perturbative methods face enormous challenges. Lattice QCD, starting from the first principle at zero and small quark chemical potential [@Aoki:2006br; @Schmidt:2006us; @Philipsen:2005mj; @Heller:2006ub], is regarded as an important tool to overcome the non-perturbative problem. Besides lattice QCD, other non-perturbative methods such as Dyson-Schwinger equations (DSEs) [@Alkofer:2000wg; @Bashir:2012fs], functional renormalization group equations (FRGs) [@Wetterich:1992yh; @Pawlowski:2005xe; @Gies:2006wv] and QCD effective models have been developed. Among QCD low energy effective models, Nambu–Jona-Lasinio (NJL) model [@Nambu:1961fr; @Klevansky:1992qe] offers the mechanism of spontaneous chiral symmetry breaking and has been widely used in describing chiral phase transition and investigating QCD phase structures under variant extreme conditions. In this model, the QCD gluon-mediated interactions are replaced by effective interactions among quarks, which are built according to the global symmetries of QCD. The NJL model does not contain dynamical gluons, which can be improved by adding the Ginzburg-Landau type potential of the traced Polyakov loop to the lagrangian to describe gluon dynamics and an interaction term of the Polyakov loop with the quarks. The improved model is usually named the Polyakov-loop improved Nambu–Jona-Lasinio(PNJL) model [@Meisinger:1995ih; @Fukushima:2003fw; @Ratti:2005jh; @Roessner:2006xn; @Ghosh:2006qh; @Schaefer:2007pw; @Ratti:2007jf; @Sasaki:2006ww; @Sasaki:2006ws; @Megias:2006bn; @Megias:2004hj; @Zhang:2006gu; @Sakai:2008py; @Ciminale:2007ei; @Fu:2007xc; @Hansen:2006ee; @Contrera:2007wu].

In recent decades, the discovery of the 