∂~z[(1+~z2)4/3h′3(~z)]+19[~M2−3κ(1+~z2)1/3]h3(~z)=0 (20) cotθ+Θ′(θ+)Θ(θ+)+Θ′′(θ+)Θ(θ+)+1sin2θ+Φ′′(ϕ−)Φ(ϕ−)=−κ, (21)

Equation (21) is the known spherical harmonics differential equation for the two-sphere

 Y(θ+,ϕ−)≡Θ(θ+)Φ(ϕ−)=Cl,mPml(cosθ+)eimϕ−withκ=l(l+1) (22)

where is the normalization constant. It is sufficient to study the lowest Kaluza-Klein state, in order to characterize the stability. Setting () in (20) we obtain:

 ∂~z[(1+~z2)4/3h′3(~z)]+19~M2h3(~z)=0 . (23)

Equation (23) can be brought to Schrödinger form via the coordinate change , where

 ∂2ξh3(ξ)+(~M2−V(ξ))h3(ξ)=0 ,   where (24) V(ξ)=6(1+~z(ξ)2)1/3>0 .

The fact that the effective potential is positive implies that there are no bound states (meson states) with negative and therefore the meson spectrum corresponding to the fluctuations along is tachyon free. We continue by solving numerically equation (23). The meson spectrum is obtained by imposing either even or odd boundary condition at the turning point of the U-shaped embedding ( in our coordinates). For the first several excited states we obtain:

 ~Meven=3.335,6.189,8.932,11.703,14.523,… (25) ~Modd=4.797,7.561,10.312,13.107,15.950,… (26)

once again we confirm that the spectrum is tachyon free.

3.2 Spectrum of δθ

The scalar modes couple to the other modes only through dependence on , however for the lowest lying Kaluza-Klein modes we can suppress the dependence and the modes decouple from the rest. To implement this we consider the ansatz:

 δθm=eiωth(z)Y(θ+). (27)

Separating variables, and defining again and , we obtain the following set of differential equations

 ∂~z((1+~z2)h′(~z))+(~M29(1+~z2)1/3−4~z29(1+~z2)+κ+49)h(~z)=0, (28) Y′′(θ+)+cotθpY′(θ+)−13(κ−2+3sin2θ+)Y(θ+)=0. (29)

Changing variables in (29) in the following way

 cosθ+=1−2x, (30)

it is possible to obtain an analytic solution

 Y(θ+)=c√x(1−x)2F1[16(9−√33−12κ),16(9+√33−12κ),2,κ]. (31)

Quantizing the first argument of the hypergeometric function we obtain

 κ=−4−3m(m+3). (32)

To verify stability it is enough to focus on the lowest lying Kaluza-Klein modes, which implies and hence . We can further bring the equation to a Schrödinger form via the change of coordinates , where

 ∂2ξh(ξ)+(~M2−Veff(ξ))h(ξ)=0 (33) Veff(ξ)=3(1+2~z(ξ)2)(1+~z(ξ)2)2/3>0 (34)

Again the positive effective potential implies that there are no states with negative and hence the spectrum of fluctuations of is tachyon free. Solving numerically (28) for and imposing separately even and odd boundary conditions at , we obtain the first several excited states

 ~Meven=2.995,6.099,8.874,11.659,14.487,… (35) ~Modd=4.668,7.49010.263,13.067,15.918,… , (36)

confirming that the spectrum is tachyon free.

3.3 Spectrum of δy and δϕp

The equations of motion for the fluctuations of and are coupled, furthermore both couple to the fluctuations of , However, the coupling to is through the dependence and is suppressed at the lowest Kaluza-Klein mode. In general it is hard to solve the coupled equations of motion for and in separated variables. However for the lowest Kaluza-Klein mode one can separate variables by considering the following ansatz:

 δy=eiωthy(z)cosθ+ ,   δϕp=eiωthϕ(z) . (37)

The result is a system of coupled differential equations for and :

 h′′y(~z)+⎛⎜⎝~M29(1+~z2)4/3−2(3+5~z2)9(1+~z2)2⎞⎟⎠hy(~z)−41+~z2h′ϕ(~z)=0 (38) h′′ϕ(~z)+2z1+~z2h′ϕ(~z)+~M29(1+~z2)4/3hϕ(~z)−2~z9(1+~z2)hy(~z)=0 (39)

We can bring (38) and (39) into a Schrödinger form, via the following transformation:

 (hyhϕ)=(1+~z2)−1/6(2(3√1+~z2−~z)√6(√1+~z2+2~z)−1√6).(Δ1Δ2) (40)

and a change of variables , such that  . The result is:

 ∂2ξ(Δ1Δ2)+[~M2^1−(V11V12V21V22)].(Δ1Δ2)=0 , (41)

where:

 V11(ξ)=3+54~z(ξ)2−24~z(ξ)√1+~z(ξ)27(1+~z(ξ)2)2/3;  V22(ξ)=18+44~z(ξ)2+24~z(ξ)√1+~z(ξ)27(1+~z(ξ)2)2/3; V12(ξ)=V21(ξ)=√6−3+2~z(ξ)2+10~z(ξ)√1+~z(ξ)27(1+~z(ξ)2)2/3; . (42)

For normalizable solutions of (41) vanishing at infinity, a sufficient condition for to be positive is the matrix potential to be positively definite. This would be the case if and . Using (3.3) this is indeed the case

 Tr^V=3+14~z(ξ)2(1+~z(ξ)2)2/3>0     and     det^V=24~z(ξ)4(1+~z(ξ)2)2/3>0 . (43)

Therefore we conclude that and the meson spectrum corresponding to and is tachyon free for normalizable solutions vanishing at infinity. In fact the only normalizable solution non-vanishing at infinity is the constant solution. We will show that such a solution has and following Kuperstein:2008cq we will identify it with the Goldstone boson of the broken conformal symmetry.

