Cold Nuclear Matter In Holographic QCD
Moshe Rozali, Hsien-Hang Shieh, Mark Van Raamsdonk, and Jackson Wu
Department of Physics and Astronomy, University of British Columbia
6224 Agricultural Road, Vancouver, B.C., V6T 1W9, Canada
Theory group, TRIUMF, 4004 Wesbrook Mall, Vancouver, BC, V6T 2A3, Canada
We study the Sakai-Sugimoto model of holographic QCD at zero temperature and finite chemical potential. We find that as the baryon chemical potential is increased above a critical value, there is a phase transition to a nuclear matter phase characterized by a condensate of instantons on the probe D-branes in the string theory dual. As a result of electrostatic interactions between the instantons, this condensate expands towards the UV when the chemical potential is increased, giving a holographic version of the expansion of the Fermi surface. We argue based on properties of instantons that the nuclear matter phase is necessarily inhomogeneous to arbitrarily high density. This suggests an explanation of the “chiral density wave” instability of the quark Fermi surface in large QCD at asymptotically large chemical potential. We study properties of the nuclear matter phase as a function of chemical potential beyond the transition and argue in particular that the model can be used to make a semi-quantitative prediction of the binding energy per nucleon for nuclear matter in ordinary QCD.
1 Introduction and Summary
QCD at finite temperature and chemical potential The phase diagram of QCD as a function of temperature and baryon chemical potential (or alternatively baryon density) displays a rich variety of phases and transitions (for reviews, see [1, 2, 3]). However, apart from the regimes of asymptotically large temperature or chemical potential, where some analytic calculations are possible, and of zero chemical potential, where reliable lattice simulations are possible, our knowledge of the phase diagram is based exclusively on extrapolations and semi-empirical toy models. For intermediate values of the chemical potential, numerical simulation is plagued by a notorious ‘sign problem’ (see for example ), while analytic calculations are not possible due to strong coupling. Thus, while there has been significant progress recently in understanding the qualitative features of the phase diagram, reliable quantitative calculations that would definitively verify the proposed phase structure or determine the locations of various transitions or properties of the various phases seem a formidable challenge at present. A better understanding of the details of the phase diagram at intermediate chemical potential would have valuable applications, for example in understanding the physics of neutron-star interiors. Holographic models of QCD With the advent of the gauge theory / gravity duality , we have a new tool for studying the properties of certain strongly coupled gauge theories. While the original and most studied examples involve highly supersymmetric conformal gauge theories without fundamental matter, much progress has been made in constructing examples without supersymmetry , with confinement , with fundamental matter  and with chiral symmetry breaking . We now have examples of gauge theories with a known gravity dual that share most of the qualitative features of QCD, and the duality permits analytic calculations that would be otherwise impossible.
It is obviously interesting to study these QCD-like theories in regimes for which neither analytic or numerical studies are currently possible in real QCD. One such regime is the near-equilibrium behavior of the theory at finite temperature . This has received a great deal of attention recently (see  for a review) since calculations in holographic
There is already a large literature on studies of gauge theories at finite chemical potential using gravity duals (see  and references therein). Many of these consider a chemical potential for R-charge in theories with only adjoint matter. There have been some some studies of the behavior of theories with fundamental matter at finite baryon chemical potential, but the early examples of holographic theories with fundamental matter had both bosonic and fermionic fields carrying baryon charge. In these cases, the physics at finite chemical potential involves Bose condensation rather than the formation of a Fermi surface. In order to get behavior similar to real QCD, it is essential to study a theory with baryon charge carried exclusively by fermionic fields. Such a model was constructed a few years ago by Sakai and Sugimoto , and it is this model that we will focus on the present work.
