A Semi-analytic solution for f\simeq 1 in the chirally broken phase

Inverse magnetic catalysis in dense holographic matter

Abstract:

We study the chiral phase transition in a magnetic field at finite temperature and chemical potential within the Sakai-Sugimoto model, a holographic top-down approach to (large-) QCD. We consider the limit of a small separation of the flavor D8-branes, which corresponds to a dual field theory comparable to a Nambu-Jona Lasinio (NJL) model. Mapping out the surface of the chiral phase transition in the parameter space of magnetic field strength, quark chemical potential, and temperature, we find that for small temperatures the addition of a magnetic field decreases the critical chemical potential for chiral symmetry restoration – in contrast to the case of vanishing chemical potential where, in accordance with the familiar phenomenon of magnetic catalysis, the magnetic field favors the chirally broken phase. This “inverse magnetic catalysis” (IMC) appears to be associated with a previously found magnetic phase transition within the chirally symmetric phase that shows an intriguing similarity to a transition into the lowest Landau level. We estimate IMC to persist up to at low temperatures.

Gauge-gravity correspondence, QCD, Chiral Lagrangians

1 Introduction

Spontaneous breaking of chiral symmetry in a system of relativistic fermions is profoundly affected by an external magnetic field. A sufficiently strong, homogeneous magnetic field results in an effective dimensional reduction of the dynamics of the system. As a consequence, an instability with respect to condensation of fermion-antifermion pairs, i.e., with respect to the formation of a chiral condensate, occurs even at arbitrarily weak coupling. This is analogous to the Bardeen-Cooper-Schrieffer (BCS) mechanism of fermion-fermion pairing in a superconductor, where the effective dimensional reduction is achieved by the presence of a Fermi surface. The enhancing effect of the magnetic field on chiral symmetry breaking has been termed magnetic catalysis (MC) and has originally been discussed in Gross-Neveu [1, 2] and Nambu-Jona-Lasinio (NJL) [3, 4, 5] models and in QED [6]. Recently, this effect has also been reproduced in holographic models with flavor branes subjected to magnetic fields [7, 8, 9, 10].

In this paper, we consider the chiral phase transition under the influence of a magnetic field at finite temperature and quark chemical potential . Our main interest is QCD where chiral symmetry is spontaneously broken for sufficiently small temperatures and chemical potentials, and where the effects of strong magnetic fields may be observable in the chiral transition at small and large (namely in ultrarelativistic heavy-ion collisions) and also at small and large in astrophysical systems (compact stars). Indeed, it has been argued that extremely strong magnetic fields of up to occur in non-central heavy-ion collisions [11] and up to at the surface of magnetars [12] (possibly even up to in the interior [13]). Our results may also be relevant for graphene [14] which is under much better experimental control. Fermion excitations in graphene are effectively relativistic and MC manifests itself in a nonzero Dirac mass induced by electron-hole pairing [15, 16, 17], analogous to the constituent quark mass induced by quark-antiquark and quark-hole pairing in the QCD context.

In both QCD and condensed matter contexts it is important to develop a strong-coupling description of MC. To this end we employ the AdS/CFT correspondence [18, 19, 20, 21], more precisely the Sakai-Sugimoto model [22, 23]. This holographic model, based on type-IIA string theory, is, in a certain (albeit inaccessible) limit dual to large- QCD. In contrast to most other holographic models, it accounts for the full chiral symmetry group by realizing left- and right-handed massless fermions through D8- and -branes in a background of D4-branes. Moreover, the model has a confined and a deconfined phase, realized by two different background geometries. In the original version of the model, where the D8- and -branes are maximally separated in a compact extra dimension, the deconfinement and chiral phase transitions are identical and happen at a certain for all values of and (provided that any backreaction on the background is neglected). Here we are interested in a different limit of the model where the distance of the flavor branes is small and where a much richer phase structure is obtained. This limit can be understood as the NJL limit of the model [24, 25, 26].

