A Review of Magnetic Phenomena in Probe-Brane Holographic Matter1footnote 11footnote 1To appear in Lect. Notes Phys. “Strongly interacting matter in magnetic fields” (Springer), edited by D. Kharzeev, K. Landsteiner, A. Schmitt, H.-U. Yee.

# A Review of Magnetic Phenomena in Probe-Brane Holographic Matter111To appear in Lect. Notes Phys. “Strongly interacting matter in magnetic fields” (Springer), edited by D. Kharzeev, K. Landsteiner, A. Schmitt, H.-U. Yee.

Oren Bergman222bergman@physics.technion.ac.il , Johanna Erdmenger 333jke@mppmu.mpg.de  and Gilad Lifschytz444giladl@research.haifa.ac.il
Department of Physics,
Technion - Israel Institute of Technology,
Haifa 32000, Israel
Max Planck Institute for Physics,
80805 Munich, Germany
Department of Mathematics and Physics,
University of Haifa at Oranim,
Tivon 36006, Israel
###### Abstract

Gauge/gravity duality is a useful and efficient tool for addressing and studying questions related to strongly interacting systems described by a gauge theory. In this manuscript we will review a number of interesting phenomena that occur in such systems when a background magnetic field is turned on. Specifically, we will discuss holographic models for systems that include matter fields in the fundamental representation of the gauge group, which are incorporated by adding probe branes into the gravitational background dual to the gauge theory. We include three models in this review: the D3-D7 and D4-D8 models, that describe four-dimensional systems, and the D3-D7’ model, that describes three-dimensional fermions interacting with a four-dimensional gauge field.

## 1 Introduction

The behavior of strongly interacting matter subject to background magnetic fields is an interesting and physically relevant problem in many different scenarios, ranging from the effective 2d electron gas in graphene, to magnetars, which are neutron stars with a strong magnetic field. Magnetic fields give rise to a rich array of phenomena. Some examples in QCD are the magnetic catalysis of chiral symmetry breaking [1, 2, 3], anomaly-driven phases of baryonic matter [4], and the chiral magnetic effect [5]. It has also been suggested that magnetic fields induce -meson condensation and superconductivity in the QCD vacuum [6, 7]. There are also many interesting examples in condensed matter physics, most notably the fractional quantum Hall effect [8, 9].

Gauge/gravity duality, also known as holographic duality, has emerged in recent years as a particularly useful approach to strong-coupling dynamics. Although it does not seem to be directly applicable to physical systems, this approach can be used to study theoretical systems that exhibit the same type of phenomena, and that capture some of the relevant physics. The techniques of holographic duality are especially efficient in addressing questions associated with finite temperature and density, background fields and transport properties, that are difficult to study using other non-perturbative methods. This approach can also lead to new ideas for constructing effective theories of the physical phenomena one is interested in.

Holographic models are divided into two main types, commonly referred to as top-down models and bottom-up models. In top-down models the bulk gravitational description of the system corresponds to a consistent solution of a well-defined quantum gravity theory, either in the context of the full string theory or in terms of the low-energy effective supergravity theory. This then defines some particular strong-coupling boundary dynamics. In bottom-up models, on the other hand, one builds into the description what one needs in order to produce the desired boundary dynamics. Each approach has advantages and disadvantages. Top-down models are more firmly grounded than bottom-up models, however they are more restrictive in terms of the variety and scope of phenomena they can exhibit.

Probe-brane models are a class of top-down holographic models, in which matter fields transforming in the fundamental representation of a gauge group are incorporated by embedding “flavor” D-branes in the gravitational background dual to the gauge theory [10]. These branes are treated as probes, in the sense that we neglect their backreaction on the background. (This corresponds to the ‘quenched’ approximation in the dual gauge theory, where matter loops are neglected in computing gluon amplitudes.) The matter fields are manifest in this construction: they correspond to the open strings between the flavor branes and the “color” branes that make up the background. In particular, one can easily design probe brane models in which the light matter degrees of freedom are purely fermionic, which is obviously a desirable feature for many physical systems, including QCD and condensed matter electron systems.

The matter dynamics is determined by the properties of the probe brane embedding. In particular, the fluctuations of the probe brane worldvolume fields correspond to gauge-invariant composite operators that describe the mesonic states of the matter system. There are generically two types of embeddings at finite temperature: “BH embeddings”, in which the brane extends to the horizon of the background, and “MN embeddings”, in which the brane terminates outside the horizon. The two embedding types describe different phases of the matter in the dual gauge theory. For example, in the MN phase the mesons are stable since they are associated with real eigenfrequencies of the probe brane fluctuations. In the BH phase, on the other hand, some of the energy of the fluctuations is dissipated into the black hole, leading to complex eigenfrequencies and damping. In this case the mesons have a finite lifetime. MN embeddings are favored at low temperature, and as the temperature is increased one generically observes a first order phase transition to a BH embedding.