The Sakai-Sugimoto model
The details of the Sakai-Sugimoto model are reviewed in section 2. Briefly, the model gives a holographic construction of a non-supersymmetric gauge theory with fundamental fermions. The gravity dual involves D8-branes in the near-horizon geometry of D4-branes wrapped on a spatial circle with anti-period boundary conditions for the fermions. In the geometry, the compact direction of the field theory together with the radial direction form a cigar-type geometry, in which the D8-branes are embedded as shown in figure 1. The other directions include an carrying units of D4-brane flux and the directions of the field theory. In addition to and , the theory has a dimensionless parameter , the ’t Hooft coupling at the field theory Kaluza-Klein scale.
For small values of , the scale where the running coupling becomes large is well below the field theory Kaluza-Klein scale, and the low-energy physics should be precisely that of pure Yang-Mills theory coupled to massless (fermionic) quarks.
We do not expect our results to agree quantitatively with real QCD (both because the Kaluza-Klein scale is not well separated from for large and because the classical calculations give only the leading terms in the expansion), but it would certainly be interesting to have a precise understanding of the phase diagram for a theory that is so similar to QCD. Indeed, at least some features of the phase structure and the qualitative behavior of certain transitions are likely to be the same as in QCD, and we might even hope for rough quantitative agreement for quantities that are relatively insensitive to and (we will discuss one such quantity below) .
The transition to nuclear matter
Our focus in this paper will be on the part of the phase diagram for zero temperature and intermediate values of the baryon chemical potential. In real QCD, as we increase the chemical potential from zero, the equilibrium state (i.e. the ground state) continues to be the vacuum until some critical value of the chemical potential at which point it becomes advantageous for baryons to condense. A first approximation to this critical value is the baryon mass, since it is at this point where it becomes energetically favorable to add single baryons to the vacuum. In fact, the critical value is somewhat lower, since the baryons have a negative binding energy. At the critical value, we have a first order transition from the vacuum state to homogeneous nuclear matter with some minimal baryon density.
to take into account the energy due to strong interactions of each nucleon in the interior of a nucleus with its neighbors plus the average kinetic energy per nucleon (non-zero due to Fermi-Dirac statistics). The best fit for this energy is
Ignoring electromagnetic interactions, this gives the binding energy per nucleon in the limit of large nuclei, and thus should be a good approximation to the value for homogeneous nuclear matter just beyond the transition. Thus, the critical chemical potential for the transition to nuclear matter in QCD should be approximately
As we increase the chemical potential further, the baryon density and the energy per baryon will increase from their values just above the transition. Eventually we hit at least one more transition, to a phase characterized by quark-quark condensates .
In this paper, we will study the physics of the transition to nuclear matter in the Sakai-Sugimoto model at large . Via classical calculations in the dual gravitational theory, we will be able to determine the critical chemical potential and calculate the baryon density and the energy per baryon for above the transition. Expectations at large Since our gravity calculations will give results corresponding to the large limit of the field theory (with a fixed ), we should briefly recall the expectations for how baryons behave for large . In this limit, baryon masses and baryon-baryon interaction energies go as , but the baryon size approaches a constant. Thus, we expect that both the baryon density above the transition and the binding energy per nucleon divided by the baryon mass to have a finite limit for large . These properties indeed follow from our calculations.
One significant difference between the large theory and ordinary QCD is the expected behavior at asymptotically large values of the chemical potential. In both cases, we have attractive interactions between excitations on the Fermi surface that result in an instability, but the nature of the resulting condensates is different. Whereas for the instability is a BCS-type instability, believed to lead to a color superconductor phase, the dominant instability at large is toward the formation of “chiral density waves” , inhomogeneous perturbations in the chiral condensate with wave number of order twice the chemical potential. This suggests that the ground state for large QCD at large enough chemical potential is inhomogeneous, however the nature of the true ground state remains mysterious (see  for a recent discussion). We believe that our analysis sheds some light on this question, as we will discuss shortly. Results for the Sakai-Sugimoto model In the Sakai-Sugimoto model, a chemical potential for baryon number corresponds to a nonzero asymptotic value of the electrostatic potential on the D8-branes, equal on both asymptotic regions of the D8-brane. Generally, this potential behaves asymptotically (for radial coordinate to be described below) as
The baryon density is proportional to the asymptotic abelian electric flux , so configurations with non-zero baryon density in the field theory correspond to D8-brane configurations with sources for the electric flux. These sources can be either string endpoints on the D8-branes which originate from D4-branes wrapped on the internal of the geometry  or (for ) configurations of the Yang-Mills field carrying instanton charge [16, 17]. The latter can be thought of as the wrapped D4-branes dissolved into the D8-branes and expanding into smooth instanton configurations.
