Noncongruent phase transitions in strongly interacting matter within the quantum van der Waals model

The noncongruent liquid-gas phase transition (LGPT) in asymmetric nuclear matter is studied using the recently developed quantum van der Waals model in the grand canonical ensemble. Different values of the electric-to-baryon charge ratio, $Q/B$, are considered. This noncongruent LGPT exhibits several features which are not present in the congruent LGPT of symmetric nuclear matter. These include a continuous phase transformation, a change in the location of the critical point, and the separation of the critical point and the endpoints. The effects which are associated with the noncongruent LGPT become negligible for the following cases: when $Q/B$ approaches its limiting values, 0.5 or 0, or if isospin dependence of nucleon-nucleon interaction and quantum statistical effects can be neglected. The latter situation is realized when the particle degeneracy attains large values, $g\gtrsim 10$.

Figure 1

(a) Congruent LGPT lines in the $({\mu }_B,T)$ coordinates and (b) congruent LGPT regions in the $(n_B,T)$ coordinates within the QvdW model for nucleons with degeneracy factors $g=4$ (symmetric nuclear matter, blue curves) and $g=2$ (pure neutron matter, black curves). The orange solid and dash-dotted curves in (b) correspond to a huge degeneracy factor $g=1000$ and to the Boltzmann approximation (with arbitrary $g$ or $Q/B)$, respectively. Critical points and ground states are represented, respectively, by the stars and the squares.

Figure 2

Dependence on the asymmetry parameter of (a) the ground state binding energy and (b) of the baryon density.

Figure 3

Dependence of the binding energy on baryon density at zero temperature for the constant values of the asymmetry parameter. The ground states are represented by the squares.

Figure 4

Noncongruent LGPT regions (a) in the $({\mu }_B,T)$ coordinates and (b) in the $(n_B,T)$ coordinates for the constant value $Q/B=0.3$. The critical point and temperature endpoint are represented by the stars and full circles, respectively. The ground state is represented by the squares. Horizontal lines depict the isotherm at $Q/B=0.3$ and $T=15\mathrm{MeV}<T_c$.

Figure 5

The zoomed in picture of the CP region for the phase diagrams shown in Fig. 4. Horizontal lines depict the isotherm at $Q/B=0.3$ and $T_c<T=18.714\mathrm{MeV}<T_{\text{TEP}}$. The pressure endpoint is represented by triangles.

Figure 6

(a) Temperature of coexistence as a function of local charge fractions in coexistent phases, ${(Q/B)}_{\mathrm{local}}$, for a fixed global charge fraction, $Q/B=0.3$. ${(Q/B)}_{\mathrm{local}}=Q/B$ in pure phases on the borders of mixed phase are presented by thick red vertical lines, while ${(Q/B)}_{\mathrm{local}}$ in correspondent liquid and gaseous infinitesimally small fractions are represented by dashed and dotted lines, respectively. (b) The zoomed in picture of the CP region. CP, TEP, pressure endpoint, and GS are shown by a star, a full circle, a triangle, and a square, respectively.

Figure 7

(a) $T=15\mathrm{MeV}<T_c$ isotherm in the $(n_B,p)$ coordinates for a constant asymmetry parameter $Q/B=0.3$ (solid curve). Metastable and unstable areas of isotherm are represented by dotted curves. Mixed phase boundaries are represented by red dashed curves. (b) The same as (a) but for the $T_c<T=18.714\mathrm{MeV}<T_{\text{TEP}}$ isotherm. The CP, the pressure endpoint, and the TEP are shown by the star, the triangle, and the full circle, respectively. The points $G1,L1,G2,L2$ characterize the phase transformation along the isotherm; see text.

Figure 8

Local baryon densities in the gaseous and in the liquid fractions of the mixed phase as functions of the total baryon density for $Q/B=0.3$ and $T=15\mathrm{MeV}$.

Figure 9

Fraction of volume occupied in the mixed phase by liquid, $\chi$, as a function of baryon density for global asymmetry $Q/B=0.3$. As a measure of density ${\xi }_B$ is taken. For all $T$ the start and the finish of the PT correspond to ${\xi }_B=0$ and ${\xi }_B=1$, respectively. The black curve corresponds to $T=15\mathrm{MeV}<T_c$ while the red curve corresponds to $T_c<T=18.714\mathrm{MeV}<T_{\mathrm{TEP}}$.

Figure 10

Congruent and noncongruent LGPT regions in the $({\mu }_B,T)$ coordinates (a) and in the $(n_B,T)$ coordinates (b) for the constant $Q/B$ values. Solid blue, red, and black lines correspond to $Q/B=0.5$, 0.3, and 0 respectively. Green lines correspond to $Q/B=0.1,{10}^{-2},{10}^{-3}$. Critical points and temperature endpoints are represented by stars and full circles, respectively. Ground states for different $Q/B$ are represented by squares.

Figure 11

The scaled variance of the baryonic (a), (b) and of the electric (c), (d) charge in the $({\mu }_B,T)$ coordinates for asymmetric nuclear matter with the asymmetry parameter $Q/B=0.3$. The critical point, the temperature endpoint, and the pressure endpoint are represented by stars, full circles, and triangles respectively. The zoomed in picture of the CP region is shown in (b) and (d).

Figure 12

Dependence of the binding energy on baryon density at zero temperature. The black line is the same as in Fig. 3. The blue line represents the QvdW model with isospin-dependent interaction parameters. The gray band shows result of chiral effective field theory [53].

Figure 13

(a) $T_c$ and (b) $n_c$ as functions of $Q/B$. Black and blue lines correspond, respectively, to the cases $a_{np}=a_{nn}=a,b_{np}=b_{nn}=b$ and $a_{np}/a_{nn}=2.5,b_{np}/b_{nn}=1.7$. Solid and dotted lines represent, respectively, critical points and temperature endpoints.