Confinement and Z_3 symmetry in three-flavor QCD

We investigate the confinement mechanism in three-flavor QCD with imaginary isospin chemical potentials $(\mu_u,\mu_d,\mu_s)=(i\theta T,-i\theta T,0)$, using the Polyakov-loop extended Nambu--Jona-Lasinio (PNJL) model, where $T$ is temperature. As for three degenerate flavors, the system has $\mathbb{Z}_{3}$ symmetry at $\theta=2\pi/3$ and hence the Polyakov loop $\Phi$ vanishes there for small $T$. As for 2+1 flavors, the symmetry is not preserved for any $\theta$, but $\Phi$ becomes zero at $\theta=\theta_{\rm conf} < 2\pi/3$ for small $T$. The confinement phase defined by $\Phi=0$ is realized, even if the system does not have $\mathbb{Z}_{3}$ symmetry exactly. In the $\theta$-$T$ plane, there is a critical endpoint of deconfinement transition. The deconfinement crossover at zero chemical potential is a remnant of the first-order deconfinement transition at $\theta=\theta_{\rm conf}$. The relation between the non-diagonal element $\chi_{us}$ of quark number susceptibilities and the deconfinement transition is studied. The present results can be checked by lattice QCD simulations directly, since the simulations are free from the sign problem for any $\theta$.