One flavor For any value of chemical potential, we always have a trivial solution for which the electrostatic potential is constant on the D8-branes and the baryon density is zero. However we can also consider translation invariant configurations with a uniform baryon density. In the single flavor case, which we consider first, the bulk description of baryons is in terms of pointlike instantons, since there are no large instanton configurations in the abelian gauge theory of a single D8-brane. In this case, configurations with a uniform baryon density correspond to having some density of these pointlike instantons on the D8-brane. For a given value of the chemical potential greater than the critical value, we find some preferred distribution of charges on the D8-brane. The total baryon density for a given value of may be read off from the asymptotic value of the electric flux, and the result increases smoothly from above the critical chemical potential, approaching an asymptotic behavior . The charge distribution in the radial direction for a given value of represents the distribution of energies in the condensate of baryons in the field theory. In particular, the distribution has a sharp edge at some value of the radial coordinate which increases for increasing chemical potential, and this gives a bulk manifestation of the (quark) Fermi surface in the field theory.
For the single flavor case, the transition to nuclear matter is continuous, unlike QCD, but it may be expected that the single flavor case is different due to the absence of pions which usually play a crucial role in interactions between nucleons. Two flavors In the case with , we can have nonsingular instantons on the coincident D8-branes, and the minimum energy configurations for large enough are should involve smooth configurations of the nonabelian gauge field carrying an instanton density. While we might expect this to be homogeneous in the field theory directions, we argue that there are no allowed configurations of the D8-brane gauge field that are spatially homogeneous in the three field theory directions such that the net energy density and baryon density in the field theory are both finite. Thus, any phase with finite baryon density is necessarily spatially inhomogeneous. This has a simple interpretation: it suggests that at large , the nucleons retain their individual identities for any value of the chemical potential. Assuming that this holds true also for small where the theory becomes 2 flavor QCD, this suggests that the chiral density wave instability of the quark Fermi surface in large QCD simply indicates that the quarks want to bind into nucleons even at asymptotically large densities. This is discussed further in section 5.
To avoid the complication of directly studying inhomogeneous configuration, we approximate these by certain singular homogeneous configurations, arguing that our approximation should become exact in the limit of large densities. Within the context of this approximation, we study the behavior of the system as a function of chemical potential.
Our model displays a first order transition to nuclear matter at some critical chemical potential that depends on the parameter , with the baryon density behaving as for large . In the limit of large , the critical value approaches the baryon mass, so the binding energy per nucleon is a vanishing fraction of the baryon mass at large .
For large but finite , we find the behavior
where is the large result for the baryon mass
It is interesting that the result for the binding energy per nucleon at the threshold for nuclear matter formation,
is actually insensitive to the value of for large . Since we also know that this binding energy approaches some constant value in the limit of small (the large QCD result with two massless flavors), then assuming a smooth behavior at intermediate values of , we can treat the large result as a prediction for the order of magnitude of the QCD result.
In order to reliably compute the the numerical coefficients and , we require knowledge of the nonabelian analogue of the Born-Infeld action, and (in the case of ) probably corrections to this involving derivatives of field strengths. However, assuming is of order one,
While this paper was in preparation, the paper  appeared, which has some overlap with the present work, in particular section 4.1.
