1 Introduction
Abstract

In this work we further extend the investigation of holographic gauge theories in external magnetic fields, continuing earlier work. We study the phenomenon of magnetic catalysis of mass generation in 1+3 and 1+2 dimensions, using D3/D7– and D3/D5–brane systems, respectively. We obtain the low energy effective actions of the corresponding pseudo Goldstone bosons and study their dispersion relations. The D3/D7 system exhibits the usual Gell-Mann–Oakes–Renner (GMOR) relation and a relativistic dispersion relation, while the D3/D5 system exhibits a quadratic non-relativistic dispersion relation and a modified linear GMOR relation. The low energy effective action of the D3/D5 system is related to that describing magnon excitations in a ferromagnet. We also study properties of general Dp/Dq systems in an external magnetic field and verify the universality of the magnetic catalysis of dynamical symmetry breaking.

arXiv:0802.????

Universal Holographic Chiral Dynamics

in an External Magnetic Field



Veselin G. Filev, Clifford V. Johnson, Jonathan P. Shock



School of Theoretical Physics, Dublin Institute For Advanced Studies

10 Burlington Road, Dublin 4, Ireland

filev@stp.dias.ie

Department of Physics and Astronomy, University of Southern California

Los Angeles, CA 90089-0484, U.S.A.

johnson1@usc.edu

Departamento de Física de Partículas

,

Universidade de Santiago de Compostela and Instituto Galego de Física de Altas Enerxías (IGFAE)

E-15782, Santiago de Compostela, Spain

shock@fpaxp1.usc.es



1 Introduction

The applications of holographic gauge/gravity correspondences to the study of more and more diverse phenomena are ever widening in scope. Over the last half a decade the links between finite temperature generalizations of AdS/CFT and experimental heavy-ion collisions have become much more concrete and the theoretical methods available to us are yielding ever deeper results concerning the properties of the quark-gluon plasma (see ref. [1, 2] for a recent review). Moreover in the last year the links between holography and condensed matter systems have also flourished, with work on superconductivity, and superfluidity, quantum phase transitions and both the classical and quantum hall effects having recent successes (e.g. refs. [3, 4] and references therein).

In the present work we extend the investigation of holographic gauge theories in the presence of external magnetic fields from the work first studied in ref. [5]. In this paper we are interested in finding both universal properties of strongly coupled gauge theories in the presence of magnetic fields, as well as in the different phenomena exhibited in such theories in a variety of space-time dimensions.

The phenomenon of dynamical flavor symmetry breaking catalysed by an arbitrarily weak magnetic field is known from refs. [6, 7] and refs. [8, 9, 10]. This effect was shown to be model independent and therefore insensitive to the microscopic physics underlying the low energy effective theory. In particular the infra-red (IR) description of the Goldstone modes associated with the dynamically broken symmetry should be universal. We therefore expect to be able to study this phenomenon using the holographic formalism. The aim of the present study will be to investigate the dynamics of the Goldstone modes and construct the low energy chiral Lagrangian of theories both in 3+1 and 2+1 dimensions in the presence of external magnetic fields, showing that the appropriate holographic models give precisely the results expected from the traditional field theory approach.

The effective dynamics of fermion pairing, in dimensions, in the presence of an external magnetic field is constrained to spatial dimensions. For this reason there are marked differences in the phenomenology of such systems in two and three spatial dimensions. In 2+1 dimensions refs. [7]-[10] Poincare symmetry is broken by the magnetic field (there is no longer any trace of the original boost invariance), removing the strong constraints on the dynamics of Goldstone modes imposed by special relativity. The naive Goldstone boson counting therefore does not hold and the resulting dispersion relation for the Goldstone modes takes a quadratic form, unlike in the case of 3+1 dimensions where an subgroup of constrains the dynamics. Although the number of Goldstone particles is no longer constrained in the non-relativistic setting, the number of Goldstone fields is fixed by the dimension of (=symmetry of the action, =symmetry of the ground state). We will show that this also holds in the AdS/CFT context.

