Erratum: Towards a holographic quark matter crystal

We have identified a sign in our paper that needed to be corrected. The main conclusion of our paper was the presence of instabilities that suggest the existence of crystalline phases in the system.


Introduction and summary
• As a consequence of the corrected sign there is no critical value of the charge density.
Thus, we rewrite the seventh paragraph of the introduction, which now reads: Our results are pictorially summarised in figure 1, in which each solution on the gravity side, or equivalently each Renormalization Group (RG) flow on the gauge theory side, is represented by a curve running from top to bottom. Each curve corresponds to a different value of the charge density. A point on a curve corresponds to a given energy scale. At zero temperature a solution is described by an entire curve. Cutting off a curve at different points would correspond to different solutions with the same charge density and different non-zero values of the temperature. In the limit T N 1/3 q the solutions were constructed perturbatively in N 1/3 q /T in [2,3]. In this limit, however, all the relevant IR physics is hidden behind the horizon.  (see  table 1) and divide the diagram into six regions. Note that Region III includes a small area below and to the left of region IV.
• When describing the main results, two paragraphs below eq. (1.3), the text and figures change in several places. The new description until the end of this section now reads: The main result of our analysis is the phase diagram of the system. This is schematically summarised in figure 2. We will refer to this figure as a "phase diagram" despite the fact that it is a slight abuse of terminology, since some of the solutions are unstable and we have not identified the putative stable solution that would be thermodynamically preferred. The signs of different physical quantities in each region of figure 2 are summarised in table 1, where C Q is the specific heat at constant charge (5.4), χ is the charge susceptibility (5.2), c 2 s is the speed of sound (5.8), and D is the charge diffusion constant (6.4). Note that the latter equation implies that the last column is the product of the first three. Region I describes the high-temperature behavior of the system. In this region the solutions are locally thermodynamically unstable because the specific heat at constant charge, C Q , is negative. The system is also afflicted by a dynamical instability associated to a negative speed of sound squared, c 2 s < 0. The behavior of C Q and c 2 s in Region I is qualitatively analogous to the behavior in the hightemperature limit of the neutral solution of [1]. In the neutral case C Q and c 2 s become negative at exactly the same point because they are related through c 2 s = s/C Q , with s the entropy density, which is of course positive. It is remarkable that this feature extends to charged solutions, at least within our numerical precision. In other words, the top curve in figure 2 indicates the locus where both C Q and c 2 s change sign simultaneously. The behavior of C Q and c 2 s in region VI is the same as in Region I. In addition, in this region the charge susceptibility χ is negative. This indicates an additional thermodynamic instability towards charge clustering, and also results in an additional dynamical instability since the charge diffusion constant becomes negative. As explained in [1], the hightemperature behavior follows directly from the properties of the LP. Since our main interest is in IR features that are independent of the UV completion of the theory, we will not discuss Regions I and VI further.

JHEP07(2019)058
Region III is both thermodynamically and dynamically unstable, since both χ and D are negative. In contrast, Region V is only dynamically unstable since c 2 s and D are negative. However, while the existence of the other regions is robust, that of Region V may be a numerical artifact due to our finite resolution in parameter space.
Region II is the only locally stable region in the phase diagram. Local thermodynamic stability is guaranteed by the fact that both C Q and χ are positive. Our analysis shows no sign of a dynamical instability either, since c 2 s and D are both positive.
Region IV is particularly interesting since it corresponds to the low-temperature, high-charge density regime. This region is locally thermodynamically unstable since χ < 0 and also dynamically unstable since c 2 s < 0. The negative values of c 2 s and/or D that we have encountered imply dynamical instabilities in the hydrodynamic sound and charge diffusion channels, respectively, towards the spontaneous breaking of translation invariance. This suggests that the putative, stable phase in the corresponding regions may be a crystalline phase. We will come back to this point in sections 5 and 6.