2 The Sakai-Sugimoto model
The basic setup for the Sakai-Sugimoto model  begins with the low-energy decoupling limit of D4-branes wrapped on a circle of length with anti-periodic boundary conditions for the fermions . Apart from , this theory has a single dimensionless parameter
the four-dimensional gauge coupling at the Kaluza-Klein scale. Because of the antiperiodic boundary conditions, the adjoint fermions receive masses of order while the scalars get masses of order due to one-loop effects. The coupling runs as we go to lower energies, becoming strong at a scale
for some numerical constant . As pointed out by Witten , for small , the dynamical scale is far below the scale of the fermion and scalar masses and the Kaluza-Klein scale, so the dynamics should be exactly that of pure Yang-Mills theory.
The field theory here is dual to type IIA string theory on the near-horizon geometry of the branes. The Lorentzian metric, dilaton, and four-form field strength are given by
where is the volume of a unit 4-sphere, is the volume form on , and
The direction, corresponding to the Kaluza-Klein direction in the field theory, is taken to be periodic, with coordinate periodicity , however, it is important to note that this circle is contractible in the bulk since the and directions form a cigar-type geometry.
The parameters and appearing in the supergravity solution are related to the string theory parameters by
while the four-dimensional gauge coupling is related to the string theory parameters as
In terms of the field theory parameters, the dilaton and string-frame curvature at the tip of the cigar (the IR part of the geometry) are of order and , so as usual, supergravity will be a reliable tool for studying the infrared physics when both and are large (in this case, with ).
Note that this is opposite to the regime of where we expect pure Yang-Mills theory at low energies. However, we may still learn about pure Yang-Mills theory by studying this regime, since many qualitative features of the theory remain the same and we might expect further that certain quantitative features may be relatively insensitive to the value of (as for example with the free energy in SYM theory).
2.1 Adding fundamental matter
Now that we have defined the adjoint sector of the theory, we would like to add fundamental quarks. We keep the number of quark flavors fixed in the large limit, but this means that the number of degrees of freedom in the fundamental fields (including the gauge field) is smaller than the number of degrees of freedom in the adjoint sector by a factor . Thus, for fixed in the large limit, the influence of the fundamental fields on the dynamics of the adjoint fields should be negligible.
The Sakai-Sugimoto construction is motivated by the observation that the light open string modes living at a 3+1 dimensional intersection of D4-branes and D8-branes give rise to chiral fermion fields on the intersection without accompanying bosons. Thus, to the original D4-branes, which we can take to lie in the 01234 directions with the direction periodic, Sakai and Sugimoto consider adding a stack of D8-branes and a stack of anti-D8 branes separated at fixed locations in the directions and extended along the remaining directions. This configuration is unstable before taking a near horizon limit
The specific embedding of the D8-branes in the bulk depends on the asymptotic separation of the stacks (and also any distribution of matter on the branes), but we will focus exclusively on the case where the two asymptotic parts of the D8-brane stack sit at opposite sides of the D8 circle, in which case each side simply extends to the tip of the cigar along a line of constant as shown in figure 1. The corresponding field theory has all flavors massless.
2.2 D8-brane action
To understand the physics of the probe D8-branes, we will need the action for the worldvolume D8-brane fields in the background above. We will begin by discussing the action for a single D8-brane before discussing the nonabelian generalization.
The Born-Infeld action for the worldvolume D8-brane fields (in the case of a single brane) is
We also have a Wess-Zumino term
Here, only the term contributes. Noting that is the derivative of the five-dimensional Chern-Simons form, and integrating by parts, we get
After integrating over the sphere, this gives
where . For a single D8-brane, .
To simplify the Born-Infeld action, we can choose to identify the worldvolume and spacetime coordinates in the sphere and the field theory directions, and parameterize the profile of the brane in the and directions by and respectively (we will soon focus on the solution where is constant).
We will be interested only in time-independent configurations homogeneous and isotropic in the spatial directions of the field theory (which we label by indices ). The most general configurations we will consider will have non-zero , , and , all functions only of .