The 2+1 dimensional model is of particular interest because, as shown in ref. [11], the low energy effective description is that of magnon excitations in a ferromagnet. Using a D3/D5 brane intersection we will be able to reproduce such a result at quadratic order in the chiral Lagrangian. Moreover such 2+1 dimensional theories may have relevance in the arenas of the quantum hall effect, and high superconductivity.

In addition to the phenomena discussed specifically in two and three spatial dimensions we show that certain universal behaviors are exhibited holographically in the present context. Here we will study holographic systems T-dual to the D3/D7 flavor model and show that the existence of an arbitrarily small magnetic field induces a spiral behaviour in the equation of state for such systems. In the limit that the chiral symmetry of the underlying theory is preserved, this equation of state can be studied analytically and such a symmetric vacuum can be shown to be unstable. This is holographically equivalent to the findings of refs. [6]-[10] – flavor symmetry breaking is induced dynamically by the presence of a magnetic field.

The outline of the present paper is as follows:

In section 2 we will return to the D3/D7 brane intersection in the presence of an external magnetic field, discussed in ref. [5]. We shall show explicitly how the magnetic catalysis of flavor symmetry breaking is realised in the holographic system, including the calculation of the chiral Lagrangian to second order in the low energy degrees of freedom. We will show that the Gell-Mann–Oakes–Renner relation holds analytically and obtain the dispersion relation for the Goldstone modes.

In section 3 we turn to the case of the D3/D5 defect theory and show here how the flavor symmetry is dynamically broken to the subgroup in the presence of a magnetic field. In this non-relativistic system we find the Goldstone modes and show that the number of massless modes is not the same as the number of broken generators, but satisfies a more general counting rule [25] applicable for non-relativistic systems. We also show that the single Goldstone mode satisfies a modified Gell-Mann–Oakes–Renner relation and a quadratic dispersion relation. Again we obtain the dispersion relation analytically in the small mass limit and find the low energy effective Lagrangian which describes magnon excitations in a ferromagnet.

In section 4 we prove that the magnetic catalysis of dynamical symmetry breaking is a universal effect in gauge theories dual to Dp/Dq intersections in the appropriate decoupling limit. This proof involves showing that all such systems exhibit a self–similar spiral behaviour in their equation of state which leads to an instability for the solution with zero dynamical mass. Just as in the work of ref. [7] this effect is independent of the magnitude of the external magnetic field.

2 Mass generation in the D3/D7 system

In this section we will review the results of refs. [5, 12, 13], where a holographic study of flavored supersymmetric Yang-Mills in an external magnetic field was studied using the D3/D7 system. We will focus on the effect of mass generation by magnetic catalysis in this theory and provide a detailed analysis of the pseudo-Goldstone mode associated to the spontaneous breaking of a global R-symmetry. In particular we will show that the Gell-Mann–Oakes–Renner relation for the mass of the corresponding meson is satisfied.

The D3/D7 system provides a dual holographic description of fundamental hypermultiplets coupled to supersymmetric Yang Mills theory in the quenched approximation  [14]. At zero separation between the D3 and D7–branes the fundamental hypermultiplets are massless and the –function of the theory is proportional to . Thus in the quenched approximation the –function vanishes and the corresponding gauge theory is conformal. The global R-symmetry of the SYM theory is broken to an R–symmetry, the corresponding to rotations in the 2-plane transverse to both the D3 and D7–branes. The left and right handed fermions of the hypermultiplet have opposite charges under this and thus the formation of a fermionic condensate would lead to the spontaneous breaking of this symmetry.

