## What is/are Crossover Line?

Crossover Line - The phase diagram features a crossover line starting from the transition temperature already determined at zero chemical potential.^{[1]}This crossover line between gas- and liquid-like behaviors is defined as the curve for which an individual property, the contribution to viscosity due to molecules' translation, is exactly equal to a collective property, the contribution to viscosity due to molecular interactions.

^{[2]}We further prove that three phase transition lines meet at a multicritical point from which a deconfinement crossover line extends into the disordered phase.

^{[3]}The curvature of the chiral crossover line decreases.

^{[4]}The behavior of the structural crossover lines as the size ratio and densities of the two species are changed is also discussed.

^{[5]}The crossover line and the location of CEP resulted from the model agree with both the lattice result and the experimental analysis from relativistic heavy-ion collisions.

^{[6]}From these results, we examine the nature of the chiral phase transition and the variation of the curvature of the crossover line as we approach the chiral limit.

^{[7]}In our criterion, Brazil nut separation (BN, large particles rise to the top), reverse Brazil nut separation (RBN, large particles sink to the bottom), and their crossover line (It can present as a MIX distribution where particles mix uniformly) are discussed.

^{[8]}Furthermore, the algorithm we used can help to illustrate the evolution of the solutions of the gap equation from the chiral limit to non-chiral limit and gives a prediction where the crossover line is located in the phase diagram for $\mu<272.

^{[9]}We find that the critical endpoint linking the crossover transition with the first-order phase transition still exists and locates at (μB,Tc)≃(390 MeV,145 MeV), which along with the crossover line are consistent with lattice result and experimental analysis from relativistic heavy ion collisions.

^{[10]}We map out the QCD crossover line $\frac{T_c(\mu_B)}{T_c(0)} = 1 - \kappa_2 \left(\frac{\mu_B}{T_c(0)} \right)^2 - \kappa_4 \left( \frac{\mu_B}{T_c(0)} \right)^4 + \mathcal{O}(\mu_B^6)$ for the first time up to $\mathcal{O}(\mu_B^4)$ for a strangeness neutral system by performing a Taylor expansion of chiral observables in temperature $T$ and chemical potentials $\mu$.

^{[11]}We provide compelling evidence that this phase is a liquid and show that it is divided by a crossover line that terminates in a quantum critical point.

^{[12]}In the common observation area, through comparing and validating the values of the polarization parameters on the crossover line to analyze the vertical structure characteristics of hail cloud.

^{[13]}78 from the temperature dependence of the crossover line and γ = 1.

^{[14]}We map out the QCD crossover line T c ( μ B ) T c ( 0 ) = 1 − κ 2 ( μ B T c ( 0 ) ) 2 − κ 4 ( μ B T c ( 0 ) ) + O ( μ B 6 ) for the first time up to O ( μ B 4 ) for a strangeness neutral system by performing a Taylor expansion of chiral observables in temperature T and chemical potentials μ.

^{[15]}Finally, we identify the critical point at the end of the first-order transition line and a crossover line.

^{[16]}By examining the relationship between the Voronoi entropy and the solidlike fraction of simple fluids, we suggest that the Frenkel line, a rigid-nonrigid crossover line, be a topological isomorphic line at which the scaling relation qualitatively changes.

^{[17]}

## Qcd Crossover Line

We map out the QCD crossover line $\frac{T_c(\mu_B)}{T_c(0)} = 1 - \kappa_2 \left(\frac{\mu_B}{T_c(0)} \right)^2 - \kappa_4 \left( \frac{\mu_B}{T_c(0)} \right)^4 + \mathcal{O}(\mu_B^6)$ for the first time up to $\mathcal{O}(\mu_B^4)$ for a strangeness neutral system by performing a Taylor expansion of chiral observables in temperature $T$ and chemical potentials $\mu$.^{[1]}We map out the QCD crossover line T c ( μ B ) T c ( 0 ) = 1 − κ 2 ( μ B T c ( 0 ) ) 2 − κ 4 ( μ B T c ( 0 ) ) + O ( μ B 6 ) for the first time up to O ( μ B 4 ) for a strangeness neutral system by performing a Taylor expansion of chiral observables in temperature T and chemical potentials μ.

^{[2]}