The NJL model in its original form approximates the fermionic interaction by a point-like four-fermion interaction. It has been employed for the chiral phase transition in the presence of a background magnetic field at finite and/or in refs. [27, 28, 29, 30, 31, 32, 33]. We shall find a phase diagram which shows a striking qualitative resemblance with some of the NJL results. In particular, we shall discuss that at finite chemical potential and not too large magnetic field the chirally broken phase becomes disfavored by increasing the magnetic field, in stark contrast to MC. We term this effect inverse magnetic catalysis (IMC) and present a simple physical explanation, employing the analogy to superconductivity. Moreover, we shall discuss a discontinuity in the quark density [34] for small temperatures in the chirally symmetric phase. Our result for this discontinuity, in particular the comparison of its location with respect to the chiral phase transition to recent NJL results, supports its interpretation as a transition to the lowest Landau level. This is remarkable since with the exception of the Sakai-Sugimoto model [34, 35], Landau-level-like structures have been discussed in the AdS/CFT literature only in bottom-up scenarios [36, 37, 38, 39]. Here we shall see that the top-down approach of the Sakai-Sugimoto model suggests the presence of a lowest Landau level, but no further de Haas-van Alphen oscillations from higher Landau levels.

Our calculation builds upon previous work within the Sakai-Sugimoto model in the presence of a magnetic field [35, 40, 41, 42], and generalizes the results for the chiral phase transition in the - plane at [43] and the - plane at [44, 45] to the entire -- space. (For recent discussions of the chiral phase transition in a magnetic field within other holographic models see for instance refs. [46, 47, 48, 49].)

The remainder of the paper is organized as follows. In sec. 2 we explain the geometry of our holographic setup, introduce our notation, and derive the on-shell action and the equations of motion. In secs. 3 and 4 we treat the chirally broken and symmetric phases separately and point out discontinuities in the quark density in both phases. The main part of the paper is sec. 5 where the chiral phase transition is discussed. After discussing several limit cases in sec. 5.15.3 (including a discussion of the split of chiral and deconfinement transitions in sec. 5.3) we present our main results in sec. 5.4 before giving our conclusions in sec. 6.

2 General setup

2.1 Context and brief summary of the model

We consider the Sakai-Sugimoto model for one flavor, , in the deconfined phase. The corresponding background geometry is given by the ten-dimensional supergravity description of D4-branes in type-IIA string theory compactified on a supersymmetry breaking Kaluza-Klein circle [21]. Fundamental flavor degrees of freedom are implemented by D8- and -branes which are separated asymptotically by a given distance in the compactified dimension [22, 23]. Employing the probe brane approximation , the background geometry will be fixed throughout the paper, while two qualitatively different embeddings of the flavor branes account for the chirally broken and chirally symmetric phases. The gauge symmetries on the D8- and -branes are interpreted as left- and right-handed global symmetries of the dual field theory which lives at the 4+1-dimensional boundary of the ten-dimensional space (including the compact extra dimension, which needs to be small to arrive at an effectively 3+1-dimensional field theory). In the case of disconnected flavor branes the system is invariant under the full chiral group while connected branes in the bulk lead to the smaller symmetry group , see fig. 1. This reflects the usual spontaneous chiral symmetry breaking of massless quarks.

Figure 1: Schematic picture of the chirally symmetric (left) and chirally broken (right) phases in the deconfined phase of the Sakai-Sugimoto model (in the confined phase, the - subspace is cigar-shaped, not cylinder-shaped). The -direction is compactified on a circle with radius , and is the holographic coordinate with being the boundary where the dual field theory lives. If the distance between the flavor branes is sufficiently small, a deconfined, chirally broken phase (right figure) becomes possible. In this phase, the connected flavor branes are embedded nontrivially in the background according to a function which has to be determined from the equations of motion which couple to the gauge fields on the flavor branes. The location of the tip of the connected branes is part of this solution. In this paper, we work in the “NJL limit” of the model, where , i.e., the tip of the branes is far away from the horizon. For a fixed temperature – which fixes – the distance between and can be made arbitrarily large by choosing a sufficiently small separation .