The most extensively studied probe-brane models are the D3-D7 model [10], in which D7-branes are added to the D3-brane background, and the D4-D8 (or Sakai-Sugimoto) model [11], in which D8-branes and anti-D8-branes are added to the background of D4-branes compactified on a circle. Both models describe a strongly-coupled gauge theory in four dimensions with fundamental matter degrees of freedom, and both exhibit a number of phenomena similar to QCD. More recently, a different D3-D7 system, more closely related to the D4-D8 system, has been used as a model of strongly-interacting fermionic matter in three spacetime dimensions [12, 13, 14, 15]. We will refer to this as the D3-D7’ model. This model exhibits several interesting phenomena that are familiar in planar condensed matter systems.

Probe-brane models are especially well-designed to study the properties of the dual matter systems at non-zero density and in background electromagnetic fields. Both are implemented by turning on specific components of the probe-brane worldvolume gauge field, and solving the resulting coupled differential equations for the embedding and the gauge fields. Here one observes another basic difference between the two types of embeddings in terms of their response to a background electric field. MN embeddings correspond to electrical insulators with a mass-gap to charged excitations, and BH embeddings describe gapless conductors.

Probe-brane models also exhibit a number of interesting phenomena in a background magnetic field, which are qualitatively similar to the phenomena listed in the beginning. In this paper we will review how each of the three models mentioned above respond at non-zero density to a background magnetic field in various situations. In particular, we will encounter the magnetic catalysis effect in both the D3-D7 and D4-D8 models. In the D3-D7 model we will also demonstrate the formation of a superfluid state. In the D4-D8 model we will describe anomaly-generated currents and baryonic states, as well as a metamagnetic-like transition. In the D3-D7’ model we will see both quantum and anomalous Hall effects, as well as how the magnetic field influences the instability to the formation of stripes, and the zero-sound mode.

This paper is divided into three main sections, reviewing each of the probe-brane models in turn. For completeness let us mention that magnetic fields also play an important role in the related D3-D5 system, where the probe D5-brane corresponds to additional -dimensional degrees of freedom in the dual gauge theory. In this model, the magnetic field leads to a phase transition of Berezinskii-Kosterlitz-Thouless (BKT) type [16, 17]. For brevity we do not discuss this model in this review.

## 2 The D3-D7 model

### 2.1 Brane construction

The starting point for this model is the usual configuration of the AdS/CFT correspondence [18] which involves a stack of D3 branes. This has an open string interpretation in which the low-energy degrees of freedom are described by Super-Yang-Mills theory. On the other hand, in the closed string interpretation of D3 branes, the low-energy near-horizon limit gives rise to the space . Identifying the two pictures leads to the AdS/CFT correspondence.

Let us now add probe D7-branes to this configuration, as first done in [10] and reviewed in detail in [19]. Within (9+1)-dimensional flat space, the D3-branes are extended along the 0123 directions, whereas the D7-branes are extended along the 01234567 directions. This configuration preserves of the total amount of supersymmetry in type IIB string theory (corresponding to 8 real supercharges) and has an isometry in the directions transverse to the D3-branes. The rotates , while the group acts on . Separating the D3-branes from the D7-branes in the (8,9) directions by a distance explicitly breaks the group. These geometrical symmetries are also present in the dual field theory: The dual field theory is an supersymmetric (3+1)-dimensional theory in which the degrees of freedom of super Yang-Mills theory are coupled to hypermultiplets of flavor fields with fermions and scalars , , , which transform in the fundamental representation of the gauge group. Separating the D7-branes from the D3-branes corresponds to giving a mass to the hypermultiplets.

For massless flavor fields, the Lagrangian is classically invariant under conformal transformations 555However note that the scale-invariance is broken at the quantum level since the beta function is proportional to and therefore non-vanishing. In the limit with being fixed, the beta function is approximately zero, i.e. we may treat the theory as being scale invariant also at the quantum level. Moreover, the theory is invariant under the -symmetries and as well as under the global , which rotates the scalars in the adjoint hypermultiplet. Note that the mass term in the Lagrangian breaks the symmetry explicitly. If all flavor fields have the same mass the field theory is invariant under a global flavor group. The baryonic symmetry is a subgroup of the flavor group. These symmetries of the field theory side may be identified with symmetries of the D3-D7 brane intersection and hence also with the dual gravity description.

For this field theory, gauge invariant composite operators may now be constructed which transform in suitable representations of the symmetry group isomorphic to the geometrical . These operators are expected to be dual to the fluctuations of the D7-brane which transform in the same representation, as worked out in detail in [20]. An example of a meson operator is given by

 MA=¯ψiσAijψj+¯qmXAqm,(i,m=1,2), (1)

with the vector of adjoint scalars associated with the directions, and a doublet of Pauli matrices. Thus (1) has charge under . It is a singlet under both and . The conformal dimension is . This operator may be viewed as a supersymmetric generalization of a mesonic operator in QCD, with the index labelling two scalar mesons.

