Interface tension and interface entropy in the $2+1$ flavor Nambu–Jona-Lasinio model

We study the QCD phases and their transitions in the $2+1$ flavor Nambu–Jona-Lasinio model, with a focus on the interface effects such as the interface tension, the interface entropy, and the critical bubble size in the coexistence region of the first-order phase transitions. Our results show that under the thin-wall approximation, the interface contribution to the total entropy density changes its discontinuity scale in the first-order phase transition. However, the entropy density of the dynamical chiral symmetry (DCS) phase is always greater than that of the dynamical chiral symmetry broken (DCSB) phase in both the heating and hadronization processes. To address this entropy puzzle, the thin-wall approximation is evaluated in the present work. We find that the puzzle can be attributed to an overestimate of the critical bubble size at low temperature in the hadronization process. With an improvement on the thin-wall approximation, the entropy puzzle is well solved with the total entropy density of the hadron-DCSB phase exceeding apparently that of the DCS-quark phase at low temperature.

Figure 1

Calculated variation behavior of the $u$- and $d$-quark condensate with respect to the quark chemical potential at several temperatures [normalized by the condensate at zero temperature and zero chemical potential ${\varphi }_{u,d}(T=0,\mu =0)=-{(242\mathrm{MeV})}^3$].

Figure 2

Calculated variation behavior of the $s$-quark condensate with respect to the quark chemical potential at several temperatures [normalized by the condensate at zero temperature and zero chemical potential ${\varphi }_s(T=0,\mu =0)=-{(258\mathrm{MeV})}^3$].

Figure 3

The grand thermodynamical potential in the coexistence region with respect to the quark chemical potential at several temperatures.

Figure 4

Calculated variation behavior of the (mixed) chiral susceptibility with respect to the quark chemical potential at various temperature values. Upper panel: for those in the high temperature region (above $T_{\mathrm{CEP}}$). Lower panel: for those in the low temperature region (below $T_{\mathrm{CEP}}$), the solid, dotted, dot-dashed, and dashed lines correspond to $T=0.01$, 0.05, 0.09, $0.11\hspace{0.17em}\hspace{0.17em}\mathrm{\Lambda }$, respectively. The susceptibilities for the Nambu phase (blue lines) and Wigner phase (red lines) are positive within the coexistence region and diverge at the higher and lower boundaries, respectively, at each temperature, while the value of the intermediate phase (green lines) is negative definite. The vertical black lines locating the divergence of the susceptibilities constitute the boundaries of the coexistence region at that temperature.

Figure 5

Obtained phase diagram in terms of the quark chemical potential and temperature. The CEP is labeled by a star. The mixed susceptibility of the Wigner phase diverges to positive infinity at the green line and becomes finite and remains positive in the higher chemical potential region (the Wigner phase is metastable or stable in this region). The mixed susceptibility of the Nambu phase remains positive (stable or metastable) and finite until it diverges at the blue dot-dashed line. The Nambu phase does not exist anymore beyond this limit.

Figure 6

Obtained phase diagram in terms of the baryon density and temperature. ${\rho }_0=0.17{\mathrm{fm}}^{-3}$ is the saturation baryon density of nuclear matter.

Figure 7

Obtained phase diagram in terms of the baryon density and energy density (the EOS). The blue lines are isothermal trajectories. ${\rho }_0=0.17{\mathrm{fm}}^{-3}$ is the saturation baryon density of nuclear matter.

Figure 8

Calculated variation behaviors of the interface tension and the interface entropy density with respect to the temperature.

Figure 9

Calculated body entropy density in terms of the temperature. Left panel: in the process $\mathrm{DCSB}\to \mathrm{DCS}$ (deconfinement). Right panel: in the process $\mathrm{DCS}\to \mathrm{DCSB}$ (hadronization).

Figure 10

Calculated critical bubble size in the two phase transition processes with respect to the temperature.

Figure 11

Calculated total entropy density in terms of the temperature. Left panel: in the process $\mathrm{DCSB}\to \mathrm{DCS}$ (deconfinement). Right panel: in the process $\mathrm{DCS}\to \mathrm{DCSB}$ (hadronization). The “V” and “A” in the brackets stand for the body and interface contribution, respectively.

Figure 12

Calculated bubble density profile (upper panel) and normalized square of the density gradient (lower panel) at several temperatures in the transition from the DCSB (Nambu) phase to the DCS (Wigner) phase.

Figure 13

Calculated bubble density profile (upper panel) and normalized square of the density gradient (lower panel) at several temperatures in the transition from the DCS (Wigner) phase to the DCSB (Nambu) phase.

Figure 14

Calculated critical bubble sizes in the two processes with the improvement (red lines) and comparison with the results within the original thin-wall approximation (blue lines).

Figure 15

Total entropy density calculated with the improved thin-wall results of $R_c$. Left panel: in the process $\mathrm{DCSB}\to \mathrm{DCS}$ (deconfinement). Right panel: in the process $\mathrm{DCS}\to \mathrm{DCSB}$ (hadronization). For comparison, the original thin-wall approximation results are also displayed for each process (thinner dotted lines).