In the original applications of the model the separation of the flavor branes is maximal, , i.e., the D8- and -branes are put on opposite ends of the circle with radius . Here, is the Kaluza-Klein mass which sets the mass scale below which adjoint scalars and fermions decouple from the dynamics of the dual field theory. For this maximal separation of flavor branes chiral symmetry is broken if and only if the system is confined. In other words, the chiral phase transition is dictated entirely by the background geometry. As a consequence, in the probe brane approximation the chiral transition is unaffected by all quantities that live on the flavor branes such as chemical potential and magnetic field. As an alternative to going beyond the probe brane approximation – which is very difficult –, such a “rigid” behavior can be softened by choosing a smaller separation of the flavor branes. This leads to a much richer phase structure, in particular a decoupling of the chiral and deconfinement phase transitions becomes possible. In fact, this decoupling is realized for values of below a critical value [50], yielding a deconfined, chirally broken phase for sufficiently small chemical potentials [43]. Now that both connected and disconnected flavor branes are possible solutions in the deconfined background geometry, also a magnetic field affects the chiral transition. It has been shown that for vanishing chemical potential, the critical temperature above which chiral symmetry is restored increases with increasing magnetic field [44, 45], in accordance with expectations from MC, as explained in the introduction. A simple consequence is that, in a certain regime of non-maximal separations , a magnetic field may induce a splitting of chiral and deconfinement phase transitions. We discuss this effect in more detail in sec. 5.3. Such a splitting has been observed in a linear sigma model coupled to quarks and Polyakov loop [51] and an NJL model with Polyakov loop (PNJL) [52] (see however ref. [53]), but has not been seen in lattice QCD calculations [54].

For a large separation of the flavor branes, i.e., of the order of , the main features of the phase diagram in the - plane are the same as for large- QCD [55]. In the opposite limit the connected flavor branes are far away from the horizon so that the effect of confinement becomes less “visible” for the fundamental fermions. Hence we expect this limit to correspond to a field theory where the dynamics of chiral symmetry breaking decouples from that of the gluons. Therefore, by varying the distance the Sakai-Sugimoto model interpolates between large- QCD () and a (non-local) NJL model () where there are no gluons and no confinement [24, 25]. Since in the former limit the many gluons dominate the phase diagram, the latter may in fact be an interesting limit for QCD at finite , at least at comparatively low temperatures and high quark number densitites. Our main results correspond to the latter limit, which indeed will show many similarities to those obtained recently in NJL model calculations.

2.2 Action on the flavor branes and equations of motion

In the deconfined phase the induced (Euclidean) metric on the D8-branes is

(1)

where are the coordinates of 3+1-dimensional space-time, is the coordinate of the compactified extra dimension, is the curvature radius of the background, and is the metric of a four-sphere. The holographic coordinate on the flavor branes is denoted by with (symmetric phase) and (broken phase), see fig. 1, and

(2)

with the temperature . We have used capital letters for the coordinates , to reserve lower-case letters for their dimensionless versions introduced below.

The action on the D8-branes has a Dirac-Born-Infeld (DBI) and Chern-Simons (CS) part,

(3)

Let us first discuss the DBI action. Its general form for one of the disconnected D8- and -branes in the chirally symmetric phase is

(4)

where with the string length , the D8-brane tension , the volume of the unit four-sphere , and the dilaton with the string coupling . For brevity we have denoted the space-time integral by although it is actually a Euclidean integral with imaginary time , such that the integral over a space-time independent integrand (which is all we need in our calculation) yields with the three-volume . The DBI action for one half of the connected branes in the chirally broken phase is given by the same expression with the lower integration boundary replaced by . (In this general section, we shall give the expressions for the symmetric phase, but the broken phase is easily obtained via this simple replacement.)

In our ansatz the only nonzero field strength components are , , . The field strength is constant in the bulk and corresponds to a homogeneous magnetic field in the spatial 3-direction. Since the gauge symmetry on the flavor branes corresponds to a global symmetry at the boundary, is not a dynamical magnetic field. However, this is not problematic in our context where we are interested in a fixed background magnetic field.

We introduce the dimensionless quantities

(5)

where , are the dimensionful gauge fields. Then, with the relation

(6)

we can write the DBI action in the convenient form, also used in refs. [34, 40],

(7)

Here the prime denotes the derivative with respect to ,

(8)

and

(9)

with the dimensionless temperature

(10)

The CS action is (in the gauge )

(11)

where and . Strictly speaking, this CS action is the action on the left-handed brane (D8-brane). The corresponding action for the right-handed brane (-brane) has an overall minus sign. To avoid complications in the notation such as introducing left- and right handed gauge fields we shall only write expressions for the left-handed brane. This is sufficient for our purpose since we are mainly interested in the free energy of the system, which does not distinguish between the left- and right-handed fermions. There are of course quantities, such as the currents, where the sign of the CS action becomes relevant.

