content_image/411135.jpg content_image/411136.jpg content_image/411137.jpg content_image/411134.jpg content_image/411128.jpg
<img src='content_image/411135.jpg'>

<img src='content_image/411136.jpg'>

<img src='content_image/411137.jpg'>

that in the case without symmetry breaking, the bulk viscosity over entropy density ζ/s decreases monotoni- cally with the increase of the temperature. In the case of 2nd-order phase transition, ζ/s decreases with T at low temperature region, then rises up at the critical tempera- ture T$_{c}$ and shows an upward cusp, and decreases further in the temperature T > T$_{c}$ . In the case of crossover, it is observed the cusp behavior of ζ/s is washed out. In the case of 1st-order phase transition, ζ/s shows divergent behavior at T$_{c}$ . 

From Refs.[50, 52], we know that η/s shows a shal- low valley in the case of crossover and a jump at T$_{c}$ for first-order phase transition. But it is hard to distinguish whether the system experiences a crossover or first-order phase transition just from the value of η/s extracted from the elliptic flow v$_{2}$ . 

From our results in the real scalar model, it is found that the ratio of ζ/s shows a very sharp peak at T$_{c}$ in the case of first order phase transition, and there is no obvi- ous change of ζ/s for crossover. As pointed out in Ref. [40] that a sharp rise of bulk viscosity near phase tran- sition induces an instability in the hydrodynamic flow of the plasma, and this mode will blow up and tear the system into droplets. Therefore, one can distinguish whether the system experiences a first order phase tran- sition or a crossover from observables at RHIC exper- iments. This result supports the idea of using ζ/s to locate the CEP as suggested in Ref. [30]. 

## The Polyakov-linear-sigma model 

We have shown the behavior of shear viscosity over entropy density η/s and bulk viscosity over entropy den- sity ζ/s near T$_{c}$ for different orders of phase transitions in a toy model, i.e, the real scalar model. In the follow- ing, we use a more realistic QCD effective model, i.e, the Polyakov-linear-sigma model (PLSM), which is described by the Lagrangian [56] 

<img src='content_image/411134.jpg'>

<img src='content_image/411128.jpg'>

