Magnetized Domain Walls in the Deconfined Sakai-Sugimoto Model at Finite Baryon Density

The magnetized pure pion gradient ($\mathcal{5}\phi$) phase in the deconfined Sakai-Sugimoto model is explored at zero and finite temperature. We found that the temperature has very small effects on the phase. The thermodynamical properties of the phase shows that the excitations behave like a scalar solitonic free particles. By comparing the free energy of the pion gradient phase to the competing multiquark-pion gradient (MQ-$\mathcal{5}\phi$) phase, it becomes apparent that the pure pion gradient is less thermodynamically preferred than the MQ-$\mathcal{5}\phi$ phase. However, in the parameter space where the baryonic chemical potential is smaller than the onset value of the multiquark, the dominating magnetized nuclear matter is the pion gradient phase.


I. INTRODUCTION
Physics of dense nuclear matter is one of the most challenging area due to the lack of appropriate theoretical modeling. On one hand, the entities in the nuclear matter strongly couple to one another and therefore the perturbative treatment cannot be applied in a straightforward manner. On the other hand, the lattice approach to the Quantum Chromodynamics (QCD) can be applied to the situations of hot nuclear matter. The lattice results predict the deconfinement phase transition at temperature around 175 MeV for the dilute nuclear matter. However, this approach also faces difficulty in describing the nuclear matter with finite density due to the fermion sign problem.
An alternative and complementary approach is the application of the holographic principle or the variations of the AdS/CFT correspondence [1][2][3] in string theory to study the behaviour of nuclear matter. The Sakai-Sugimoto (SS) model [4,5] is a holographic model which mimics the QCD at low energy most accurately. Starting with a type IIA string background with D4-branes as the source. Take the near-horizon limit and add the black hole horizon to generate Hawking-Page temperature to be identified with the temperature of the dual gauge matter. Since we need an approximately 4 dimensional QCD, one of the 5 dimensional subspace is compactified into a circle whose radius is chosen so small that the Kaluza-Klein states are much heavier than the relevant energy scales and temperatures.
The quarks and antiquarks are introduced as open-string excitations on the stack of U(N f ) flavour D8 and D8 branes located at fixed separation distance in the compactified coordinate. The boundary conditions of the sparticles in the circle are chosen to be antisymmetric at the location of the flavour branes and the zeroth modes are thus eliminated.
Consequently, the gauge theory at the flavour branes is a SUSY-broken 5 dimensional Yang-Mills theory with quarks and antiquarks in the fundamental representation. The effective theory has the same particle content as the QCD. Using the AdS/CFT correspondence, the bulk theory of this brane configuration is conjectured to be dual to the QCD-like gauge theory at the boundary. The striking feature of the SS model is that it provides a natural geometric realization of the chiral symmetry breaking. When the D8 and D8 merge at certain location in the radial coordinate, the quarks and antiquarks do not transform independently under the chiral transformation and therefore the chiral symmetry is broken in the connected brane configuration. A chiral symmetric configuration occurs when the two flavour branes are parallel and the dual gauge matter will be in the chiral symmetric phase. Subsequent investigation reveals that the SS model accommodates the exotic possibility that the chiral symmetry restoration and the deconfinement can occur separately [6] when the distance between the D8 and D8 branes in the compactified dimension is not too large. The deconfinement could occur at relatively low temperature while the chiral symmetry would be restored at larger temperature. Even though both the chiral symmetry breaking and the confinement are results of the strong coupling of the gauge theory, they are independent of one another as far as we know. It is thus possible that the real QCD also has distinctive chiral symmetry restoration and deconfinement.