Within the above ansatz and using our dimensionless quantities, the CS part can be written as

(12)

Here we have kept all terms that contribute to the equations of motion, although the two terms vanish in our on-shell action since and do not depend on .

If we worked with the action as given by eqs. (7) and (12), we would encounter an ambiguity in the currents: in the presence of a homogeneous magnetic field the currents defined via the usual AdS/CFT dictionary would deviate from the currents defined through thermodynamic relations. This problem was pointed out in refs. [40, 42]. In ref. [40] a modified action was suggested,

(13)

with the additional contribution

(14)

This term does not change the equations of motion since it is a boundary term, but it is not a usual holographic renormalization because the first term in the curly brackets is a term at the spatial boundary, not the holographic boundary. The new action is invariant under residual gauge transformations which do not vanish at the spatial boundary [40] and removes the ambiguity in the currents. However, the correspondingly modified currents do not satisfy correct anomaly equations and reproduce the expected anomalous conductivities only up to a factor [42]. Here we are interested in the phase diagram, and not primarily in the anomalous conductivities, so we do not attempt to resolve this subtle issue; we simply follow the prescription with the modified action . The on-shell contribution of is given solely by the term at the spatial boundary. This term becomes simply one half of the original CS part, and thus adding effectively amounts to multiplying the original CS action by 3/2. Therefore, we expect our results to differ quantitatively, but not qualitatively, when we use the original action instead. We can write our on-shell action for the left-handed flavor brane as

(15)

where we used , and where the trivial space-time integral has been performed.

The equations of motion for , , and can be derived from eqs. (7) and (12),

(16a)
(16b)
(16c)

The left-hand (right-hand) sides of these equations originate from the DBI (CS) part of the action. Since the CS part does not depend on , the right-hand side of eq. (16c) vanishes.

According to the AdS/CFT dictionary, the current is defined as

(17)

Consequently, the 0- and 3-components of the (left-handed) current are

(18a)
(18b)

where, in the second equality of each line, we have used the integrated form of the equations of motion (16) with the integration constants and . Due to our use of , these results are identical to the ones which are obtained by taking the derivative of the free energy with respect to the corresponding source. For example, is the charge density which is also obtained by the negative of the derivative of with respect to the chemical potential. Since will turn out to be very complicated in general, eq. (18a) yields a simple alternative way to compute the density.

In the subsequent sections we shall solve the equations of motion. For all , , and there are two classes of solutions. One with , corresponding to straight, disconnected flavor branes and thus the chirally symmetric phase, and one with , corresponding to curved, connected flavor branes and thus the chirally broken phase, see fig. 1. The general solution of the equations has to be found numerically, but we shall discuss various limits where semi-analytic solutions can be found. The solutions will then be inserted into the action in order to compare the free energies of the chirally broken and symmetric phase. This will lead us to our main result, the chiral phase transition as a critical surface in the -- parameter space.

3 Chirally broken phase

3.1 Solution in the approximation

In general, the case of connected flavor branes is the more complicated one since besides the gauge fields and the equations of motion also contain the nontrivial function . We simplify this case by approximating

(19)

for all on the flavor branes. This approximation is valid for sufficiently large since for all on the flavor branes we have , see fig. 1. In principle, the approximation can – at a fixed temperature and thus fixed – be made arbitrarily good by decreasing the asymptotic distance of the flavor branes . However, we have to keep in mind that we are interested in the critical temperature for the chiral phase transition. Suppose we choose very small such that is a good approximation at some small temperature. Then, increasing the temperature and keeping fixed tends to invalidate our approximation because increases and approaches . But at some critical temperature the chirally symmetric phase takes over and thus our approximation only needs to be valid at temperatures below this (a priori unknown) critical temperature. At , where the full treatment is simple, we have checked that our result for the critical temperature deviates by about 10% from the full result, see sec. 5.1. In the general case we have only solved the equations of motion in the limit , and thus have no quantitative comparison with the full result, but we have checked that the transition takes over before severe artifacts such as occur in our approximation. Note also that within the approximation the broken phase becomes independent of . (The chiral phase transition will still depend on due to the -dependence of the chirally restored phase.)

