Towards a holographic realization of the quarkyonic phase

Large-$N_c$ QCD matter at intermediate baryon density and low temperatures has been conjectured to be in the so-called quarkyonic phase, i.e., to have a quark Fermi surface and on top of it a confined spectrum of excitations. It has been suggested that the presence of the quark Fermi surface leads to a homogeneous phase with restored chiral symmetry, which is unstable towards creating condensates that break both the chiral and translational symmetry. Motivated by these exotic features, we investigate properties of cold baryonic matter in the single-flavor Sakai-Sugimoto model, searching for a holographic realization of the quarkyonic phase. We use a simplified mean-field description and focus on the regime of parametrically large baryon densities, of the order of the square of the ’t Hooft coupling, as they turn out to lead to new physical effects similar to the ones occurring in the quarkyonic phase. One effect—the appearance of a particular marginally stable mode breaking translational invariance and linked with the presence of the Chern-Simons term in the flavor-brane Lagrangian—is known to occur in the deconfined phase of the Sakai-Sugimoto model, but turns out to be absent here. The other, completely new phenomenon that we, preliminarily, study using strong simplifying assumptions are density-enhanced interactions of the flavor-brane gauge field with holographically represented baryons. These seem to significantly affect the spectrum of vector and axial mesons and might lead to approximate chiral symmetry restoration in the lowest part of the spectrum, where the mesons start to qualitatively behave like collective excitations of the dense baryonic medium. We discuss the relevance of these effects for holographic searches of the quarkyonic phase and conclude with a discussion of various subtleties involved in constructing a mean-field holographic description of a dense baryonic medium.

Figure 1

The plot of the lowest position of the flavor brane $\xi$, as a function of the asymptotic separation $L$. The ratio $L/\delta \tau =0.5$ corresponds to antipodal embedding.

Figure 2

The relative mass difference, given by Eq. (29), for the three lowest axial and vector mesons. The continuous curve corresponds to the lowest axial and vector mesons, the dashed curve to the second lowest, and the dotted one to the third lowest. The difference is attributed to the axial symmetry breaking, which is the $N_f=1$ analogue of chiral symmetry breaking.

Figure 3

The relative mass-squared difference, given by Eq. (30), for the three lowest axial and vector mesons. The notation is as in Fig. 2.

Figure 4

The electric field ${\tilde{A}}_0^{\prime }(\tilde{z})$ as a function of the radial coordinate $\tilde{z}$ in the antipodal case ($\tilde{\xi }=1$) for three different rescaled baryon densities: $\tilde{E}=1$ (dotdashed), $\tilde{E}=100$ (dotted), and $\tilde{E}=500$ (dashed). The envelope (solid curve) is the electric field inside the charge distribution. The discontinuity in the derivative corresponds to the radial position at which the holographic charge density terminates. The electric field never gets large, as the matching condition bounds it from above. Note that the volume occupied by holographic charges increases with the baryon density on the field theory side, but the distribution of the charge in the core is not altered by the outer layers, which is why we used the envelope.

Figure 5

The electric field ${\tilde{A}}_0^{\prime }(\tilde{z})$ as a function of the radial coordinate $\tilde{z}$ at fixed baryon density ($\tilde{E}=500$) for four different embeddings, ranging from the antipodal ($L/\delta \tau =0.5$) to very nonantipodal ($L/\delta \tau =0.05$): $L/\delta \tau =0.5$ (solid), $L/\delta \tau =0.2$ (dotted), $L/\delta \tau =0.1$ (dotdashed), and $L/\delta \tau =0.05$ (dashed). As in Fig. 4, a discontinuity in the derivative corresponds to the radial position at which the holographic charge density terminates. The electric field also never gets large in this case due to the matching condition at the boundary of the bulk charge distribution.

Figure 6

Asymptotic values of the gauge field perturbation $\tilde{h}$ at $\tilde{z}=-\infty$ (${\tilde{h}}_L$ given by the solid black curve) and $\tilde{z}=\infty$ (${\tilde{h}}_R$ given by the dashed gray curve) as functions of the momentum of a mode for baryon density $\tilde{E}=1000$, ${\tilde{h}}^{\prime }(0)=0$, and antipodal embedding. No marginally stable mode is present. The inset plot shows that functions quickly achieve large values. Note that the plot is symmetric with respect $\tilde{z}\leftrightarrow -\tilde{z}$ only because ${\tilde{h}}^{\prime }(0)=0$.