Gravitational description
• Equations (2.48) and (2.49) now read and Figure 3 changes slightly to the one provided in this erratum.
• Three paragraphs before eq. (3.14) we eliminate a reference to the critical value of the charge density, which is not present anymore. This happens again later in the section, but we will not mention it explicitly any further.
• The behaviour of the IR coefficients in figures 4 and 5, and the corresponding description of these plots in the two paragraphs below eq. (3.15), change to: We see that for large values of r h all the curves converge to a single one, reflecting the fact that the leading UV behavior is controlled by the neutral solution. In contrast, for sufficiently small values of r h the charged curves deviate from the neutral one and approach straight lines with slopes that agree precisely with JHEP07(2019)058  those predicted by eqs. (3.14) and (3.15). For the dilaton the normalization also agrees with that in (3.15), whereas for the b h parameter we have performed a fit to (3.14). As expected, the value of r h at which the IR behavior sets in decreases as the charge density decreases. In particular, this means that solutions with small • Figures 6 and 7 change slightly, but the qualitative behaviour, and therefore our discussion of the plots, does not get altered.
• The description of figure 8 in the last two paragraphs of page 21, and the figure itself, change slightly: Finally, figure 8(left) shows the horizon value B h . We see that at small r h the slope is that determined by the leading term in (3.13), i.e. B h ∼ r 10 h , whereas at large r h the horizon value of B approaches the negative constant corresponding to the leading term in (3.6).
To summarise so far, we have seen that the large-and small-r h values of the IR parameters of our numerical solutions match precisely those predicted by the LP and Lifshitz geometries, respectively. This strongly supports the fact that our configuration interpolates between the UV LP geometry of the neutral solution and the IR Lifshitz one. Moreover

Thermodynamics
• Figure 13 is modified, and the last paragraph of this section is changed accordingly to: We close this section with plots in figure 13 of the energy density, E, the pressure, P , the enthalpy, E +P , and the chemical potential, µ. Note that these quantities are positive for all values of T and N q .

Instabilities
• The following two paragraphs right below eq.  Figure 13. Dimensionless energy density, pressure, enthalpy and chemical potential, as defined in (4.26) and (4.28).
In figure 14(left) we show a contour plot of the specific heat of our solutions as a function of T and √ 2 N q , while in figure 14(right) we show slices at several different values of the charge. We see that C Q is positive at low temperature but becomes negative at high temperature. This property is also illustrated by figure 15, where the negative slope of the entropy density curves as a function of T is evident at high T . In figures 14(right) and 15 we see that all curves converge to the same one at high T . The common curve is the same as in the neutral case studied in [1]. As explained in that reference, the negativity of C Q in this region is an UV effect associated to the presence of the LP. Since in this paper we are interested in IR physics that is safe from LP effects, we will not elaborate further on this instability.
In figure 16 we show a contour plot of the inverse charge susceptibility of our solutions as a function of T and  heat, we see that χ −1 is negative at arbitrarily low r h and T for charge densities √ 2 N q 4.5. The negativity of χ signals an instability towards charge clustering, which suggests that the putative, stable phase in this region may break translational invariance spontaneously. We will come back to this point in section 6.
• Figure 17 is modified, and its description after equation (5.9) reads: In figure   in contrast to C Q and similarly to χ, c 2 s is also negative in the IR region of arbitrarily low temperature and √ 2 N q 4.5.
• Figure 18 changes slightly, with g being monotonically increasing with the charge density.
• The region where c 2 s < 0, mentioned right before equation (6.3), changes in the new nomenclature of figure 2, most interestingly in Region IV.
• The paper ends after equation (6.4) with the following two paragraphs: In this expression σ is the electrical conductivity, which must be positive in order for the divergence of the entropy current of first-order hydrodynamics to be non-negative, i.e. in order for the second law of thermodynamics to hold. A negative value of D indicates a dynamical instability towards charge clustering (anti-diffusion), thus also generating inhomogeneities. Our numerical results show that this is the situation in several regions of the phase diagram 2, most interestingly in Regions III and V.
To conclude, we reiterate that some of the instabilities that we have identified suggest the possible existence of quark matter crystalline phases in our model, but establishing this definitively requires further analysis. We hope to report on these issues in the near future.

Appendices A-F
• In appendix B the last term of equation (B.2) gets a sign The terms in (B.3) multiplying the A and E terms also switch sign • Appendix F is removed.