With eq. (19) the integrated version of the equations of motion becomes

(20a)
(20b)
(20c)

with integration constants , , and . Our boundary conditions are

(21a)
(21b)
(21c)

where we have introduced the dimensionless separation

(22)

These boundary conditions arise as follows. First, we require the temporal component of the gauge field to approach the quark chemical potential at the holographic boundary. Since the chemical potential is the same for left- and right-handed quarks, must be symmetric, i.e., if is a smooth function along the entire connected branes, its derivative at the tip of the brane must vanish, . It turns out, however, that in the given choice of coordinates the more general version of this boundary condition is This takes into account that for finite magnetic field has a cusp at , i.e., approaches two values with the same magnitude but opposite sign depending on whether one approaches from the left or right. At the same time becomes infinite. This apparent singularity can be removed by a coordinate change, for instance to the variable used in ref. [42], defined through (and vice versa, i.e., the smooth solutions in terms of of ref. [42] acquire the same cusp in after changing the coordinate to ).

For the spatial component in the direction of the magnetic field we must allow for a nonzero value at the holographic boundary which has to be determined dynamically by minimization of the free energy. It has been shown for the technically simpler cases of maximally separated branes and/or the Yang-Mills approximation of the DBI action [35, 40, 41] that assumes a nonzero value in the presence of a magnetic field. This corresponds to an anisotropic chiral condensate and thus, viewing this condensate as a superfluid, corresponds to a supercurrent [41]. As a consequence, the system acquires nonzero baryon number, even for baryon chemical potentials smaller than the baryon mass [35, 56]. Since the chiral condensate carries axial, not vector, charge, is an axial supercurrent. Consequently, the boundary value at the other asymptotic end of the connected branes (where the right-handed fermions live) must be , leading to an antisymmetric gauge field and thus to the boundary condition .

Finally, the first boundary condition in eq. (21c) says that the connected branes “turn around” smoothly at while the second one says that the asymptotic (dimensionless) separation of the branes is .

We can solve eqs. (20) semi-analytically by generalizing the method introduced in ref. [40] to the case . In this way, the differential equations can be reduced to two coupled algebraic equations which have to be solved numerically. We defer the details of this procedure to appendix A. The result can be written as follows. We introduce the constant via

(23)

A nonzero implies that not only , but also the derivative of becomes infinite at the tip of the branes . Moreover, we define the new variable through

(24)

with

(25)

In terms of these quantities, the solution for the gauge fields is

(26a)
(26b)

where , and the embedding of one half of the connected flavor branes is given by

(27)

The functions , , are written in terms of , (which are the externally fixed physical parameters), (which has to be determined from minimizing the free energy), and the constants , , which are functions of , , and and given by the coupled equations

(28a)
(28b)

The dependence on the separation can be eliminated by rescaling

(29)

( and are invariant under rescaling with .) Employing these rescalings and changing the integration variable is equivalent to simply setting in eq. (28a).

We could now proceed by solving eqs. (28) for all , , and , insert the result into the solutions (26) and (27), these solutions into the on-shell action (15) and minimize the resulting free energy with respect to . However, there is a simpler way to determine the supercurrent . We recall that the total axial current is obtained by taking the derivative of the free energy with respect to the corresponding source. Here, plays the role of that source and thus we conclude that extremizes the free energy if , which implies, using eq. (18b),

(30)

This result can now be inserted into eq. (28b) which eliminates from the numerical calculation. Written in this way, is the same as in ref. [40], but note that is different in this reference. The reason is that there maximally separated flavor branes were considered, and thus the chirally broken phase was discussed in the confined geometry.

The free energy is

(31)

where the factor 2 takes into account both halves of the connected branes and where is given in eq. (15). After inserting the solutions of the equations of motion and after some algebra we can write the free energy as

(32)

where the result below the curly bracket eliminates and has been obtained by using eqs. (28b) and (30). Inserting the rescaled quantities from eq. (29) into shows that the free energy and the chemical potential scale as

(33)

In the remainder of the paper we shall set in all plots (except for fig. 7) for convenience. The dependence of all curves can easily be recovered with the rescalings (29) and (33).

Finally, we can compute the quark number density . With eq. (18a) we find

(34)

(Recall that eqs. (18) only take into account one half of the branes, hence the factor 2 in the definition of the total density .)

