Chiral phase transition in the presence of spinodal decomposition

The thermodynamics of a first-order chiral phase transition is considered in the presence of spinodal phase separation within the Nambu–Jona-Lasinio model. The properties of the basic thermodynamic observables in the coexistence phase are discussed for zero and nonzero quark masses. We focus on observables that probe the chiral phase transition. In particular, the behavior of the specific heat and entropy as well as charge fluctuations are calculated and analyzed. We show that the specific heat and charge susceptibilities diverge at the isothermal spinodal lines. We determine the scaling behavior and compute the critical exponent $\gamma$ of the net quark number susceptibility at the isothermal spinodal lines within the Nambu–Jona-Lasinio model and the Ginsburg-Landau theory. We show that in the chiral limit the critical exponent $\gamma =1/2$ at the tricritical point as well as along the isothermal spinodal lines. On the other hand, for finite quark masses the critical exponent at the spinodal lines, $\gamma =1/2$, differs from that at the critical end point, $\gamma =2/3$, indicating a change in the universality class. These results are independent of the particular choice of the chiral Lagrangian and should be common for all mean-field approaches.

Figure 1

The dynamical quark mass at fixed temperature ($T=30\mathrm{MeV}$) as a function of the quark chemical potential in the chiral limit (left panel) and for a finite quark mass $m=5.6\mathrm{MeV}$ (right panel). The dashed line indicates the equilibrium first-order phase transition while the dotted lines show the isothermal spinodal points.

Figure 2

The net quark number density at $T=30\mathrm{MeV}$ computed in the chiral limit (left panel) and for a finite quark mass $m=5.6\mathrm{MeV}$ (right panel). The dashed lines and the dotted lines have the same meaning as in Fig. 1.

Figure 3

In the left panel we show the pressure as a function of inverse quark number density for fixed temperature $T=30\mathrm{MeV}$ obtained in the chiral limit, while in the right panel we show the results for a finite quark mass $m=5.6\mathrm{MeV}$.

Figure 4

The phase diagram of the NJL model in $(T,n_q)$ plane. The left figure corresponds to the chiral limit. The dot indicates the position of the tricritical point (TCP) and the full line above the TCP shows the second-order transition line. The full lines below the TCP represent the phase boundaries of the first-order chiral transition. Finally, the dashed (dotted) lines show the isothermal (isentropic) spinodal lines. The right figure was computed with a finite quark mass $m=5.6\mathrm{MeV}$. Above the CEP, indicated by a dot, there is crossover transition.

Figure 5

The ratio of the entropy to the net quark number density, i.e., the entropy per quark, at fixed temperature as a function of the net quark density in the chiral limit (left panel) and for a finite quark mass $m=5.6\mathrm{MeV}$ (right panel). The dashed lines and the dotted lines have the same meaning as in Fig. 1.

Figure 6

The ${\chi }_{\mu \mu }$ susceptibility at fixed temperature $T=30\mathrm{MeV}$ in the chiral limit (left panel) and for a finite quark mass $m=5.6\mathrm{MeV}$ (right panel). The vertical dashed lines and the dotted lines have the same meaning as in Fig. 1.

Figure 7

The specific heat $C_V$ at constant volume and at fixed temperature $T=30\mathrm{MeV}$ in the chiral limit (left panel) and for a finite quark mass $m=5.6\mathrm{MeV}$ (right panel). The vertical dashed lines indicate the location of the first-order phase transition. The dotted lines show the position of the isothermal spinodal points.

Figure 8

The specific heat $C_P$ at constant pressure and at fixed temperature $T=30\mathrm{MeV}$ in the chiral limit (left panel) and for a finite quark mass $m=5.6\mathrm{MeV}$ (right panel). The vertical dashed lines indicate the location of the first-order phase transition. The dotted (dash-dotted) lines show the position of the isothermal (isentropic) spinodal points.

Figure 9

The net quark number density fluctuations ${\chi }_{\mu \mu }/{\Lambda }^2$ normalized to momentum cutoff as a function of the quark number density $n_q$. The full line represents the fluctuations at the CEP at constant temperature $T_{\mathrm{CEP}}\simeq 80\mathrm{MeV}$. The dashed line shows the ${\chi }_{\mu \mu }$ at fixed $T=70\mathrm{MeV}$ corresponding to the 1st order transition.

Figure 10

The net quark number susceptibility in the vicinity of the TCP (left panel) and CEP (right panel) as a function of the reduced quark chemical potential $t=(\mu -{\mu }_c)/{\mu }_c$ at fixed $T$. The filled circle (●) denotes the results calculated at $T=T_{\mathrm{TCP}/\mathrm{CEP}}$ and the filled triangle (▴) at $T=30\mathrm{MeV}<T_{\mathrm{TCP}/\mathrm{CEP}}$ corresponding to the first-order transition.

Figure 11

The chiral susceptibility in the vicinity of the TCP and spinodals in the chiral limit (left panel) and at finite quark mass (right panel) as a function of the reduced quark chemical potential $t=(\mu -{\mu }_c)/{\mu }_c$ at fixed $T$. The filled circle (●) denotes the results calculated at $T=T_{\mathrm{TCP}/\mathrm{CEP}}$ and the filled triangle (▴) at $T=30\mathrm{MeV}<T_{\mathrm{TCP}/\mathrm{CEP}}$ corresponding to the first-order transition.