Integrating the determinant from the sphere directions over the sphere, we get a factor
while the remaining five-dimensional determinant is
Note that we are using here to refer to the spacetime metric and for the worldvolume metric. The final result (in the Abelian case) is
This action is manifestly invariant under reparametrizations of . The nonabelian generalization of this action is known only up to terms. Up to order , we symmetrize all of the nonabelian field strengths in expanding the square root and take an overall trace. However, this symmetrized trace prescription is known to fail beyond order .
2.3 Chemical potential for baryon charge
We would like to study the theory at finite chemical potential for baryon charge or alternatively, the theory with a modified Hamiltonian density
where is the baryon charge density operator
This is equivalent to adding a term to the action since there are no time derivatives in . Turning on the operator in the boundary gauge theory with real coefficient should correspond to turning on some (real) non-normalizible mode in the gravity picture. From the original brane setup, we know that the operators and couple to the time-components of the and brane gauge fields respectively. We will see below that the equations of motion for these fields require them to approach some constant values in the UV part of the geometry. If we describe the probe branes as above with a single gauge field for the whole configuration, then we have two such constant values,
These two values give the chemical potentials for the operators and .
2.4 Asymptotic solutions
In the simple case where the D8-brane is at constant and we assume that only the electrostatic potential is turned on, the Born-Infeld action above reduces to
The reparametrization invariance allows us to chose to be whatever we like. For a given choice of , the equation of motion for away from any sources (which we assume are localized in the infrared part of the geometry) is
The quantity in round brackets is analogous to the conserved electric flux. Integrating and rearranging, and choosing (valid for either half of the brane), we get
where is an integration constant proportional to the conserved flux. Solving this, we find
valid in the region outside the sources. The constant is the normalizible mode of in the asymptotic solution, so the values of for the two sides of the brane correspond to the expectation values for and in the field theory.
In general, the sum of the s for the two halves of the brane (times ) is equal to the total charge density on the brane,
If we fix as we have argued corresponds to a chemical potential for baryon number, and we assume that the sources are symmetric under a reflection in the direction, then for continuous we must have , and
Since the charge density in the bulk (divided by ) corresponds to the baryon density in the field theory, we obtain
We have seen that configurations with non-zero baryon charge density (as measured by the asymptotic electric flux ) require sources for on the D8-branes. The basic source for is the endpoint of a fundamental string. In order to have some net charge, we need the number of string endpoints of one orientation to be unequal to the number of string endpoints of the other orientation. So we need a source for fundamental strings in the bulk. In our background, such a source is provided by D4-branes wrapped on . These necessarily have string endpoints, since the background D4-brane flux gives rise to units of charge on the spherical D4-branes, so we need units of the opposite charge (coming from the string endpoints) to satisfy the Gauss law constraint. Thus, we can get a density of charge on the D8-brane by having a density of D4-branes wrapped on in the bulk, with strings stretching between each D4-brane and the D8-brane.
In the case where we have D8-branes, there is another possible picture of the configurations with baryons [16, 17]. To see this, note that a D4-brane / D8-brane system with four common worldvolume directions is T-dual to a D0-D4 system. In that case, it is well known that the D0-branes can “dissolve” in the D4-branes, where they show up as instanton configurations of the spatial non-abelian gauge field. Similarly, our baryon branes can dissolve in the D8-branes (if we have ) and show up as instantons. Indeed, the Chern-Simons term (2) gives rise to a coupling
between the instanton charge density and the abelian part of the gauge field, showing that instantons act as a source for the electrostatic potential on the branes.
The question of which of these two pictures is more appropriate is a dynamical one, but it turns out that the dissolved instantons give rise to a lower energy configuration since the electrostatic forces prefer the instanton density to be delocalized [16, 17].