3.2 Discontinuity in the density

Figure 2: Left panel: free energy as a function of the supercurrent in the chirally broken phase for a fixed chemical potential and several values of the magnetic field [in the dimensionless units given in eq. (5), and with which is equivalent to using the rescaled quantities from eqs. (29) and (33)]. Since is dynamically determined from minimization of , we see that for the given there is a first-order phase transition where is discontinuous. Right panel: supercurrent as a function of for three values of . The solid lines are the full numerical result while the dashed lines are the linear approximations for small according to eq. (38). The middle curve corresponds to the potentials shown in the left panel. All three curves approach the asymptotic limit for large magnetic field, i.e., the upper two curves change their curvature to approach this limit on a scale much larger than shown here.

Figure 3: First-order phase transition (solid line) in the - plane for the chirally broken phase with , in the same units as in fig. 2. The supercurrent (and thus the baryon density) is discontinuous across this line, as shown in fig. 2. The dashed line is the chiral phase transition at (see sec. 5). On the right-hand side of the dashed line, the ground state turns out to be chirally symmetric (disconnected flavor branes) such that the solid line is only relevant for a metastable state.

Although we can compute the supercurrent directly from eq. (30), let us first discuss the form of the free energy as a function of . Solving eqs. (28) shows that there are parameter regions where there is a unique solution for the pair and parameter regions where there are three solutions. This is reflected in the free energy shown in the left panel of fig. 2. To obtain this plot we have renormalized by subtracting the vacuum contribution

(35)

where we have used that for , see eq. (28b). The curves of the renormalized potential as a function of show that there is a first-order phase transition where is discontinuous and thus, due to eq. (34), also the baryon density . The discontinuity is shown explicitly in the right panel where is plotted as a function of for three different values of . The full result has been obtained numerically, but we can easily find analytic approximations for small and large values for the magnetic field. For small magnetic fields and small , eqs. (28) give

(36)

with

(37)

We can thus approximate and eq. (30) becomes

(38)

This simple linear form for is compared to the full result in the right panel of fig. 2. For large magnetic fields, and thus approaches .

In fig. 3 we show the discontinuity in the baryon density in the - plane. As suggested from the right panel of fig. 2, the discontinuity is only present for sufficiently large chemical potentials. The first-order phase transition line terminates in a critical point and approaches the axis for small magnetic fields. The figure also shows the chiral phase transition at , to be computed and discussed in sec. 5. On the right-hand side of this line, chiral symmetry is restored. Therefore, the discontinuity in the density only occurs in a metastable phase and is probably of little physical relevance. We shall thus not display it in the phase diagrams in the subsequent sections.

4 Chirally symmetric phase

The chirally symmetric phase has been considered in ref. [34] within the same setup as discussed here. Nevertheless we shall discuss some of the details of this phase before we come to the chiral phase transition. One reason is that we work at fixed chemical potential, while in ref. [34] the density was held fixed. Furthermore, we shall elaborate on a discontinuity in the charge density within this phase, which resembles a transition to the lowest Landau level. A physical understanding of this discontinuity will turn out to be useful in the comparison of our phase diagrams with NJL model calculations.

In the case of disconnected flavor branes the integrated form of the equations of motion (16) becomes

(39a)
(39b)

with integration constants and . Since the branes are straight, we have set . In this sense, the equations are simpler than for the case of connected branes. However, now we cannot use the approximation because the branes extend all the way down to , see fig. 1. In this sense, the equations are more difficult than the ones for the connected branes. In general, we have to solve these equations numerically. Our boundary conditions are

(40)

As in the chirally broken phase, the value of at the holographic boundary is identified with the chemical potential. In contrast to the broken phase, the boundary value of vanishes because there are no Goldstone modes without spontaneous symmetry breaking, and thus there cannot be any supercurrent of these modes. We also require to vanish at the horizon, which is a regularity constraint1 [43]. For , there is a priori no condition at the horizon. Because of , eq. (39b) immediately yields

(41)

(Provided that is finite, which is true in all solutions we consider.) The numerical evaluation of eqs. (39) can be done with the “shooting method”: we consider the two differential equations as an initial value problem by imposing the initial values at the boundary according to eq. (40). Then we solve the equations by letting the gauge fields evolve from to for all from an appropriately chosen interval. (It turned out to be useful to implement this procedure by promoting the ordinary differential equations to partial differential equations with the additional variable .) Then we determine the value(s) of for which the gauge field is “shot” to its correct value at the horizon, . (Since in some cases two of these values for are very close to each other, it is more convenient to reparametrize in the numerics, motivated by the zero-temperature solution, see below.)