2.1 Spontaneous symmetry breaking

There are various ways in which one can study the breaking of the chiral symmetry holographically. This has been studied in the past by the deformation of AdS by a field corresponding to a marginally irrelevant operator on the gauge theory side refs. [17, 18, 19]. In the present case however we will stimulate the formation of a condensate by turning on the magnetic components of the gauge field of the D7–branes (equivalent to exciting a pure gauge field in the supergravity background). This gauge field corresponds to the diagonal of the full gauge symmetry of the stack of D7–branes. Since the D7–branes wrap an infinite internal volume, the dynamics of the gauge field is frozen in the four dimensional theory and the gauge symmetry becomes a global flavor symmetry . Therefore the gauge field that we consider corresponds to the gauged baryon symmetry and the magnetic field that we introduce couples to the baryon charge of the fundamental fields [27].

The problem thus boils down to studying embeddings of probe D7–branes in the AdS background parameterized as follows:

(1)

where and are polar coordinates in the transverse and planes respectively.

Here parameterize the world volume of the D7–brane and the following ansatz is considered for its embedding:

leading to the following induced metric on its worldvolume:

(2)

The D7–brane probe is described by the DBI action:

(3)

Here is the D7–brane tension, and are the induced metric and -field on the D7–brane’s world volume, while is its world–volume gauge field. A simple way to introduce a magnetic field is to consider a pure gauge –field along the directions:

(4)

Since and appear on equal footing in the DBI action, the introduction of such a -field is equivalent to introducing an external magnetic field of magnitude to the dual gauge theory.

Though the full solution of the embedding can only be calculated numerically, the large behaviour (equivalently the ultraviolet (UV) regime in the gauge theory language) can be extracted analytically:

(5)

As discussed in ref. [18], the parameters (the asymptotic separation of the D7- and D3- branes) and (the degree of bending of the D7–brane in the large region) are related to the bare quark mass and the fermionic condensate respectively. It should be noted that the boundary behavior of really plays the role of source and vacuum expectation value (vev) for the full hypermultiplet of operators. In the present case, where supersymmetry is broken by the gauge field configuration, we are only interested in the fermionic bilinears and this will refer only to quarks, and not their supersymmetric counterparts.

At this point it is convenient to introduce dimensionless parameters and . By performing a numerical shooting method from the infrared while varying the small boundary value, , we recover the parametric plot presented in figure 1, the main result explored in ref. [5].

Figure 1: Parametric plot of against for fundamental matter in the presence of an external magnetic field. The lower (black) line represents the curve , fitting the large behavior. It is also evident that for the outer branch of the spiral, for the condensate, is non-zero. The corresponding value of the condensate is .

The lower (black) curve corresponds to the analytic behavior of for large . The most important observation is that at there is a non-zero fermionic condensate:

(6)

Where is the ’t Hooft coupling and is a numerical constant corresponding to the -intercept of the outer spiral from figure 1. Equation (6) is telling us that the theory has developed a negative condensate that scales as . This is not surprising, since the theory is conformal in the absence of the scale introduced by the external magnetic field. The energy scale controlled by the magnetic field, , leads to an energy density proportional to . In order to lower the energy, the theory responds to the magnetic field by developing a negative fermionic condensate.

Another interesting feature of the theory is the discrete–self–similar structure of the equation of state ( vs. ) in the vicinity of the trivial embedding, namely the origin of the plot from figure 1 presented in figure 2.

Figure 2: A magnification of figure 1 shows the spiral behavior near the origin of the -plane. The second (left) spiral arm represents the symmetry of the theory.

This double logarithmic structure has been analyzed in ref. [12], where a study of the meson spectrum revealed that only the outer branch of the spiral is tachyon free and corresponds to a stable phase having spontaneously broken chiral symmetry. In Section 3 of this paper we will show that an identical structure is also present for the D3/D5 system and in Section 4 we will demonstrate that this structure is a universal feature of the magnetic catalysis of mass generation for gauge theories holographically dual to Dp/Dq intersections.