The standard AdS/CFT duality relates the Super Yang-Mills degrees of freedom to supergravity on . In addition, there are new degrees of freedom associated to the D7-brane worldvolume fields originating from the open strings on the D7-brane. The additional duality maps these to the mesonic operators in the field theory. This is an open-open string duality, as opposed to the standard AdS/CFT correspondence, which is an open-closed string duality. The dynamics of the D7-brane is described by the Dirac-Born-Infeld (DBI) action

 SD7=−μ7gs∫d8ξ√−det(Gab+B(2)ab+2πα′Fab), (2)

where . and are the induced metric and two-form field on the probe brane worldvolume, and is the worldvolume field strength. The D7-brane action also contains a fermionic term . In addition there may also be contributions of Wess-Zumino form. An example for this will be discussed below.

Let us write the metric in the form

 ds2=r2R2ηijdxidxj+R2r2(dρ2+ρ2dΩ23+dx28+dx29), (3)

with , , and the AdS radius. Since the D7-brane is transverse to , in flat space, we see that it extends along and wraps an inside in the near-horizon background. The action for a static D7-brane embedding, for which may be consistently set to zero on its world-volume, is given from (2) up to angular factors by

 SD7=−μ7gs∫d8ξρ3√1+˙x28+˙x29, (4)

where a dot indicates a derivative (e.g. ). The ground state configuration of the D7-brane then corresponds to the solution of the equation of motion

 ddρ[ρ3√1+˙x82+˙x92dxdρ]=0, (5)

where denotes either or . Clearly the action is minimized by being any arbitrary constant. Therefore the embedded D7-brane is flat. According to string theory, the choice of the position in the plane corresponds to choosing the quark mass in the gauge theory action. The fact that are constant at all values of the radial coordinate , which corresponds to the holographic renormalization scale, may be interpreted as non-renormalization of the mass in the dual field theory.

In general, the equations of motion have asymptotic () solutions of the form

 x=l+cρ2+..., (6)

where is related to the quark mass by

 m=l2πα′. (7)

In agreement with the standard AdS/CFT result about the asymptotic behaviour of supergravity fields near the boundary, the parameter must correspond to the vev of an operator with the same symmetries as the mass and of dimension three, since carries energy dimension. is therefore a measure of the quark condensate . is obtained from which in addition to the fermion bilinear also includes scalar squark terms. We may consistently assume that the squarks have zero vev. Moreover, supersymmetry requires that a vev for must be absent since is an F-term of a chiral superfield: is the F-term of . Supersymmetry is broken if . This is reflected also in the supergravity solution: The solutions to the supergravity equations of motion with non-zero are not regular in AdS space and are therefore excluded.

We therefore consider the regular supersymmetric embeddings of the D7-brane for which the quark mass may be non-zero, but the condensate vanishes. For massive embeddings, the D7-brane is separated from the stack of D3-branes in either the or directions, where the indices refer to the coordinates given in (3). In this case the radius of the becomes a function of the radial coordinate in . At a radial distance from the deep interior of the AdS space given by the hypermultiplet mass, the radius of the shrinks to zero. From a five-dimensional AdS point of view, this gives a minimal value for the radial coordinate beyond which the D7-brane cannot extend further. This is in agreement with the induced metric on the D7-brane world-volume, which is given by

 ds2 =ρ2+l2R2ηijdxidxj+R2ρ2+l2dρ2+R2ρ2ρ2+l2dΩ32, (8) dΩ32 =dψ2+cos2ψdβ2+sin2ψdγ2, (9)

where and are spherical coordinates in the 4567-space. For , this is the metric of . When (i.e. ), the radius of the shrinks to zero.

### 2.2 Finite temperature

The finite temperature system is realized holographically by placing the D7-brane in an AdS-Schwarzschild black hole background with metric given by

 ds2 = r22R2⎛⎝d→x2(rh4+r4)r4−dt2(r4−rh4)2r4(rh4+r4)⎞⎠ (10) +R2r2(dL2+dρ2+L2dϕ2+ρ2dΩ23),

where . The temperature is given by .

In this metric we have introduced polar coordinates in the plane and consider solutions for D7-brane embeddings with , . The asymptotic near-boundary behaviour of these brane embeddings is given by

 L(ρ)=l+cρ2+…, (11)

where is the bare quark mass and is proportional to the condensate . At finite temperature, supersymmetry is broken and brane embeddings with non-zero are possible, in contrast to the supersymmetric case discussed above.

Depending on the quark mass, there are two different types of embeddings: Those which reach the black hole, and those which do not since the they wrap shrinks to zero outside the black hole horizon. The first type of branes is referred to as ‘black hole’ (BH) embeddings, while the second type is referred to as ‘Minkowski’ (MN) embeddings. In the BH case, fluctuations of the probe brane have complex eigenfrequencies or quasi-normal modes, which means that the mesons associated with these fluctuations decay. In the MN phase, the mesons are stable. The phase transition between the two types of embeddings is first order.