Chiral condensate of the QCD-like dense matter is explored in Ref. [7] using the Wess-Zumino-Witten induced anomalous term in the chiral perturbation theory and in Ref. [8] using the bottom-up AdS/QCD based on the confined SS model. When the magnetic field is applied, the condensate will respond by developing a gradient in the direction of the applied field. This gradient also carries the baryonic charge density proportional to the applied field and the gradient of the condensate. Holographic studies of the chiral condensate response to the magnetic field is investigated in Ref. [9] for the confined SS model. In Ref. [10], the pure pion gradient phase is explored and compared with the chiral symmetric quark-gluon plasma phase in the zero temperature approximation of the deconfined SS model. In Ref. [11], it is roughly compared with the mixed phase of the multiquark-pion gradient (MQ-▽ϕ) using a zero-instanton limit of the multiquark configuration. The preliminary results suggest that the pure pion gradient phase might be thermodynamically less preferred than the MQ-▽ϕ phase. In this article, we perform a thorough investigation into the pure pion gradient phase at finite temperature as well as its thermodynamical comparison to the MQ-▽ϕ in order to obtain a more definitive quantitative result. It is found that the pure pion gradient phase is insensitive to the change of temperature in the range T = 0 − 0.16. It is also shown that the pure pion gradient phase is generically less preferred than the MQ-▽ϕ phase except when the baryon chemical potential is smaller than the onset value of the multiquarks. In that region of the phase diagram, the dominating phase is the pure pion gradient.
The article is organized as the following. In Section II, we setup the holographic model of the magnetized chirally broken nuclear phase without an instanton. A zero temperature solution is obtained and relevant dual physical quantities as well as their relationships are discussed. Thermodynamical properties of the pure pion gradient phase at finite temperature and the comparison with the multiquark phase are discussed in Section III. Section IV concludes the article.

II. HOLOGRAPHIC SETUP OF THE MAGNETIZED CHIRALLY BROKEN PHASE
In the non-antipodal SS model, a stack of N c D4-branes generates a curved 10 dimensional spacetime in type IIA string theory. The near-horizon limit of this background is then taken and the black hole horizon is added by introducing the factor f (u) [3,12] into the background.
The x 4 direction is compactified with certain radius to obtain an effective (1+3) dimensional subspace in the low energy limit. The resulting spacetime of the Sakai-Sugimoto model is in the form . T is the Hawking-Page temperature of the black hole which is identified with the temperature of the dual gauge matter at the boundary.
R D4 is the curvature of the background which is generically different from the compactified radius R of the x 4 coordinate. φ is the dilaton field, a function of u in this background.
We then introduce stacks of N f D8 and D8 flavour branes with separation L on the circle of compactified x 4 at the boundary u → ∞. Open string excitations with one end on these branes behave like chiral "quarks" and "antiquarks" in the fundamental representation of the U(N f ). In the brane configuration where D8 and D8 are parallel, open-string excitations on each stack of branes transform independently under the chiral transformation and thus we have a chiral symmetric background. The dual gauge matter will be in the chiral symmetric phase. On the other hand, in the connecting brane configuration, chiral symmetry is broken at the tip and the corresponding gauge matter will be in the chirally broken phase [6].
To add the baryonic density to the boundary gauge matter, the non-normalizable mode of the a V 0 component of the U(1) ⊂ U(N f ) field is turned on. The baryon chemical potential µ of the corresponding gauge matter is identified with the non-normalizable mode of the DBI gauge field at the boundary by [13] To turn on the magnetic field, another component a V 3 is used as the vector potential generating the magnetic field. The direction of the magnetic field is chosen so that the vector potential is The Chern-Simon action in the background couples these two components to the third component a A 1 of the U(1), generating the response to the external magnetic field. The response appears as the gradient of the chiral condensate along the direction of B at the boundary which is defined to be a A 1 (u → ∞) ≡ ▽ϕ. Here and henceforth, we will call ▽ϕ a pion gradient.
The DBI and the Chern-Simon actions are then given by where N = N c R 2 D4 /(6π 2 (2πα ′ ) 3 ) defines the brane tension. The factor 3/2 in the Chern-Simon action comes from addition of surface term in order to maintain the gauge invariance of the total action in the situation when the gauge transformation does not vanish at the boundary (see Ref. [9] for details).