Next we proceed by solving numerically the coupled system of equations (38) and (39). Again the modes can be either even or odd depending on the boundary conditions at the turning point of the U-shaped embedding. It turns out that the even modes of couple to the odd modes of and the odd modes of couple to the even modes of .

δy even and δϕp odd:

Solving numerically equations (38) and (39) for the spectrum of the even, odd modes we obtain:

 Meven−odd=2.474,4.354,6.096,7.340,8.931,… (44)

δy odd and δϕp even:

Solving numerically equations (38) and (39) for the spectrum of the odd, even modes we obtain:

 Modd−even=2.637,4.558,5.89075,7.529,8.753,… (45)

One can also check that the constant solution and is a solution to the equations of motion (38) and (39) for . Following Kuperstein:2008cq we associate this Goldstone mode to the spontaneously broken conformal symmetry.

We conclude that there are no tachyons in the meson spectrum of and .

3.4 Fluctuation along the worldvolume gauge fields

Another set of modes that decouples from the rest are the worldvolume gauge fields. Following the analysis of Kuperstein:2008cq , we are interested only on the two-sphere independent modes with coordinates dependence and . We also ignore the components of the gauge field along the directions. The reduced action for the fluctuations of the gauge field is

 S=−(2πα′)2N∫d3xdz(C(z)FμνFμν+2D(z)FμzFμ  z) (46)

where:

 C(z)=πL49(r60+z2)2/3 ,   D(z)=π(r60+z2)2/3 . (47)

 ξ(z)=z∫0dz′√C(z′)D(z′)=L2z3r402F1(12,23,32,−z2r60) (48)

and using that444Note that this is not the case for the Kuperstein-Sonnenschein model considered in Kuperstein:2008cq . , we arrive at

 S=−T′∫d3xξ∗∫−ξ∗dξ(14FμνFμν+12FμξFμ  ξ) , (49)

where:

 T′=43πL2(2πα′)2N   and   ξ∗=π1/2L26r0Γ(1/6)Γ(2/3) . (50)

Next we follow refs. Kuperstein:2008cq ; Bayona:2010bg ; Ihl:2010zg ; Sakai:2004cn and expand the components of the gauge field in terms of the complete sets , :

 Aμ(x,ξ)=∑nanμ(x)αn(ξ) ,   Aξ(x,ξ)=∑nbn(x)βn(ξ) . (51)

After substituting in equation (50) we obtain:

 Saa = −T′∫d3xξ∗∫ξ∗dξ∑m,n(14fnμνfμνmαnαm+12anμaμm∂ξαn∂ξαm) (52) Sbb = −T′∫d3xξ∗∫ξ∗dξ∑m,n12∂μbn∂μbmβnβm (53) Sab = +T′∫d3xξ∗∫ξ∗dξ∑m,nanμ∂μbm∂ξαnβm (54)

Since the functions are defined in the finite interval , a simple choice of basis (which proves useful) is:

 αn=1ξ1/2∗cos(Mnξ), (55) Mn=nπξ∗=6√πΓ(2/3)Γ(1/6)n (56)

The functions (55) satisfy:

 (αn,αm)≡ξ∗∫−ξ∗dξαnαm=δnm,   and   ξ∗∫−ξ∗dξ∂ξαn∂ξαm=M2nδnm . (57)

Note that the zero mode , corresponding to is normalizable. This is different from the analysis of the vector mesons considered in refs. Kuperstein:2008cq ; Bayona:2010bg ; Ihl:2010zg ; Sakai:2004cn and as we are going to show leads to the presence of a massless vector field in the meson spectrum.

The second equation in (57) as well as the fact that , suggests the following choice for the functions :

 βn=⎧⎪⎨⎪⎩1Mn∂ξαn=−1ξ1/2∗sin(Mnξ)for n≥1α0=1ξ1/2∗for n=0 . (58)

One can easily check that for and hence using the second equation in (57) one concludes that:

 (βn,βm)≡ξ∗∫−ξ∗dξβnβm=δnm . (59)

With the choice of basis functions and given in equations (55) and (58) the total action for the meson modes becomes:

 S=−T′∫d3x{12∂μb0∂μb0+14f0μνfμν0+∞∑n=1[14fnμνfμνn+12M2n(anμ−1Mn∂μbn)2]} . (60)

After the gauge transformation (for ), we obtain:

 S=−T′∫d3x{12∂μb0∂μb0+14f0μνfμν0+∞∑n=1[14fnμνfμνn+12M2nanμaμn]} , (61)

where