At finite temperature, the solution with also has . However, in gravity backgrounds corresponding to confining field theories, brane embeddings with , are possible. These realize spontaneous chiral symmetry breaking [21].

### 2.3 Magnetic catalysis

Let us now consider, as was first done in [22], a magnetic field induced by a pure gauge B-field in the worldvolume direction of the D3-branes,

 B(2)=Bdx2∧dx3, (12)

which satisfies . This contributes to the DBI action (2). Since and enter (2) in the same way, we may trade for a gauge field on the probe brane via , which justifies interpreting of (12) as a magnetic field.

In addition, there is a non-trivial Wess-Zumino contribution to the action, which at first order in is of the form

 SWZ=2πα′μ7∫F∧C(6). (13)

In the presence of the B-field, this leads to an additional non-trivial contribution to the action, as explained in detail in [22]. This gives rise to a non-trivial which breaks supersymmetry on the worldvolume of the D7-brane.

Since supersymmetry is broken, the D7-brane now has a profile which depends on as in (11), even at zero temperature. The Lagrangian corresponding to (2) takes the form

 L=−μ7gsρ3sinψcosψ√1+L′2√1+R4B2(ρ2+L2)2. (14)

For , the brane embedding solution obtained from this Lagrangian has non-zero in (11). The magnetic field therefore induces spontaneous chiral symmetry breaking, a phenomenon known as magnetic catalysis [1, 2, 3]. For large , the condensate may be calculated analytically [22] and is found to be

 ⟨¯ψψ⟩∝−c=−R44lB2. (15)

For small , has to be evaluated numerically. The result is shown in figure 1. By evaluating the free energy, it has been shown that when there is more than one solution, the one with the larger condensate is preferred.

At small values of the magnetic field, it is possible to analytically evaluate the shift of the meson masses due to its presence. For fluctuations of the embedding scalar in particular, a Zeeman splitting is observed [22]. While in the absence of a magnetic field, in the supersymmetric case described here, the scalar meson mass obtained from the fluctuations is [20], for non-zero magnetic field there is a mass splitting

 M±=M0±1√λBm. (16)

A review of magnetic catalysis in probe D7-brane systems is given in [24]. Magnetic catalysis is also found in systems involving D7 branes where the backreaction of the metric on the background geometry is taken into account [25, 26, 27]. Out-of-equilibrium dynamics associated with the phase transition induced by magnetic catalysis has been investigated in [28].

In the finite temperature case, there is a competition between two mechanisms: The black hole attracts the D7-brane, while it is repelled at small radii by the magnetic field. This implies a phase transition between a phase where the D7-brane reaches the black hole and one where it does not. This is shown in figure 2 for different values of the magnetic field, where the dimensionless ratio is used. A detailed discussion of the normalization is found in [23].

There is a critical value for above which the probe brane is repelled from the black hole for all values of the bare quark mass: In this case, chiral symmetry is spontaneously broken and the mesons are stable. The phase diagram is shown in figure 3.

More involved phase diagrams are obtained if a chemical potential and density are turned on in addition to the magnetic field by considering a non-trivial profile for the gauge field on the D7-brane [29, 30, 31]. An example of a phase diagram is shown in figure 4.

### 2.4 Superfluid

At finite isospin density, the D3-D7 model realizes a holographic superfluid [32, 33]. Finite isospin density is obtained by considering two coincident D7-branes, and using an ansatz for solving the equations of motion, which involves a non-trivial profile for the the temporal component of the worldvolume gauge field, with asymptotic behavior

 A3t(ρ)∝μ3+d3ρ2. (17)

breaks the symmetry explicitly to a residual . In the presence of this background, the energetically favored solution also involves a non-trivial spatial component of the worldvolume gauge field,

 A1x(ρ)∝d1xρ2. (18)

Here the leading contribution is absent in the asymptotic behavior, so the symmetry is spontaneously broken. is dual to a condensate of the form

 d1x∝⟨¯ψσ1γxψ+¯ϕσi∂xϕ⟩, (19)

which is the supersymmetric equivalent of the meson. The calculation of the frequency-dependent conductivity for this solution shows that it describes a superfluid: displays a gap. For the Sakai-Sugimoto model discussed below, a similar condensation mechanism has been found in [34] and superfluidity has been discussed in [35].

As discussed in [36, 37], a similar condensation process also happens when the profile (17) for the temporal component of the gauge field is replaced by a non-trivial profile for a spatial component of the form

 A3x=By, (20)

which corresponds to a background magnetic field. In this case a similar condensation as above takes place. This has been demonstrated by analyzing the fluctuations about the magnetic field background [37]: The quasi-normal modes of particular fluctuations cross into the upper half of the complex frequency plane above a critical value of the magnetic field, indicating an instability. This is shown in figure 5. Finally let us note that the Hall conductivity has been calculated for the D3/D7 model in [38]. Unlike the isospin case, the ground state involving the condensate is spatially modulated for the magnetic field background, leading to an Abrikosov lattice [39]. A similar meson condensation mechanism in a background magnetic field has been found in the context of field theory in [6, 7, 40], based on similar earlier results in electroweak theory [41]. For the Sakai-Sugimoto model which we discuss below, a similar mechanism has been discussed in [42, 43].

