Functional renormalization group study of rho condensate at a finite isospin chemical potential in the quark meson model

We investigate the effect of an isospin chemical potential ($μ_{I}$) within the quark-meson model, which approximates quantum chromodynamics (QCD) by modeling low energy phenomena such as chiral symmetry breaking and phase structure under varying conditions of temperature and chemical potential. Using the functional renormalization group (FRG) flow equations, we calculate the phase diagram in the chiral limit within the two-flavor quark-meson model in a finite $μ_{I}$ with $ρ$ vector meson interactions. Fluctuation effects significantly decrease the critical chemical potential from the mean-field (MF) value $μ_{I, MF} > m_ρ$ to lower value, at which point the $ρ$ vector meson condensates alongside the chiral condensate once the isospin chemical potential exceeds the critical value $μ_{I}^{\text{crit}}$. This $ρ$ condensation is investigated numerically for different meson coupling strengths. The $ρ$ meson dominated region is delineated from other phases by a second-order phase transition at lower $μ_{I}$ and a first-order transition at slightly higher $μ_{I}$.