4.1 Zero-temperature limit and ”Landau level” transition

For , we have and thus . In this case the equations (39) can be solved semi-analytically. The solution is (see appendix B for details)

(42a)
(42b)

with the new variable

(43)

where has to be determined numerically from the relation

(44)

Inserting the solution (42) into the on-shell action (15) yields the free energy

(45)

The quark number density is obtained from eq. (18a),

(46)

One solution of eq. (44) is . In this case, the density becomes

(47)

where is the dimensionful quark chemical potential, , see eq. (5) for the corresponding relation for the gauge fields.

The numerical calculation shows that in certain regions of the , parameter space there are two additional nontrivial solutions for . Also for nonzero temperatures, where we solve the differential equations purely numerically, one or three solutions are found.

Figure 4: Critical lines in the - plane across which there is a discontinuity in the density of the chirally symmetric phase for three different (dimensionless) temperatures (to recover the dependence, has to be replaced by ). For all nonzero temperatures the critical line ends at a critical point. We discuss in the text and with the help of figs. 5 and 6 that the critical line is reminiscent of a Landau level transition, i.e., a population solely in the lowest Landau level above the line and populated higher Landau levels below the line (no further transitions between the higher Landau levels are seen in our model). The thin (blue) dashed line is the same line as in fig. 3 and indicates the chiral phase transition at (to be computed in sec. 5), i.e., on the left-hand side of this line the (blue) critical line has no physical meaning. The thick (black) dashed line is the analytic approximation (48) to the numerical result.

When we find three solutions , where for , the intermediate solution is never a global minimum of the free energy, while and compete for the lowest free energy. Where the global minimum jumps from to , a first-order critical surface appears in the -- parameter space. This surface is bounded by a critical line such that two-dimensional cuts through this parameter space, say at fixed temperature, show a critical line which, for nonzero temperatures, ends at a critical point. This is shown in fig. 4. For zero temperature, the critical line is given by the approximate critical magnetic field

(48)

This result is derived in appendix C and compared to the full solution in fig. 4. The ground state above this critical line is given by the solution and thus the corresponding density by eq. (47). Below the critical line the state with a nontrivial solution (which depends on and ) has the lowest free energy. In this case, the density is more complicated. Only for , we find the approximate behavior , because in this limit , see appendix C. For nonzero temperatures, all solutions for are finite and they continuously merge into each other for sufficiently small .

Figure 5: Quark number density as a function of the magnetic field for several temperatures at a fixed chemical potential . There is a first-order phase transition at a critical magnetic field for sufficiently small temperatures. At , the density for magnetic fields above this critical value is exactly that of a non-interacting Fermi gas in a magnetic field, see eq. (47).

Figure 6: Number density for a system of non-interacting, massless fermions as a function of the magnetic field for several temperatures (all in appropriate units of the quark chemical potential ). The oscillations are caused by the successive population of Landau levels and are smeared out for large temperatures. This plot should be compared with the holographic result in fig. 5, which shows some features comparable to a Landau level structure.

In fig. 5 we show the density as a function of for several temperatures at a fixed . There are interesting parallels and differences to the case of free massless fermions in a magnetic field. The free energy of non-interacting spin- fermion species of charge 1 in a homogeneous magnetic field is

(49)

where labels the Landau levels. (In general, has to be replaced by in this expression, where is the charge of the fermions.) The factor takes into account that the lowest Landau level (LLL) is occupied by a single spin degree of freedom, while all other Landau levels are degenerate with respect to both spin projections. The single-particle excitations are , where is the projection of the momentum on the direction of the magnetic field. The density follows immediately by taking the derivative with respect to ,

(50)

where is the Fermi distribution function. At , the distribution acquires a sharp Fermi surface and the density can be written as

(51)

Here we have separated the contribution from the LLL which is populated for arbitrarily large . The higher Landau levels are, for a given chemical potential, only populated for sufficiently small magnetic fields which is reflected in the upper limit of the sum over .