## 3 The D4-D8 (Sakai-Sugimoto) model

### 3.1 Basics

D4-branes on with anti-periodic boundary conditions for fermions provide a holographic model for the low energy behavior of 4d Yang-Mills theory with [44]. The near-horizon background at zero temperature is given by (we work with dimensionless coordinates rescaled by )

 ds2con = u32(−dx20+dx2+f(u)dx24)+u−32(du2f(u)+u2dΩ24) eΦ = gsu3/4,F4=3π(α′)3/2NcdΩ4, (21)

where , and . The IR “wall” at implies that the dual gauge theory is confining. At nonzero temperature there is another possible background with a metric

 ds2dec=u32(−f(u)dx20+dx2+dx24)+u−32(du2f(u)+u2dΩ24), (22)

where and . This background becomes the dominant one when . The presence of a horizon at in this background indicates that the gauge theory undergoes a (first order) deconfinement transition at this temperature.

Quarks are added to the model by including D8-branes and anti-D8-branes that are localized on the circle [11]. With D8-branes at one point and anti-D8-branes at another point, the model has flavors of massless right-handed and left-handed fermions, and a chiral symmetry. The 8-branes are treated as probes in the near horizon background of the D4-branes. The flavor dynamics is thus encoded in the 5d effective worldvolume theory of the D8-branes, which includes a DBI term and a CS term (in Lorentzian signature)666For simplicity, we will consider the single flavor case with one D8-brane and one anti-D8-brane. This does not affect any of the results qualitatively.

 SDBI = −N∫d4xduu1/4√−det(gMN+fMN) (23) SCS = −N8∫d4xduϵMNPQRaMfNPfQR. (24)

The dimensionless worldvolume gauge field and field strength are defined as and , and the overall normalization is given by . The anti-D8-brane has a similar action in terms of its worldvolume gauge field . The DBI term is identical to that of the D8-brane, and the CS term has the opposite sign. We define the vector combination as , and the axial combination as .

In the low-temperature confining background (3.1) the D8-brane and anti-D8-brane connect at into a smooth U-shaped configuration (Fig. 6a), reflecting the spontaneous breaking of the chiral symmetry to the diagonal . The embedding is determined by the DBI action (setting )

 SconDBI=−N∫d4xduu4[f(u)(x′4(u))2+1u3f(u)]12, (25)

which implies an asymptotic behavior

 x4(u)≈L2−29u40√f(u0)u9/2, (26)

where is the asymptotic brane-antibrane separation.

The normalizable fluctuations of the D8-brane worldvolume fields in this embedding correspond to the (low spin) mesons of the model. Their mass scale is set by , which we can think of as the mass of a “constituent quark” described by an open open string from to . There is one massless pseudoscalar field , precisely as one expects from the broken chiral symmetry. This is related to the meson in QCD. It appears (in a gauge with ) as the zero mode :777Note that, although the boundary value is non-zero, this is a normalizable mode since the field strength is normalizable. Ordinarily, boundary values of bulk fields correspond to parameters in the boundary theory. But in this case there is a possible ambiguity, since the boundary value of can also describe a non-trivial gradient of the pseudoscalar field.

 aAμ(xμ,u)=−∂μφ(xμ)ψ0(u)+higher modes, (27)

where

 ψ0(u)=2πarctan√u3u3KK−1. (28)

Baryons are described by D4-branes wrapped on inside the D8-brane. Their charge comes from the strings which must be attached to the wrapped D4-brane to cancel a tadpole due to the background RR field. These strings end on the D8-brane, giving units of charge.888For the baryons correspond to instantons in the non-abelian D8-brane theory [11, 45]. This reproduces the known description of baryons as Skyrmions in the chiral Lagrangian. In this description the baryon charge comes from the CS term coupling the field to the instanton density in the part.

In the high-temperature deconfining background (22) the D8-branes and anti-D8-branes can be either connected (Fig. 6b) or disconnected, with (Fig. 6c), the latter corresponding to the restoration of the chiral symmetry [46]. The DBI action in the high-temperature deconfining background is very similar:

 SdecDBI=−N∫d4xduu4[f(u)(x′4(u))2+1u3]12. (29)

Consequently the properties of the U embedding in this background are qualitatively similar to those of the embedding in the confining background, for example in terms of the spectrum of mesons. In the disconnected embedding there are no normalizable fluctuations corresponding to mesons, as one expects in a chiral-symmetric phase. Comparing the (Euclidean) actions of the two embeddings shows that the disconnected one becomes dominant when . In particular, for small () the gauge theory has an intermediate phase of deconfinement with broken chiral symmetry.