3.1 Baryon mass
The baryon mass was estimated originally by Sakai and Sugimoto  as the energy of a D4-brane wrapped on and located at the tip of the cigar. Since we will also need to know the potential energy for such branes, we briefly recall the calculation. Starting with the Born-Infeld action for a D4-brane wrapping ,
and integrating over the sphere, we get
as the velocity independent term in the action (the negative of the potential energy). The minimum energy occurs for , and this gives the baryon mass
This agrees with the Yang-Mills action for a pointlike instanton configuration on the D8-brane . Both of these calculations ignore the energy from the electric flux sourced either by the string endpoints coming from the wrapped D4-brane or by the instanton density. To take this into account, the authors of  and  considered more general smooth instanton configurations with varying scale factor, inserting these into the Yang-Mills approximation to the D8-brane action. They found that the optimal size for the instanton behaves as , and that the baryon mass is
This method ignores the effects of the non-trivial geometry on the Yang-Mills configuration and also does not include effects from the corrections to the D8-brane effective action, which should be important, since for large , the instanton is small so that derivatives of the Yang-Mills field strength are large. Thus, as the authors point out, the numerical coefficient should probably not be trusted. On the other hand, an analysis of the effects of Born-Infeld corrections  indicates that at least the power of in the correction to the mass and in the instanton size should be reliable.
3.2 Critical Chemical Potential
We have seen that turning on a chemical potential in the gauge theory corresponds to including boundary conditions for the two asymptotic regions of the D8-brane. For any , one solution consistent with these boundary conditions is to have constant everywhere on the brane. This represents the vacuum configuration in the field theory. However, beyond a certain critical chemical potential, this solution is unstable to the condensation of baryons.
The critical value of the chemical potential should not be larger than the baryon mass. At this value, a zero-momentum baryon has effectively negative energy in the modified hamiltonian, so it is advantageous to add baryons to the vacuum. If there were no interactions between the baryons, the critical chemical potential would be exactly the baryon mass. Note that even in the absence of interactions, the baryon density above the transition is limited by the Fermi statistics for the baryons for odd or in any case by the Fermi statistics of the quarks. The condensate will have occupied all states whose Fermi energy is less than the chemical potential. In this case, the baryon density will rise smoothly from zero above the critical chemical potential and the transition will be second order.
With short range repulsive interactions, the story would be qualitatively similar, with a slower growth in the baryon density as the chemical potential is increased. In QCD, however, we have attractive interactions, and this lowers the critical chemical potential below the baryon mass. With the repulsive interactions, there is a specific nonzero value of the baryon density for which the energy per baryon is lowest, and when the chemical potential is increased to this value the baryon density jumps from zero to this density.
In the next sections, we will study this transition to nuclear matter in the Sakai-Sugimoto model for one flavor (section 4) and two flavors (section 5). In the first case, it appears that the transition is second order, unlike QCD, while in the multi-flavor case, we find some evidence for a more realistic first-order transition.
4 One flavor physics
In this section, we study the physics of the Sakai-Sugimoto model at finite chemical potential in the simpler case of a single quark flavor. Here, we have only a single D8-brane in the bulk, and we can use the abelian Born-Infeld action for our analysis. Since the abelian gauge theory does not support large instantons, the wrapped D4-branes cannot dissolve into the D8-branes, so the baryons are pointlike charges on the D8-brane that source the electrostatic potential. For chemical potential larger than the baryon mass, it is favorable for some of these baryons to condense, and we would now like to determine the baryon density as a function of chemical potential for above the critical value.
4.1 Localized source approximation
As a first approximation, we make the simplifying assumption that all the pointlike instantons sit at . More realistically, the charge should spread out dynamically, via electrostatic repulsion; we will include this effect in section 4.2.
In our simple approximation, the relevant action is the Abelian Born-Infeld action (5), together with the action taking into account the baryon masses and their interaction with the electromagnetic field on the brane.
where the terms in the last line are the potential terms taking into account the energy from the charges in the electrostatic potential and the masses of the pointlike instantons.
To obtain the energy, we perform a Legendre transform, but it is convenient first to rewrite the first term in the second line as
since we will be holding fixed. Performing the Legendre transform (which amounts to taking the negative of the action, since we are only looking at static configurations), and rewriting everything in terms of the electric flux (7), we find
where we have defined and
In the first term, we have included a factor of 2 to take into account the energy f rom both halves of the D8-brane.