A further result of refs. [5, 12, 13] was the detailed analysis of the light meson spectrum of the theory. In ref. [5] it was shown that the introduction of an external magnetic field breaks the degeneracy of the spectrum studied in ref. [15]. This manifests itself as Zeeman splitting of the energy levels. In the limit of zero quark mass, the study also revealed the existence of a massless “ meson” corresponding to the spontaneously broken symmetry. In the next subsection we will revisit the study of the meson spectrum of the theory and provide an analytic proof of the Gell-Mann–Oakes–Renner relation [24]:

(7)

in the spirit of the analysis performed in ref. [18].

2.2 The Gell-Mann–Oakes–Renner relation - an analytic derivation

In order to study the light meson spectrum of the theory one needs to consider the quadratic fluctuations of the D7–brane embedding and study the corresponding normal modes [15]. Technically one should consider the full supergravity action for the D7–branes:

(8)

where is given by equation (3) and the relevant part of the Wess-Zumino term is given by [5]:

(9)

The next step is to consider fluctuations of the D7–brane in the transverse :

(10)

and expand equation (8) to second order in . Note that with such an expansion we should also consider fluctuations of the gauge field on the D7–brane. As demonstrated in refs. [5, 20] the effect of the magnetic field will be to mix the equations of motion for the scalar and vector fluctuations. In particular couples to the and components of the gauge field, while couples to the and components. The rest of the components of the vector field decouple and can be consistently set to zero. This splitting of the meson spectrum is a manifestation of the broken Lorentz symmetry. Indeed the external magnetic field breaks the Lorentz symmetry down to corresponding to boosts in the plane and rotations in the plane. Since the massless “pion” that we are interested in corresponds to fluctuations along , we will excite only the fields. The relevant terms of the expansion are [5]:

(11)

where:

(12)

Here corresponds to the classical embedding of the D7–brane and are the components of the background metric equation (1).

The equations of motions for and are calculated from the quadratic action, resulting in:

(13)

where and the gauge constraint is imposed (note that this is the usual Lorentz gauge, corresponding to the unbroken ) and we have defined:

(14)

Once again the broken Lorentz symmetry is manifest in equation (14). The definition of the spectrum is now a subtle issue in the presence of the broken space-time symmetry. We will define the spectrum as the energy of a particle as measured in its rest frame. In fact because we retain the symmetry we may consider fluctuations propagating in the direction. Since we are interested in describing the lowest lying modes (“pions” in particular) we will focus on modes that have no dependence. Therefore we consider the ansätze:

(15)

and define:

(16)

The equations (13) simplify to:

(17)

Note that for large bare masses (and correspondingly large values of ) the term proportional to the magnetic field is suppressed and the meson spectrum should approximate to the result for the pure AdS space-time case studied in ref. [15], where the authors obtained the following relation:

(18)

between the eigenvalue of the excited state and the bare mass . If one imposes the boundary conditions:

(19)

the coupled system of differential equations can be solved numerically. Then by requiring the functions and to be regular at infinity one can quantize the spectrum of the fluctuations. It is also convenient to define the following dimensionless parameter . The resulting plot for the first three excited states is presented in figure 3. There is Zeeman splitting of the states due to the magnetic field. (In the absence of the field there are three straight lines emanating from the origin; these are split to form six curves.) Also, at zero bare quark mass there is indeed a massless Goldstone mode, appearing at the end of the lowest curve. Furthermore the plot in figure 4 shows that for small bare quark mass one can observe a characteristic dependence. In the next section we shall provide an analytic proof of that relation and obtain an integral expression for the numerical coefficient presented above the plot in figure 4.

Figure 3: There is Zeeman splitting of the states due to the magnetic field. In the absence of the field there are three straight lines emanating from the origin; these are split to form six curves. At zero bare quark mass (the end of the lowest curve) there is indeed a massless Goldstone mode. The straight lines correspond to the asymptotic AdS results.
Figure 4: There is a characteristic behavior at small bare quark mass.