### 3.2 Finite density and background fields

The D8-brane worldvolume vector and axial gauge fields are dual to conserved vector and axial currents in the gauge theory, and therefore999The axial symmetry is broken by an anomaly. However this is a subleading effect at large which we can neglect. In particular, we will assume that the one-flavor pseudoscalar is massless. For a discussion of the anomaly and the mass in the context of the Sakai-Sugimoto model see [11, 47].

 jμV,A=1NV4∂SD8|on−shell∂aV,Aμ(u→∞). (30)

The chemical potentials are defined by101010We would like to stress that this is a gauge invariant definition. The standard boundary condition on the gauge field in AdS/CFT fixes the value of . In this case only the transformations that vanish at are gauged in the bulk. In particular, these transformations do not change the asymptotic value of .

 μV=aV0(u→∞)andμA=aA0(u→∞). (31)

In our conventions quarks carry one unit of vector charge and baryons carry units. Nevertheless we will refer to the vector current as the “baryon number current”. We are also interested in studying the effects of background “electromagnetic” fields that couple to this current, which correspond to turning on spacetime dependent boundary values of the worldvolume gauge field, in particular

 ei=f0i(u→∞),bi=ϵijkfjk(u→∞). (32)

In some situations one may be required to add boundary terms to the action. These are especially relevant if there is a CS term in the bulk. In deriving the equations of motion from the variational principle one usually assumes that the surface terms vanish. However in some instances one has to be more careful. The surface terms (in the gauge) are given in general by

 δS∣∣on−shell=∫d4x∂L∂a′νδaν∣∣∞umin+∫d3xdu∂L∂(∂μaν)δaν∣∣xμ→∞xμ→−∞. (33)

In holography the boundary values of the fields at are fixed, so . However and need not vanish. Therefore a surface term may be non-trivial if the fields extend to these boundaries. In order to have a well-defined variational principle we must therefore add boundary terms , whose variation cancels the surface terms in the variation of the bulk action.

These boundary terms also allow one to derive an alternative and useful definition of the conserved currents. We do this by varying the off-shell action, now allowing to vary, and then going on-shell by applying the equations of motion. Due to the boundary terms, only the surface term at remains. Thus

 jμ=1N∂L∂a′μ(u→∞)∣∣on−shell. (34)

In particular this relates the charge density in the boundary theory to the bulk radial electric field. The boundary term for should then be interpreted as a source term for this field. One could in principle also add boundary terms at . These have no effect on the derivation of the equations of motion, but may change the value of the on-shell action, and therefore may lead to additional contributions to (34).

A state with a non-zero baryon number density corresponds to an embedding with a radial electric field. In particular, for a U embedding (in both the confining and deconfining backgrounds) this requires the addition of a baryonic source at the tip, corresponding to a uniform spatial distribution of wrapped D4-branes (Fig.6d,e) [48] (see also [49, 50]). We assume that the distribution is dilute enough so that we can ignore interactions between the D4-branes. We should therefore include the D4-brane action, which is given by (in the confining background)

 SD4=−NV4nD4Nc(13u0−aV0(u0)), (35)

where is the density of D4-branes. The first term is the D4-brane DBI action, and corresponds to the baryon mass, and the second term comes from the strings that connect each D4-brane to the D8-brane. The second term is precisely the boundary term at that was discussed above.

The resulting asymptotic behavior of the gauge field is

 aV0(u)≈μV−23du3/2, (36)

where is the baryon number density. On the other hand, extremizing the action with respect to fixes the value of the gauge field at the tip to . This implies, as expected, that a non-zero density configuration exists only when the chemical potential is above the baryon mass. In fact the non-zero density state is always the dominant one. The transition to “nuclear matter” occurs at . Near the critical point the density scales linearly with the chemical potential . The D4-brane action also sources the embedding field , creating a cusp at . This can be understood in terms of a force balance condition between the D4-branes pulling down and the D8-brane pulling up.

At high temperature the preferred embedding is parallel and there is a different finite density solution. In this case the gauge fields and are independent, and the field theory has a conserved axial current as well as a baryon current. For a baryonic solution we take and . In addition, since the D8-brane and anti-D8-brane reach the horizon we must impose . Therefore the radial vector electric field, and thus the baryon number density, is non-zero when . In this phase for small .

### 3.3 Magnetic catalysis of chiral symmetry breaking

A strong magnetic field in QCD is believed to catalyze the spontaneous breaking of chiral symmetry [1, 2, 3]. The basic mechanism for this is that in a strong magnetic field all the quarks sit in the lowest Landau level, and the dynamics is effectively 1+1 dimensional. This phenomenon has been exhibited in the Sakai-Sugimoto model in [51, 52].111111At non-zero baryon number density the magnetic field can actually induce an inverse magnetic catalysis in this model [53, 54].