Figure 7

Negative scan for the marginally stable mode at $\tilde{E}=1000$, ${\tilde{h}}^{\prime }(0)=1$, and antipodal embedding with the same notation as in Fig. 6. Note that the plot is no longer symmetric with respect to $\tilde{z}\leftrightarrow -\tilde{z}$ as ${\tilde{h}}^{\prime }(0)\ne 0$.

Figure 8

Negative scan for the marginally stable mode at $\tilde{E}=1000$, ${\tilde{h}}^{\prime }(0)=1.5$, and antipodal embedding with the same notation as in Fig. 6.

Figure 9

Negative scan for the marginally stable mode at $\tilde{E}=1000$, ${\tilde{h}}^{\prime }(0)=3$, and antipodal embedding with the same notation as in Fig. 6.

Figure 10

Negative scan for the marginally stable mode at $\tilde{E}=1000$, ${\tilde{h}}^{\prime }(0)=50$, and the antipodal embedding with the same notation as in Fig. 6.

Figure 11

The appearance of the marginally stable mode (${\tilde{h}}_{L/R}=0$ at $\tilde{k}\approx 1.23$) when the baryon mass is increased fivefold ($\tilde{E}=100$, ${\tilde{h}}^{\prime }[0]\approx 1.98$, and $\beta =20{\pi }^2$). The notation is the same as in Fig. 6.

Figure 12

The Schrödinger potential and the three lowest axial and vector mesons in the antipodal case ($L/\delta \tau =0.5$) at vanishing (top plot) and large (bottom plot, $\tilde{E}=1000$) density. Green dashed and red dotted lines correspond to the squares of masses of the lowest three vector and axial mesons, respectively. The most interesting finding is the appearance of an emergent potential well due to the holographic baryons on the D8-brane. It can be clearly seen that in the presence of a dense baryonic medium, the lowest axial and vector mesons are bound states not of the master potential well—which is also present in the vacuum—but rather of the emergent one. This leads to a tempting connection with a quasiparticle picture in which some of the lowest in-medium mesons are collective excitations of the underlying Fermi surface.

Figure 13

The Schrödinger potential and the three lowest axial and vector mesons for a very nonantipodal embedding ($L/\delta \tau =0.05$) at zero (top plot), medium (middle plot, $\tilde{E}=100$), and high (bottom plot, $\tilde{E}=1000$) density. Because of the small potential barrier residing in the vicinity of $\tilde{z}=0$, the emergent potential well does not give rise to bound states and effectively acts as a potential barrier. On the other hand, the emergent barrier leads to an approximate chiral symmetry restoration for the two lowest states, as their relative mass difference [given by Eq. (29)] reduces from about 40% in the vacuum to 10% in this case.

Figure 14

The Schrödinger potential in the antipodal case in the vacuum (solid curve) and at densities $\tilde{E}=100$ (dashed curve), $\tilde{E}=500$ (dotdashed curve), and $\tilde{E}=1000$ (dotted curve). One clearly sees that the emergent potential barrier arising because of the charges on the D8-brane affects more and more of the lowest lying axial and vector mesons as the density increases.

Figure 15

Masses of the three lowest vector (top) and axial mesons (bottom) as a function of the rescaled baryon density $\tilde{E}$ for the antipodal embedding of the D8-brane. One can see that within the adopted approximations the masses obtain significant contributions from interactions with the dense baryonic medium. The saturation of masses with density follows from the fact that the charge distribution is not altered by adding new layers, which leads to the Schrödinger potential, as in Fig. 14.

Figure 16

Schrödinger potential in the antipodal case for density $\tilde{E}=1000$ in the absence of direct holographic meson-baryon interactions, i.e., $C_0(\tilde{z})$ from Eq. (77) encoding the holographic baryons’ response to the electric field perturbations is set, by hand, to zero. Comparing this with the second plot within Fig. 12 we see that the emergent potential well gets much more shallow and lacks the barriers present when direct holographic baryon-meson interactions are taken into account. The discontinuity of the potential at the boundary of the charge distribution follows from the discontinuous first derivative of the background radial electric field.