Chiral density wave versus pion condensation at finite density and zero temperature

The quark-meson model is often used as a low-energy effective model for QCD to study the chiral transition at finite temperature $T$, baryon chemical potential ${\mu }_B$, and isospin chemical potential ${\mu }_I$. We determine the parameters of the model by matching the meson and quark masses, as well as the pion decay constant to their physical values using the on shell (OS) and modified minimal subtraction ($\overline{\mathrm{MS}}$) schemes. In this paper, the existence of different phases at zero temperature is studied. In particular, we investigate the competition between an inhomogeneous chiral condensate and a homogeneous pion condensate. For the inhomogeneity, we use a chiral-density wave ansatz. For a sigma mass of 600 MeV, we find that an inhomogeneous chiral condensate exists only for pion masses below approximately 37 MeV. We also show that due to our parameter fixing, the onset of pion condensation takes place exactly at ${\mu }_I^c=\frac{1}{2}{m}_{\pi }$ in accordance with exact results.

Figure 1

Chiral condensate (blue line) and pion condensate (red line) as functions of the isospin chemical potential ${\mu }_I$ for $\mu =0$ (upper panel) and $\mu =260\mathrm{MeV}$ (lower panel) at the physical point and $T=0$.

Figure 2

Phase diagram in the $\mu –{\mu }_I$ plane at the physical point in the homogeneous case at $T=0$. See main text for details.

Figure 3

Chiral condensate (blue line) and pion condensate (red line) as functions of the chemical potential $\mu$ for ${\mu }_I=0$ (upper panel) and ${\mu }_I=90\mathrm{MeV}$ (lower panel) at the physical point and $T=0$.

Figure 4

Lower and upper limits of the chemical potential $\mu$ where an inhomogeneous phase exists as a function of $m_{\pi }$ for ${\mu }_I=0$.

Figure 5

Phase diagram in the $\mu –{\mu }_I$ plane at $T=0$ in the chiral limit. See main text for details.

Figure 6

Pion condensate as a function of the chemical potential ${\mu }_I$ in the chiral limit for $\mu =0$.

Figure 7

Chiral condensate $\mathrm{\Delta }$ (blue line) and wave vector $q$ (red line) as a function of the isospin chemical potential ${\mu }_I$ in the chiral limit for $\mu =325\mathrm{MeV}$.

Figure 8

Chiral condensate (blue line), wave vector (red line), and pion condensate (green line) as functions of the chemical potential $\mu$ in the chiral limit for ${\mu }_I=5\mathrm{MeV}$.

Figure 9

Pion condensate $\rho$ as a function of the chemical potential $\mu$ in the chiral limit for ${\mu }_I=75\mathrm{MeV}$.