Consequently, the equations of motion with respect to each gauge field The corresponding density and current density, d, j A , at the boundary(u → ∞) are defined as They are related to the components of the U(1) gauge field by For the phase of pure pion gradient where chiral symmetry is broken, the axial current j A is set to zero and the density d = 3 2 B▽ϕ is the definition adapted from the Wess-Zumino-Witten action of the boundary gauge theory [7].
The constant of motion with respect to x 4 (u) for the pure pion gradient phase yields where where . u c is the position where the D8 and D8 branes connect. Since there is no instanton in this case, the branes connect smoothly at u c . We also have D(u c ) = 0 from a A 1 (u c ) = 0, and x ′ 4 (u c ) = ∞. Since the DBI action, Eqn. (3), is divergent from the limit u → ∞, we would need the action of the magnetized vacuum for the regularization. For the magnetized vacuum, we can let the non-normalizable modes, a V 0 , a A 1 = 0 and d, j A = 0. The vacuum action then takes the following form Again, the position of the tip of the brane configuration is denoted by u 0 . The temperature and field dependence of the position u 0 are given in Fig. 1 of Ref. [14]. It saturates approximately at 1.23 for all temperatures at high magnetic field. The action of the vacuum will be used to regulate the infinity of the DBI action from the limit u → ∞ when we calculate the free energy of the dual gauge matter in the subsequent section.
A. Zero Temperature case f (u) = 1 We can numerically solve the equations of motion, Eqn. (5),(6) by using the shooting algorithm. However, it is illustrative to consider first the limiting case of zero temperature where f (u) = 1 and the equations of motion are simplified so much that they yield exact analytic solutions. Later on we will actually find from the numerical solutions that most physical properties of the pion gradient phase are insensitive to the change of temperature.
In the zero temperature case, starting from the equations of motion, Eqn. (5),(6), rewritten as From Eqn. (15), for f = 1 we can solve to obtain Using Eqn. (16), direct integration leads to where by using the boundary conditions a A 1 (u c ) = 0 and Eqn. (17) at u c . Additionally, there are two constraints which need to be satisfied, where I ∞ ≡ I(u → ∞). In the zero temperature case x ′ 4 is given by The pion gradient is thus In order to obtain the solutions, we numerically solve for u c from the constraints Eqn. (23) and Eqn. (24) simultaneously by fixing two parameters among (B, d, ▽ϕ, µ). The solutions always have µ 2 > ▽ϕ as a reality condition. As is found in Ref. [10] for the pure pion gradient and Ref. [11,15] for the model with instantons, there are 2 possible brane configurations satisfying the scale fixing condition L 0 = 1, one with small and one with large u c . The brane configuration with small u c has longer stretch in the u-direction and therefore has higher energy than the configuration with large u c . The excess energy makes this configuration less preferred thermodynamically. For the pure pion gradient phase, there is also the small-u c configuration (for sufficiently large µ and small B) which we found to be less preferred thermodynamically even than the vacuum.
Therefore this configuration will not be considered in this article.
The solutions can be explored by slicing through the plane in the parameter space at fixed magnetic field (B), fixed density (d), and fixed chemical potential (µ) respectively. At fixed B, the solutions are shown in Fig. 1 for the position u c and the chemical potential as a function of the density. For small B, the position u c has certain variation with respect to the density. As B increases, u c saturates to an almost constant curve with a slight density dependence. The chemical potential at fixed B(≥ 0.1) is found to be an exact linear function of the density. The slope of the linear function is inversely proportional to B. The relation can be summarized into the following simple form This implies from d = 3B▽ϕ/2 that µ = 2▽ϕ for B ≥ 0.1. The linear relation between µ and d can be interpreted as the absence of self-interaction among the pion gradient excitations.