In the following section we shall demonstrate that for small bare quark mass, , the spectrum exhibits the characteristic dependence. Once we have illustrated that the functional dependence is correct we will show that the constant of proportionality is also that expected from the GMOR relation. Furthermore we shall generalize the ansätze (15) to consider fluctuations depending on both the momentum along the magnetic field and the transverse momentum :

(20)

We shall also show that for small and the following dispersion relation holds:

(21)

where is a constant that we shall determine.

2.2.1 The dependence

Using an approach similar to the one employed in ref. [18] we define:

(22)

The equations of motions (13) can then be written in the compact form:

(23)

Let us remind the reader that for large , has the behavior:

(24)

Let us denote by the classical embedding corresponding to . It is relatively easy to verify that at and correspondingly the choice:

(25)

is a solution to the system (23). Next we consider embeddings corresponding to a small bare quark mass . This will correspond to small nonzero values of and . It is then natural to consider the following variations:

(26)

where and are of order . Note that corresponds to the mass of the ground state at and we are assuming that the variations of the wave functions and are infinitesimal for infinitesimal . After expanding in equation (23) we get the following equations of motion:

(27)

where . The second equation in (27) can be integrated to give:

(28)

From the boundary conditions that and we see that the constant of integration is zero and arrive at:

(29)

Next we multiply the first equation in (27) by and integrate along to obtain:

(30)

where the last term on the right-hand side of equation (30) has been integrated by parts using the fact that should be regular at infinity. From the definition of it follows that as and as . This together with the requirement that is regular at and vanishes at infinity, suggests that the first term on the right-hand side of equation (30) vanishes. For the next term, we use the fact that:

(31)

and therefore obtain:

(32)

The second term in equation (30) then becomes:

(33)

Finally using the equality in equation (29) we arrive at the result:

(34)

Now we define:

(35)

and solve for from equation (21) to obtain:

(36)

Equation (36) suggests that the mass of the “pion” associated to the softly broken global symmetry satisfies the Gell-Mann–Oakes–Renner relation [24]:

(37)

In order to prove equation (37) we need to evaluate the effective coupling of the “pion” . Noting that and , we conclude that:

(38)

At this point is useful to verify the consistency of our analysis by comparing the coefficient in equation (36) to the numerically determined coefficient from the plot in figure 4. Indeed from equation (36) we obtain:

(39)

where we have defined the dimensionless quantities:

(40)

There is excellent agreement with the fit from figure 4.

Next we will obtain an effective four dimensional action for the “pion” and from this derive an exact expression for .

2.2.2 Effective chiral action and

In this section we will reduce the eight dimensional world-volume action for the quadratic fluctuations of the D7–brane to an effective action for the massless “pion” associated to the spontaneously broken global symmetry. Note that our effective action should be describing a single “pion” mode, while the 8D action given by equation (11) describes the dynamics of two independent degrees of freedom, namely and coupled by the magnetic B-field via the second equation in equation (11). As rigid rotations along correspond to chiral rotations, (the asymptotic value of at infinity corresponds to the phase of the condensate in the dual gauge theory) the spectrum of at zero quark mass contains the Goldstone mode that we are interested in. This is why we first integrate out the gauge field components and and then dimensionally reduce to four dimensions.

Furthermore as mentioned earlier, because of the magnetic field the Lorentz symmetry is broken down to symmetry. This is why in order to extract the value of we consider excitations of depending only on the directions and read off the coefficient in front of the kinetic term. The resulting on-shell effective action for is:

(41)

where is given by:

(42)

We refer the reader to Appendix A for a detailed derivation of the 4D effective action .

