Competition of inhomogeneous chiral phases and two-flavor color superconductivity in the NJL model

We study the phase structure of the two-flavor Nambu–Jona-Lasinio (NJL) model in the chiral limit, extending a previous study of the competition of an inhomogeneous chiral phase and a two-flavor color-superconducting (2SC) phase [M. Sadzikowski, Phys. Lett. B 553, 45 (2003); M. Sadzikowski, Phys. Lett. B 642, 238 (2006)]. There, an analytic expression for the dispersion relations for quasiparticle excitations in the presence of both a particular inhomogeneous chiral condensate, the so-called chiral density wave (CDW), and a homogeneous 2SC condensate was found. In this work we show how to determine the dispersion relations for arbitrary modulations of the chiral condensate in the presence of a homogeneous 2SC condensate, if the dispersion relations in the absence of color superconductivity are known. In our calculations, we employ two different Ansätze for the inhomogeneous chiral condensate, the CDW as well as the real-kink crystal (RKC). Depending on the value of the diquark coupling we find a region of the phase diagram where the inhomogeneous chiral and the 2SC condensates coexist, confirming results of M. Sadzikowski [Phys. Lett. B 553, 45 (2003); 642, 238 (2006)]. Decreasing the diquark coupling favors the inhomogeneous phase over the coexistence phase. On the other hand, increasing the diquark coupling leads to a larger 2SC phase, while the inhomogeneous chiral and the coexistence phases become smaller. In agreement with previous studies the RKC Ansatz is energetically preferred over the CDW Ansatz. Both Ansätze lead to a qualitatively similar phase diagram, however, the coexistence phase is smaller for the RKC Ansatz.

Figure 1

The phase boundaries for different diquark couplings $G_{\mathrm{\Delta }}=0,0.3G,G/2,3G/4$. The solid lines correspond to the phase transition line for the CDW Ansatz, while the dashed lines are associated with the RKC Ansatz. Where only solid lines are visible, the phase boundaries for both Ansätze coincide. The ancillary files with the plot data are found in Ref. [31].

Figure 2

Comparison of the grand potentials for different scenarios at $T=0\mathrm{MeV}$ and for $G_{\mathrm{\Delta }}=0.3G$. Two intersecting lines represent first-order transitions between the two respective phases, while converging lines characterize second-order phase transitions. For each case we subtract the potential of the fully restored ($M=q=\mathrm{\Delta }=0$) solution.

Figure 3

A detailed view of the order parameters for the CDW and the RKC Ansätze for $G_{\mathrm{\Delta }}=0.3G$.