Each pion gradient excitation seems to behave as free entity for B ≥ 0.1.   Consistent with what previously found, µ is found to be inversely proportional to B as is shown in Fig. 3.
For fixed µ, Fig. 4 shows interesting transition between 2 regions of the parameter space.
In saturation at large B occurs around ▽ϕ = µ/2, the slope of the linear region, B ≤ 0.2, is therefore proportional to µ. Consequently, for small B, ▽ϕ ∼ µB. The behaviour at small B is similar to the behaviour found in Ref. [7,9] for the confined phase. The result in the deconfined phase of the SS model at zero temperature was first obtained in Ref. [10].
It should be noted that the nonlinear effects of the DBI action become apparent for

III. THERMODYNAMICAL PROPERTIES OF THE PURE PION GRADIENT PHASE
In the pure pion gradient phase, since x ′ 4 → ∞ at u c , the integrand of the action diverges at u c in addition to the limit u → ∞. This also occurs with the magnetized vacuum where x ′ 4 is divergent at u 0 . However, the limit which makes the integral and consequently the action divergent comes only from u → ∞ (the divergences at u 0,c are weaker than a simple pole and thus finite over integration). We can therefore regulate the action by subtracting the total action with the action of the magnetized vacuum in the usual manner. The regulated free energy is thus given by where Ω(µ, We can calculate the total action satisfying the equation of motion For zero temperature the total action reduces to We can compute this action by substituting Eqn. (18) into the expression. The free energy at fixed chemical potential F (µ, B) of the pure pion gradient phase at zero temperature is shown in Fig. 6. Once d, µ > 0, the free energy becomes smaller than the free energy of the magnetized vacuum (being negative) and thus thermodynamically preferred. The The pressure of the pure pion gradient as a function of the density can be calculated using Eqn. (27) and d = ∂P ∂µ (see Ref. [16]), where k(B) = 4/3B for B ≥ 0.1. The quadratic dependence of the pressure on the density without higher order term reveals that the pion gradient excitations behave like free particles without either repulsive or attractive interaction among themselves.
For the parameter space in the region d ≪ 1, B ≪ 1 such as the regions shown in Fig. 4 Ref. [9,10] for the confined and deconfined SS model at zero temperature.
The energy density can be calculated straightforwardly where k(B) = 1/αB 2 , 4/3B for small and large B respectively. The results are remarkably similar to the results from the bottom-up AdS/QCD model considered in Ref. [8]. The equation of state then becomes simply P = ρ representing free gas of the solitonic excitations of the the pion gradient. The adiabatic index, Γ, and the sound speed, c s , are then calculated to be the typical behaviour of the free gas.
For nonzero temperature, the full equations of motion, Eqn.

A. Comparison to the Multiquark-▽ϕ phase
We would like to consider whether the pure pion gradient phase is thermodynamically preferred than the other possible nuclear phases exist in the same parameter space. In the deconfined SS model in the presence of the magnetic field, there are generically 3 possible phases in addition to the vacuum; the chiral symmetric QGP phase, the chirally broken phase of multiquark-▽ϕ (MQ-▽ϕ), and the pure pion gradient (▽ϕ) phase. The multiquark nuclear phase has been studied in Ref. [11,14,17] and found to be the most preferred phase for the dense deconfined nuclear matter under moderate external magnetic fields. The multiquark phase actually has certain mixture of the pion gradient as the source for the baryon density. This is inevitable since the response of the nuclear matter to the external magnetic field is in the form of the spatial variation of the chiral condensate in the direction of the applied field which we call the pion gradient.
However, the ratio of the pion gradient population with respect to the multiquark decreases as d grows [14]. It is thus suggestive that the multiquark phase is likely to be more thermodynamically preferred than the pion gradient phase. In this subsection we directly compare the two phases at zero temperature using the free energy at fixed density d = 1.0.
