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The quark-meson model is often used as an effective low-energy model for QCDto study the chiral transition at finite temperature $T$, baryon chemicalpotential $\mu_B$, and isospin chemical potential $\mu_I$. The parameters ofthe model are determined by matching the meson and quark masses, as well as thepion decay constant to their physical values using the on-shell and modifiedminimal subtraction schemes. In this paper, we study the possibility ofdifferent phases at zero temperature. In particular, we investigate thecompetition between an inhomogeneous chiral condensate and a 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 exist only for pionmasses below approximately 37 MeV. We also show that due to our parameterfixing, the onset of pion condensation takes place exactly at$\mu_I={1\over2}m_{\pi}$ in accordance with exact results.

Chiral density wave versus pion condensation at finite density