For , the combined energy from the string endpoints (or Chern-Simons action) and the D4-brane mass (or Born-Infeld energy of the instantons) is negative and should be proportional to , while the energy from the flux is a positive function of which behaves as for small and for large . Thus, there will be some positive value of where the total energy is minimized.
so that corresponds to , and using the relation (8) between and , the total energy may be written as
From this, we find that the energy is minimized when
This can be inverted to determine the relationship between (proportional to ) and above the transition. For small , we find
For large we have
4.2 Dynamical charge distribution
The analysis of the previous section assumed that all charges were localized at . Presumably, the charges would prefer to spread out dynamically. To take this into account, we can define a charge distribution which we would like to determine. For a given , the action is given in terms of a Lagrangian density
Here, the second term is the action arising from the string endpoints, while the third term takes into account the potential energy from the baryon masses (recalling that the action for a wrapped D4-brane at location is proportional to ).
For a given , the electric flux is determined by solving the equation of motion for ,
where we have defined an electric flux
We can now reexpress all terms in the action in terms of and Legendre transform (which again amounts to switching the sign) to find the energy. We obtain
where we have included an extra factor of 2 in the denominator on the left side since we are integrating over only half the brane on the right side. To maximize this, we can first minimize over all such that , , and to determine . Then we can minimize over .
Varying the energy functional with respect to , we find that the energy functional is locally stationary if and only if
This satisfies for as desired but approaches arbitrarily large values for large . On the other hand, our constraints and imply that can never exceed . It is straightforward to check that the local contribution to the energy from a point is a function of that decreases from to the optimal value (13) and then increases again, so when the value (13) exceeds , the best we can do to minimize the energy is to set . We conclude that the minimum energy configuration for fixed and fixed is
Here represents the extent of the charge distribution in the radial direction, and is related to as
We can now write the energy as a function of , or more conveniently, as follows. We define a function by
Then in terms of and , the energy is given by
where as in the previous section, we define
We can now minimize this as a function of . The result is
To compare with the results of the previous section, we note that (using (15)) the dimensionless variable proportional to the baryon mass is related to by
From these, we find that for small ,
where we have used (15). Thus, for small we obtain the same result as in the previous approximation, with
for small , where the critical value of is as before. For large , we find
Again, we find that
Thus, the qualitative behavior of is the same as in the simplified model of the previous section, though the numerical coefficients come out different. We also found the behavior of the energy density:
for small, and
when is large.
It is interesting that (in this approximation) the charge distribution has a sharp edge at which progresses further and further towards the UV in the radial directions as the chemical potential is increased. In the field theory picture, the radial direction represents an energy scale, so the charge distribution we find in the bulk should be related to the spectrum of energies for the condensed baryons. The edge of the distribution is then a bulk manifestation of the Fermi surface.
Since our large calculation does not distinguish between even and odd values of , it is insensitive to whether or not the baryons are fermions or bosons. Thus, the Fermi surface that we see should probably be thought of as the quark Fermi surface. It is interesting that the fermionic nature of the quarks in the field theory arises in the bulk from the classical electrostatic repulsion between the instantons.
5 Two massless flavors
For , the authors of [16, 17] argued that single instantons on the D-brane prefer to grow to some finite size on the baryon in order to balance the electrostatic forces which tend to make the instanton spread out with the gravitational forces which prefer the instanton to be localized as much as possible near the IR tip of the D8-branes. From these considerations, we also expect that the minimum energy configurations with nonzero baryon density will involve some smooth configuration of the nonabelian gauge field on the D8-brane locally carrying an instanton density . In this section, we consider such configurations.