The MQ-▽ϕ phase imposes the boundary conditions (see Ref. [11,14] for details); where n s is the number of colour strings (hanging from the baryon vertex down to the horizon) in fractions of 1/N c .
The result is shown in Fig. 7. Clearly, the multiquark-pion gradient (MQ-▽ϕ) phase is more preferred than the pure pion gradient phase. Similar behaviours are confirmed for small d ≃ 0.1 and large d ≫ 1. Especially at large densities, since the baryon chemical potential of the MQ-▽ϕ phase increases slower than a linear function [11] whilst it is linear for the ▽ϕ phase, the dominant term µd in the free energy for the MQ-▽ϕ phase becomes much smaller and thus more stable thermodynamically.
However, there is a region of parameter space where the ▽ϕ phase is dominant. When the baryon chemical potential µ < µ onset ≡ µ(d = 0) = 1 3 u 0 f (u 0 ) + n s (u 0 − u T ) of the multiquark, the multiquarks cease to exist and the pion gradient which can be constructed at arbitrarily small µ (since µ ∼ d) will be dominating. The corresponding transition line in the (µ, T ) diagram for B = 0 is shown in Fig. 8 of Ref. [17]. For B > 0, dependence of u 0 on B affects the transition line accordingly as shown in Fig. 9. The dotted line represents schematic transition to the chiral symmetric quark-gluon plasma (χS-QGP) phase. The chiral symmetry restoration between the magnetized vacuum and the χS-QGP has been studied in Ref. [18]. The transition between the pure pion gradient phase and the χS-QGP has been explored in Ref. [10] with f = 1 approximation for the pure pion gradient. Since we found that the ▽ϕ phase is insensitive to the change of temperature, the results in Ref. [10] should be justified to be a good approximated phase diagram. The chiral symmetry restoration between the MQ-▽ϕ phase and the χS-QGP phase has been investigated in Ref. [14].

IV. CONCLUSIONS AND DISCUSSIONS
The behaviour of the chirally broken pure pion gradient phase in the deconfined SS model is studied at zero temperature and subsequently finite temperature. The magnetic response of the chirally broken phase is linear ▽ϕ ∼ µB for small field and saturates to constant value ▽ϕ ∼ µ/2 for large field.
Relationship between µ and d is also linear µ = k(B)d where k(B) ∼ 1/B 2 , 1/B for small and large B respectively. This implies that the excitations of the pion gradient behave like a free gas with no interaction among each other. The equation of state is thus simply P = ρ with the sound speed equal to the speed of light. The free energies at fixed µ and d are obtained numerically. Magnetization at fixed µ increases with B for small field and drops to constant value for large field. Magnetization at fixed d is a decreasing function with respect to the magnetic field.
Using the free energy at fixed density, we show that the pure ▽ϕ phase is less preferred thermodynamically than the MQ-▽ϕ phase at zero temperature. The configuration of the pure pion gradient phase is found to be insensitive to the change of temperature, the difference of the free energy at fixed µ for T = 0 and T = 0.14 is minimal, only about 2 × 10 −4 .
On the other hand, the free energy of the MQ-▽ϕ phase is a decreasing function in the temperature [14]. Therefore, we can conclude that the pure pion gradient phase is generically less preferred than the MQ-▽ϕ for general situation.
However, there is an exception for small chemical potential, µ ≃ 0.175 − 0.41. When µ < µ onset of the multiquarks, the multiquarks simply cannot exist while the pion gradient can be induced at arbitrarily small µ. Therefore, in this region of the parameter space, the pure pion gradient phase is dominating over any other phases. The transition lines are given by µ = µ onset in the (µ, T ) plane. The interior of certain classes of the dense astrophysical objects such as the magnetars [19] would have the corresponding regions where the chemical potential (and the density) and temperature fall into this range. In those regions, the dominating nuclear phase which governs physics of the stars would be the pure pion gradient phase.