We have defined via , where corresponds to rotations in the transverse plane and is the angle of chiral rotation in the dual gauge theory. The chiral Lagrangian is then given by:

(43)

and therefore:

(44)

The D7–brane charge in equation (44) is given by and the overall prefactor in equation (44) can be written as . Now, recalling the expressions for the fermionic condensate, equation (6), and the bare quark mass, , one can easily verify that equation (36) is indeed the Gell-Mann–Oakes–Renner relation:

(45)

It turns out that for small momenta and small mass one can obtain the following more general effective 4D action (see appendix A for a detailed derivation):

(46)

where is defined in equation (132). As one can see, the action (46) is the most general quadratic action consistent with the symmetry and suggests that pseudo Goldstone bosons satisfy the dispersion relation (21).

3 Mass generation in the D3/D5 system

In this section we provide a holographic description of the magnetic catalysis of chiral symmetry breaking in dimensional supersymmetric Yang-Mills theory coupled to  fundamental hypermultiplets confined to a dimensional defect. Recently this theory received a great deal of attention and emphasis has been made of the potential application of this brane configuration in describing qualitative properties of dimensional condensed matter systems (see for example refs. [28, 29, 30]). In this section we will study the effect of an external magnetic field on the theory and demonstrate that the system develops a dynamically generated mass and negative fermionic condensate leading to a spontaneous breaking of a global symmetry down to a symmetry. On the gravity side this symmetry corresponds to the rotational symmetry in the transverse . Naively there should be two massless Goldstone bosons corresponding to the generators of the coset . As we will show the 1+2 dimensional nature of the defect theory leads to a coupling of the transverse scalars corresponding to the coset generators and as a result there is only a single Goldstone mode. Furthermore the characteristic Gell-Mann–Oakes–Renner relation is modified to a linear behavior. It turns out that these features can be understood from a low energy effective theory point of view. Indeed in dimensions the effect of the magnetic field is to break the Lorentz symmetry down to rotational symmetry and as a result the theory is non-relativistic. A single time derivative chemical potential term is allowed (there is no boost symmetry) and interestingly the supergravity action generates such a term through the Wess-Zumino contribution of the D5–brane. It is this term that is responsible for the modified counting rule of the number of Goldstone bosons [25] and leads to a quadratic dispersion relation as well as to the modified linear Gell-Mann–Oakes–Renner relation. Another interesting feature of the model is that to quadratic order the effective low energy action is the same as the effective action describing spin waves in a ferromagnet [11] in an external magnetic field. We comment briefly on the possible applications of this similarity.

3.1 Generalities

Let us consider the AdS supergravity background (1) and introduce the following parameterization:

(47)

We have split the transverse to and introduced spherical coordinates and in the first and second planes respectively. Next we introduce a stack of probe D5–branes extended along the directions, and filling the part of the geometry parameterized by . As mentioned above on the gauge theory side this corresponds to introducing fundamental hypermultiplets confined on a dimensional defect. The asymptotic separation of the D3 and D5 –branes in the transverse space parameterized by corresponds to the mass of the hypermultiplet. In the following we will consider the following anzatz for a single D5–brane:

(48)

The asymptotic separation is related to the bare mass of the fundamental fields via . If the D3 and D5 branes overlap, the fundamental fields in the gauge theory are massless and the theory has a global symmetry. Clearly a non-trivial profile of the D5–brane in the transverse would break the global symmetry down to , where is the little group in the transverse . If the asymptotic position of the D5–brane vanishes () this would correspond to a spontaneous symmetry breaking, the non-zero separation on the other hand would naturally be interpreted as the dynamically generated mass of the theory.

Note that the D3/D5 intersection is T–dual to the D3/D7 intersection from the previous section and thus the system is supersymmetric. The D3 and D5 –branes are BPS objects and there is no attractive potential for the D5–brane, hence the D5–brane has a trivial profile . However a non-zero magnetic field will break the supersymmetry and as we are going to demonstrate, the D5–brane will feel an effective repulsive potential that will lead to dynamical mass generation. In order to introduce a magnetic field perpendicular to the plane of the defect, we consider a pure gauge -field in the plane given by:

(49)

This is equivalent to turning on a non-zero value for the