With a uniform background magnetic field , the D8-brane action in the deconfining background becomes

 SdecD8=−N∫d4xduu4√(f(u)(x′4(u))2+1u3)(1+b2u3). (37)

The U embedding has the same form as before (26), but now depends on the magnetic field, as shown in Fig. 7a. The mass scale associated with chiral symmetry breaking is seen to increase with the magnetic field. One therefore expects that chiral symmetry breaking becomes more favored as the magnetic field increases. This is indeed the case, as can be seen by comparing the Euclidean actions of the U and parallel embeddings as the temperature and magnetic field are varied. The resulting phase diagram is shown in Fig. 7b. We observe that in this model the critical temperature approaches a finite value at infinite magnetic field.

A qualitatively similar effect was observed in the D3-D7 model above. However in the D3-D7 model there is a critical value of above which the chiral symmetry is always broken, whereas in the D4-D8 there is a critical temperature above which the chiral symmetry is always broken.

It is also instructive to study the effect of the background magnetic field on the mesons. This was partly done in [55], in which the high spin mesons were studied. It was shown that the magnetic field enhances their stability by increasing their angular momentum, and thereby increasing the dissociation temperature at which they fall apart into their quark constituents. This is consistent with the above results.

### 3.4 Anomalous currents

The chiral anomaly leads to two interesting phenomena when both the magnetic field and chemical potential are non-zero. The first is the generation of anomalous currents in the chiral symmetric phase of QCD. In [56, 57] it was shown that the combination of a magnetic field and a non-zero baryon chemical potential generates an axial current

 →JA=e2π2μphysB→B. (38)

Since the source for this current is the anomaly it is an exact result, and should be valid in particular at strong coupling. Similarly, an anomalous vector current is generated in a non-zero axial chemical potential:

 →JV=e2π2μphysA→B. (39)

This is known as the “chiral magnetic effect”, and may have some relevance to heavy ion physics at RHIC [5]. Within the D3/D7 model, this effect has been discussed in [58].

In the Sakai-Sugimoto model, the chiral-symmetric phase corresponds to the parallel D8- embedding in the deconfined background, and the chiral anomaly is encoded in the five-dimensional CS term (24). The background magnetic field and chemical potentials correspond to different components of the worldvolume gauge field. Through the five-dimensional CS term these source a third component, which corresponds to a current in the four-dimensional theory [59]. Let us review the calculation of the anomalous axial current in the Sakai-Sugimoto model. The calculation of the anomalous vector current is virtually identical.

To be specific, we will consider a background magnetic field in the direction by turning on a background gauge field . A non-trivial boundary value of will then source, via the CS term, a non-trivial . In general and can depend on both and in this case, although on-shell they will depend only on . We will take and . The D8-brane DBI and CS actions in this case become

 SdecDBI = −N∫∞uTd4xduu5/2√(1−(aV′0)2+f(u)(aA′1)2)(1+b2u3) (40) SCS = −N∫d4xdu[b(aV0aA′1−aV′0aA1)+aV3(aV′0∂2aA1−∂2aV0aA′1)]. (41)

The necessary boundary terms are

 S∂=−12N∫d3xduaV3(aV0aA′1−aV′0aA1)∣∣∣x2→∞x2→−∞. (42)

By integrating by parts the last two terms in the CS action one can show that up to a surface term at , the bulk CS and boundary actions combine into a bulk action

 SCS+S∂=−N∫d4xdu[32b(aV0aA′1−aV′0aA1)−12aV′3(aV0∂2aA1−∂2aV0aA1)]. (43)

One can get rid of the remaining surface term by adding a boundary term at [59], however this particular term does not contribute to the on-shell action, so we might as well ignore it.121212Other boundary terms at could affect the on-shell action, and therefore the currents. See for example [60, 61].

The equations of motion for and can be integrated once to yield

 √u5+b2u2aV′0(u)√1−(aV′0(u))2+f(u)(aA′1(u))2 = −3baA1(u)+d (44) √u5+b2u2f(u)aA′1(u)√1−(aV′0(u))2+f(u)(aA′1(u))2 = −3baV0(u), (45)

where is the baryon number charge density. The integration constant in the equation vanishes since and . Using (34) and (45) we can then evaluate the axial current:

 j1A=32baV0(∞)=32bμV. (46)

The correctly normalized physical currents are given by , where the factor of 2 comes from adding the anti-D8-brane contribution, and the physical chemical potentials are . Thus in terms of the physical variables our result translates to

 →JA=Nc4π2μphysV→B. (47)

Similarly, for an axial chemical potential we would get

 →JV=Nc4π2μphysA→B. (48)

Interestingly, our results are half of the weak-coupling results (where in the holographic model). It has been argued that the discrepency in the axial current is due to a different treatment of the triangle anomaly, consistent vs. covariant, and can be corrected by adding an appropriate Bardeen counterterm on the boundary [61]. However, the same counterterm leads to a vanishing vector current. This issue is still under investigation.