We plot as a function of the magnetic field in fig. 6. At , there are cusps in the density curve (i.e., discontinuities in the second derivative of the thermodynamic potential) which are caused by the Landau levels. Coming from large , where only the LLL is occupied, contributions from higher Landau levels set in successively at each of these cusps. At small , the sum over discrete levels can be approximated by an integral, and the result approaches the constant plus a highly oscillatory contribution with amplitude proportional to . For arbitrarily small nonzero temperature the cusps are smeared out. The oscillatory behavior survives for small and then completely disappears for large . This is due to the smearing of the Fermi surface, i.e., at any nonzero strictly speaking all Landau levels are occupied.

We can summarize the comparison of our holographic result for the chirally symmetric phase to the particle picture as follows.

  • Zero temperature.– For large magnetic fields, the holographic density behaves exactly (i.e., all geometric constants of the model drop out) like that of a system of non-interacting fermions; this can be seen by comparing eqs. (47) and (51). In the particle picture, all fermions sit in the LLL in this limit.

    At a certain value of the magnetic field, namely , the non-interacting system starts to populate the first Landau level. This manifests itself in a cusp in the density curve corresponding to a second order transition, with infinitely many more as is lowered. In the holographic system there is instead a single first-order phase transition at the point where, coming from large , the apparent LLL behavior ends. The critical value of at which this transition happens cannot be directly compared to the one in the particle picture since it involves the geometric constants of the model such as the curvature radius . In dimensionless quantities, this value is , i.e., it goes with a different power of than in the case of free fermions. In other words, the effective mass through the magnetic field seems to behave as , not as .

    At small magnetic fields, the density in both systems becomes approximately constant in , for free particles , while in the Sakai-Sugimoto model . Whereas the free fermion system shows an oscillatory behavior due to the Landau levels, the holographic result does not seem to know about Landau levels other than .

  • Nonzero temperature.– While the cusps in the density of the ordinary fermionic system are smeared out at any nonzero temperature, the first order phase transition in the holographic result survives for small temperatures (the larger the chemical potential, the larger the temperature below which the discontinuity persists, see fig. 4). Eventually, for sufficiently large temperatures, in both cases the density becomes monotonically increasing with increasing magnetic field, i.e., the transition in the holographic result disappears.

5 Chiral phase transition

Since the chiral phase transition has to be determined numerically in general, the next three subsections are devoted to some limit cases where the calculation is more transparent. These subsections also serve to discuss the approximation in the chirally broken phase and the possible split of chiral and deconfinement phase transitions.

5.1 Zero magnetic field

In the chirally broken phase at vanishing magnetic field , the location of the tip of the connected flavor branes is given by eq. (36),

(52)

We recall that here we have employed the approximation . In this case, we see that there is a unique solution for for any given . This solution can become arbitrarily small. We need to ensure, however, that in order to avoid the artifact of the flavor branes hanging farther down than they are allowed to by the geometry, see fig. 1. With the result (52) and the definition of in eq. (9) this condition is equivalent to , which yields a temperature limit at for the applicability of our approximation.

In the full treatment, there is a critical value for the separation above which there is no solution for (the branes must be disconnected then). For separations smaller than this maximal value there are in fact two solutions, one of which is unstable [58] and which approaches for (i.e., when the connected flavor branes are very close together they stretch down almost to the horizon). This unstable solution does not exist in the approximation, where the unique solution is an approximation to the stable solution of the full calculation.

At zero magnetic field we have and thus the free energy of the chirally broken phase (32) becomes

(53)

with given by eq. (52).

In the chirally symmetric phase, the equations of motion in the limit are obtained by setting in eqs. (39). This yields a simple differential equation for , which, when evaluated at , relates the integration constant to the chemical potential,

(54)

The free energy can be obtained from eq. (15). Using the equation of motion for we have

(55)

The chiral phase transition is now obtained by finding the zero of the free energy difference

(56)

(While each of the free energies is divergent, their difference is finite.) Even in the case , the zero of has to be found numerically in general. Our result for zero (and nonzero) magnetic fields is shown in the next subsection in the lower panel of fig. 9. For vanishing chemical potential we find the analytic result

(57)

which, using eq. (52), yields the critical temperature