The absence of homogeneous configurations We first consider static, spatially homogeneous configurations, such that is translation invariant in the 3+1 directions of the field theory and rotationally invariant (up to a gauge transformation) in the three spatial directions (which we denote by an index ). The general configuration of the spatial gauge field with these symmetries is
for an arbitrary function . These give
From these, we find that
We see that unless both and vanish for , the Yang-Mills action density integrated over will diverge, corresponding to an infinite energy density in the field theory. On the other hand, we find
In order that we have a configuration with finite baryon density in the field theory, we require that this instanton density, integrated over the sigma direction be non-zero
The apparent conclusion for the dual field theory is that there are no spatially homogeneous configurations with finite non-zero baryon density and finite energy density. Now, there certainly are non-homogeneous configurations with finite average energy density and finite average baryon density: we can simply take a periodic array of individual instantons. For large enough chemical potential (greater than the energy density divided by the baryon density), such configurations are favored over the vacuum, so we will certainly have a phase transition to a phase with nonzero baryon density as the chemical potential is increased. However, our observation suggest that this phase cannot be spatially homogeneous.
Interpretation of the inhomogeneity and origin of the chiral density wave
The inhomogeneity of nuclear matter is not unexpected, and indeed is what we have for real nuclear matter at low densities (e.g. in the interior of large nuclei). It simply reflects the fact that the individual nucleons retain their identities (and therefore that the baryon density is clumped
Our approach is motivated by the observation that in the limit of infinite baryon density, the bulk configuration should become homogeneous. Such homogeneous configurations are singular at the core, corresponding to a divergence of the instanton charge density. For example, we can have a self-dual configuration of the form (16) if we choose
This should arise from the limit of a periodic array of instantons for which the separation is taken to zero while adjusting the scale factors to yield a non-trivial configuration in the limit. We expect that some similar configuration
As we move away from infinite density, the minimum energy configuration will only be approximately homogeneous. We expect, however, that the averaged field strengths and instanton density should be qualitatively similar to those for the configuration (18) but with finite values at . This behavior can be achieved in a configuration of the form (16) for which is an odd function like (18) but with some finite limit at . Such configurations are singular at , but we will ignore any effects associated with the singularity at since we are using our configurations to approximate non-singular inhomogeneous configurations that do not have any pathologies at .
5.1 Energy density for approximate configurations
We would now like to analyze the behavior of the model as a function of chemical potential in the approximation where we consider only configurations of the form (16), taking to be a monotonically increasing function for that takes some finite (negative) value at and vanishes for . In practice, we work with the action for half the brane, assuming that is an odd function so that all the field strengths are symmetric about . As we mentioned above, such configurations are singular at but we ignore any effects of the singularity, motivated by the expectation that the nonsingular contributions may provide a good approximation to the averaged quantities for the non-singular inhomogeneous configuration that we should really be studying.
The configuration of the spatial Yang-Mills field carries instanton density, and therefore acts as a source for the abelian electrostatic potential on the D8-branes. In order to determine the potential for a given , we need the equation of motion for , which should come from the non-abelian generalization of the Born-Infeld action (4) and the Chern-Simons action (10).
As we have noted, the nonabelian generalization of the Born-Infeld action (4) is known only up to terms. In the absence of the full result, we will work with a naive ordering prescription in which we simply insert our ansatz into the abelian expression (4) and (noting that each product of s above gives an identity matrix) evaluate the trace. This will give us results that are precisely correct in the limit where the field strengths are small and only the Yang-Mills terms in the action are important, but we should not trust numerical coefficients whose calculation depends on the higher order terms in the Born-Infeld action.
while the Chern-Simons term (2) gives:
If we define
then the action takes the form
The equations of motion for the electrostatic potential are
From this, we conclude that
where we have determined the integration constant by demanding that vanish as , as is required for finite energy configurations. Since vanishes by symmetry at (assuming that there is no delta function charge distribution at ) we see that the asymptotic value of is related directly to the value of at by
We may therefore rewrite (23) as
Using this result, the electrostatic potential may be determined in terms of by inverting (22).
We may now write an expression for the energy density of a configuration for a given value of .
where we have subtracted off the energy density of the unexcited brane such that the vacuum state is normalized to zero energy. We can now rewrite the energy in terms of , assuming that the equation of motion for is obeyed. We have first
Now writing in terms of as above, changing variables to , defining
we finally have