### 3.5 The pion gradient phase

In the broken chiral symmetry phase the chiral anomaly leads to a novel finite density phase that dominates over nuclear matter at large magnetic fields [4]. In this phase the baryon charge is carried not by baryons but rather by a non-zero pion gradient background:

 D=e4π2fπ→B⋅→∇π0, (49)

where

 →∇π0=e4π2fπμphysB→B. (50)

In the Sakai-Sugimoto model (in the gauge) the pseudoscalar meson appears in the zero mode of (27), so

 ∂μφ(xμ)=−aAμ(xμ,u→∞). (51)

As in the chiral-symmetric phase, the presence of a vector chemical potential together with a background magnetic field sources a component of the axial gauge field, which in this case corresponds to a non-trivial gradient of the pseudoscalar field [59, 62]. Since there is only one flavor this field should really be thought as the meson. For simplicity, we will consider only the confined phase with the anti-podal D8-brane embedding, namely . (The results do not change qualitatively for more general U-shape embeddings, or for U-shape embeddings in the deconfined phase.)

Following [59], the D8-brane DBI action in this case is

 SconDBI=−N∫∞uKKd4xduu5/2 ⎷(1f(u)−(aV′0)2+(aA′1)2)(1+b2u3), (52)

and the CS plus boundary actions are the same as in the deconfined phase (43). The equations of motion integrate to

 √u5+b2u2aV′0(u)√1f(u)−(aV′0(u))2+(aA′1(u))2 = −3baA1(u)+NcnD4 (53) √u5+b2u2aA′1(u)√1f(u)−(aV′0(u))2+(aA′1(u))2 = −3baV0(u)+c, (54)

where we have explicitly included the baryon sources in the equation. Our boundary conditions are now and . In particular is a field rather than a parameter in the boundary theory, and we must minimize the action with respect to its value. This simply sets and therefore sets the integration constant in the equation to .131313This is also consistent with the fact that there are no quarks in this phase to carry such a current.

We can now compute the total baryon number charge density using the same procedure as in the previous section for the current. In the absence of sources and we find

 d=−32baA1(∞)=32b∇φ. (55)

Let us express this in terms of the physical variables. First we must define a field with a canonically normalized kinetic term. Inserting (27) into the action (52) we find that the canonically normalized field is given by

 η′(xμ)=R22πα′fη′φ(xμ),f2η′=Ncu3/2KK4π4α′. (56)

Converting to physical variables we then find

 D=Nc4π2fη′→B⋅→∇η′, (57)

in agreement with (49). We would like to stress that this agreement did not depend on the specific value of required for canonical normalization, since it cancels out when we express the result in terms of . The correct numerical factor of is a direct consequence of including the proper boundary terms in the action, leading to the “3/2” in (55).

To find the value of the gradient we need to solve (53) and (54). The result will not be as simple as (50). In particular it is not linear in the magnetic field, since we are using the full non-linear DBI action. It turns out that a closed form solution can be found in terms of a new variable

 y=∫uuKK3bd~u√f(~u)√~u5(1+b2~u−3)−(32bμV)2+(3b∇φ)2. (58)

The solution is

 aV0(y)=μV2(coshycoshy∞+1),aA1(y)=−μV2sinhycoshy∞, (59)

where . The pseudoscalar gradient is then given by

 ∇φ=−aA1(∞)=μV2tanhy∞. (60)

The dependence on is shown in Fig. 8a. For small the behavior is linear in :

 ∇φ≈π2u3/2KKμVb, (61)

and in terms of the physical quantities:

 →∇η′≈Nc4π2fη′μphysV→B, (62)

in agreement with the single flavor version of (50).

As in the case with no magnetic field, an embedding that includes sources is possible above a critical value of the chemical potential, which then becomes the dominant configuration. This describes a “mixed phase” that includes both “pion-gradient” matter and nuclear matter, with a total baryon number density

 d=32b∇φ+NcnD4. (63)

As before, one can find a closed form solution for and , and from it determine the values of and in terms of the magnetic field and baryon number chemical potential . The resulting phase diagram is shown in Fig. 8b. In particular, the critical value of is determined by setting . The relative proportion of baryons in the mixed phase increases with and decreases with .

### 3.6 Magnetic phase transition

The high-temperature chiral-symmetric phase of this model also exhibits an interesting magnetic phenomenon associated with the distribution of the baryonic charge [63]. In what follows we analyze the situation for the D8-brane. As in the broken chiral symmetry phase above, the distribution of baryonic charge along the radial direction changes with . We can therefore identify two types of baryonic charge, one originating from the horizon, and the other from outside the horizon. The latter , corresponds to D4-branes that are radially smeared inside the D8-brane. This can be best seen from the longitudinal and transverse conductivities [64]

 σL = √u8T+b2u5T+u3T(d−d∗)2u3T